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IMA Newsletter #406

August 2010

News and Notes
2010-2011 IMA Participating Institutions Conferences
IMA Events

IMA Workshop

Integral Equation Methods, Fast Algorithms and Applications

August 2-5, 2010

Organizers: Jingfang Huang (University of North Carolina), Shidong Jiang (New Jersey Institute of Technology), Per-Gunnar J. Martinsson (University of Colorado), Mark Tygert (New York University), Hong Xiao (University of California)
Schedule

Sunday, August 1

2:00pm-8:00pm Mathematical Modeling in Industry XIV Registration [OFFSITE] Complete ScheduleCIMAT in Guanajuato, Gto., México

Monday, August 2

All Day [OFFSITE]Mathematical Modeling in Industry XIV - August 2-11, 2010 Schedule CIMAT in Guanajuato, Gto., México MM8.2-11.10
8:00am-8:45am Registration and coffee Keller Hall 3-176 SW8.2-5.10
8:45am-9:00am Welcome to the IMAFadil Santosa (University of Minnesota)Keller Hall 3-180 SW8.2-5.10
9:00am-9:30am Methods to solve multi-scale electromagnetic problemsWeng Cho Chew (University of Hong Kong)Keller Hall 3-180 SW8.2-5.10
9:30am-10:00am Simulation of diffusion MRI signal in biological tissue Jing-Rebecca Li (INRIA )Keller Hall 3-180 SW8.2-5.10
10:00am-10:30am Multiscale system-level electromagnetic simulations Qing Huo Liu (Duke University)Keller Hall 3-180 SW8.2-5.10
10:00am-10:30am [OFFSITE]Team 1: Methods for CVaR optimization for a portfolio of equitiesChristopher Bemis (Whitebox Advisors) CIMAT in Guanajuato, Gto., México MM8.2-11.10
10:30am-11:00am BreakKeller Hall 3-176 SW8.2-5.10
10:30am-11:00am [OFFSITE]Team 2: Modelling of a novel solution potash mining processHarvey Haugen (Beechy Industries) CIMAT in Guanajuato, Gto., México MM8.2-11.10
11:00am-12:00pm Fast solvers in practice: Lessons learned and persistent challengesJacob K. White (Massachusetts Institute of Technology)Keller Hall 3-180 SW8.2-5.10
11:00am-11:30am [OFFSITE]Team 3: Production planning for water supply networksMichael Hofmeister (Siemens) CIMAT in Guanajuato, Gto., México MM8.2-11.10
12:00pm-1:30pm Lunch SW8.2-5.10
12:00pm-12:30pm [OFFSITE]Team 4: Mathematical modeling of two-phase flow in a PEM fuel cellAndreas Putz (Automotive Fuel Cell Corporation) CIMAT in Guanajuato, Gto., México MM8.2-11.10
12:30pm-1:00pm [OFFSITE]Team 5: Gravimetric measurements on moving and non-inertial platformsRaul Quiroga-Barranco (Center of Investigations in Mathematics (CIMAT)) CIMAT in Guanajuato, Gto., México MM8.2-11.10
1:00pm-1:30pm [OFFSITE]Team 6: Aware system for aerial supervision of forest/suburban firesMariano Rivera Meraz (Center of Investigations in Mathematics (CIMAT)) CIMAT in Guanajuato, Gto., México MM8.2-11.10
1:30pm-2:00pm Boundary integral equations for elliptic systems John A. Strain (University of California, Berkeley)Keller Hall 3-180 SW8.2-5.10
2:00pm-2:30pm Spectrally accurate fast summation for periodic Stokes potentialsAnna-Karin Tornberg (Royal Institute of Technology (KTH))Keller Hall 3-180 SW8.2-5.10
2:30pm-3:00pm Fast integral equation methods for the heat equation and the modified Helmholtz equation in two dimensionsMary-Catherine Andrea Kropinski (Simon Fraser University)Keller Hall 3-180 SW8.2-5.10
3:00pm-3:30pm BreakKeller Hall 3-176 SW8.2-5.10
3:30pm-4:00pm Generalized fast multipole methodEric Felix Darve (Stanford University)Keller Hall 3-180 SW8.2-5.10
4:00pm-4:30pm The discrete counterpart of Gauss' theoremXiaobai Sun (Duke University)Keller Hall 3-180 SW8.2-5.10

Tuesday, August 3

All Day [OFFSITE]Mathematical Modeling in Industry XIV - August 2-11, 2010 ScheduleCIMAT in Guanajuato, Gto., México MM8.2-11.10
8:30am-9:00am CoffeeKeller Hall 3-176 SW8.2-5.10
9:00am-9:30am A parallel adaptive fast-multipole method on heterogeneous architecturesGeorge Biros (Georgia Institute of Technology)Keller Hall 3-180 SW8.2-5.10
9:30am-10:00am Simulating vesicle flowsDenis Zorin (New York University)Keller Hall 3-180 SW8.2-5.10
10:00am-10:30am EMX: a commercial full-wave 3D Electromagnetic simulator Sharad Kapur (Integrand Software, Inc.)Keller Hall 3-180 SW8.2-5.10
10:30am-11:00am BreakKeller Hall 3-176 SW8.2-5.10
11:00am-12:00pm Simple FMM libraries for electrostatics, elastostatics and frequency-domain wave propagationLeslie F. Greengard (New York University)Keller Hall 3-180 SW8.2-5.10
12:00pm-2:00pm Lunch SW8.2-5.10
2:00pm-2:30pm Higher-order finite difference schemes via minimum Sobolev norm techniques and fast solversShivkumar Chandrasekaran (University of California, Santa Barbara)Keller Hall 3-180 SW8.2-5.10
2:30pm-3:00pm FMM on multicore processorsNikos P. Pitsianis (Duke University)Keller Hall 3-180 SW8.2-5.10
3:00pm-3:30pm BreakKeller Hall 3-176 SW8.2-5.10
3:30pm-4:00pm Group Photo SW8.2-5.10
4:00pm-5:00pm Reception and Poster Session
Poster submissions welcome from all participants
Instructions
Lind Hall 400 SW8.2-5.10
Frechet differentiation of boundary integral representationsMichael Andrew Epton (Boeing)
Treecode accelerated electrostatic calculation in implicit solvated biomolecular systems Weihua Geng (University of Michigan)
Second kind integral equations for the first kind Dirichlet problem of the biharmonic equation in three dimensionsShidong Jiang (New Jersey Institute of Technology)
Cartesian treecodesRobert Krasny (University of Michigan)
AFMPB: An adaptive fast multipole Poisson-Boltzmann solver for calculating electrostatics in biomolecular systemsBenzhuo Lu (Chinese Academy of Sciences)
A massively parallel butterfly algorithm for applying Fourier integral operatorsJack Poulson (University of Texas)
Fast integral equation methods for the modified Helmholtz equationBryan Quaife (Simon Fraser University)
Periodic density functional theory solver using multiresolution analysis with MADNESSWilliam Scott Thornton (University of Tennessee)
Directional fast multipole method for electromagneticsPaul Hikaru Tsuji (University of Texas)

Wednesday, August 4

All Day [OFFSITE]Mathematical Modeling in Industry XIV - August 2-11, 2010 ScheduleCIMAT in Guanajuato, Gto., México MM8.2-11.10
8:30am-9:00am CoffeeKeller Hall 3-176 SW8.2-5.10
9:00am-9:30am Approximations and fast algorithms for Helmholtz Green’s functionsGregory Beylkin (University of Colorado)Keller Hall 3-180 SW8.2-5.10
9:30am-10:00am Recent progress on efficient solution of the Helmholtz equationLexing Ying (University of Texas)Keller Hall 3-180 SW8.2-5.10
10:00am-10:30am The butterfly algorithm for radar imagingLaurent Demanet (Massachusetts Institute of Technology)Keller Hall 3-180 SW8.2-5.10
10:30am-11:00am BreakKeller Hall 3-176 SW8.2-5.10
11:00am-12:00pm Accurate randomized algorithms of numerical analysisVladimir Rokhlin (Yale University)Keller Hall 3-180 SW8.2-5.10
12:00pm-1:30pm Lunch SW8.2-5.10
1:30pm-2:00pm Preconditioners based on Calderon's formulae for FMMs in periodic wave problemsNaoshi Nishimura (Kyoto University)Keller Hall 3-180 SW8.2-5.10
2:00pm-2:30pm A new approach to the numerical solution of Maxwell's equationsCharles L. Epstein (University of Pennsylvania)Keller Hall 3-180 SW8.2-5.10
2:30pm-3:00pm A new integral representation for quasi-periodic scattering problems in two dimensionsAlex Barnett (Dartmouth College)Keller Hall 3-180 SW8.2-5.10
3:00pm-3:30pm BreakKeller Hall 3-176 SW8.2-5.10
3:30pm-4:00pm Efficient field computation using Gaussian beams for both transmission and receptionThorkild Hansen (Seknion Inc.)Keller Hall 3-180 SW8.2-5.10
4:00pm-4:30pm A fast and stable method for rotating spherical harmonic expansionsZydrunas Gimbutas (New York University)Keller Hall 3-180 SW8.2-5.10
4:30pm-5:00pm Fast direct solvers for elliptic PDEsPer-Gunnar J. Martinsson (University of Colorado)Keller Hall 3-180 SW8.2-5.10
6:30pm-8:30pm Workshop dinner at Golden Bowl Golden Bowl
313 Oak Street SE
Minneapolis, MN 55414
612-367-8118
SW8.2-5.10

Thursday, August 5

All Day [OFFSITE]Mathematical Modeling in Industry XIV - August 2-11, 2010 ScheduleCIMAT in Guanajuato, Gto., México MM8.2-11.10
8:30am-9:00am CoffeeKeller Hall 3-176 SW8.2-5.10
9:00am-9:30am Solving integral equations on non-smooth domainsJohan Helsing (Lund University)Keller Hall 3-180 SW8.2-5.10
9:30am-10:00am Laplace and Helmholtz boundary integral equations on domains with corners James Bremer (University of California, Davis)Keller Hall 3-180 SW8.2-5.10
10:00am-10:30am Sparse solution of integral equations Yu Chen (New York University)Keller Hall 3-180 SW8.2-5.10
10:30am-11:00am BreakKeller Hall 3-176 SW8.2-5.10
11:00am-11:30am TBAEric Michielssen (University of Michigan)Keller Hall 3-180 SW8.2-5.10
11:30am-12:00pm Rapid evaluation of the Fokker-Planck collision operatorAndras Pataki (New York University)Keller Hall 3-180 SW8.2-5.10
12:00pm-1:30pm Lunch SW8.2-5.10
1:30pm-2:00pm Cartesian treecodesRobert Krasny (University of Michigan)Keller Hall 3-180 SW8.2-5.10
2:00pm-2:30pm On accurate methods for field interpolation in particle mesh calculationsAnita Mayo (Bernard M. Baruch College, CUNY)Keller Hall 3-180 SW8.2-5.10
2:30pm-3:00pm A numerical technique for time dependent differential equationsJingfang Huang (University of North Carolina)Keller Hall 3-180 SW8.2-5.10
3:00pm-3:00pm Closing remarksKeller Hall 3-180 SW8.2-5.10

Friday, August 6

All Day [OFFSITE]Mathematical Modeling in Industry XIV - August 2-11, 2010 ScheduleCIMAT in Guanajuato, Gto., México MM8.2-11.10

Saturday, August 7

All Day [OFFSITE]Mathematical Modeling in Industry XIV - August 2-11, 2010 ScheduleCIMAT in Guanajuato, Gto., México MM8.2-11.10

Sunday, August 8

All Day [OFFSITE]Mathematical Modeling in Industry XIV - August 2-11, 2010 ScheduleCIMAT in Guanajuato, Gto., México MM8.2-11.10

Monday, August 9

All Day [OFFSITE]Mathematical Modeling in Industry XIV - August 2-11, 2010 ScheduleCIMAT in Guanajuato, Gto., México MM8.2-11.10

Tuesday, August 10

All Day [OFFSITE]Mathematical Modeling in Industry XIV - August 2-11, 2010 ScheduleCIMAT in Guanajuato, Gto., México MM8.2-11.10

Wednesday, August 11

All Day [OFFSITE]Mathematical Modeling in Industry XIV - August 2-11, 2010 ScheduleCIMAT in Guanajuato, Gto., México MM8.2-11.10
Abstracts
Alex Barnett (Dartmouth College) A new integral representation for quasi-periodic scattering problems in two dimensions
Abstract:

Boundary integral equations are an efficient approach for the scattering of acoustic and electromagnetic waves from periodic arrays (gratings) of obstacles. The standard way to periodize is to replace the free-space Green's function kernel with its quasi-periodic cousin. This idea has two drawbacks: i) the quasi-periodic Green's function diverges (does not exist) for parameter families known as Wood's anomalies, even though the scattering problem remains well-posed, and ii) the lattice sum representation of the quasi-periodic Green's function converges in a disc, thus becomes unwieldy when obstacles have high aspect ratio.

We bypass both problems by means of a new integral representation that relies on the free-space Green's function alone, and adds auxiliary layer potentials on the boundary of the unit cell strip while expanding the linear system to enforce quasi-periodicity. The result is a (slightly larger) 2nd kind system that achieves spectral accuracy, is immune to Wood's anomalies, avoids lattice sums, handles large aspect ratios, and couples to existing scattering code. By careful summing of neighboring images, obstacles may intersect the unit cell walls.

If time, we will discuss a similar robust new approach to the photonic crystal band structure eigenvalue problem.

Joint work with Leslie Greengard (NYU).

Christopher Bemis (Whitebox Advisors) Team 1: Methods for CVaR optimization for a portfolio of equities
Abstract: Project Description:

In mathematical finance, financial risk is nearly always a statistic of the underlying assets in question. That is, some salient property of the joint density function of a basket of assets is chosen to proxy risk. One of the first such formulations was the Capital Asset Pricing Model, wherein assets were assumed to be jointly normal, and volatility was therefore assumed as a risk metric. As time has informed us, though, the normal distribution may not be the best choice to describe equity returns on all time scales. Alternate risk measures have been created, all with varying degrees of success and at least a modicum of failure.

One promising risk metric is Conditional Value at Risk, or CVaR. This measure has many appealing properties, one of which seems paramount: it leaves the task of defining the joint density to the practitioner. We will study CVaR as an objective function in a set of optimization problems. Rockafellar and Uryasev (1999) suggested a linear programming formulation to solve such a problem. However, we note a curse of dimensionality issue with their proposal. We will examine alternative methods for solving the basic CVaR optimization problem with linear constraints; viz., we will consider a smooth approximation to the objective as well as a reformulation of the problem. We will also, of necessity, be concerned with modeling a joint density for the assets we will concern ourselves with. In the end, we will test our methodology ex-post on real market data.



References:

R.T. Rockafellar and S. Uryasev, Optimization of Conditional Value-At-Risk. The Journal of Risk, Vol. 2, No. 3, pp. 21-41, 2000.

S. Alexander, T. F. Coleman, and Y. Li, Minimizing VaR and CVaR for a Por tfolio of Derivatives, Journal of Banking and Finance, Vol. 30, no. 2, pp. 583-605, 2006.

G. Iyengar and A.K.C. Ma, Fast gradient descent method for mean-CVaR optimization, 2009, preprint.

Prerequisites:

Familiarity with mean-variance optimization, constrained optimization methods, and some statistics.

Desired: Coursework in mathematical finance, statistics and optimization; Matlab programming.
Gregory Beylkin (University of Colorado) Approximations and fast algorithms for Helmholtz Green’s functions
Abstract: In an approach similar to Ewald’s method for evaluating lattice sums, we split the application of Helmholtz Green’s functions between the spatial and the Fourier domains and, for any finite accuracy, approximate their kernels. In the spatial domain we use a near optimal linear combination of decaying Gaussians with positive coefficients and, in the Fourier domain, a multiplication by a band-limited kernel obtained by using new quadratures appropriate for the singularity in the Fourier domain. Applying this approach to the free space and the quasi-periodic Green’s functions, as well as those with Dirichlet, Neumann or mixed boundary conditions on simple domains, we obtain fast algorithms in dimensions two and three for computing volumetric integrals involving these Green's functions.

This is a joint work with Chris Kurcz amd Lucas Monzon.

George Biros (Georgia Institute of Technology) A parallel adaptive fast-multipole method on heterogeneous architectures
Abstract: The fast multipole method (FMM) is an efficient algorithm for what is known as "N-body problems." I will present a new scalable algorithm and a new implementation of the kernel-independent fast multipole method, in which both distributed memory parallelism (using MPI) and shared memory/SIMD parallelism (via GPU acceleration) are employed. I will conclude my talk by discussing the direct numerical simulation of blood flow in the Stokes regime using the FMM. I will describe simulations with 200 million red blood cells, an improvement of four orders of magnitude over previous results.

Bio:

George Biros holds Associate Professor appointments with the Schools of Computational Science and Engineering at Georgia Tech and The Wallace H. Coulter Department of Biomedical Engineering at Georgia Tech and Emory University. Prior to joining Georgia Tech, he was an assistant professor in Mechanical Engineering and Applied Mechanics, Bioengineering and Computer and Information Science at the University of Pennsylvania. He received his BS in Mechanical Engineering from Aristotle University Greece (1995), his MS in Biomedical Engineering from Carnegie Mellon (1996), and his PhD in Computational Science and Engineering also from Carnegie Mellon (2000). He was a postdoctoral associate at the Courant Institute from 2000 to 2003.

James Bremer (University of California, Davis) Laplace and Helmholtz boundary integral equations on domains with corners
Abstract: We describe a collection of techniques which allow for the fast and accurate solution of boundary integral equations on two-dimensional domains whose boundaries have corner points. Our approach has two key advantages over existing and recently suggested schemes: (1) it does not require a prior analytic estimates for solutions, and (2) many aspects of the scheme generalize readily to singular three-dimensional domains.
Shivkumar Chandrasekaran (University of California, Santa Barbara) Higher-order finite difference schemes via minimum Sobolev norm techniques and fast solvers
Abstract: We show how to construct higher-order (10 and above) finite-difference schemes using Minimum Sobolev Norm techniques that lead to O(N2) condition number discretizations of second-order elliptic PDEs. Numerical experiments comparing with standard FEM solvers show the benefit of the new approach. We then discuss fast numerical methods to convert the discrete equations into discretized integral equations that can have bounded condition number, and hence achieve even higher numerical accuracy.
Yu Chen (New York University) Sparse solution of integral equations
Abstract: Sparse solution to an underdetermined linear integral equation is the central problem for a broad range of applications - scattering, sensing, imaging, machine learning, signal and image processing, data analysis and compression, model reduction, optimal control and design. We will introduce a weak formulation of the problem and construct its sparse solution by a nonlinear process - the design of a Gaussian quadrature for the kernel of the integral equation. We will present a systematic method to solve the resulting quadrature problem by eigen decomposition.
Weng Cho Chew (University of Hong Kong) Methods to solve multi-scale electromagnetic problems
Abstract: Solving electromagnetics problem is a challenging task, especially when the structure is multi-scale. These structures are often encountered in circuits in electronic packaging, small antenna designs, as well as small sensor designs. A challenging problem in computational electromagnetics is in solving multi-scale problems in the low frequency regime, especially in the regime between statics and electrodynamics. When the wavelength is much longer than the size of the structure, the physics of the electromagnetic field resembles those of circuits, and hence, this is the circuit physics regime. When the wavelength is sizeable compared to the structure, than wave physics becomes important, and it is important that a simulation method can capture the wave physics interaction. When a structure is multi-scale, and has parts that are small compared to wavelength, but at the same time, is on the order of wavelength, then both circuit physics and wave physics are important. A simulation method has to capture both physics. We will discuss the use of the equivalence principle algorithm (EPA) to capture the multi-scale physics of complex structures. In this method, complex structures are partitioned into parts by the use of equivalence surfaces. The interaction of electromagnetic field with structures within the equivalence surface is done through scattering operators working via the equivalence currents on the equivalence surfaces. The solution within the equivalence surface can be obtained by various numerical methods. Then the interaction between equivalence surfaces is obtained via the use of translation operators. When accelerated with the mixed-form fast multipole method, large multi-scale problems can be solved in this manner. We will also discuss the augmented electric field integral equation (A-EFIE) approach in solving the low-frequency breakdown problem as encountered in circuits in electronic packaging. In this method, the EFIE is augmented with an additional charge unknown, and an additional continuity equation relating the charge to the current. The resultant equation, after proper frequency normalization, is frequency stable down to very low frequency. This belongs to a generalized saddle-point method, and apparently it does not suffer from the low-frequency breakdown, but it does have the low-frequency inaccuracy problem. We will discuss the use of the perturbation method to derive accurate solutions when the low-frequency inaccuracy problem occurs. We will also discuss the hybridization of EPA and A-EFIE to tackle some multi-scale problems.
Eric Felix Darve (Stanford University) Generalized fast multipole method
Abstract: Joint work with Cris Cecka and Pierre-David Letourneau (Stanford University).

The fast multipole method (FMM) is a technique allowing the fast calculation of sums in O(N) or O(N ln N) steps with some prescribed tolerance. Original FMMs required analytical expansions of the kernel, for example using spherical harmonics or Taylor expansions. In recent years, the range of applicability and the ease of use of FMMs has been considerably extended by the introduction of black box or kernel independent techniques. In these approaches, the user only provides a subroutine to numerically calculate the kernel K. This allows changing the definition of K with practically no change to the computer program. In this talk we will present a novel method that attempts to address some of the shortcomings of existing kernel independent FMMs. This method is applicable to all kernels that are analytic in some appropriate region. An important property of the method is the fact that the multipole-to-local operator is diagonal, that is for p multipole coefficients, the cost of applying the operator is O(p). The computational cost is O(N) for non-oscillatory kernels and O(N ln N) for oscillatory kernels. The approach is based on Cauchy's integral formula. We will present a numerical analysis of the convergence, and methods to find the optimal parameters in the FMM. Numerical results will be presented for some benchmark calculations to demonstrate the accuracy as a function of the number of multipole coefficients, and the computational cost of the different steps.

Laurent Demanet (Massachusetts Institute of Technology) The butterfly algorithm for radar imaging
Abstract: The butterfly algorithm is a robust alternative to the FFT for computing certain oscillatory integrals in a fast and accurate manner. In this approach low-rank interactions are updated in a hierarchical fashion up and down quadtrees. We review the method, its expected accuracy, and present an application to synthetic aperture radar imaging. Joint work with Lexing Ying.
Charles L. Epstein (University of Pennsylvania) A new approach to the numerical solution of Maxwell's equations
Abstract: I will discuss recent joint work with Leslie Greengard on a new representation of solutions to the time harmonic Maxwell Equations. Using this representation we reduce the solution of the classical problems of scattering off of a smooth bounded interface to the solution of Fredholm integral equations of second kind on the interface. What distinguishes our representation is that it does not have any spurious "interior" resonances, or suffer from low frequency breakdown. It also reveals some interesting topological features of the time harmonic equations at non-zero frequencies.
Michael Andrew Epton (Boeing) Frechet differentiation of boundary integral representations
Abstract: The general form of the boundary integral representation formula for a first order linear system in convervation law form is reviewed. The Frechet derivative of this representation is then studied with the aid of the parametric derivative formulae for the normal boundary element, n ds (resp. n dS or n dV) for boundaries in R2 (resp. R3 or R4). It is shown how the proper handling of the surface variation enables the recovery of the obvious representation formula the parametric derivative of the field vector.
Weihua Geng (University of Michigan) Treecode accelerated electrostatic calculation in implicit solvated biomolecular systems
Abstract: Poisson-Boltzmann (PB) equation based implicit solvent model can greatly reduce the computational cost in simulating solvated biomolecular systems by applying the mean field approximation in permittivities and capturing the mobile ions with Boltzmann distribution. However, solving PB equation suffers many numerical difficulties ranging from discontinuous permittivities and electrostatic field across the dielectric interface, the infinite boundary conditions, and charge singularities. In this project, we provide an efficient and accurate numerical algorithm, which adopts a well-conditioned boundary integral equation to handle these difficulties while accelerates the Krylov subspace based iterative methods such as GMRES with treecode. This Cartesian coordinates based treecode is an O(N*log(N)) scheme with properties of easy implementation, efficient memory usage O(N) , and straightforward parallelization. Benchmark testing results on Kirkwood sphere plus simulations of biomoleculars in various sizes are provided to demonstrate the accuracy, efficiency and robustness of the present methods.
Zydrunas Gimbutas (New York University) A fast and stable method for rotating spherical harmonic expansions
Abstract: In this talk, we present a simple and efficient method for rotating a spherical harmonic expansion. This is a well-studied problem, arising in classical scattering theory, quantum mechanics and numerical analysis, usually addressed through the explicit construction of the Wigner rotation matrices. Existing fast algorithms, based on recurrence relations, are subject to a variety of instabilities, limiting the effectiveness of the approach for expansions of high degree.

We show that rotation can be carried out easily and stably through "pseudospectral" projection, without ever constructing the matrix entries themselves. In the simplest version of the method, projection is carried out on the equator of the rotated sphere. If only the lowest angular modes are required, the algorithm can be further accelerated by using a sequence of constant latitude circles.

This is joint work with Leslie Greengard.

Leslie F. Greengard (New York University) Simple FMM libraries for electrostatics, elastostatics and frequency-domain wave propagation
Abstract: We have developed new versions of the FMM in three dimensions for both static and dynamic problems that are reasonably efficient, easy to implement, and fully adaptive. By combining these schemes with suitable quadrature codes, we have constructed "triangle-based" libraries for layer potentials that permit the rapid deployment of robust integral equation solvers in complex geometry. We will describe some of the core algorithmic ideas as well as some applications in electrodynamics and acoustics. This is joint work with Zydrunas Gimbutas.
Thorkild Hansen (Seknion Inc.) Efficient field computation using Gaussian beams for both transmission and reception
Abstract: An exact representation is presented for the field inside a sphere (the observation sphere) due to primary sources enclosed by a second sphere (the source sphere). The regions bounded by the two spheres have no common points. The field of the primary sources is expressed in terms of Gaussian beams whose branch-cut disks are centered in the source sphere. The expansion coefficients for the standing spherical waves in the observation sphere are expressed in terms of the output of Gaussian-beam receivers, whose branch-cut disks are centered in the observation sphere. In this configuration the patterns of the transmitting and receiving beams "multiply" to produce a higher directivity than is usually seen with Gaussian beams. This leads to a fast method for computing matrix-vector multiplications in scattering calculations, as will be illustrated for a Dirichlet square plate.
Harvey Haugen (Beechy Industries) Team 2: Modelling of a novel solution potash mining process
Abstract: Project Description: Our project is a flow modelling project. It would be a project using similar calculations to those used in Computational Fluid Dynamics or Sedimentation Engineering. We have a solution mining process, that has specific application to potash mining. The process uses interconnected horizontal wells drilled through a high grade potash zone to develop a horizontal development that should eventually dissolve out all the potash in the zone, while leaving the salt in the caverns created.
Potash will be dissolved out of the ore using a brine saturated in sodium chloride, but substantially undersaturated in potassium chloride. Sodium Chloride will collect in the well bore and the intersections similar to sand and rock in a developing meandering stream. In fact the whole process is similar to the mechanics of a meandering stream, except that the flow is bounded on the vertical surface by ore (rather than air). The calculations involve consideration of the dissolution rate (we can provide dissolution rate curves related to temperature, flow velocity, and brine saturation). Actual flow patterns will need to be modelled based on the tendency of a fluid to flow in a sinusoidal pattern even in a clean pipe, but amplified as the sine flow erodes the cavern wall at the extremities of the sine wave. Sedimentation occurs inside the curve on the point bar.
One well will intersect the next well at angles of 30 to 45 degrees. This will cause a larger case of the meander in the well bore. This will be the major extension of the cavern action. Sedimentation occurs in the vortex created. While the meander and intersections will add to the horizontal extension, the low flow area over the point bars will continue to add to the cavern height and will eventually be the major production zone. Salt liberated will simply fall to the point bar as the cavern height increases.
Note: The project can be handled as a simple computational problem, determining the amount of potash dissolved down the length of the horizontal well bore and around the intersection cavern based on the changing cross sectional area over time, the change accompanying change in flow velocity in the meander, the change in saturation of the brine and change in temperature of the brine (due to heat of solution of the potash). I will supply curves and data from literature to provide equations and experimental curves for these factors. The more complicated project would involve predicting the form of the flow from CFD engineering models as well as the calculations described.
References: National Centre For Computational Hydroscience And Engineering, University of Mississippi http://www.ncche.olemiss.edu/ COMPUTATIONAL METHODS FOR FLUID DYNAMICS
Ferziger, Joel H., Peric, Milovan
3rd, rev. ed., 2002, XIV, 423 p. 128 illus., Softcover
ISBN: 978-3-540-42074-3 Sedimentation Engineering Theory, Measurements, Modeling, and Practice (ISBN: 0784408238 /0-7844-0823-8)
Vanoni Vito A. Prerequisites: Required: differential equations, computing skills. Keywords: Stream meander modelling, sedimentation engineering
Johan Helsing (Lund University) Solving integral equations on non-smooth domains
Abstract: First I discuss low-threshold stagnation problems in the GMRES iterative solver. I show how to alleviate them in certain situations.

Then I turn to the main topic of the talk – a method to enhance the efficiency of integral equation based schemes for elliptic PDEs on domains with corners, multi-wedge points, and mixed boundary conditions. The key ingredients are a block-diagonal inverse preconditioner 'R' and a fast recursion, 'i=1,...,n', where step 'i' inverts and compresses contributions to 'R' from the outermost quadrature panels on level 'i' of a locally 'n'-ply refined mesh. From an efficiency point of view, this method converts a non-smooth boundary into a smooth boundary and mixed boundary conditions into pure boundary conditions. The spectral properties of the system matrices in the linear systems that eventually have to be solved are essentially the same. The "corner difficulties" are gone.

The talk is available as a pdf-file:
http://www.maths.lth.se/na/staff/helsing/corners.pdf

Michael Hofmeister (Siemens) Team 3: Production planning for water supply networks
Abstract: Project Description:

The management of water resources is one of the big future challenges of our world. Besides the "big picture" how to provide the humankind with a sufficient amout of high quality potable water, there are a lot of problems to be solved on an operational level. For example, local water suppliers have to make sure that water is available when needed in sufficient quantity and quality. They have to deal both the requirement for the lowest utility prices and the fiscal requirements of their municipalities which might possibly be in a precarious financial situation. Usually, the energy expended for pumps in wells and pumping stations is their major cost factor. The complex rates of electricity providers have high-cost and low-cost periods as well as limitations of total energy consumption; violations result in considerable additional fees.

The goal of the project is to develop a software which is able to find an tradeoff between mimimum energy costs and the reduction of pump switches, where this last aspect minimizes service and maintenance efforts. The degree of freedom for the optimization is essentially an intelligent water storage management.

The project requires basically three steps for the team to be performed:

  1. Modelling of water nets as well as the corresponding production and delivery problem, based on resources and demand forecasts
  2. A concept how to deal conflicting goals, like cost optimization and reduction of pump switches
  3. Definition and implementation of solution algorithms. This includes to decide which libraries of basic optimization algorithms should be used.

Besides mathamatics, it is intended to introduce some basic aspects of project management. For example, there will be assigned "roles" of the team members, like project manager, systems architect, or sales (I am the customer). A project schedule has to be set up in the beginning.

References:

G.L. Nemhauser, L.A. Wolsey, Integer and Combinatorial Optimization, John Wiley & Sons, 1988

R.K. Ahuja, T.L. Magnanti, J.B. Orlin, Network Flows - Theory, Algorithms, and Applications, Prentice Hall, 1993

http://zibopt.zib.de/ The optimization suite of the Zuse Zentrum Berlin is one possibility to attack the optimization problems of the project. The advantage is that it is free for academic purposes. However, the team members are free to choose any other tool for linear and integer optimization, provided they will have solved the licensing requirements. (I strongly recommend not to start from scratch, by implementing the Simplex algorithm.)

Prerequisites:

Methods of Linear Optimization as well as programming skills are a must (preferably in C/C++, especially if the team will decide to choose the ZIB Optimization Suite), experience in Discrete Optimization and Graph Theory a real advantage.

Jingfang Huang (University of North Carolina) A numerical technique for time dependent differential equations
Abstract: In this talk, we discuss a numerical scheme for the accurate and efficient solution of time dependent partial differential equations. The technique first discretizes the temporal direction using Gaussian type nodes and spectral integration, and applies low-order time marching schemes to form a preconditioned elliptic system. The better conditioned system is then solved iteratively using Jacobi-Free Newton–Krylov techniques, and each function evaluation is simply one low-order time-stepping approximation of the error by solving a decoupled system using available fast elliptic equation solvers. Preliminary numerical experiments show that this technique can be unconditionally stable and spectrally accurate in both temporal and spatial directions.
Shidong Jiang (New Jersey Institute of Technology) Second kind integral equations for the first kind Dirichlet problem of the biharmonic equation in three dimensions
Abstract: No Abstract
Sharad Kapur (Integrand Software, Inc.) EMX: a commercial full-wave 3D Electromagnetic simulator
Abstract: We describe the implementation of EMX, a commercial full-wave 3D Electromagnetic simulator used for Integrated Circuit (IC) design. EMX uses a volume integral equation formulation and a specialized iterative solver that is accelerated using a new 3D kernel independent Fast Multipole Method. The code has been implemented to take advantage of multiple-core machines. In addition, domain specific implementation issues are discussed. For example, in advanced IC processes, the physical properties of wires vary depending on the surrounding wiring. EMX automatically modifies the drawn layout to mimic the IC fabrication process. We compare simulation results to measurements over a large variety of IC structures.
Robert Krasny (University of Michigan) Cartesian treecodes
Abstract: We present recent work on treecode algorithms for computing integrals that arise in particle simulations. Our approach uses a Cartesian Taylor series for the far-field expansion and is suitable for a variety of kernels including multiquadric radial basis functions and the screened Coulomb potential. A sample of results will be presented for vortex sheet motion in 3D fluid flow and boundary integral simulations of the Poisson-Boltzmann equation for solvated biomolecules.
Robert Krasny (University of Michigan) Cartesian treecodes
Abstract: We present recent work on treecode algorithms for computing integrals that arise in particle simulations. Our approach uses a Cartesian Taylor series for the far-field expansion and is suitable for a variety of kernels including multiquadric radial basis functions, the screened Coulomb potential, and a regularized Biot-Savart kernel. Sample results are shown.
Mary-Catherine Andrea Kropinski (Simon Fraser University) Fast integral equation methods for the heat equation and the modified Helmholtz equation in two dimensions
Abstract: We present an efficient integral equation approach to solve the heat equation in a two-dimensional, multiply connected domain, and with Dirichlet boundary conditions. Instead of using integral equations based on the heat kernel, we take the approach of discretizing in time, first. This leads to a non-homogeneous modified Helmholtz equation that is solved at each time step. The solution to this equation is formulated as a volume potential plus a double layer potential. The volume potential is evaluated using a fast multipole-accelerated solver. The boundary conditions are then satisfied by solving an integral equation for the homogeneous modified Helmholtz equation. The integral equation solver is also accelerated by the fast multipole method (FMM). For a total of N points in the discretization of the boundary and the domain, the total computational cost per time step is O(N) or O(N log N).



Jing-Rebecca Li (INRIA ) Simulation of diffusion MRI signal in biological tissue
Abstract: Water diffusion magnetic resonance imaging (DMRI) is a method which uses a combination of applied magnetic fields to measure, statistically, the diffusion of water molecules due to Brownian motion. Its spatial resolution is on the order of millimeters.

The cellular structure inside the human brain varies on the scale of micrometers, which is much smaller than the size of a voxel. There may be thousands of irregularly-shaped cells within a voxel, and they all contribute to the environment seen by water molecules whose displacement is measured by the MRI scanner. In the typical DMRI experiment, the time interval over which water diffusion is measured is in the range of 50-100 milliseconds. Using the diffusion coefficient of 'free' water at 37 degrees Celsius, D = 3e-9 m2/s, we get an estimated diffusion distance of 15-25 micrometers. Clearly, in a DMRI experiment, water molecules encounter numerous times inhomogeneities in the cellular environment, such as cell membranes, fibers, and macromolecules. We want to simulate the DMRI signal at the scale of a single voxel, while taking into account cellular structure and the shape and duration of the diffusion gradients.

I will describe the Bloch-Torrey partial differential equation which is a phenomenological equation that describes the time evolution of the nuclear magnetization in a sample. I will talk about the numerical solution of the Bloch-Torrey PDE using Green's functions and alternatively by finite elements discretization. Finally, the measured diffusion MRI signal is the sum of the magnetization in the sample and I give some analytical results on the aggregate signal under some simplying assumptions.

Qing Huo Liu (Duke University) Multiscale system-level electromagnetic simulations
Abstract: System-level electromagnetic design problems are multiscale and very challenging to solve. They remain a significant barrier to system design optimization for a foreseeable future. Such multiscale problems often contain three electrical scales, i.e., the fine scale (geometrical feature size much smaller than a wavelength), the coarse scale (geometrical feature size greater than a wavelength), and the intermediate scale between the two extremes. Existing commercial tools are based on single methodologies (such as finite element method or finite-difference time-domain method), and are unable to solve large multiscale problems. For example, no commercial software package is known to be able to model a simple package (say at 0.1 mm resolution) inside a reverberation chamber (1 m) for RF interference testing.

We will present our recent work in solving realistic multiscale system-level EM design simulation problems in both frequency domain and time domain. In frequency domain, we combine the spectral element method for coarse scales and finite element method for fine scales with a fast spectral integral method. In time domain, we hybridize the spectral element time-domain method and finite-element time-domain method together with the perfectly matched layer, with explicit and implicit time integration schemes for different subdomains. The discontinuous Galerkin method is used as the subdomain interface condition. In time domain, we further incorporate a nonlinear circuit solver, making it possible to perform nonlinear circuit simulation with RF interactions in a seamless manner. Several previously intractable multiscale problems will be illustrated.

Benzhuo Lu (Chinese Academy of Sciences) AFMPB: An adaptive fast multipole Poisson-Boltzmann solver for calculating electrostatics in biomolecular systems
Abstract: Poisson-Boltzmann (PB) electrostatics is a well established model in biophysics, however its application to the study of large scale biosystem dynamics such as the protein-protein encounter is still limited by the efficiency and memory constraints of existing numerical techniques. In this poster, we present an efficient and accurate scheme which incorporates recently developed novel numerical techniques to further enhance the present computational ability. In particular, a boundary integral equation approach is applied to discretize the linearized PB (LPB) equation; the resulting integral formulas are well conditioned and are extended to systems with arbitrary number of biomolecules; a robust meshing method is developed for molecular surface meshing; the solution process is accelerated by the Krylov subspace methods and an adaptive new version of fast multipole method (FMM). The Adaptive Fast Multipole Poisson-Boltzmann (AFMPB) solver is released under open source license agreement, and the meshing tool TMSmesh will also be released.

Per-Gunnar J. Martinsson (University of Colorado) Fast direct solvers for elliptic PDEs
Abstract: The talk will describe recently developed fast solvers for the linear systems arising upon the discretization of elliptic PDEs. While most existing fast methods tend to be based on iterative solvers such as GMRES, the new techniques directly construct an approximate inverse (or LU factorization) of the coefficient matrix. This makes the techniques robust and particularly fast for problems involving multiple right hand sides. Such "fast direct solvers" have been developed both for the sparse (and often very large) matrices that arise upon finite element discretizations of elliptic PDEs, and for the dense matrices arising upon discretization of the associated integral equations.
Anita Mayo (Bernard M. Baruch College, CUNY) On accurate methods for field interpolation in particle mesh calculations
Abstract: In this talk we will present accurate two and three dimensional methods for interpolating singular or smoothed force fields. The methods are meant to be used in particle mesh or particle-particle particle-mesh calculations so that the resulting schemes conserve momentum. The interpolation weights use a discretization of the differential equation the interaction potential satisfies, and the methods are most accurate when the force satisfies a homogeneous elliptic differential equation or system of equations away from the sources. We will show why the precise accuracy of the interpolation formulas depends on the accuracy of certain corresponding quadrature formulas, and give results of numerical experiments.
Naoshi Nishimura (Kyoto University) Preconditioners based on Calderon's formulae for FMMs in periodic wave problems
Abstract: Wave problems in periodic domains have many interesting applications such as photonic crystals and metamaterials in Maxwell's equations and phononic crystals in elastodynamics, etc. Fast multipole methods are considered to be effective as solvers of such problems, particularly when the problems under consideration are of scattering nature. In view of this, we have developed periodic FMMs for various wave problems. We are now interested in further enhancing the performance of the periodic FMMs with the help of effective preconditioners. In this presentation we shall discuss our recent efforts on the use of preconditioners based on Calderon's formulae in periodic transmission problems in Helmholtz' equation and elastodynamics. In Helmholtz, we shall show that the matrix of the discretised integral equations itself serves as an effective preconditioner. This fact leads to a very simple preconditioning scheme as one uses GMRES as the solver. We shall also see that a similar approach is possible in elastodynamics, but either with some restrictions on the material constants or with more complicated formulations. We shall present a number of numerical examples to test the performance of the proposed approaches.
Andras Pataki (New York University) Rapid evaluation of the Fokker-Planck collision operator
Abstract: In plasma physics, the Boltzmann equation, which describes the evolution of the plasma over time, has a nonlinear term representing the collisions of various species of the plasma. Current plasma edge simulations do not take this collision effect into account, because of the difficulties in the accurate evaluation of this term. Using the Rosenbluth potential formalism, the collision operator can be written in terms of solutions of a Poisson and a biharmonic free space PDE. Due to the inherent axisymmetry of the input data, cylindrical coordinate solvers are preferred for efficient computation. Standard numerical techniques (based typically on finite differences and finite element approximations) encounter difficulties in achieving high order accuracy, especially in the computation of derivatives of the solution (required in the collision operator formulation), and in imposing radiation conditions at infinity. Our new solver achieves arbitrary order accuracy in cylindrical coordinates based on a combination of separation of variables, Fourier analysis and the careful solution of the resulting radial ODE. A weak singularity arises in the the continuous Fourier transform of the solution that can be handled effectively with special purpose quadrature rules and spectral accuracy can be achieved in derivatives without loss of precision.

This is joint work with Leslie Greengard.
Nikos P. Pitsianis (Duke University) FMM on multicore processors
Abstract: We present preliminary results of parallelizing the fast multipole method (FMM) for multicore processors using POSIX threads.

Short-range interactions are straight forward to parallelize. We invoke multiple threads per compute core to alleviate partition and load imbalances. For the calculation of long-range interactions, we assign the multipole subtrees below a certain level to compute threads with affinity settings that conform to the interaction lists of the tree nodes and exploit the memory hierarchy.

On a Sun SunFire X4600 with 8 AMD Opteron 885 processors (16 cores) running at 2.6 GHz clock rate and 64 GB of memory, we observe a better than 15x speedup compared to the sequential version of FMM-Yukawa for two sample benchmark problems of 10 to 100 million charges uniformly distributed inside a unit box or on the surface of a unit sphere and require six-digit accuracy.

Jack Poulson (University of Texas) A massively parallel butterfly algorithm for applying Fourier integral operators
Abstract: In August of 2008, Candes, Demanet, and Ying introduced a fast algorithm for applying Fourier integral operators of the form ∫ eiΦ(x,k)f(k)dk, where Φ(x,k) is the so-called phase function and is required to be real-valued and linear in the second argument. Using a resolution of 1/N in each dimension, the transform of arbitrary sources in the unit d-dimensional cube of the frequency domain may be naively evaluated over the unit cube of the spatial domain with computational complexity O(N2d). The algorithm of Candes et al. yields the near-optimal complexity O(Ndlog2(N)).

The contribution of the author is a new method for parallelizing the above fast algorithm on distributed-memory machines. The method assumes only a power-of-two number of processes and has been shown to strong scale up to thousands of cores of Blue Gene/P with 90% efficiency even for extremely small problems. This is achieved by carefully peeling off partitions of the frequency domain and applying them to the spatial domain as the butterfly algorithm progresses. The result is that, using 2P processes in d dimensions, the only communication required by each process is P reduce-scatter summations over a team of 2d processes. Results are presented for a generalized Radon transform in two-dimensions, though the implementation has been shown to work in arbitrary dimensions. The source code is available at http://code.google.com/p/butterflyfio.

Andreas Putz (Automotive Fuel Cell Corporation) Team 4: Mathematical modeling of two-phase flow in a PEM fuel cell
Abstract: The correct and accurate modeling of two phase-flow inside a Proton-Exchange Membrane Fuel Cell (PEMFC) is one of the key elements of a realistic simulation. A schematic drawing of the cross section of a PEMFC can be seen in Figure 1.

Figure 1: Schematic cross-section of a PEMFC.

The reactant gases are supplied to the fuel cell through the gas channels and diffuse into the layered porous medium to the chemically active catalyst layers on the anode and cathode side, where they are consumed. The Proton-Exchange Membrane is (PEM) as gas tight as possible, and supposed to function purely as a proton conductive medium. In the cathode catalyst layer, oxygen reacts with electrons (supplied through the electrically conductive porous medium) and protons (transported through the membrane and the catalyst layer) and water is produced. The water is produced in either liquid or vapor form, depending on the local conditions, and is transported through the porous network structure and removed by the flow in the gas channels. The optimal performance of a fuel cell depends to a high degree on maintaining optimal saturation (liquid water content) and relative humidity (water vapor content). A certain humidity level is required to guarantee high proton conductivity of the Proton- Exchange Membrane (PEM), but if the humidity is too high, the pores of the network begin to fill with water, and the transport of the gases becomes more and more difficult. One way to mitigate the flooding problem is to assign part of the pore network to the liquid phase and the other part to the gas phase by the introduction of hydrophobic and hydrophilic compounds the the porous structure, and by the introduction of multi-layer diffusion media with varying material properties. Simple model simulation do not agree well with experimental values of the liquid water saturation, see Figure 2.

Figure 2: Experimental vs. theoretical water content (Image taken from Adam Z. Webers conference presentation).

It is therefore necessary to describe the porous diffusion media in more detail and with greater understanding of their structural and chemical properties. The function of the porous structure of a PEMFC can be split in a structural part, i.e. the distribution of the pore sizes, and a chemical part, i.e. the wettability of the material depending on the contact angle. The chemical treatment leads to multiple contact angles, idealized by a prescribed contact angle distribution. A further complication arises from the fact that the contact angle is different for the advancing and the receding edge of an interface, and therefore different whether water is drained or produced. Given a pore size distribution and the contact angle distribution, the saturation profile depending on the capillary pressure can be calculated and compared to experimental values. Due to the complex interaction in this kind of porous network, a hysteresis in the capillary pressure vs. saturation curve can be observed, as shown in Figure 3.

Figure 3: Contact angle hysteresis (Image taken from Adam Z. Webers conference presentation).

Figure 4: Lattice Boltzmann simulation of liquid water flow through a porous medium (Movie obtained from Palabos image galery).

The objective of this project is to study the two-phase flow through a porous network with the previously stated structural and chemical properties, to obtain the characteristic saturation curves and to properly quantify the energy stored in the hysteresis loop. References:
  1. Fuel cell systems explained (2nd edition), Larminie, James; Dicks, Andrew, 2003 John Wiley & Sons
  2. Wettability and capillary behavior of fibrous gas diffusion media for polymer electrolyte membrane fuel cells, J.T. Gostick et al., Journal of Power Sources 194/1(2009), dx.doi.org/10.1016/j.jpowsour.2009.04.052
  3. Characterization of internal wetting in polymer electrolyte membrane gas diffusion layers, P. Cheung et al., Journal of Power Sources, 187/2, 2009, 487-492, dx.doi.org/10.1016/j.jpowsour.2008.11.036
Prerequisites: Required: 1 semester of ordinary differential equations, 1 semester of partial differential equations, 1 semester numerical analysis, computing skills (python/scipy/sage or Matlab or C/C++). Desired: 1 semester of physics or fluid dynamics in general. Keywords: Fuel cell, two-phase flow, hysteresis, porous media
Raul Quiroga-Barranco (Center of Investigations in Mathematics (CIMAT)) Team 5: Gravimetric measurements on moving and non-inertial platforms
Abstract: Keywords: Geodesy, special and general relativity, gravimetry, GPS.

Background:

The exact computation of the gravitational field of the Earth is the point of departure to obtain a complete three-dimensional image of the surface of the Earth, which is used in various applications. The latter include geology, hydrology, civil engineering, global positioning systems, among others. The ideal surface obtained through these techniques is called the geoid, and it provides the best reference model for the above mentioned applications.

Geoid determination depends on accurately measuring the values of gravitational forces at different points on Earth. To understand the precision of existing instruments let it be known that phenomena as the following have to be taken into account:

  1. The rotation of the Earth that generates a centrifugal force that reduces the gravity force,
  2. The tides which change the distribution of the planet's mass and thus the values of gravitational forces.


Problem:

It is a fairly recent technique to perform gravimetric measurements on moving platforms, for example with gravimeters mounted on aircrafts and satellites. Considering the above mentioned required accuracy, one has to consider the following factors:
  1. Even the smallest external forces on aircraft mounted gravimeters,
  2. The non-inertial nature of the reference frames that correspond to moving platforms which is due to the Earth's curvature.


There already exist some models and formulas that take into account such effects, but we do not know of a complete model that fully considers them. Also, it does not seem to exist a model that has taken care of the relativistic effects of a non-inertial platform. We recall that global positioning systems (GPS) already incorporate relativistic effects to achieve the required accuracy of about 5 meters for non-military users. In contrast, an important goal is to determine the geoid with high precision (centimeters).

Project:

Our main goal is to study the models used for gravimetric measuring and geoid computation on moving and non-inertial platforms. We will first estimate the effective accuracy of the results obtained with the use of classical mechanics. Next, we will consider the models that take into account special and general relativistic effects. Our main tool will be the already known theory and formulas as well as some that we can develop as the project advances. Another relevant component will be the use of software and graphing tools to understand and visualize the phenomena involved. These should suggest corrections and improvements to the current techniques.

Feasibility

As mentioned above, we already have at our disposal theoretical models in the classic mechanics setup. Also, we have obtained some preliminary theoretical computations that evidence the influence of relativistic effects in the gravimetric measurements on moving platforms. Finally, we can compare the modeling results obtained with the measurements available at some government instances.

References:

  1. Geodesy 3rd Edition, Torge, Wolfgang, 2001, de Gruyter, ISBN 3110170728
  2. Gravitation, Misner, C., Thorne, K., Wheeler, J., 1973, Freeman, ISBN 0716703440
  3. Satellite Geodesy 2nd Edition, Seeber, Gunter, 2003, de Gruyter, ISBN 3110175495


Prerequisites:

Required: 1 semester course of differential geometry, computing skills (e.g. C/C++).

Desired: (one or more of the following) 1 semester course of introductory physics or classical mechanics, 1 semester course of Riemannian geometry, 1 semester course of special relativity.
Mariano Rivera Meraz (Center of Investigations in Mathematics (CIMAT)) Team 6: Aware system for aerial supervision of forest/suburban fires
Abstract:

Keywords: Image stabilization, Image segmentation, Classification, Tracking, Particle filters.

Project Description:

The project consists in detecting moving objects in suburban and forest firing areas that could indicate people in danger. Automatic monitoring systems can help in this situation, either, making a fully automatic analysis or sending a pre-alarm, to indicate situations that require operator's attention. The data are video sequences acquired, supposedly, with an unmanned aerial vehicle (UAV), or remotely piloted vehicle.

There are several advantages of using UAV in fire monitoring with respect to manned planes (or helicopters); they have a relative the lower cost of operation; they can fly in dangerous situations; they can fly for longer period of time and they can be equipped with video cameras for performing autonomous scene analysis.

Challenges of the project are:

  1. We have an unstable source video. The UVS motion introduces shifts and shakes of the scene in the video. Hence, a preprocessing for video stabilization could be useful [2].
  2. The fire's smoke is moving and must not be confused with objects' motion. We will require classifying moving objects between smoke and interesting blobs. Color, texture and motion features can support such a classification [4,5].
  3. Static objects in the scene, as trees or houses, can occlude the moving objects (MOs). Moreover smoke can partially or fully occlude such MOs. For managing this situation, a motion analysis could alert us for occlusions and allows tracking partially occluded object [3].
  4. Noise may produce false positive. Thus, the motion analysis could provide us from the information for classifying among noise, smoke and MOs and then to eliminate false positives [5].

During the development of our algorithms, we need to take into account that the monitoring system is pretended to be implemented in real time and executed autonomously in the UAV, i.e. we will prefer computationally efficient methods. References:
  1. Handbook of Mathematical Models in Computer Vision, N. Paragios, Y Chen, O. Faugeras Eds. Springer NY, 2006
  2. R. Szeliski, "Image Alignment and Stitching," Chap 17 in [1], pp 273-292.
  3. A. Blake, "Visual tracking: a short research roadmap," Chap 18 in [1], pp 293-307.
  4. M. Rivera, O. Ocegueda, J.L.Marroquin, "Entropy controlled quadratic Markov measure fields for efficient image segmentation," EEE Trans. on Image Process., vol 16.(12), 3047-3057, 2007.
  5. C.M. Bishop Pattern Recognition and Machine Learning, Springer NY, 2006. In particular chapters: 7,9,12
Prerequisites:

Required: 1 semester of image processing, 1 semester numerical methods, 1 semester numerical optimization, computing skills (Matlab or C/C++). Desirable: 1 semester pattern recognition.
Vladimir Rokhlin (Yale University) Accurate randomized algorithms of numerical analysis
Abstract: Randomized algorithms are ubiquitous in computer science and computer engineering. Many problems that are intractable when viewed deterministically can be effectively solved with probabilistic techniques. Perhaps the most important aspect of most randomized procedures in current use is the fact that they produce the correct result with (practically speaking) 100% reliability, and with (essentially) machine precision.

Historically, randomized techniques have been less popular in numerical analysis. Most of them trade accuracy for speed, and in many numerical environments one does not want to add yet another source of inaccuracy to the calculation that is already sufficiently inaccurate. One could say that in numerical analysis probabilistic methods are an approach of last resort.

I will discuss several probabilistic algorithms of numerical linear algebra that are never less accurate than their deterministic counterparts, and in fact tend to produce better accuracy. In many situations, the new schemes have lower CPU time requirements than existing methods, both asymptotically and in terms of actual timings. I will illustrate the approach with several numerical examples, and discuss possible extensions.

John A. Strain (University of California, Berkeley) Boundary integral equations for elliptic systems
Abstract: We present "locally-corrected spectral boundary integral methods" for accurate discretization and fast solution of linear elliptic systems of PDEs. Arbitrary elliptic systems are transformed to overdetermined first-order form, and a boundary integral equation is derived. Ewald summation separates the boundary integral equation into a low-rank system with regular spectral structure, followed by a simple local correction formula. Geometric nonuniform fast Fourier transforms produce accurate Fourier coefficients of piecewise-polynomial data on a d-dimensional simplicial tessellation in RD, for arbitrary dimensions d and D.

Xiaobai Sun (Duke University) The discrete counterpart of Gauss' theorem
Abstract: We introduce numerical study on the discrete counterpart of Gauss' theorem. The purpose is to seek and establish a third approach, beside the analytical and the kernel-independent approaches, for efficient dimension reduction and preconditioning of equations initially in differential form. Integration is done locally, or globally, using analytical/symbolic rules as well as numerical rules and utilizing geometric information.
William Scott Thornton (University of Tennessee) Periodic density functional theory solver using multiresolution analysis with MADNESS
Abstract: We report the first all electron adaptive refinement multiresolution band structure solver for crystalline systems. Most current software implementations of density functional theory for crystalline systems either use plane waves or a split representation such as linear augmented plane waves (LAPW) or linear muffin tin orbitals (LMTO) for their basis set. MADNESS, on the other hand, uses a fully numerical basis set which resides on an adaptive mesh. This choice has introduced numerous challenges in the implementation of a solid state band structure code. Instead of explicitly diagonalizing the one-particle Hamiltonian to update the single particle orbitals, we reformulate the problem into a bound state Helmholtz integral equation. Another challenge has been implementing the bound state Helmholtz Green's function and the Coulomb Green's function to include a lattice sum over the periodic images in the unit cell. To ensure timely convergence, we wrap the main Kohn-Sham loop inside a Krylov accelerated inexact Newton solver. We present all of these items and some preliminary results in our poster.
Anna-Karin Tornberg (Royal Institute of Technology (KTH)) Spectrally accurate fast summation for periodic Stokes potentials
Abstract: A spectrally accurate method for the fast evaluation of N-particle sums of the periodic Stokeslet is presented. Such sums occur in boundary integral- and potential methods for viscous flow problems. Two different decomposition methods, leading to one sum in real space and one in reciprocal space, are considered. An FFT based method is applied to the reciprocal part of the sum, using convolutions with a Gaussian function to place the point sources on a grid. Due to the spectral accuracy of the method, the grid size needed is low and also in practice, for a fixed domain size, independent of N. The leading cost therefore arise from the to-grid and from-grid operations, that are linear in N. Combining this FFT based method for the reciprocal sum with the direct evaluation of the real space sum, a spectrally accurate algorithm with a total complexity of O(N log N) is obtained.
Paul Hikaru Tsuji (University of Texas) Directional fast multipole method for electromagnetics
Abstract: No Abstract
Jacob K. White (Massachusetts Institute of Technology) Fast solvers in practice: Lessons learned and persistent challenges
Abstract: Joint work with H. Reid, L. Zhang, C. Coelho, Z. Zhu, T. Klemas, and S. Johnson.

Despite decades of zealous effort, there are remarkably few applications where fast integral equation solvers dominate. To help explain why, we will describe effective strategies, assess performance, and present remaining challenges associated with applying fast solvers to problems in interconnect model extraction, biomolecular electrostatics, bodies in microflows, nanophotonics, and Casimir force calculation.

Lexing Ying (University of Texas) Recent progress on efficient solution of the Helmholtz equation
Abstract: Numerical solution of the Helmholtz equation in the high frequency regime is a challenging computational problem because of the indefiniteness of the operator and the size of the discrete system (due to the high frequency nature of the problem). In this talk, we will discuss some recent progress on the numerical solution of the Helmholtz equation in two and three dimensions.
Denis Zorin (New York University) Simulating vesicle flows
Abstract: Vesicles are locally-inextensible fluid membranes that can sustain bending. We consider the dynamics of flows of vesicles suspended in Stokesian fluids. We use a boundary integral formulation for the fluid that results in a set of nonlinear integro-differential equations for the vesicle dynamics. The motion of the vesicles is determined by balancing the nonlocal hydrodynamic forces with the elastic forces due to bending and tension. Numerical simulations of such vesicle motions are quite challenging. On one hand, explicit time-stepping schemes suffer from a severe stability constraint due to the stiffness related to high-order spatial derivatives and a milder constraint due to a transport-like stability condition. On the other hand, an implicit scheme can be expensive because it requires the solution of a set of nonlinear equations at each time step. We present a set of numerical techniques for efficient simulation of vesicle flows. The distinctive features of these numerical methods include (1) using boundary integral method accelerated with the fast multipole method (2) spectral (spherical harmonic) discretization of deforming surfaces in space (3) an algorithm for surface reparameterization ensuring stability of the time-stepping scheme and spectral filtering accuracy while minimizing computational costs. By introducing these algorithmic components, we obtain a time-stepping scheme that experimentally is unconditionally stable and has a low cost per time step. We present numerical results to analyze the cost and convergence rates of the scheme.
Visitors in Residence
Bradley K. Alpert National Institute of Standards and Technology 8/1/2010 - 8/5/2010
Nusret Balci University of Minnesota 9/1/2009 - 8/31/2011
Alex Barnett Dartmouth College 8/1/2010 - 8/6/2010
Joanna Bechtel University of California, Berkeley 8/2/2010 - 8/5/2010
Rosalie Belanger-Rioux Massachusetts Institute of Technology 8/1/2010 - 8/5/2010
Gregory Beylkin University of Colorado 8/2/2010 - 8/5/2010
George Biros Georgia Institute of Technology 8/2/2010 - 8/5/2010
James Bremer University of California, Davis 8/1/2010 - 8/5/2010
Sun Young Bu University of North Carolina 8/1/2010 - 8/6/2010
Aycil Cesmelioglu Rice University 8/31/2010 - 8/30/2011
Chi Hin Chan University of Minnesota 9/1/2009 - 8/31/2011
Shivkumar Chandrasekaran University of California, Santa Barbara 8/1/2010 - 8/6/2010
Xianjin Chen University of Minnesota 9/1/2008 - 8/31/2010
Yu Chen New York University 8/1/2010 - 8/6/2010
Weng Cho Chew University of Hong Kong 8/1/2010 - 8/5/2010
Robert Crone Seagate Technology 8/2/2010 - 8/5/2010
Jintao Cui Louisiana State University 8/31/2010 - 8/30/2011
Qing Cui University of Minnesota 1/1/2010 - 12/1/2010
Eric Felix Darve Stanford University 8/1/2010 - 8/3/2010
Laurent Demanet Massachusetts Institute of Technology 8/1/2010 - 8/5/2010
Andrew Dienstfrey National Institute of Standards and Technology 8/1/2010 - 8/5/2010
Charles L. Epstein University of Pennsylvania 8/1/2010 - 8/5/2010
Michael Andrew Epton Boeing 8/1/2010 - 8/5/2010
Bela Erdelyi Northern Illinois University 8/1/2010 - 8/3/2010
Randy H. Ewoldt University of Minnesota 9/1/2009 - 8/31/2011
Oscar E. Fernandez University of Michigan 8/31/2010 - 8/30/2011
Weihua Geng University of Michigan 8/1/2010 - 8/5/2010
Adrianna Gillman University of Colorado 8/1/2010 - 8/6/2010
Zydrunas Gimbutas New York University 8/1/2010 - 8/5/2010
Anne Greenbaum University of Washington 8/1/2010 - 8/6/2010
Leslie F. Greengard New York University 8/1/2010 - 8/5/2010
Thorkild Hansen Seknion Inc. 8/1/2010 - 8/5/2010
Robert J. Harrison Oak Ridge National Laboratory 8/2/2010 - 8/4/2010
Johan Helsing Lund University 7/31/2010 - 8/5/2010
Yulia Hristova Texas A & M University 7/31/2010 - 8/5/2010
Jingfang Huang University of North Carolina 8/1/2010 - 8/6/2010
Yunkyong Hyon University of Minnesota 9/1/2008 - 8/31/2010
Vikram Jandhyala University of Washington 8/1/2010 - 8/5/2010
Jun Jia Oak Ridge National Laboratory 8/1/2010 - 8/5/2010
Lijian Jiang University of Minnesota 9/10/2008 - 8/31/2010
Shidong Jiang New Jersey Institute of Technology 7/31/2010 - 8/5/2010
Sharad Kapur Integrand Software, Inc. 8/1/2010 - 8/5/2010
Hyejin Kim University of Minnesota 9/1/2009 - 8/31/2010
Pawel Konieczny University of Minnesota 9/1/2009 - 8/31/2011
Robert Krasny University of Michigan 8/1/2010 - 8/6/2010
Mary-Catherine Andrea Kropinski Simon Fraser University 8/1/2010 - 8/6/2010
Harun Kurkcu Simon Fraser University 8/1/2010 - 8/6/2010
June-Yub Lee Ewha Women's University 8/1/2010 - 8/5/2010
Hengguang Li Syracuse University 8/16/2010 - 8/15/2011
Jing-Rebecca Li INRIA 8/1/2010 - 8/6/2010
Yongfeng Li University of Minnesota 9/1/2008 - 8/31/2010
Zhi (George) Lin University of Minnesota 9/1/2009 - 8/31/2011
Chun Liu University of Minnesota 9/1/2008 - 8/31/2010
Qing Huo Liu Duke University 8/1/2010 - 8/6/2010
Benzhuo Lu Chinese Academy of Sciences 8/1/2010 - 8/6/2010
Kara Lee Maki University of Minnesota 9/1/2009 - 8/31/2011
Yu (David) Mao University of California, Los Angeles 8/31/2010 - 8/30/2011
Per-Gunnar J. Martinsson University of Colorado 7/31/2010 - 8/6/2010
Anita Mayo Bernard M. Baruch College, CUNY 8/1/2010 - 8/5/2010
Eric Michielssen University of Michigan 8/1/2010 - 8/5/2010
Sonia Mogilevskaya University of Minnesota 8/2/2010 - 8/5/2010
Arje Nachman Air Force Office of Scientific Research (AFOSR) 8/1/2010 - 8/5/2010
Naoshi Nishimura Kyoto University 8/1/2010 - 8/6/2010
Cecilia Ortiz-Duenas University of Minnesota 9/1/2009 - 8/31/2011
Andras Pataki New York University 8/1/2010 - 8/5/2010
Nikos P. Pitsianis Duke University 8/1/2010 - 8/6/2010
Jack Poulson University of Texas 8/2/2010 - 8/5/2010
Bryan Quaife Simon Fraser University 8/1/2010 - 8/5/2010
Abtin Rahimian Georgia Institute of Technology 7/31/2010 - 8/5/2010
Fernando Reitich University of Minnesota 8/2/2010 - 8/5/2010
Vladimir Rokhlin Yale University 8/1/2010 - 8/6/2010
Panu Sam-ang University of Washington 8/1/2010 - 8/6/2010
Fadil Santosa University of Minnesota 7/1/2008 - 6/30/2011
Francisco-Javier Sayas University of Minnesota 8/2/2010 - 8/6/2010
Tsvetanka Sendova University of Minnesota 9/1/2008 - 8/10/2010
Shuanglin Shao University of Minnesota 9/1/2009 - 8/31/2011
Josef Aaron Sifuentes Rice University 8/1/2010 - 8/5/2010
Jiming Song Iowa State University 8/1/2010 - 8/5/2010
John A. Strain University of California, Berkeley 8/1/2010 - 8/6/2010
Jiguang Sun Delaware State University 8/1/2010 - 8/6/2010
Xiaobai Sun Duke University 8/1/2010 - 8/5/2010
William Scott Thornton University of Tennessee 8/2/2010 - 8/4/2010
Anna-Karin Tornberg Royal Institute of Technology (KTH) 8/1/2010 - 8/5/2010
Paul Hikaru Tsuji University of Texas 8/1/2010 - 8/5/2010
Shravan Veerapaneni Courant Institute of Mathematical Sciences 8/1/2010 - 8/5/2010
Felipe Vico-Bondía Polytechnical University of Valencia 8/2/2010 - 8/5/2010
Haitao Wang Qinghua (Tsing Hua) University 8/1/2010 - 8/6/2010
Jacob K. White Massachusetts Institute of Technology 8/1/2010 - 8/5/2010
Hong Xiao University of California, Davis 8/1/2010 - 8/6/2010
Liwei Xu Rensselaer Polytechnic Institute 8/1/2010 - 8/5/2010
Lexing Ying University of Texas 8/3/2010 - 8/4/2010
Tsuyoshi Yoneda University of Minnesota 9/4/2009 - 8/31/2010
Patrick McKendree Young University of Colorado 8/1/2010 - 8/5/2010
Bo Zhang University of North Carolina 8/1/2010 - 8/6/2010
Weigang Zhong University of Minnesota 9/8/2008 - 9/30/2010
Denis Zorin New York University 8/1/2010 - 8/5/2010
Legend: Postdoc or Industrial Postdoc Long-term Visitor

IMA Affiliates:
Arizona State University, Boeing, Corning Incorporated, ExxonMobil, Ford, General Motors, Georgia Institute of Technology, Honeywell, IBM, Indiana University, Iowa State University, Korea Advanced Institute of Science and Technology (KAIST), Lawrence Livermore National Laboratory, Lockheed Martin, Los Alamos National Laboratory, Medtronic, Michigan State University, Michigan Technological University, Mississippi State University, Northern Illinois University, Ohio State University, Pennsylvania State University, Portland State University, Purdue University, Rice University, Rutgers University, Sandia National Laboratories, Schlumberger Cambridge Research, Schlumberger-Doll, Seoul National University, Siemens, Telcordia, Texas A & M University, University of Central Florida, University of Chicago, University of Delaware, University of Houston, University of Illinois at Urbana-Champaign, University of Iowa, University of Kentucky, University of Maryland, University of Michigan, University of Minnesota, University of Notre Dame, University of Pennsylvania, University of Pittsburgh, University of Tennessee, University of Wisconsin, University of Wyoming, US Air Force Research Laboratory, Wayne State University, Worcester Polytechnic Institute