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Abstract forthcoming.
Consider a VietorisRips complex of the circle with the geodesic metric. This simplicial complex has an infinite number of vertices, one for each point in the circle. A theorem of JeanClaude Hausmann implies that for small connectivity parameter, the VietorisRips complex is homotopy equivalent to a circle. What happens as the connectivity parameter increases? We show that the VietorisRips complex obtains the homotopy type of the circle, the 3sphere, the 5sphere, the 7sphere, …, until finally it is contractible.
In particular, we describe the persistent homology of the VietorisRips complex of the circle of unit circumference. The persistence diagram for (2k+1)dimensional homology consists of the open interval with birth time k/(2k+1) and death time (k+1)/(2k+3).
Joint workinprogress with Michal Adamaszek, Christopher Peterson, and Corrine Previte.The solutions of compressible Euler equations in one space dimension (1D) with large amplitude (large data) are still not well understood, where the large data global existence is still open for the full system. The major difficulty comes from the lack of regularity due to shock formation. In this talk, we discuss the shock formation and structure of shockfree solutions for 1D compressible Euler equations, where our results are for large data problems, i.e. there is no restriction on the size of the solutions.
We study the effects of fluid elasticity on interfacial instabilities created by applied AC and DC electric fields. In the case of DC fields, asymptotic methods are employed to analyse the instability, and in the case of AC fields, we apply a numerical procedure based on Floquet theory to determine the maximum growth rate and corresponding wavenumber of the instability, as a function of the AC oscillation amplitude and frequency. Elasticity is found to increase both the maximum growth rate and the corresponding wavenumber, with the effects being the most pronounced when the AC oscillation period is comparable to the fluid relaxation time.
In the past decades, electrode recordings and optical imaging techniques have revealed a large variety of different cortical coherent structures ranging from localized regions of activity to traveling waves (fronts, pulses and spiral waves). One natural way to model these structures is to use neural field equations which are nonlocal differential equations describing the coarsegrained activity of large ensembles of cortical neurons in a spatially continuous way. In this talk, we will give an overview of recent mathematical studies related to the analysis of some of these coherent structures with an emphasis on applications.
Queueing networks arise as models in various areas including computer systems, telecommunications, manufacturing, finance, and service industry. One of the key objectives in the queueing network settings is to obtain the “good” (or nearly optimal) control policies for scheduling, sequencing, and routing of jobs in the system. In this talk, I'll present a recent study on scheduling control problem for a singleserver multiclass queueing network in heavy traffic, operating in a changing environment. The changing environment is modeled as a finite state Markov process that modulates the arrival and service rates in the system. Various cases are considered: fast changing environment, fixed environment and slow changing environment. In each of the cases, using weak convergence analysis, in particular functional limit theorems for renewal processes and ergodic Markov processes, it is shown that an appropriate “averaged” version of the classical cμpolicy (the priority policy that favors classes with higher values of the product of holding cost c and service rate μ) is asymptotically optimal for an infinite horizon discounted cost criterion.
We consider convex optimization problems with stochastic order constraint. In these problems, utility functions play the role of Lagrange multipliers associated with the stochastic order constraint. We propose a penalty based method where the penalty function is defined on the space of utility functions. This is an ongoing project with Darinka Dentcheva. I will present some preliminary results.
In this talk, I will present recent progress of our research on the periodic‐parabolic logistic equation. First, I will give a brief review of the history of the logistic equation and the existing results. Then I will report our work on the periodic‐parabolic logistic equation with spatial and temporal degeneracies. More specifically, we examine the effects of various natural spatial and temporal degeneracies of the carrying capacity function on the long‐time dynamical behavior of the positive solution. Our analysis leads to a new eigenvalue problem for periodic‐parabolic operators over a varying cylinder and certain parabolic boundary blow‐up problems not known before. Our investigation shows that the temporal degeneracy causes a fundamental change of the dynamical behavior only when spatial degeneracy also exists; but in sharp contrast, whether or not temporal degeneracy appears in the equation, the spatial degeneracy always induces fundamental changes of the dynamical behavior, though such changes differ significantly according to whether the temporal degeneracy is present or not.
Abstract: Resonances are important in the study of transient phenomena associated with the wave equation, especially in understanding the large time behavior of the solution to the wave equation when radiation losses are small. In this talk, I will present recent studies on the scattering resonances for photonic structures and Schrodinger operators. In particular, for a finite one dimensional photonic crystal with a defect, it is shown that the near boundstate resonances converge to the point spectrum of the infinite structure with an exponential rate. A similar result holds for the Schrodinger operator with a potential that is a lowenergy well surrounded by a thick barrier. I will also discuss non near boundstate resonances for both cases briefly. This is a joint work with Prof. Fadil Santosa at the IMA.
Abstract: For some geophysical flows in which computational efficiency is of paramount importance, simplified mathematical models are central. For example, in climate modeling the QuasiGeostrophic Equations (QGE), are commonly used in the numerical simulation of large scale winddriven ocean circulations. Due to the requisite of long time integration in climate modeling, even for the simplified model, fast and accurate numerical algorithms are still desired. In this talk, we will pursue this direction and discuss efficient numerical approaches for QGE.
Abstract: Many basic equations in physics can be written in the form of conservation law. However, as it is common in the field of PDEs and SPDEs, classical or strong solutions do not exist in general and, on the other hand, weak solutions are not unique. The notion of kinetic formulation and kinetic solution turns out to be a very convenient tool to overcome these difficulties.
In this talk, I will present a wellposedness result for degenerate parabolic stochastic conservation laws and, in the case of hyperbolic stochastic conservation laws, discuss the hydrodynamic limit of an approximation in the sense of BhatnagarGrossKrook (BGKlike for short).
Abstract: Identifying clusters of similar objects in data plays a significant role in a wide range of applications such as information retrieval, pattern recognition, computational biology, and image processing. A classical approach to clustering is to model a given data set as a graph whose nodes correspond to items in the data and edges indicate similarity between the corresponding endpoints, and then try to decompose the graph into dense subgraphs. Unfortunately, finding such a decomposition usually requires solving an intractable combinatorial optimization problem. In this talk, I will discuss a convex relaxation approach to identifying these dense subgraphs with surprising recovery properties: if the data is randomly sampled from a distribution of “clusterable” data, then the correct partition of the data into clusters can be exactly recovered from the optimal solution of the convex relaxation with high probability.
Abstract: Geometric singular perturbation theory is used to analyze systems of ordinary differential equations
with much different time scales. Those kinds of systems are largely used in math biology. It is interesting to
see what happens if noise being taken into consideration.
In this talk, we show the existence of random invariant manifolds and families of
random invariant manifolds of a singular perturbation systems of ordinary di erential equations with
uniformly small real noise. Then apply these to prove a random version of inclination theorem.
Abstract: One of the original motivations for the study of Stochastic PDEs traces it's origins to the study of turbulence. As such a better understanding of stochastic versions of the fundamental equations of fluid flow remain an important direction in applied analysis.
In this talk we discuss some recent results concerning the Stochastic Euler equations as well as some related issues concerning inviscid limits.
The periodic generalized Kortewegde Vries equation (gKdV) can be interpreted as an infinitedimensional Hamiltonian system. Some properties of finitedimensional Hamiltonian dynamics can be extended to infinite dimensions; for example, the invariance of the Gibbs measure under the flow. In this talk we present invariance of the Gibbs measure for the (gaugetransformed) periodic quartic gKdV. The proof relies on probabilistic arguments which exhibit nonlinear smoothing when the initial data are randomized.
In this talk we will discuss qualitative properties of positive solutions of general parabolic partial differential equations. Specifically, we will indicate how the shape of the domain influences the asymptotic properties of solutions of parabolic equations, that is, their behavior for large times. We show that regardless of the initial conditions, the functions in the omegalimit set have specific shapes, determined by the symmetry properties of the domain. As an application we prove that the shape of the domain simplifies the dynamics of solutions and it enables us to prove nontrivial convergence results. We will see that nonnegative, nontrivial equilibria which are not positive play an important role in these results. If time permits, we will discuss existence and nonexistence results for such equilibria.
We discuss a numerical approach for simulating inertial manifolds for a class of stochastic evolutionary equations with multiplicative noise. After splitting the stochastic evolutionary equations into a backward and a forward part, a numerical scheme is devised for solving the backwardforward stochastic system, and an ensemble of graphs representing the inertial manifold is consequently obtained. By a simple illustrative example, we then show the dimension of the original full system can be reduced through this technique.
In this talk we will discuss asymptotically asymmetric slowly nondissipative reactiondiffusion equations, also known as growup equations, and recent developments with regards to the longtime behavior of solutions and the noncompact global attractor structure for these PDEs. I will present some of the background of this problem, and then introduce recent research utilizing Fučik spectra, nodal property methods, the time map, numerical continuation methods, and Conley index at infinity to determine both the bounded portion of the noncompact global attractor as well as the portion contained within the socalled Sphere at Infinity.
Additionally, we will discuss the transfinite heteroclinic structures which connect these two regions and why asymmetry adds significant complexity to the task of solving the connection problem for these heteroclinics. This talk is based on joint work with Kristen Moore (U. Mich.) and Juliette Hell (F.U. Berlin).
We consider ultrashort pulse propagation in nonlinear metamaterials characterized by a weak Kerrtype nonlinearity in their dielectric response. We will derive shortpulse equation (SPE) u_{xt} = u+1/6 (u^3)_{xx} in frequency band gaps. We will discuss the connection of SPE with the nonlinear Schrodinger equation, SPE in its characteristic coordinate and the transformation to sineGordon equation, and robustness of various solutions emanating from the sineGordon equation and their periodic generalizations. Finally, we will discuss the wellposedness of generalized Ostrovsky equation u_{xt} = u+1/6 (u^p)_{xx} with small initial data.
Loosely speaking, a canard is a special solution of a singularly perturbed system which closely follows an attracting slow manifold and then subsequently follows a repelling slow manifold for a long time. Canard orbits are ubiquitous in slowfast systems where normal hyperbolicity is lost at a fold in the critical manifold, and often serve as intermediaries in parameter regimes connecting qualitatively different dynamical behavior. Techniques based on the blowup method for singular points in the plane have proved useful for the study of canards, and have been extended in a variety of ways for higher dimensional systems.
In this talk, I'll briefly present the classical theory of planar canards and then go on to introduce recent results for a generalization called the torus canard. The torus canard was first discovered in the transition region from bursting to rapid spiking in a mathematical model of a cerebellar Purkinje cell, and has since been observed in many other models from the field of mathematical neuroscience. These and other applications of canard or generalized canard phenomena will be highlighted throughout the talk.
Conformally symplecti systems send a symplectic form into a multiple of itself. The conservation of this geometric structure provides identities that we use to prove "aposteriori" theorems that show the existence of a solution close to an approximate solution. We also use the identities to obtain very efficient algorithms with small storage and operation counts.
It is a classical problem to study incidences between a finite number of points and a finite number of geometric objects. In this presentation, we see that continuous incidence theory can be used to derive a number of results in geometry, geometric measure theory, and analytic number theory.
The applications to geometry include a fractal variant of the regular value theorem. The applications to geometric measure theory include a generalization of Falconer distance problem in which we prove that a compact subset of R^d of sufficiently large Hausdorff dimension determines a positive proportion of all (k+1)configurations described by certain restrictions.
The applications to Number Theory include counting integer lattice points neighborhoods of variable coefficient families of surfaces.
Understanding the conformation of proteinbound DNA is extremely important for biological and medical research, including improvement of drug creation and administration methods. Many proteinbound DNA conformations have been catalogued in the Protein Data Base; however the process of cataloging larger complexes can prove extremely difficult or unsuccessful. When standard lab techniques fail to determine a conformation, we turn to a branch of mathematical knot theory, tangle analysis, used in conjunction with difference topology experiments to analyze the topology of protein bound DNA. In this talk, we will discuss these techniques and explain why topology alone is not enough. We will introduce preliminary software which can be used to determine likely DNA geometries consistent with proteinbound DNA topologies. Combining geometric and topological solutions will allow us to more accurately describe conformations for large proteinbound DNA complexes.
Mathematical knot theory is an exciting branch of topology that is relevant to physics, chemistry and biology. After a brief introduction to knot theory, we will explore several applications where computational methods play a useful role. When experimenting with knots on a computer, it is helpful to have powerful tools for the creation of initial conformations, for performing simulations on those conformations and also to compute topological and geometric properties of any resulting products. One such tool is the software KnotPlot, developed by the speaker over a period of many years. KnotPlot permits a wide range of interesting experiments to be performed. Several of these will be discussed, one important area being the tangling and untangling of DNA. Finally, we will dip into the more esoteric realm of higher dimensional knotting.
We consider twostage riskaverse stochastic optimization problems with a stochastic ordering constraint on the recourse function. Two new characterizations of the increasing convex order relation are provided. They are based on conditional expectations and on integrated quantile functions: a counterpart of the Lorenz function. We propose two decomposition methods to solve the problems and prove their convergence. Our methods exploit the decomposition structure of the riskneutral twostage problems and construct successive approximations of the stochastic ordering constraints. Numerical results confirm the efficiency of the methods
The JohnsonLindenstrauss (JL) Lemma states that any set of p points in high dimensional Euclidean space can be embedded into O<(δ^{ −2} log(p)) dimensions, without distorting the distance between any two points by more than a factor between 1 − δ and 1 + δ . We establish a new connection between the JL Lemma and the Restricted Isometry Property (RIP), a wellknown concept in the theory of sparse recovery often used for showing the success of ℓ_{1}minimization.
Consider an m X N matrix satisfying the (k, δ_{k})RIP and an arbitrary set E of O(e^{k}) points in R^{N}. We show that with high probability, such a matrix with randomized column signs maps E into R^{m} without distorting the distance between any two points by more than a factor of 1±4δ_{k}. Consequently, matrices satisfying the Restricted Isometry of optimal order provide optimal JohnsonLindenstrauss embeddings up to a logarithmic factor in N. Moreover, our results yield the best known bounds on the necessary embedding dimension m for a wide class of structured random matrices. In particular, for partial circulant, partial Fourier, and partial Hadamard matrices, our method optimizes the dependence of m on the distortion δ: We improve the recent bound m = O(δ^{−4} log(p) log^{4}(N)) of Ailon and Liberty (2010) to m = O(δ^{−2} log(p) log^{4}(N)), which is optimal up to the logarithmic factors in N. Our results also have a direct application in the area of compressed sensing for redundant dictionaries.
This is joint work with Rachel Ward.
Global Dimension Minimization is a new geometric method for Subspace Clustering. In this talk we will introduce subspace clustering (also called Hybrid Linear Modeling) and explore a motivating example. We will investigate some of the new concepts involved in Global Dimension Minimization, including an exciting new (and flexible) technique for estimating the dimension of an arbitrary data set. We will tour the Global Dimension Minimization Algorithm and see stateoftheart results in a realworld application.
In this talk we discuss a nonconforming finite element approximation
of the vibration modes
of an acoustic fluidstructure interaction. Displacement variables are used
for both the fluid and the solid. The numerical scheme is based on
the irrotational
fluid displacement formulation; hence it is free of spurious eigenmodes.
The method uses weakly continuous P_{1} vector fields for the fluid and
classical piecewise linear elements for the solid; and it satisfies optimal
order error estimates on properly graded meshes. The theoretical results
are confirmed by numerical experiments.
This is joint
work with Susanne Brenner, Aycil Cesmelioglu and Liyeng Sung.
It is well known that standard Monte Carlo estimates of small probabilities are unreliable in the sense that under fixed computational effort, the relative error (i.e., the sample standard deviation divided by the sample average) is expected to become unbounded as the probability being estimated approaches zero. We consider the problem of estimating small probabilities of the Langevin stochastic differential equation in various asymptotic regimes. As an example, we consider the probability of exiting a deep potential well.
Graphical models use graphs to compactly capture stochastic dependencies amongst a collection of random variables. Inference over graphical models corresponds to finding marginal probability distributions given joint probability distributions. In general, this is computationally intractable, which has led to a quest for finding efficient approximate inference algorithms. We propose a framework for generalized inference over graphical models that can be used as a wrapper for improving the estimates of approximate inference algorithms. Instead of applying an inference algorithm to the original graph, we apply the inference algorithm to a blockgraph, defined as a graph in which the nodes are nonoverlapping clusters of nodes from the original graph. This results in marginal estimates of a cluster of nodes, which we further marginalize to get the marginal estimates of each node. Our proposed blockgraph construction algorithm is simple, efficient, and motivated by the observation that approximate inference is more accurate on graphs with longer cycles. We present extensive numerical simulations that illustrate our blockgraph framework with a variety of inference algorithms (e.g., those in the libDAI software package). These simulations show the improvements provided by our framework. More details
I will talk about the problems of denoising images corrupted by impulsive noise and blind inpainting (i.e., inpainting when the deteriorated region is unknown). Our basic approach is to model the set of patches of pixels in an image as a union of lowdimensional subspaces, corrupted by sparse but perhaps large magnitude noise. For this purpose, we develop a robust and iterative method for single subspace modeling and extend it to an iterative algorithm for modeling multiple subspaces. I will also cover the convergence for the algorithm and demonstrate state of the art performance of our method for both imaging problems.
Finding a densely connected subset of nodes in a graph is a fundamental problem in combinatorial optimization, with many practical applications. We consider the problem and of identifying the most densely connected set of k nodes of a given graph. This problem is known to be NPhard. We propose a new convex relaxation of this problem using principal component pursuit. In the special case that the input graph contains a single dense subgraph plus noise in the form of edge additions and deletions, this relaxation is exact: the densest ksubgraph can be recovered directly from the unique optimal solution of this tractable relaxation. We provide two analyses of when the algorithm succeeds. In the first, the diversionary edges are added or deleted deterministically by an adversary. In the second, edges are placed or removed independently at random. These results can be thought of generalizations of existing recovery guarantees for the maximum clique problem. In both cases, the bounds ensuring perfect recovery match the best existing in the literature for the maximum clique problem.
We introduce Tyler's Mestimator to robustly recover the underlying linear model from a data set contaminated by outliers. We prove that the objective function of this estimator is geodesically convex on the manifold of all positive definite matrices and have a unique minimizer. Besides, we prove that when inliers (i.e., points that are not outliers) are sampled from a subspace and the percentage of outliers is bounded by some number, then under some very weak assumptions a commonly used algorithm of this estimator can recover the underlying subspace exactly. We also show that empirically this algorithm compares favorably with other convex algorithms of subspace recovery.
Identifying communities of densely connected nodes in a network is an important problem in the analysis of social and biological networks. These communities may overlap in many practical applications. For example, an individual may belong to many demographic groups within a social network or a particular gene may serve several biological functions within a genegene interaction network. However, most popular heuristics for community identification implicitly exploit the assumption that the underlying communities do not intersect.
I will present the recent paper Finding Overlapping Communities in Social Networks: Toward a Rigorous Approach by Arora et al. In this work, Arora et al. propose a quasipolynomial time algorithm for recovering the underlying community structure of networks satisfying particular assumptions. I will also discuss ongoing research by myself on this subject. In particular, I will discuss how a combination of Aroras algorithm and standard clustering and graph partitioning techniques might lead to a polynomial time algorithm for networks sampled from a particular generative model.
An undirected graphical model is a joint probability distribution defined on an undirected graph $G = (V,E)$, where the nodes in the graph index a collection of random variables and the edges encode conditional independence relationships amongst random variables. The undirected graphical model selection (UGMS) problem is to estimate the graph $G$ given observations drawn from the undirected graphical model. This paper proposes a framework for decomposing the UGMS problem into multiple subproblems over clusters and separators of a junction tree. Under certain conditions, we show that the junction tree framework significantly weakens the sufficient conditions on the number of observations required for highdimensional consistent graph estimation. When the conditions on the graphical model do not hold, we recover the standard conditions for highdimensional consistency. This motivates the use of our framework as a wrapper around algorithms for more accurate graph estimation. Further, we show that the subproblems over the clusters and separators can be solved independently, which allows for using different regularization parameters or different UGMS algorithms to learn different parts of the graph. Finally, the junction tree framework motivates active learning for UGMS that sequentially draws observations from the graphical model based on prior observations. In the highdimensional setting, we identify conditions under which the sufficient conditions on the number of scalar observations needed for an active algorithm is significantly less than that needed for a nonactive algorithm. Intuitively, the active learning algorithm draws more observations from parts of the graph that are difficult to learn and less measurements from parts of the graph that are easy to learn. Our numerical results clearly identify the advantages of using the junction tree framework for both nonactive and active learning for UGMS. This is joint work with Prof. Robert Nowak.
Consider the set X of all possible arrangements of nonoverlapping disks of radius r in the plane. There is a probability measure mu_d on X which assigns equal probabilistic weight to all arrangements of the same density d. What are the qualitative properties of typical samples from mu_d when d is large? One might expect that for density d near maximum, a typical sample is a perturbation of a perfect crystalline arrangement. This turns out not to be the case (T. Richthammer, 2006). We ask a simpler question: Can one expect to see an infinite connected cluster of disks, where pairs of disks are connected if they are within distance epsilon? We give a positive response to the question in the case where epsilon > r. In particular this implies percolation of excluded volume, which has been heuristically connected to the liquidgas phase transition in statistical mechanics (KW Kratky, 1988). The proof extends to 3 dimensions.
In this talk, we show how alternating direction methods are used recently in sparse and lowrank optimization problems. We also show the iteration complexity bounds for two specific types of alternating direction methods, namely alternating linearizaton and multiple splitting methods. We prove that the basic versions of both methods require $O(1/eps)$ iterations to obtain an epsoptimal solution, and the accelerated versions of these methods require $O(1/\sqrt{eps})$ iterations, while the computational effort needed at each iteration is almost unchanged. To the best of our knowledge, these complexity results are the first ones of this type that have been given for splitting and alternating direction type methods. Numerical results on problems with millions of variables and constraints arising from computer vision, machine learning and image processing are reported to demonstrate the efficiency of the proposed methods.
We consider the problem of testing isomorphism of groups of order n given by Cayley tables. The $n^{\log n}$ bound on the time complexity for the general case has not been improved over the past four decades. We demonstrate that the obstacle to efficient algorithms is the presence of abelian normal subgroups; we show this by giving a polynomialtime isomomrphism test for "semisimple groups,'' defined as groups without abelian normal subgroups (joint work L. Babai, Y. Qiao, in preparation). This concludes a project started with with L. Babai, J. A. Grochow, Y. Qiao (SODA 2011). That paper gave an $n^{\log\log n}$ algorithm for this class, and gave polynomialtime algorithms assuming boundedness of certain parameters. The key ingredients for this algorithm are: (a) an algorithm to test permutational isomorphism of permutation groups in time polynomial in the order and simply exponential in the degree; (b) the introduction of the "twisted code equivalence problem," a generalization of the classical code equivalence problem (which asks whether two sets of strings over a finite alphabet are equivalent by permuting the coordinates), by admitting a group action on the alphabet; (c) an algorithm to test twisted code equivalence; (d) a reduction of semisimple group isomorphism to permutational isomorphism of transitive permutation groups under the given time limits via twisted code equivalence.
The twisted code equivalence problem and the problem of permutational isomorphism of permutation groups are of interest in their own right. In this talk I will give the definitions of the problems that make up the overall plan, show how they fit together, and then give some details on the algorithm to test permutational isomorphism of transitive groups.
This talk is based on joint work with: Laszlo Babai (University of Chicago), Joshua A. Grochow (University of Chicago) and Youming Qiao (Tsingua University).
We study maximum likelihood estimation in Gaussian graphical models from a geometric point of view. In current applications of statistics, we are often faced with problems involving a large number of random variables, but only a small number of observations. An algebraic elimination criterion allows us to ?nd exact lower bounds on the number of observations needed to ensure that the maximum likelihood estimator exists with probability one. This is applied to bipartite graphs and grids, leading to the ?rst instance of a graph for which the maximum likelihood estimator exists with probability one even when the number of observations equals the treewidth of the graph.
Graphical model selection, where the goal is to estimate the graph underlying a distribution, is known to be computationally intractable in general. An important issue is to study theoretical limits on the performance of graphical model selection algorithms. In particular, given parameters of the underlying distribution, we want to find a lower bound on the number of samples required for accurate graph estimation. When deriving these theoretical bounds, it is common to treat the learning problem as a communication problem where the observations correspond to noisy messages and the decoding problem infers the graph from the observations. Current analysis of graphical model selection algorithms is limited to studying graph estimators that output a unique graph. In this paper, we consider graph estimators that output a set of graphs, leading to setbased graphical model selection (SBGMS). This has connections to listdecoding where a decoder outputs a list of possible codewords instead of a single codeword. Our main contribution is to derive necessary conditions for accurate SBGMS for various classes of graphical models and show reduction in the number of samples required for consistent estimation. Further, we derive necessary conditions on the cardinality of the setbased estimates given graph parameters.
We analyze nonlinear stochastic optimization problems with joint probabilistic constraints using the concept of a $p$efficient point of a probability distribution. If the problem is described by convex functions, we develop two algorithms based on first order optimality conditions and a dual approach to the problem. The algorithms yield an optimal solution for problems involving $\alpha$concave probability distributions. For arbitrary distributions, the algorithms provide upper and lower bounds for the optimal value and nearly optimal solutions. When the problem is described by continuously differentiable nonconvex functions, we describe the tangent and the normal cone to the level set of the underlying probability function. Furthermore, we formulate first order and second order conditions of optimality based on the notion of $p$efficient points. For the case of discrete distribution functions, we developed an augmented Lagrangian method based on progressive inner approximation of the level set of the probability function by generation of $p$efficient points. Numerical experience is provided.
Identifying clusters of similar objects in data plays a significant role in a wide range of applications such as information retrieval, pattern recognition, computational biology, and image processing. We consider as a model problem for clustering the average weight kdisjoint clique problem (WKDC), whose goal is to identify the collection of k disjoint cliques of a given weighted complete graph maximizing the sum of the average edge weights over the complete subgraphs induced by these cliques. We show that this problem can be formulated as a nonconvex quadratic maximization problem and subsequently relaxed to a semidefinite program using symmetric matrix lifting. Although the WKDC problem is NPhard, we show that this relaxation is exact under certain assumptions on the input graph. That is, the optimal solution for the original hard combinatorial problem can be recovered directly from the solution of the relaxed problem for certain program inputs. In particular, the semidefinite relaxation is exact for input graphs corresponding to data consisting of k large, distinct clusters and a small number of outliers.
This approach also yields a semidefinite relaxation for the biclustering problem with similar recovery guarantees. Given a set of objects and a set of features exhibited by these objects, biclustering seeks to simultaneously group the objects and features according to their expression levels. We pose this problem as partitioning of a weighted complete bipartite graph such that the edge weight within the resulting bicliques is maximized. As in our analysis of the WKDC problem, we consider a nonconvex quadratic programming formulation for this problem, and relax to semidefinite programming using matrix lifting. As before, we show that the correct partition of the objects and features can be recovered from the optimal solution of the semidefinite relaxation, in the case that the input instance consists of several disjoint sets of objects exhibiting similar features.
We study a sequence of nearly critically loaded queueing networks, with time varying arrival and service rates and routing structure. The n^{th} network is described in terms of two independent finite state Markov processes {X^{n}(t): t = 0} and {Y^{n}(t) : t = 0} which can be interpreted as the random environment in which the system is operating. The process X^{n} changes states at a much higher rate than the typical arrival and service times in the system, while the reverse is true for Y^{n}. The variations in the routing mechanism of the network are governed by X^{n}, whereas the arrival and service rates at various stations depend on the state process (i.e. queue length process) and both X^{n} and Y^{n}. Denoting by Z^{n} the suitably normalized queue length process, it is shown that, under appropriate heavy traffic conditions, the pair Markov process (Z^{n}, Y^{n}) converges weakly to the Markov process (Z, Y), where Y is a finite state continuous time Markov process and the process Z is a reflected diffusion with drift and diffusion coefficients depending on (Z, Y) and the stationary distribution of X^{n}. We also study stability properties of the limit process (Z, Y).
We formulate a convex minimization to robustly recover a subspace from a contaminated data set, partially sampled around it, and propose a fast iterative algorithm to achieve the corresponding minimum. We establish exact recovery by this minimizer, quantify the effect of noise and regularization, explain how to take advantage of a known intrinsic dimension and establish linear convergence of the iterative algorithm. We compare our method with many other algorithms for Robust PCA on synthetic and real data sets and demonstrate stateoftheart speed and accuracy.
I will review a standard object recognition architecture, and discuss each of the components: SIFT features, sparse coding, pooling, and linear classifiers. I will then mention recent work of myself and collaborators on a fast version of this machine.
The eigenvalues of a symmetric operator play a significant role in many areas of mathematics and engineering. As such, it is important to understand the behaviour of the spectrum of a matrix under small perturbations. This talk is divided into two parts. In the first, I will review several important results from the literature regarding the sensitivity of the eigenvalues of a symmetric matrix. In particular, I will provide formulas for the directional derivatives of the spectrum of a symmetric matrix, and discuss how these formulas can be extended to obtain a series approximation of the spectrum of a symmetric matrix under small perturbations. The second half of the talk will focus on the sensitivity of functions of the spectrum of a symmetric matrix. If F(X) is a function that depends smoothly on the eigenvalues of X, I will provide conditions for when this smoothness is inherited by F. As examples, I will provide conditions for differentiability of spectral functions and primary matrix functions, as well as formulas for their gradients.
The IMA Postdoc Seminar met at 11:1512:15 every Tuesday
during the semester in Lind Hall 305, except during IMA workshop
weeks. The Fall semester postdoc
seminar was organized by
OrtizDuenas and
Dimitar Trenev was the
organizer for the Winter and Spring semesters postdoc seminars.
September 21, 2010 Randall J. Leveque (Founders Term Professor of Applied Mathematics and Adjunct Professor of Mathematics, University of Washington, Seattle) The GeoClaw software for tsunamis and other hazardous flows Abstract: Many geophysical flows over topography can be modeled by twodimensional depthaveraged fluid dynamics equations. The shallow water equations are the simplest example of this type, though it is often necessary to incorporate nonhydrostatic pressures, more complicated rheologies (e.g. for avalanches, landslides, or debris flows), or to use multilayer models, e.g. for capturing internal waves or to model a landslideinduced tsunamis. These equations are generally hyperbolic and can be modeled using highresolution finite volume methods designed for such problems. However, several features of these flows lead to new algorithmic challenges, such as the fact that the depth goes to zero at the edge of the flow and that vastly differing spatial scales must often be modeled, making adaptive mesh refinement essential. I will discuss some of these algorithms and the GeoClaw software, a specialized version of Clawpack that is aimed at solving realworld geophysical flow problems over topography. In particular I will show results from some recent tsunamis and potential future events. For information about the software, see www.clawpack.org/geoclaw. 
September 28, 2010 On detection of low emission sources in the presence of a large random background Abstract: One of the missions of the Department of Homeland Security is to prevent smuggling of weapongrade nuclear materials. It is expected that such materials, unlike those needed for a "dirty bomb," will have low emission rates and will be well shielded, so that very few gamma photons or neutrons could escape, and even less would be detected. An additional hurdle for the detection of illicit nuclear substances is the strong natural radiation background. In this talk I will discuss when detection of such sources is feasible and how Compton camera type detectors could be used for this purpose. 
October 5, 2010
Multigrid methods for Maxwell’s equations Abstract: In this work we study finite element methods for twodimensional Maxwell’s equations and their solutions by multigrid algorithms. We first introduce two types of nonconforming finite element methods on graded meshes for a twodimensional curlcurl and graddiv (CCGD) problem that appears in electromagnetics. The first method is based on a discretization using weakly continuous P_{1} vector fields. The second method uses discontinuous P_{1} vector fields. Optimal convergence rates in the energy norm and the L_{2} norm are established for both methods on graded meshes. Then we consider a class of symmetric discontinuous Galerkin methods for a model Poisson problem on graded meshes that share many techniques with the nonconforming methods for the CCGD problem. We establish the uniform convergence of Wcycle, Vcycle and Fcycle multigrid algorithms for the resulting discrete problems. Finally, we propose a new numerical approach for twodimensional Maxwell’s equations that is based on the Hodge decomposition for divergencefree vector fields, and present multigrid results. 
October 12, 2010 Hierarchical approximations, coarsegraining and fast lattice Monte Carlo simulations Abstract: We shall discuss numerical analysis aspects of coarsegraining stochastic particle systems and the connection to acceleration of kinetic Monte Carlo simulations. Mathematical tools developed for error control in microscopic simulations using the coarsegrained stochastic processes and reconstruction of microscopic scales will be presented in connection with accelerating (kinetic) Monte Carlo simulations. On specific examples of lattice as well as offlattice dynamics we demonstrate that computational implementation of constructed hierarchical algorithms results in significant speed up of simulations. The developed framework also leads to new parallel kinetic Monte Carlo algorithms that will be briefly described. 
October 26, 2010 Simulating nonholonomic mechanics using variational integrators through Hamiltonization Abstract: Although it is well known that nonholonomic mechanical systems are not Hamiltonian, recent research has uncovered a variety of techniques which allow one to express the reduced, constrained dynamics of certain classes of nonholonomic systems as Hamiltonian. In this talk I will discuss the application of these methods to develop alternative geometric integrators for nonholonomic systems with perhaps more eacuteciency than the known nonholonomic integrators. After showing how variational integrators theoretically preserve conserved mechanical quantities (such as momentum and energy), I will discuss how Hamiltonization can be used to apply these variational integrators to certain classes of nonholonomic systems. Finally, I will discuss some current research utilizing time reparameterizations. 
November 9, 2010 Robust numerical solution of singularly perturbed problems Abstract: Singularly perturbed differential equations are usually posed with a small positive (perturbation) parameter multiplying the highest derivative. Their solutions typically exhibit boundary or interior layers. In recent years much effort has been directed towards constructing and analysing socalled "parameter robust" methods. Such methods should yield solutions whose accuracy does not depend on the perturbation parameter, and should resolve any layers present. In this talk I will survey some of these methods, and the mathematics behind them, with particular emphasis on finite differences for coupled systems. 
November 16, 2010 Discontinuous Galerkin approximation for the VlasovPoisson system Abstract: One of the simplest model problems in the kinetic theory of plasmaphysics is the VlasovPoisson (VP) system with periodic boundary conditions. Such system describes the evolution of a plasma of charged particles (electrons and ions) under the effects of the transport and selfconsistent electric field. In this talk, we construct a new family of semidiscrete numerical schemes for the approximation of the VlasovPoisson system. The methods are based on the coupling of discontinuous Galerkin (DG) approximation to the Vlasov equation and several finite element (conforming, nonconforming and mixed) approximations for the Poisson problem. We present the error analysis in the case of smooth solutions. The issue of energy conservation is also analyzed for some of the methods. If time allows, I will also comment on the issue of approximating nonsmooth solutions of the VP system. The talk is based on joint works with J.A. Carrillo (Universidad Autonoma de Barcelona) and CW. Shu (Brown University). 
November 23, 2010 The Poincaré lemma and the computation of domain integrals in BEM Abstract: An effective technique to evaluate domain integrals appearing in a boundary element method (BEM) has been developed [1]. The proposed approach first converts a domain integral with continuous or weaklysingular integrand into an equivalent boundary integral. The resulting surface integral is then calculated via standard quadrature rules commonly used for boundary elements. This transformation of a domain integral into a boundary counterpart is accomplished through a systematic and rigorous generalization of the fundamental theorem of calculus to higher dimension. Moreover, it is shown that the higherdimensional version of the first fundamental theorem of calculus corresponds to the classical Poincaré lemma. Employed together with the singular treatment of surface integrals that is well established in the literature [2, 3, 4], the proposed method can be utilized to ectively solve boundaryvalue problems involving nonhomogeneous source terms by way of a collocation or a Galerkin BEM without partitioning the problem domain into volume cells. Several key features of this study, including the extension of the fundamental theorem of calculus to higher dimension, are highlighted. In addition, numerical examples dealing with mixed boundaryvalue problems for the Poisson equation on representative test geometries are carried out successfully to validate the proposed method. References: [1] S. Nintcheu Fata, Treatment of domain integrals in boundary element methods, Appl. Numer. Math., doi:10.1016/j.apnum.2010.07.003, 2010. [2] S. Nintcheu Fata, Explicit expressions for 3D boundary integrals in potential theory, Int. J. Num. Meth. Eng., 78(1), pp. 32—47, 2009. [3] S. Nintcheu Fata, L. J. Gray, Semianalytic integration of hypersingular Galerkin BIEs for threedimensional potential problems, J. Comput. Appl. Math., 231(2), pp. 561—576, 2009. [4] S. Nintcheu Fata, Semianalytic treatment of nearlysingular Galerkin surface integrals, Appl. Numer. Math., 60(10), pp. 974—993, 2010. This research was supported by the Office of Advanced Scientific Computing Research, U.S. Department of Energy, under contract DEAC0500OR22725 with UTBattelle, LLC. 
December 7, 2010 Discontinuous Galerkin methods for radiative transfer: Some old results and some new results Abstract: In the 1970s, discontinuous Galerkin (DG) methods were invented as a means to solve neutron/radiation transport problems. Their convergence analysis had been developed by the early 1980s. Between 1989 and 2000 several publications in nuclear engineering suggested, that the method does not converge in scattering dominated regimes. In this presentation, we will review these seemingly contradicting results. A first robustness result requires that the DG finite element space contains an approximating continuous space. Since this result is not sufficient for applications, we use the information contained in the analysis to devise a new DG method, which will converge to the correct solution independent of the model parameters. 
January 25, 2011 On dispersive effect of the Coriolis force for the stationary NavierStokes equations Abstract: The dispersive effect of the Coriolis force for the stationary and nonstationary NavierStokes equations is investigated. Existence of a unique stationary solution is shown for arbitrary large external force provided the Coriolis force is large enough. In addition to the stationary case, counterparts of several classical results for the nonstationary NavierStokes problem have been proven. The analysis is carried out in a new framework of the FourierBesov spaces. 
February 1, 2011 On the coupling of surface/subsurface flow with transport Abstract: The coupling of porous media flow with free flow arises in many applications an example of which is groundwater contamination through rivers. The free flow is characterized by the NavierStokes equations whereas the porous media flow is described by the Darcy equations. BeaversJosephSaffman interface condition is prescribed at the interface separating two regions. A transport equation for the contaminant concentration is fully coupled to the flow problem via the velocity field and the viscosity. First, we discuss the existence result to the related weak formulation of the full coupling problem. Second, we analyze numerical schemes based on classical finite element methods and discontinuous Galerkin methods for the special case where the coupling is only oneway, that is, the velocity field from the NavierStokes/Darcy problem is an input for the transport equation. Numerical solutions for non homogeneous porous media are also presented. 
February 15, 2011 FEMs and MG methods for axisymmetric problems Abstract: We shall discuss finite element and multigrid techniques solving the axisymmetric Poisson's equation and the azimuthal Stokes problem on polygonal domains with possible singular solutions. In particular, we construct stable interpolation operators and establish the wellposedness and regularity in some weighted Sobolev space, which in turn, leads to special finite element spaces to approximate the solutions in the optimal rate. With a careful formulation, we also obtain uniform convergence of the MG methods. These estimates can also be used to show the stability of the TaylorHood elements for the axisymmetric Stokes problem and to precondition the indefinite system from the axisymmetric Stokes equations. 
February 22, 2011 An hp DPG method for linear elasticity with symmetric stresses Joint work with Jamie Bramwell^{ 3}, Leszek Demkowicz ^{2}, and Jay Gopalakrishnan ^{1}. Abstract: In this research, we present two Discontinuous PetrovGalerkin (DPG) finite element methods for linear elasticity. For the first method, we consider asymmetric test tensors for the constitutive equation and compute infinitessimal rotations, while in the second method we only use symmetric test tensors and therefore have fewer unknowns. We define optimal test functions which are shown to deliver the best approximation error if an optimal global test norm is used. To make the method practical, we show a localizable test norm is equivalent to the global optimal norm. The majority of this proof is the verification that the infsup condition holds for our DPG formulations using the localizable test space norm. From DPG theory, this proves our methods are quasioptimal with constants independent of the mesh. We can then use results from approximation theory to show h and p convergence for both methods. Since the quasioptimal test space norm is localizable, we have implemented practical finite element codes that show h and p convergence of both methods at optimal rates. Additionally, the DPG framework provides an a priori error estimator determined by a local auxilliary variational problems. We use this estimator as the basis for various 'greedy' adaptive schemes. We test our adaptive algorithm using a manufactured smooth solution as well as a singular solution Lshape domain problem and observe adaptive h and hp convergence. The principal contributions of this research are proving p convergence for the dualmixed elasticity system, particularly without the need for a discrete exact sequence or commuting diagram, as well as a practical adaptive 2D elasticity code with a priori error estimation. We will present an overview of the theoretical DPG framework, the convergence proofs for both methods, and the numerical results for both a singular and smooth solution. References
^{1} Professor, Mathematics, University of Florida ^{2} Professor, Institute for Computational Engineering and Sciences, ^{3} Graduate Research Assistant, Institute for Computational Engineering and Sciences, University of Texas at Austin 
March 22, 2011 Reconstruction of binary functions and shapes from incomplete frequency information Abstract: Binary functions are a class of important functions that appears in many applications, e.g. image segmentation, bar code recognition, shape detection and so on. In this research we proved that under certain conditions the binary function can be reconstructed from very limited frequency information by using only simple linear programming. Numerical results and applications will be discussed. 
March 29, 2011 On the use of the finitefault solution for tsunami generation problems Abstract: We present a new approach to describe accurately the generation of a tsunami wave due to an underwater earthquake. The main goal of this work is twofold. First of all, we propose a simple and computationally inexpensive model for the description of the sea bed displacement during an underwater earthquake, based on the finite fault solution for the slip distribution under some assumptions on the dynamics of the rupturing process. Once the bottom motion is reconstructed, we study waves induced on the free surface of the ocean. For this purpose we consider three different models approximating the Euler equations of the water wave theory. Namely, we use the linearized Euler equations (we are in fact solving the CauchyPoisson problem), a Boussinesq system and a novel weakly nonlinear model. An intercomparison of these approaches is performed. The developments of the present study are illustrated on the July 27, 2006 Java event, where an underwater earthquake of magnitude 7.7 generated a tsunami that inundated the southern coast of Java. 
April 5, 2011 Data Analysis and Uncertainty Quantification of Inverse Problems Abstract: We present exploratory data analysis methods to assess inversion estimates using simple examples based on classic l2 and l1regularization. These methods can be used to reveal the presence of systematic errors such as bias and discretization effects, or to validate assumptions made on the statistical model used in the analysis. The methods include: bound for randomized trace estimators, confidence intervals and bounds for the bias, resampling methods for model validation, and construction of training sets of functions with controlled local regularity. 
April 26, 2011 A factorization method for nonsymmetric linear operator: enlargement of the functional space while preserving hypocoercivity Abstract: We present a factorization method for nonsymmetric linear operators: the method allows to enlarge functional spaces while preserving spectral properties for the considered operators. In particular, spectral gap and related convergence towards equilibrium follow easily by hypocoercivity and resolvent estimates. Applications of this theory on several kinetic equations will be presented. 
May 10, 2011, Tuesday Smoothness of Nonlinear Subdivision Curves Abstract: Subdivision Curves were discovered by Georges de Rham in 1947. He noticed that the sequence of polygonal lines obtained by a "corner cutting" (or "subdivision") process converges to a C1 curve. Bsplines are obtained by a generalization of this procedure, which depends on the affine structure of Euclidean space. A "linear subdivision curve" is the limiting curve obtained by such a linear subdivision process. The theory of such curves, is wellunderstood. Because subdivision curves also support multiresolution structure, various wavelet based compression algorithms apply to them. Consequently, subdivision curves offer a way to compress data related to curves in Euclidean space. But not all data live in Euclidean space (rotations, rigid motions, deformation tensors are examples), and the quantity of such data is proliferating. In "Multiscale representations of manifoldvalued data", Rahman, Rori, Stodden, Donoho, and Schroder, therefore, considered manifoldvalued subdivision curves, and their multiresolution structure. These curves are all based on corresponding linear subdivision curves, and Rahmanetal. conjectured that they enjoy the same regularity properties. In my talk, I will review linear subdivision curves, explain the generalization to manifoldvalued curves, and close with some surprising results concerning their regularity properties. The talk is based on joint work with Gang Xie and Thomas Yu. 
May 24, 2011, Thursday An introduction to DPG methods Abstract: We will discuss a new class of discontinuous PetrovGalerkin (DPG) methods that achieves stability by automatically finding stable pairs of trial and test spaces in a PetrovGalerkin framework. The key is that the use of discontinuous finite element spaces (as in DG methods) allows local computation of the test spaces that guarantee stability. (This is joint work with L. Demkowicz.) 
The IMA Postdoc Seminar meets at 11:1512:15 every Tuesday during the semester in Lind 305, except during IMA workshop weeks. This seminar provides opportunities for postdocs to give talks not necessarily related to the thematic program, and for invited IMA visitors (or U of MN faculty) to give talks on subjects of interest to the IMA postdocs. This year the postdoc seminar is organized by Tsvetanka Sendova and Erkan Tüzel.
Fall 2008  
September 16  Organizational meeting  
September 23  Xianjin Chen (IMA) "Two Stable Methods for Multiple Unstable Solutions to Semilinear Variational Elliptic Systems"  
September 30  No seminar (Workshop: Mathematical and Algorithmic Challenges in Electronic Structure Theory)  
October 7  Ridgway Scott (University of Chicago) "The Mathematical Basis for Molecular van der Waals Forces"  
October 14  Yongfeng Li (IMA) "Model reference control in the biological systems"  
October 21  Mark Herman (IMA) "BornOppenheimer Corrections Near a RennerTeller Crossing"  
October 28  No seminar (Workshop: MultiManifold Data Modeling and Applications)  
November 4  No seminar (Workshop:Development and Analysis of Multiscale Methods)  
November 11  Tsvetanka Sendova (IMA) "A Theory of Fracture Based Upon Extension of Continuum Mechanics to the Nanoscale"  
November 18  No seminar (Workshop:MixedInteger Nonlinear Optimization: Algorithmic Advances and Applications)  
November 25  Eray S. Aydil (University of Minnesota) "Challenges in Efficient and Inexpensive SolartoElectric Energy Conversion"  
December 2  Yunkyong Hyon (IMA) "Maximum Dissipation Principle for Numerical Methods in Complex Fluids"  
December 9  No seminar (Workshop:Solvation)  
December 16  Srividya Jeyaraman (IMA) "Time lags in signaling cascades"  
December 23  No seminar  
December 30  No seminar  
Spring 2009  
January 6  Mark Iwen (IMA) "Interpolation with Sparsity Assumptions: From Syphilis Testing to Sparse Fourier Transforms "  
January 13  No seminar (Workshop:Chemical Dynamics: Challenges and Approaches)  
January 20  Zhian Wang (IMA) "Multiscale approach in chemotaxis models"  
January 27  Jozsef Z. Farkas (University of Stirling) "Analysis of structured populations in aquaculture "  
February 3  Wei Xiong (IMA) "Hydrodynamics of particles immersed in a thermally fluctuating, viscous, incompressible fluid"  
February 10  Peter Hinow (IMA) "Predicting the Drug Release Kinetics of Matrix Tablets "  
February 17  Olivier Dubois (IMA) "Minimizing the energy for efficient linear solvers in reservoir simulation "  
February 24  Stephen Wiggins (University of Bristol) "Recent advances in the high dimensional Hamiltonian dynamics and geometry of reaction dynamics "  
March 3  No seminar (Workshop:Coherence, Control, and Dissipation)  
March 10  Heinz Siedentop (LMU Munich and IMA) "The Ground State Energy of Heavy Atoms: Relativistic Lowering of the Leading Energy Correction "  
March 17  Wei Xiong (IMA) "A Model for Liver Homeostasis Using a Modified Meanreverting OrnsteinUhlenbeck Process "  
March 24  No seminar (Workshop:Higher Order Geometric Evolution Equations: Theory and Applications from Microfluidics to Image Understanding)  
April 2  Mary Ann Horn (NSF) (rescheduled to 4:00 pm on Thursday) "To Fund or Not to Fund: Finding Funding Opportunities and the Challenges of Writing a Competitive Grant Proposal"  
April 7  Vasileios Maroulas (IMA) "Sequential Monte Carlo MultiObject Second Moment Approximation: An Application to Ecology"  
April 14  Christoph Ortner (University of Oxford) "Introduction to Quasicontinuum Methods: Formulation, Classification, Analysis"  
April 21  William Hancock (Penn State University) "Coordination of the two motor domains in the motor protein Kinesin2"  
April 28  Abdul Rehaman Moughal Shahi (Universite de Geneve) "II order Perturbation over Restricted Active Space Wavefunction : An Answer to Chemical Problems Requiring Larger Active Spaces"  
May 5  No seminar (Special Workshop: MOLCAS)  
May 12  Erkan Tuzel (IMA) "Motor mediated deformation of microtubules in living cells and gliding assays"  
May 19  No seminar (Workshop:Molecular Simulations: Algorithms, Analysis, and Applications)  
May 26  Xiaoyang Zhu (University of Minnesota) "The Coulomb Barrier in Excitonic Charge Separation" 
The IMA Postdoc Seminar meets at 11:1512:15 every Tuesday during the semester in Lind 409, except during IMA workshop weeks. This seminar provides opportunities for postdocs to give talks not necessarily related to the thematic program, and for invited IMA visitors (or U of MN faculty) to give talks on subjects of interest to the IMA postdocs. This year the postdoc seminar is organized by Erkan Tüzel and Jason Gower.
Fall 2007  
September 11  Organizational Meeting 
September 18  No seminar (Workshop: Mathematics of DNA Structure, Function and Interactions) 
September 25  Peter Hinow (IMA) "Inverse problems in nanobiology and analysis of an agestructured population model" 
October 2  Benjamin Nill (Freie Universität Berlin) "Fast food, square magic, and polyhedra" 
October 9  Timothy Newman (Arizona State University) "Modeling multicellularity: from cell rheology to gastrulation" 
October 16  Duane Nykamp (University of Minnesota) "Estimating connectivity in neuronal and other networks" 
October 23  Andy Stein (IMA) "Optimization in mathematical biology" 
October 30  No seminar (Workshop: RNA in Biology, Bioengineering and Nanotechnology) 
November 6  Deena Schmidt (IMA) "Waiting for k mutations: with applications to DNA regulatory sequence evolution and cancer" 
November 13  Zhian Wang (IMA) "Models for mesenchymal motion in tissue" 
November 20  No seminar 
November 27  Victor Barocas (University of Minnesota) "Multiscale mechanical modeling of bioartificial tissues" 
December 4  Hannah Callender (IMA) "Using mathematical modeling to make testable predictions of cellular signaling pathways" 
December 11  David Odde (University of Minnesota) "Models for mitosis: microtubules and motors" 
December 18  Erkan Tüzel (IMA) "Mesoscopic model for the fluctuating hydrodynamics of binary and ternary mixtures" 
December 25  No seminar (Christmas Day) 
Spring 2008  
January 1  No seminar (New Year's Day) 
January 8  Benjamin Stottrup (Augsburg College) "An experimental perspective on miscibility phase transitions and their biological applications" 
January 15  No seminar (Workshop: Protein Folding) 
January 22  Satish Kumar (University of Minnesota) "Brownian dynamics simulations of polymer behavior in nanofluidic and microfluidic systems" 
January 29  Milena Hering (IMA) "Syzygies of algebraic varieties" 
February 5  Laura Lurati (IMA) "Design under uncertainty using stochastic collocation" 
February 8  Anton Leykin (IMA) "Applications of numerical algebraic geometry" 
February 12  Ludovica CottaRamusino (IMA) "Looping probability densities of elastic rods" 
February 19  Sujeet Bhat (IMA) "Homogenization of a nonlinear elliptic boundary value problem related to corrosion modeling" 
February 26  David Umulis (University of Minnesota) "Computational analysis of BMPmediated embryonic patterning in Drosophila melanogaster" 
March 4  No seminar (Workshop: Organization of Biological Networks) 
March 11  Juan Latorre (Rensselaer Polytechnic Institute) "Effective transport properties of stochastic biophysical models" 
March 18  Kevin Dorfman (University of Minnesota) "TaylorAris dispersion in entropy barriers" 
March 25  Joachim Mueller (University of Minnesota) "Quantifying the assembly of optically encoded proteins with fluorescence fluctuation spectroscopy" 
April 1  Chad Myers (University of Minnesota) "Discovering biological networks by integrating diverse genomic data" 
April 8  Theoden Netoff (University of Minnesota) "Dynamical approaches to understanding epilepsy" 
April 15  Banu Baydil (Rensselaer Polytechnic Institute) "Parameterization for mesoscale ocean transport through random flow models" 
April 22  No seminar (Workshop: Design Principles in Biological Systems) 
April 29  Thomas Hillen (University of Alberta) "A user's guide to PDE models for chemotaxis" 
May 6  Peter Hinow (IMA) "A model for transfer phenomena in structured populations" 
May 13  No seminar (Workshop: Stochastic Models for Intracellular Reaction Networks) 
May 20  Duncan Clarke (Univeristy of Minnesota) "The cohesin ring is required for centrosome integrity" 
May 27  No seminar (Workshop: Quantitative Approaches to Cell Motility and Chemotaxis) 
Fall 2006  
September 12 
Organizational meeting 
September 19 
No seminar ("Algorithms" workshop) 
September 26  No seminar (talk of Peter Lax) 
October 3 
No seminar ("Negative index materials" workshop) 
October 10  Hannah Markwig (IMA): Introduction
to tropical geometry 
October 17  Anders Jensen (Århus): Tropical
varieties (slides) 
October 24  No seminar ("Software" workshop) 
October 31 
Jan Verschelde (UIC): Polyhedral
Homotopy Methods to Solve Polynomial Systems 
November 7 
Jason Gower (IMA): Algebraic
torusbased cryptography (slides) 
November 14  John Voight (IMA): Zeta
functions of varieties over finite fields 
November 21  No seminar (CANCELLED) 
November 28  Bjarke Roune (Århus): The F4 Algorithm  Speeding Up Groebner Basis Computation Using Linear Algebra 
December 5 
Ben Howard (IMA): Equations for the moduli space of n labelled points on the projective line 
December 12  John Voight (IMA): Computing
zeta functions of varieties over finite fields 
Spring 2007:  
January 16  No seminar ("Optimization" workshop) 
January 23  Jiawang Nie (IMA) On the quality bound of low
order SOS relaxations 
January 30  Milena Hering (IMA): Syzygies
of toric varieties 
February 6 
Philipp Rostalski (ETH Zurich): Characterization and Computation of
RealRadical Ideals using Semidefinite Programming Techniques 
February 13  Tristram Bogart (University of Washington): The small Chvatal rank of an integer matrix 
February 20  Hannah Markwig (IMA): Counting rational curves in P^2 using Kontsevich's formula 
February 27  Michael Kerber (Technische Universitaet Kaiserslautern): Counting rational curves in P^2 using tropical geometry 
March 6 
No seminar ("Biostats" workshop) 
March 13  Laura Matusevich (Texas A&M):Combinatorics of binomial primary decomposition 
March 20  Daniel Robertz (RWTH Aachen University): Janet's algorithm for modules over polynomial rings 
March 27  Gennady Lyubeznik (IMA, University of Minnesota): The minimum number of settheoretic defining equations of algebraic varieties 
April 3 
Josephine T. Yu (Berkeley): Tropical
Implicitization and Elimination 
April 10  Uwe Nagel (University of Kentucky): Hilbert functions of standard graded algebras 
April 17  No seminar ("Complexity" workshop) 
April 24  Anton Leykin (IMA): F...? 
May 1 
Ben Howard (IMA): Weight varieties, and projective point set 
FALL 2005:  
September 20  No seminar (Radar and Optical Imaging tutorial) 
September 27  JungHa An (IMA): A Variational PDE Based Level Set Method for a Simultaneous Segmentation and NonRigid Registration 
October 4  Changfeng Gui (University of Connecticut): Some entire solutions related to phase transitions 
October 11  Evgeniy Bart (IMA): Object recognition and classification with limited training data 
October 18  No seminar (Imaging from Wave Propagation workshop) 
October 25  Anne Gelb (Arizona State): High Order Reconstruction Methods for Piecewise Smooth Functions (full talk here). 
November 1  Steven Damelin (IMA and Georgia Southern): Numerical integration, Energy and Weighted Approximation 
November 8  No seminar (Frontiers in Imaging workshop) 
November 15  Taufiquar Khan (Clemson University): Inverse Problem in Refractive Index Optical Tomography 
November 22  No seminar (Thanksgiving Week) 
November 29  Alison Malcolm (IMA): Multiple Scattering: Exploiting it Lab Data and Attenuating it in Seismic Data 
December 6  No seminar (Integration of Sensing and Processing workshop) 
December 13  Anne Gelb (Arizona State): Region Extraction Using Edge Detection and BSplines 
SPRING 2006:  
January 10  No seminar (3D Image Analysis workshop) 
January 20 (Friday)  Afra Zomorodian (Stanford): Persistent Homology and its Applications 
January 24  Douglas Arnold (IMA): Finite element exterior calculus and its applications 
January 31  ChangOck Lee (KAIST): A variational approach to blending based on warping for nonoverlapped images 
February 7  No seminar (Film Editing and Restoration workshop) 
February 14 (changed from initial announcement)  Martin Welk (Saarland University): Analysis of SpaceDiscrete Image Filters by Dynamical Systems 
February 21  Brad Lucier (Purdue): Image processing using (mostly) functional arrays 
February 28  Gloria Haro Ortega (IMA): Image restoration: irregular to regular sampling, deconvolution, denoising and zooming 
March 7  No seminar (Natural Images workshop) 
March 14  No seminar (Spring break) 
March 21  Arnd Scheel (IMA): Periodic patterns: modulations, perturbations, and bifurcation 
March 28  Francisco BlancoSilva (Purdue University): Curvelets and Approximation Theory 
April 4  No seminar (Shape Spaces workshop) 
April 11  
April 18  No seminar, Scientific Visualization 2006 symposium 
April 25  Steven Damelin (IMA and Georgia Southern): Some open problems in Dimension Reduction, Inverse Scattering and Power Systems 
May 2  Tatiana Soleski (IMA): Prolate Spheroidal Sampling in Computerized Tomography 
May 9  Peter Philip (IMA and Corning): A Quasistatic Crack Propagation Model Allowing for Cohesive Forces and Crack Reversibility Using Local Energy Minimization 
May 16  No seminar (SIAM Imaging conference) 
May 23  No seminar (Visual Learning and Recognition workshop) 
May 30  TBD 
The seminar talks of previous years are archived here.
Complete List of Industrial Postdoc Seminar
The IMA Postdoc Seminar is intended for expository talks from the visitors on their current research interests. For the year 2003  2004 it is organized by Antar Bandyopadhyay and Gerard Awanou. The seminar meets at 409 Lind Hall on Tuesdays at 11:15 only on the weeks when there is no IMA workshop or tutorial. If you are interested to give a talk please contact the organizers at antar@ima.umn.edu or awanou@ima.umn.edu. Note that Balaji Gopalakrishnan was a coorganizer from OctoberDecember, 2003.


May
18, 2004 (Tuesday)
11:15 am12:15 pm, Room 409, Lind Hall
Speaker:Professor
Scot Adams (IMA Associate Director)
Title: Some Remarks on Finite Element Method
April
20, 2004 (Tuesday)
11:15 am12:15 pm, Room 409, Lind Hall
Speaker: Professor
Yuhong Yang (Iowa State University)
Title: Multiarmed Bandits with Covariates
Abstract: Suppose that there are K (K>1) possibly biased coins available for play. At each time, you can choose one and only one coin to flip, and win a dollar if you get a head and receive nothing otherwise. Your goal is to obtain as much money as possible. Such a problem is called a multiarmed bandit problem.
Bandit problems have applications in clinical trials, scheduling, and automated problemsolving in machine learning. In the literature, covariates, while available in most applications, are rarely considered. In this talk, I will present a nonparametric approach with a randomization technique for effectively utilizing the information in the covariates. A consistency property of the proposed approach is established.
April
6, 2004 (Tuesday)
11:15 am12:15 pm, Room 409, Lind Hall
Speaker: Professor
Peter Bank (Humboldt University of Berlin, Germany)
Title: American Options, FiniteFuel Problems, and the MultiArmed Bandit
Abstract: We show how the optimal stopping, certain singular control problems, and dynamic allocation problems all can be reduced to the same kind of stochastic representation problem. This unifying approach allows for some new (and comparably easy) proofs for classical results such as Gittins' index theorem, and also offers some new insights into the common structure of these optimization problems.
March
23, 2004 (Tuesday)
11:15 am12:15 pm, Room 409, Lind Hall
Speaker: Dr.
Noam Berger (California Institute of Technology)
Title:
Nonuniqueness for Specifications in 1^{2+}
Slides: pdf
Abstract: Keane, Berbee and others have studied the question of which specifications (aka gfunctions) admit a unique Gibbs measure. Bramson and Kalikow constructed the first example of a regular and continuous specification which admits multiple measures. For every p > 2, we construct a regular and continuous specification, whose variation is in ^{ p}, that admits multiple Gibbs measures. This shows that a recent condition of Oberg and Johansson is tight. Joint work with Christopher Hoffman and Vladas Sidoravicius.


March
16, 2004 (Tuesday)
11:15 am12:15 pm, Room 409, Lind Hall
Speaker: Professor
Ilze Ziedins (Department of Statistics, University
of Auckland, New Zealand, ilze@stat.auckland.ac.nz)
Title: Optimal Routing in Parallel Tandem Queues with Loss
Abstract: In many queueing systems, individually optimal and socially optimal polices (whether for admission or routing) can be very different. This talk will look at a system of parallel finite tandem queues with loss. For this system, when customers choose routes that minimize their individual loss probability it can sometimes be optimal to choose queues with more customers already present and/or with greater residual service requirements (where preceding customers are further from their final destination). These individually optimal policies will be compared with socially optimal routing policies obtained in the limit as the number of possible routes becomes large. This is joint work with RuShuo Sheu and Scott Spicer.
February
24, 2004 (Tuesday)
[Joint with the Random Matrix Seminar]
12:45  2:15 at 570 Vincent Hall
Speaker: Dr.
Tim Garoni (IMA Postdoc)
Title: Absolute Moments of Products of Characteristic Polynomials, and Impenetrable Bosons  II
Abstract: I will discuss the links between the moments of products of characteristic polynomials of random matrices, with the asymptotics of a certain class of Hankel determinants, and also with certain 1D quantum models. Some exact results from the literature will be discussed, as well as some interesting and as yet unproven conjectures.
February
17, 2004 (Tuesday)
[Special meeting jointly with Random Matrix Seminar]
12:45  2:15 at 570 Vincent Hall
Speaker: Dr.
Tim Garoni (IMA Postdoc)
Title: Absolute Moments of Products of Characteristic Polynomials, and Impenetrable Bosons  I
Abstract: I will discuss the links between the moments of products of characteristic polynomials of random matrices, with the asymptotics of a certain class of Hankel determinants, and also with certain 1D quantum models. Some exact results from the literature will be discussed, as well as some interesting and as yet unproven conjectures.
February
17, 2004 (Tuesday)
[Joint with the Brown
Bag Seminar]
11:15 am12:15 pm, Room 409, Lind Hall
Speaker: Dr.
Lili Ju (IMA Postdoc)
Title: Finite Volume Method on Spherical Voronoi Meshes
Abstract: We first develop and analyze a finite volume scheme for the discretization of partial differential equations on the sphere; the scheme uses Voronoi tessellation of the sphere. For a model convectiondiffusion problem, the finite volume scheme is shown to produce firstorder accurate approximations with respect to a meshdependent discrete firstderivative norm. Then, we introduce the notion of constrained centroidal Voronoi tessellation (CCVTs) of the sphere; these are special Voronoi tessellation of the sphere for which the generators of the Voronoi cells are also the constrained centers of mass, with respect to a prescribed density function, of the cells. After discussing an algorithm for determining CCVT meshes on the sphere, we discuss and illustrate several desirable properties possessed by these meshes. In particular, it is shown that CCVT meshes define very high quality uniform and nonuniform meshes on the sphere. Finally, we discuss, through some computational experiments, the performance of the CCVT meshes used in conjunction with the finite volume scheme for the solution of simple model PDEs on the sphere. The experiments show, for example, that the CCVT based finite volume approximations are secondorder accurate if errors are measured in discrete L^{2}norms.February
3, 2004 (Tuesday)
11:15 am12:15 pm, Room 409, Lind Hall
Speaker: Professor
Greg Rempala (University of Louisville, visiting
IMA)
Title: Bootstrapping Parametric Models of Mortality
Abstract: We consider a general problem of modeling a mortality law of a population of failing units with some parametric function. In this setting we define a mortality table of crude rates as a statistical estimator with multinomial distribution and show its consistency as well as asymptotic normality. We further derive the statistical properties of parameter estimators in a parametric mortality model based on a weighted square loss function. We use the obtained results to study consistency and appropriateness of the parametric bootstrap method in our setting. We derive the conditions on the assumed parametric mortality law and the loss function, under which the bootstrap is consistent for estimating the model parameters, their standard errors and corresponding confidence intervals. We apply our results to a model of Aggregate US Mortality Table based on a so called mixture of extreme value distributions suggested by Carriere (1992).
January
27, 2004 (Tuesday)
11:15 am12:15 pm, Room 409, Lind Hall
Speaker: Dr.
Olga Brezhneva (IMA Postdoc)
Title: Modified Methods for Nonlinear Optimization and Complementarity Problems in the Absence of Strict Complementarity
Abstract: For nonlinear constrained optimization and complementarity problems, we consider the case when the strict complementarity condition does not hold. In this situation, only a linear rate of convergence can be guaranteed for most classical algorithms. In this talk, we consider a LagrangeNewton method and the modified Lagrangian method for nonlinear constrained optimization, and propose an approach that allows us to obtain modifications of these methods. The obtained modifications attain superlinear convergence even when the strict complementarity condition does not hold and subsume the case when this condition holds. Moreover, the proposed approach to modifying the methods can be applied to a variety of problems with some kind of degeneracy. We illustrate this by constructing a method for nonlinear complementarity problems in the absence of strict complementarity.
January
20, 2004 (Tuesday)
11:15 am12:15 pm, Room 409, Lind Hall
Speaker: Dr.
Gerard Awanou (IMA Postdoc)
Title: Trivariate Spline Approximations of 3D NavierStokes Equations
Abstract: We present numerical approximations of the 3D steady state NavierStokes equations in velocitypressure formulation. We use trivariate splines of arbitrary degree d and arbitrary smoothness r < d. Using functional arguments, we derive the discrete NavierStokes equations in terms of Bcoefficients of trivariate splines over a tetrahedral partition of any given polygonal domain. Smoothness conditions, boundary conditions and the divergencefree conditions are enforced through Lagrange multipliers. The discrete equations are solved by a variant of the augmented Lagrangian algorithm for which we prove a linear algebraic convergence rate. We have implemented this approach in MATLAb and present numerical evidence of the convergence rate as well as experiments on the lid driven cavity flow problem.


December
9, 2003 (Tuesday) 11:15
am12:15 pm, Room 409, Lind Hall
Speaker: Dr.
Soohan Ahn (Seoul National University
(SRCCS), Visiting IMA)
Title: Transient Analysis of Fluid Flow Models via Stochastic Coupling to a Queue
Abstract: Markovian fluid flow models are used extensively in performance analysis of communication networks. They are also instances of Markov reward models that find applications in several areas like storage theory, insurance risk and financial models, and inventory control. This paper deals with the transient analysis of such models. Given a Markovian fluid flow, we construct on the same probability space a sequence of queues that are stochastically coupled to the fluid flow in the sense that at certain selected random epochs the distribution of the fluid level and the phase (the state of the modulating Markov chain) is identical to that of the work in the queue and the phase. The fluid flow is realized as a stochastic process limit of the processes of work in the system for the queues, and the latter are analyzed using the matrixgeometric method. These in turn provide the needed characterization of transient results for the fluid model.
December
2, 2003 (Tuesday) 11:15
am12:15 pm, Room 409, Lind Hall
Speaker: Dr.
Amir Niknejad (University of Illinois at Chicago,
visiting IMA)
Title: Missing Data Imputation for Gene Expression Arrays : An Algebraic Approach
Abstract: We introduce an algebraic framework for missing data imputation through Fixed Rank Approximation Algorithm(FRAA). Preliminary test indicates that FRAA is Robust and might be a feasible alternative in cases when the number of columns are small in Genome  Wide Matrix.
November
25, 2003 (Tuesday)
11:15 am12:15 pm, Room 409, Lind Hall
Speaker: Dr.
Karen Ball (IMA Postdoc)
Title: Factors
of Processes on Groups and Graphs
Slides: pdf
Abstract: Let X be a set with a group G acting on it. We will consider the question of when there exists a Ghomomorphism between two i.i.d. processes which are indexed by X. In the case where X=G, we will see that amenable and nonamenable groups are characterized by very different behavior with respect to this question. We will also consider the case where X is a graph. The proofs involve applications of interesting ideas from percolation theory.
November
11, 2003 (Tuesday)
11:15 am12:15 pm, Room 409, Lind Hall
Speaker: Professor
Thomas G. Kurtz (Center for Mathematical Sciences,
University of WisconsinMadison) kurtz@math.wisc.edu
Title: Introduction
to Martingale Problems
Slides: pdf
Abstract: The generator for a Markov process is a linear operator that characterizes infinitesimally the evolution of the distribution of the process. Classically, the HilleYosida theory of operator semigroups was used to connect the Markov process to its generator. In the context of diffusion processes, Stroock and Varadhan showed that Markov processes can be characterized by the requirement that certain functionals of the process, determined by the generator, must be martingales, that is, the Markov process can be characterized as the unique solution of a martingale problem. For example, Brownian motion is the unique solution of the martingale problem for the Laplacian.
Martingale problems provide a powerful approach to the study of Markov processes. The basic theory of martingale problems will be reviewed and several illustrative examples of its application will be given.
NOTE: This talk is intended to give some useful background for the November 12 talk in the Complex Systems Seminar.
October
28, 2003 (Tuesday)
11:15 am12:15 pm, Room 409, Lind Hall
Speaker: Professor
Mohammad Kazim Khan (Kent State University)
Title: On SPRT and CUSUM Procedures and Some Open Problems Slides: pdf
Abstract: Let Y_{1}, Y_{1}, ··· , Y_{1} be iid random variables with a distribution function (df) F_{0}(y), and let Y_{} , Y_{+1} , ··· be iid random variables with a df F_{1}(y), where is an unknown time index of a change in distribution with the change in parametric value. For a suitable function let X_{j} = (Y_{j}) denote some convenient data reduction or it may be defined by certain optimality consideration such as the Sequential Probability Ratio Test. For detecting a changepoint in the distribution, Page (1954) defined his famous cusum (cumulative sum) procedure. From the inherent renewal property of the cusum, Page noted that EN = EM/P(S_{M} h), where N is the cusum stopping rule and M is the SPRT and S_{n} represents the partial sum of the information X_{1}, X_{2},... . The constant h is the trigering constant for the cusum. This link between the SPRT and the cusum is quite useful in approximating and/or evaluating EN. However, a deeper connection between N and M is known that I will try to present. The purpose of this exposition is to further exploit such a relationship between N and M to study the properties of some one sided and two sided cusums with several applicable examples. We will see how exact results can be computed in discrete time settings. There are some open problems which I will outline.
October 14, 2003
(Tuesday)
11:15 am12:15 pm, Room 409, Lind Hall
Speaker:
Professor Simon Morgan (Visiting Assistant Professor,
Department of Mathematics, University of Minnesota)
Title: Applications of Geometric Measure Theory to Variational Problems with Surfaces
I will introduce the objects of geometric measure theory; rectifiable sets, currents and varifolds. Key examples of each and their associated topologies will be given.
The primary application of geometric measure theory was minimal surface theory, but I will also explain my own research into geometric measure theory techniques and my motivating applications.
Title:
Tree Structured Prognostic Model for Hepatocellular Carcinoma
Slides: html pdf ps ppt
Abstract: Recent progress in both diagnostic and therapeutic technique of hepatocellular carcinoma appears to improve the prognosis. The purpose of this study was designed to evaluate the prognosis of HCC in relation to treatment methods and their affecting factors by the tree structured model. This paper attempt to identify and quantify the effect of prognostic factors namely, patient characteristics that related to the prognosis of Hepatocellular Carcinoma. Our proposed survival tree model uses statistical tests for the modified CoxSnell residuals derived from fitting a Cox proportional hazards model, and therefore it can detect curvature of covariates and interactions among them. As a result, split covariate selection bias is negligible. Furthermore, our proposed piecewise constant modeling can be generalized with piecewise multiple linear modeling without much computational burden.
January 15  Gregory Duane, IMA Postdoctoral Associate  Applications of Synchronized Chaos, Part II  In Part II, I will start by reviewing the many forms of loose coupling of chaotic systems that give rise to synchronized motion, to argue for the ubiquity of the phenomenon, as illustrated by predicted relationships between largescale weather phenomena at distant points on the globe. The same phenomenon of synchronization of fluiddynamical channel systems, coupled through only mediumscale Fourier components, also forms the basis for a new approach to data assimilation with an interpretation of one system as "truth," and the other as "model". It is argued that the sufficiency of coupling the mediumscale components for synchronizing the entire systems follows in part from the existence of inertial manifolds for the two systems separately. I will conclude by suggesting that the synchronizationbased approach to data assimilation is in harmony with a theory that human consciousness is a manifestation of brief periods of synchronized activity among widely spaced neurons. Related neuralsynchronization models of auditory segmentation in the cocktailparty problem motivate potential applications to image segmentation and combinatorial optimization problems generally. 
May 28  Gregory Duane, IMA Postdoctoral Associate  Applications of Synchronized Chaos, Part I  While the synchronization of regular oscillators with limit cycle attractors is ubiquitous in Nature, the synchronization of loosely coupled chaotic oscillators has been studied only recently. In this twopart talk, I will discuss applications and potential applications of synchronized chaos to the foundations of quantum theory, atmospheric dynamics, and biologicallymotivated neural network architectures. In Part I, starting with a review of Bell's Theorem, I will explore an analogy between quantum cryptography and a form of cryptography based on synchronized chaos. In one such scheme, two variableorder Generalized Rossler Systems will synchronize when coupled through only one of many variables. But in the infiniteorder limit, the dynamical parameters of the driving system cannot be extracted in finite time. The phenomenon supports the possibility of an interpretation of quantum mechanics in which quantum nonlocality is mediated by supraluminal connections that are real, but perfectly disguised. 
May 20  Don Aronson, Director of Postdoctoral Program in the IMA  Transmission in Inhomogeneous Excitable Media  In the first part of the talk, I will review some aspects of the transmission of signals in a homogeneous excitable medium such as a nerve axon. There is a hierarchy of mathematical models ranging from the ultra precise (and Nobel Prize worthy) HodgkinHuxley system to the McKeanFitzHughNagumo caricature. I will be mostly concerned with the latter. The second part of the talk will be a highly speculative discussion of what happens to transmission when one introduces inhomogeneities in the form of passive gaps. The object here is to pose some potentially interesting problems. 
May 13  Beth Allen, Professor of Economics, University of Minnesota  Some Theoretical Approaches to Product Development  This talk will survey some recent research focusing on the early stages of the product life cycle: Design (choice of product or product line) and manufacturing (in contrast to the later stages in the supply chain of, for instance, scheduling, distribution, inventory strategy, postpurchase service/maintenance/repair, and endoflife reuse/recycling/disposal). First, the mathematical structure of the design space (in particular, for homogeneous geometric objects or shapes, so that the set of potential product designs consists of equivalence classes of nonempty closed subsets of a Euclidean space), will be examined. Implications for constrained optimization will be analyzed. Relations to quality control and manufacturing will be considered. 
April 22  Luis Goddyn, Mathematics, Simon Fraser University  The WorstCase Euclidean Traveling Salesman Problem  How should a finite set X of points be arranged within a finite region R in Euclidean dspace, so as to maximize the length L(X) of a TSP tour through X? It turns out that, if the size n of X is large, then the shape of R becomes irrelevant, and if R has unit volume, then the assymptotic growth is L(X) ≈ α_{d} n^{(d1) /d} (α_{d} depends only on the dimension d). We discuss methods and issues involved in estimating the value of the fundamental coefficient α_{d}. This talk involves a mix of geometry, sphere packing, quantizers, heuristics and algorithm analysis. 
April 15  Douglas N. Arnold, IMA Director  Talking Math to People Who Don't Know Any  Mathematicians are often criticizedby themselves and othersfor being incapable of describing their work to nonmathematicians. In this seminar I will discuss ways to communicate mathematics to the public using as a case study a talk I gave recently on optimization for the IT Quarterly, a "forum for friends of the Institute of Technology." 
April 1  Tamon Stephen, IMA Postdoctoral Associate  The LiftandProject Approach to Integer Programs  In this survey talk, we discuss the ``lift and project'' approach for solving 01 integer programs. Our main example is the method of semidefinite matrix relaxations proposed by Lovasz and Schrijver. 
March 25  C. Roos, Delft University of Technology, Netherlands and IMA visitor, March 2003  What is special with the logarithmic barrier function in optimization? Slides:PDf  After its introduction by Frisch in 1955, the logarithmic barrier function (LBF) has played a major role in the field optimization. The revolutionary developments of the past two decades in this field, which gave rise to the subfield of interiorpoints (IP) methods, has reemphasized its importance. The search directions in all stateoftheart IPsolvers for linear, and also for secondorder cone and semidefinite optimization problems are explicitly or implicitly based on an LBF, and the analysis of these methods strongly depends on properties of such functions. Other barrier functions have been proposed, but both from a theoretical and computational viewpoint LBF's always were winning, at least surviving. It has often been asked what makes LBF so special. In this talk we deal with this question. We focus on primaldual methods for linear optimization. It is probably for the first time that alternative barrier functions have been found that in some cases provide better theoretical complexity results than the LBF. The results can be extended to other conic optimization problems; it is an open question if the new barrier functions can be adapted to primal methods and dual methods, respectively.
This is joint work with Y. Bai (Shanghai University, China) and M. Elghami (Delft University of Technology, Netherlands). 
February 25  Luis N. Vicente, Department of Mathematics, University of Coimbra, Portugal IBM T.J. Watson Research Center, Yorktown Heights, New York and IMA, University of Minnesota  Space Mapping: Models, Algorithms and Applications  A number of new techniques have been developed to deal with optimization problems which involve expensive function evaluations from simulation or experimentation.
One of the techniques that has been recently considered in the engineering community is the socalled spacemapping approach. Space mapping assumes the existence of two models for the same physical phenomenon: a fine model, accurate and expensive, and a coarse model, significantly cheaper and considerably less accurate. The idea behind space mapping is to "construct" a mapping between the finemodel space of parameters or variables and the coarsemodel space that allows to defer the optimization process to the coarse model, where most function evaluations should take place. Spacemapping techniques are typically iterative as the mapping is unknown a priori and it is calculated for a sequence of points in the fine space. One of the goals of this talk is to organize some of the model and algorithmic aspects of space mapping in a mathematical framework which allows us to look at the properties of the space mapping and to fit the convergence analysis of the algorithms into the existence convergence theory of nonlinear optimization. We will also introduce new ways of building the space mapping and deriving the algorithms. We will report recent numerical testing with the application of the spacemapping methodology to optimal control problems governed by partial differential equations. We will show how the space mapping technique can be used in the context of this class of problems. This is joint work with Michael Hintermuller. 
February 18 (Tuesday)  Moritz Diehl, University of Heidelberg, Germany & IMA visitor JanuaryMarch 2003  Nominal Stability for Nonlinear Model Predictive Control  Nonlinear Model Predictive Control (NMPC) is a technique for the design of feedback controllers that works by online minimization of an objective depending on the predicted system behaviour. Typically, the system shall be kept in some desired steady state, and the objective penalizes deviations from it. A basic requirement for the resulting feedback controller is that it stabilizes the system at the given operating point at least in the case that the model is perfect  this property is called nominal stability.
In the talk, I will introduce the basic ideas underlying the standard stability proofs for NMPC. I will shortly discuss how the stability problem changes if numerical optimization errors are taken into account, and give a sketch of recent results to address this problem. 
February 11  Peh Ng, Dept of Mathematics, University of Minnesota  Morris & IMA Visitor 20022003  A Commodity Family Extended Formulation Approach to Solving Uncapacitated Fixed Charge Network Flow Problems  In general, uncapacitated fixed charge network flow problem, (UF), is NPHard. However, previous research on a few NPHard problems has shown that improved linear programming relaxations can be obtained by using extended reformulations. In this research, we develop a theory of extended formulations for (UF) by reformulating these problems in terms of an extended variable set corresponding to flow commodities defined by arbitrary demand subsets. In particular, we show how to produce an extended formulation for any suitable commodity family and isolate simple axioms characterizing the families that yield the most useful reformulations. 
February 4  Michael J.D. Powell, University of Cambridge  Radial Basis Function Methods for Global Optimization  PDF PS 
January 28  John Dennis, Rice University and the IMA  Working With Industry or My Life as an Evangelist  In this talk, I will discuss my work with various industrial groups, including my first real experience as a consultant to the National Bureau of Economic Research starting in the early `70s. I will try to explain the research each collaboration motivated and how it affected my work with graduate students. I will also try to draw some conclusions on how to collaborate successfully with industrial or semiindustrial groups  in case you might want to try it.
My hope is to convince you that you can do interesting work and help the rather dismal image of mathematicians by seeking such collaborations yourself. To this end, there will not be many technical details, and I hope for an interactive presentation rather than a lecture. . 
January 21  Dacian N. Daescu, IMA Postdoctoral Associate  Adjointbased Techniques for the Analysis of Largescale Uncertain Systems  Despite the increased complexity in the model representations, it is often the case that comprehensive dynamical models show poor results when compared to observational data. To address this problem we need to consider various factors that may contribute to the uncertainties in the analysis of dynamical systems. Data assimilation techniques (e.g. Kalman filter, variational methods) combine observations of a dynamical system with a dynamical model of the system to provide an optimal estimate of the evolving state of the system.
In fourdimensional variational data assimilation (4DVar) a minimization algorithm is used to find the set of control variables such that an optimal fit between the model forecast and observations, scattered in time, is achieved. For largescale models, the minimization of the cost functional is a very intensive computational process. At the same time, the development and validation of dynamical models require a systematic sensitivity analysis to evaluate the effects of parameter variations on model prediction. The adjoint modeling is presented as an efficient tool to evaluate the sensitivity of a scalar response function with respect to initial conditions and model parameters. Using the adjoint method, the necessary gradient may be computed at the expense of few function evaluations, making the optimization process very efficient. The sensitivity field obtained through a single backward integration of the adjoint model may be used to identify parameters that play an essential role in determining the model forecast. The intimate connection between measurements and models can be illustrated through the example of largescale field experiments. The difficulty in planning such experiments lies in the fact that the features of interest are usually transient in nature. Expensive fielddeployed resources (facilities and people) can be employed/utilized more effectively and science success can be maximized by an optimal allocation of the observational resources. The problem of optimal deployment of additional observational resources is presented and adjointbased adaptive observations strategies using the singular value decomposition and gradient sensitivity are discussed in detail. Numerical illustrations are shown for nonlinear chemical reactions systems and atmospheric circulation models. 
December 3  Maurice Queyranne, Professor of Management Science, Faculty of Commerce and Business Administration, University of British Columbia, Vancouver, Canada  On Optimum SizeConstrained Set Partitions with Submodular Costs PDF  Given a finite set N, a function f associating a cost f(S) to each subset S of N, and integers h and k, we consider the problem of finding a partition of N into at least h and at most k nonempty parts S1,..,Sm, and with minimum total cost f(S1)+...+ f(Sm). Such problems arise in VLSI design (netlist partitioning), clustering, statistical mechanics (the Potts model of spin systems), network design, and graph connectivity. When the cost function f is submodular, we identify important cases that can be solved to optimality in polynomial time in the value oracle model. We also present a simple approximation algorithm with a performance guarantee better than 2 for the case when f is also symmetric (i.e., every subset has the same cost as its complement, as happens with network and hypergraph cuts). This approximation algorithm is purely combinatorial and uses O(N^4) oracle calls. Its analysis relies on the existence of cut trees (aka GomoryHu trees) for symmetric submodular set functions. Some of the results presented in this talk extend to general (symmetric) submodular functions earlier results known only for network cut functions. 
November 26  Lili Ju, IMA Industrial Postdoc  Some Topics on Centroidal Voronoi Tessellations  Centroidal Voronoi tessellations (CVT) are Voronoi tessellations of a region such that the generating points of the tessellations arealso the centroids of the corresponding Voronoi regions. Such tessellations and their extension for general surfaces are of use in very diverse applications, including data compression, clustering analysis, cell biology, territorial behavior of animals, optimal allocation of resources, grid generations and optimization, meshless computing, and interpolation. Some detreminstic and probabilistic algorithms for computing CVTs will also be discussed. 
November 19  Marshall Hampton, NSF Postdoctoral Fellow, School of Mathematics, U of M  Celestial Mechanics: Recent Progress and Open Problems  In the last few years there have been some exciting developments in celestial mechanics. A new type of orbit (a "choreography") in the 3body problem was discovered which leads to many open questions. The study of relative equilibria in the nbody problem has blossomed and led to applications in energy efficient orbits for interplanetary spacecraft. Several weeks ago a solution was announced for the simplest case of a longstanding conjecture ("Saari's Conjecture") in the 3body problem. An attempt will be made to survey some of these results, applications, and open problems. 
November 5  Michael Ball, R H Smith School of Business and Institute for Systems Research University of Maryland  Mathematical Models for Supporting AvailabletoPromise (ATP) Slides:html pdf  The Available to Promise (ATP) business function is the set of capabilities that support responding to customer order requests. Traditionally ATP refers to a simple database lookup into the Master Production Schedule. With the advent of ebusiness, maketoorder production and high variety product offerings, ATP functionality has become a critical component of many business? strategies and also now requires much more complex model and IT support. In this talk, we provide an overview of ATPrelated research and of ATP business practice. We classify ATP research into two categories: pushbased models, which allocate resources and prepare information based on forecasted demand and pullbased models, which generate responses to actual customer orders. A variety of relevant research will be covered, including mixed integer programming models for resource allocation, stochastic models of uncertain customer demand, online scheduling algorithms for generating order delivery dates and strategies for searching for available inventory. 
October 29  Daniel Kern, IMA Postdoctoral Associate  Multispecies Competition and Traveling Waves  The consideration of spatial factors in ecological modeling has led to a variety of interesting problems in the current literature. Besides better explaining the development of biological phenomena, the resulting models can lead to interesting mathematics. Here, the spread of two invasive plant species and the corresponding replacement of a single native species is examined as a competition model with spatial considerations. The general model is a system of three nonlinear reactiondiffusion equations of the LotkaVolterra type. A model is developed for a specific case involving cottonwoods and two invasive plants in New Mexico. The existence of a traveling wave solution is then examined, leading to possible restrictions on the propagation speed of the exotic species. 
October 22  Lisa Evans, IMA Postdoctoral Associate  An Overview of Gomory's Group Approach to Solving Integer Programs  This talk will give an overview of Gomory's group approach to solving integer programs, including some of the key theorems. It will also describe how facets of master cyclic group problems can be used to generate cutting planes for general IP's. A related method that generates cutting planes from piecewiselinear subadditive functions that approximate the facets of master cyclic group problems will also be presented. Some new classes of facets for the master cyclic group problem will be described, as well as preliminary computational results using subadditive functions to generate cutting planes. 
October 17  Igor Vasilév,Universite di Salerno, Italy/ Institute of System Dynamics and Control Theory, Russia  Computational Experience with LargeScale pMedian Problems  Given a directed graph, the pMedian problem consist of determining p nodes (the median nodes) minimizing the total distance to the other nodes of the graph. We present a BranchandCut algorithm yielding provably good solutions for instances up to 3795 nodes of complete graphs, proving in most the cases their optimality. The key ingredients of our approach are: lagrangian relaxation, a simple procedure to choose the "promising variables," preprocessing, a columnandrow generation strategy to solve LPrelaxation, cutting planes. 
October 8  Miaojung Yvonne Ou, IMA  Object Inverse Scattering in an Ocean with Sloping Seabed  This talk considers an obstacle inverse scattering problem in a sloping seabed. The incident waves are sent from point sources along a straight line parallel to the sea surface, and the corresponding scattered fields are measured from a line above the unknown object. We prove a uniqueness theorem for the inverse problem, and describe a generalized dual space indicator method for numerical solution. Numerical results will be presented. 
October 1  Professor Mike Siddoway, Colorado College (Visiting scholar in School of Math., UMN)  RModules with the KrullSchmidt Property  A fundamental question mathematicians ask is whether a given structure decomposes in a nice way. For instance, in the ring of integers we know that any number can be written uniquely as a product of primes. If we expand the integers in some simple way we sometimes lose uniqueness. We also know that any finite dimensional vector space is completely characterized by its dimension. That is, there is only one way to write a vector space (up to isomorphism) as a direct sum of indecomposable vector spaces. The indecomposable vector spaces are simply the one dimensional vector spaces. An Rmodule is just a vector space with the scalar field replaced by a commutative ring. An abelian group can be viewed as a Zmodule, a module over the integers. In module theory, we say that a class of modules has the KrullSchmidt Property if every module in the class is uniquely (up to isomorphism) the direct sum of indecomposable members of the class. By our earlier comments we see that finite dimensional vector spaces have the KrullSchmidt Property. This idea was first formulated by Krull in the 20's for finite groups. In this talk I will explore decompositions of modules with some finiteness conditions over various rings, and say a few things about when these modules have the KrullSchmidt Property. 
January 15  Anna Mazzucato, IMA and Yale  Function Spaces and Nonlinear PDE's  I will discuss some applications of the theory of function spaces to nonlinear PDEs, specifically the NavierStokes equation. Besov, Zygmund classes, along with BMO, appear rather naturally in connection with pseudodifferential calculus for nonregular symbols and the paraproduct of J. M. Bony. Morrey spaces model interesting physical examples, such as vortex rings. 
January 22  Jianliang Qian, IMA  Eulerian Methods for Viscosity and NonViscosity Solutions of HamiltonJacobi Equations Arising from Wave Propagations  The geometric optics approximation to wave propagation reduces the anisotropic wave equation to a static HamiltonJacobi equation, the anisotropic eikonal equation. This equation has as solutions three different coupled wave modes, i.e., quasilongitudinal (qP) and two transverse (qSV and qSH) waves. Viewed in the slowness space, this eikonal equation describes a sextic surface consisting of three sheets for the three waves. Observing that the qP sheet is always convex, we propose a paraxial formulation for computing the viscosity solution to quasiP wave propagation. This formulation use partial information about characteristic directions to guarantee a stable evolution in any preferred spatial direction; furthermore, this incorporated in downnout (DNO) and postsweeping (PS) update yields an Eulerian method with O(N) complexity, where N is the number of points in the computational domain. Since the qSV slowness sheet is nonconvex, the qSV waves have cusps, which form a class of multivalued, nonviscosity solutions. We introduce a level set based Eulerian approach which is able to accurately capture the qSV waves with cusps. The work on qP waves is joint with W. W. Symes, and the work on qS waves is joint with L.T. Cheng and S. Osher. 
January 29  Lev Truskinovsky, Aerospace Engineering and Mechanics  UMN  Dynamics of Phase Boundaries in Discrete Lattices  By using a prototypical discrete model we study the motion of a generic crystalline defect and explicitly compute the functional relation between the macroscopic driving force and the velocity of the defect. Although the adopted model is purely conservative and contains information only about elasticity of the constitutive elements, the resulting kinetic relation provides a quantitative description of a dissipative process. The apparent dissipation is due to the microinstabilities and induced radiation of the high frequency waves, which are invisible at the macro level. The mechanism is believed to be generic and accounting for a considerable fraction of inelastic irreversibility in deforming solids. Joint work with Olga Martynenko. 
February 5  MiaoJung Ou, IMA  A Uniqueness Theorem of the 3Dimensional Acoustic Scattering Problem in a Shallow Ocean with a Fluidlike Seabed  We show that under the assumption of outgoing radiation conditions at infinity, the timeharmonic acoustic scattered field off a soundsoft solid in a shallow ocean with a fluidlike seabed is unique in $C^2(M_1)\cap C^2(M_2)\cap C(R_h^3 \setminus \Omega)$. Here $M_1$ is the water part, $M_2$ the seabed, $R_h^3$ the waveguide and $\Omega$ is the solid object. The associated modal problem is studied and a representation formula for the solution in terms of the Green's function is derived. 
February 19  Daniel Kern, IMA  Compartmental Model for Cancer Evolution  A model is presented that examines the role of drug resistance in the evolution of cancer subject to treatment with a single cytotoxic (chemotherapeutic) agent. The model starts from a single cell and generates the evolution of the cancer. The roles of natural occurring, and acquired, resistance from chemotherapy can be seen throughout the development of the cancer or treatment history. Numerical examples illustrate the effects of resistance on chemotherapy treatment scheduling. 
February 26  Prof. Peter Linch, Met Eireann  Resonant Triads and Swinging Springs 

March 12  Jamylle Carter, IMA  A Dual Optimizer for Total VariationBased Image Restoration  This talk will describe a computational technique for the inverse problem of edgepreserving image restoration. We solve an equivalent dual form of a variational partial differential equation. Images restored using this dual approach will have crisp edges (discontinuities), whereas images recovered under earlier primal methods may contain blurred edges. Joint work with Tony Chan, Pep Mulet, and Lieven Vandenberghe. 
March 26  Prof. O'Malley, University of Washington and IMA  Shock motion for certain advectionreactiondiffusion equations.  The talk will report on joint work with Karl Knaub, generalizing earlier work with Jacques Laforgue and Michael Ward. It considers the long time asymptotic solutions for some singularly perturbed parabolic equations in one bounded spatial dimension. We show, in particular, how the tail behavior of travelling wave solutions to the stretched problem predicts shock motion featuring either exponential or algebraic asymptotics. 
April 2  Vittorio Cristini, IMA and UMN Dept. of Chem. Eng.  A mathematical and computer model of cancer growth  I will present and discuss the current status of a mathematical and computer model of cancer growth under development in my research group. The motivation of this work is to identify and analyze diagnostic and treatment strategies through direct in silico simulations. Once completed, this sophisticated computer model will provide a physician with a tool that simulates the development of a tumor corresponding to a specific patient's clinical history. In particular, the efficacy of different treatment strategies will be assessed by direct simulation. The development of a realistic computer model is now possible due to the recent advances by our group in adaptive numerical modeling and simulation techniques for complex evolving microstructures. These new techniques are capable of describing, for example, the complex shape of a solid carcinoma characterized by invasive fingering and metastasization. The model will include all phases of growth that have been identified in the biological and biomedical literature:

April 9  Prof. John Lowengrub, UMN School of Mathematics  Variational representations, small noise large deviations and applications.  Microstructured materials, such as emulsions and polymer blends, crystals and metallic alloys, blood and biological tissues, are fundamental to many industrial and biomedical applications. These diverse materials share the
common feature that the microscale and macroscale are linked. The phenomena at microscopic scale, such as the morphological instability of crystalline precipitates and drop deformation, breakup and coalescence determine the microstructure and its time evolution; thus affecting the rheology and mechanical properties of the materials on the macroscale.
In this talk, I will focus on mathematical and numerical modeling at the microscale. In particular, I will present a class of physicallybased models of complex (multicomponent) fluid flows which incorporate buoyancy, viscosity, compressibility and surface tension at interfaces. The models are capable of describing systems with both miscible and immiscible components. In addition, the models allow topological transitions such as pinchoff and reconnection of interfaces to occur without relying on ad hoc 'cut and connect' or smoothing procedures. Results will be presented for a variety of physically interesting flows. To validate the model and numerical algorithms, we examine the pinchoff of liquid/liquid threads (Rayleigh instability) and jets and compare the numerical results to theory and experiments. We then consider the development of a complex, three dimensional microstructure in which the flow components fully interpenetrate one another to yield a spongelike microstructure. Such cocontinuous microstructures have many important industrial applications. Finally, I will demonstrate how these models and numerical techniques originally developed for fluid flows may be adapted to also investigate the behavior of complex materials and biological systems. 
April 16  Toshio Yoshikawa, IMA  Transition from Mechanical Equilibrium to Thermal Equilibrium of a Chain with NonMonotone StressStrain Relation  I will discuss a mechanical property of a chain wich consists of unit masses and identical springs. The spring is governed by the stressstrain relation: F(r)=rH(r1), where H(x) is the Heaviside function.
If a heavy mass is attached to this chain and the springs are stretched equally, the chain starts to contract. The system stays in the state of mechanical equilibrium until some point of time. After this time, the light masses start to oscillate with high frequency. I will show that after this time the chain is in the state of thermal equilibrium. By using Takahashi's method for onedimensional gas with nearest neighbor interactions, I will derive the formulae for Gibbs free energy, entropy and stressstrain relation. These relations lead to the stressstrain relation in adiabatic change. I will show that numerical experimental data satisfy the condition of adiabatic change (constant entropy) and the theoretical stressstrain relation. 
May 7  Valeriy Shcherbakov  Rock and paleomagnetism: 1. How the rocks record the geomagnetic field. 2. Geomagnetic field behaviour for the last 3 billions years.  The geomagnetic field is recorded in rocks essentially in the same way, as the sound or visual images are recorded on magnetic tapes. However, a rock is not a subject of high technology, so the task is hard enough to record even a single vector within required accuracy and to ensure an enormous time stability of the natural remanence majnetization (NRM). The signal must survive over millions or even billions of years. Both major types of rocks, volcanics and sediments, may carry the NRM. But the mechanisms of their acquisition, and properties of NRM are quite different. Igneous rocks acquire NRM through cooling from well above 1000 C to the Earth surface temperature. Sediments get the remanence simply by physical orientation of the magnetic grains during deposition and compaction. The evolution of the virtual dipole moment (VDM) over the last 3 billions years is analysed on the basis of Paleointensity Database. The C_2 test showed, at a 90% significance level, a bimodal VDM distribution over the 400 millions years. The behaviour of the VDM is characterised by a succession of high and low field. 
May 14  Santiago Betelu, IMA  Singularities at the free surface of viscous fluids  We study the flow in the neighborhood of singular points (such as cusps) of the free surface of a viscous fluid. The objective is to know when singularities may appear starting from smooth initial conditions. At the free surface, the tangential viscous stress is zero and the normal viscous stress is balanced with the Laplace pressure. We compute the asymptotic flow in corners and cusps, and we construct exact time dependent solutions using complex variable techniques. We also consider 2D singularities on thin films of fluid spreading on a plane substrate. We show physically meaningful solutions describing how a plane film separated in two halves is gradually joined by a singular point travelling at finite speed. 
May 21  Prof. Rachel Kuske, UMN School of Mathematics  Stochastic dynamics in models sensitive to noise  Many systems which are sensitive to noise exhibit dynamical features from both the underlying deterministic behavior and the stochastic elements. Then the stochastic effects are obscured in this mix of dynamics. Several different methods have recently been applied to separate the "deterministic" and "stochastic" dynamics. These approaches lead to simplified approximate models which can be analyzed or simulated efficiently, providing useful measures of the noise sensitivity. The methods combine projection methods and the identification of important scaling relationships to exploit features common to these systems, such as the presence of multiple time scales, limited regions of strong noisesensitivity, and resonances. These approaches are valuable for studying a variety of problems, including stochastic delaydifferential equations, noisy bursters, and metastable interfaces. The approach will be outlined in one or two of these applications, and the generalization to other areas will be discussed. The results have interesting connections to classical probabilistic methods, dynamical systems analysis, and singular perturbation theory. 
June 4  Prof. Mark Bebbington, Massey University  Some Stochastic Models for Seismicity  The lecture will first outline how point processes can be used to model earthquake occurrence data and the sorts of results that can be achieved from such models. The stress release process is a stochastic version of the simple elasticrebound hypothesis. For this model it is possible to calculate an upper bound on its forecasting performance using entropy gains. An example will be given for north China. The stress release model can be extended spatially to provide quantitative estimates of linkages between regions. This will be illustrated using data from central Japan. The linked stress release process gives rise to a number of problems, only partially resolved, with features such as fitting, robustness, and stability. As a final example, it will be shown that the stress release process can provide cycles of accelerating strain release. 
June 18  Prof. Gerard Schuster, University of Oregon  Similarities between Optical and Seismic Imaging  
March 9  Kim, Hyejin (IMA)  Asymptotic Problems for Stochastic Processes and Related Differential Equations.  Optical imaging has its roots in the inventions of the telescope (1608 by Lippershey) and microscope (1595 by Jansen) nearly 4 centuries ago. It's first major success was Gallileo's discovery of the Gallilean moons of Jupiter in 1609, and since then has propelled major advances in the physical, medical and engineering sciences. Luckily, the governing equations of electromagnetic wave propagation, namely the Helmholtz equation, is the same asthe governing equation of acoustic wave propagation. This means that the imaging methods used by seismologists are very similar to those used by optical physicists, and so both communities can greatly benefit by drawing from their overlapping pools of knowledge. As examples, I show the following similarities between seismic and optical imaging.

June 25  Prof. Steven Jaume, College of Charleston  Earthquake Communication 101: Elastic Stress Transfer and Its Role in Future Earthquake Occurrence  For the past 10 years numerous case studies have illuminated the large and potentially dominant role of elastic stress transfer from past earthquakes on the location and timing of future earthquakes. First I will review the basic physics of elastic stress transfer and how it is commonly implimented in seismology. I will then present the results of some of these case studies illustrating stress transfer's role in the location of aftershocks and the location and timing of future large earthquakes. Finally I will present some problems and limitations of this model and make (and take) suggestions for future studies. 
August 20  Prof. John Donaldson, University of Tasmania  Global existence and long time behavior of the general EricksenLeslie system.  The Louie, Wake et al continuous mathematical model for the combined growth of Rye and Clover includes distributed delay terms. A search for numerical solutions raises questions on the relationship between the continuous and discrete analogues of differential equations and more generally between the analytic and numerical solutions of differential equations. In the latter situation, it is shown how invariants can be used to make the numerical solutions more faithful to the analytic solution. Observations on the discrete analogues of the logistic and logistic delay equation indicate quite different behaviour patterns and indicate a need for a deeper understanding of the relationships. 
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