September 15 - 16, 2006
This two part tutorial shall introduce you to algorithmic algebraic geometry
methods of manipulating algebraic (polynomial) splines necessary for
the solution of multivariate integral and partial differential equations
emanating
from science and engineering applications.
In the first part, we shall discuss well known (classical) algorithms for
modeling
the structure and energetics of physical domains
(molecules to airplanes to oil fields), at multiple scales,
with implicit algebraic splines (A-patches), and their
rational parametric splines (NURBS) approximations. Algebraic geometry
methods include computing global and local parameterizations using Newton
factorization
and Hensel lifting, computation of adjoints by interpolating through
singularities,
and low degree curve and surface intersections via Resultants and/or Groebner
Basis calculations. The second part shall focus on more current research and
attempts
at using algebraic geometry methods for faster and more accurate calculations of
multivariate definite
integrals of algebraic functions arising from the solution of polarization
energetics
and forces (Generalized Born, Poisson Boltzmann) and Green function solutions of
two-phase Stokesian
flows. These algebraic geometry methods include the use of ideal theory and
solutions of polynomial systems of equations for more accurate cubature formulas
and their
faster evaluation.
This talk will discuss how many problems in control theory may be
formulated as problems of algebraic geometry. Topics covered include
realization theory, homotopy methods for pole placement, construction
of Schur functions, stabilization via coprime factorization, systems
over rings, controllability, and multidimensional systems. We also
discuss more recent research in the area of decentralized control.
This talk will discuss how many of the basic questions in mechanism
science are naturally formulated as systems of polynomial equations.
These questions include both analysis problems (how does this
mechanism move?) and synthesis problems (which mechanisms will move
the way I require?). Once such a problem has been formulated as a
system polynomials, the irreducible decomposition of the solution set
of the system becomes a powerful tool for describing the answer. We
will discuss how some questions can be approached in stages by
computing irreducible decompositions of subsets of the original
equations and then intersecting solution components. Finally, we will
see how some questions about the existence of special mechanisms are
nicely formulated as fiber products of algebraic sets.
This talk will discuss how numerical polynomial continuation can be
used to solve the problems formulated in the first lecture. In doing
so, we will describe the basic constructs and algorithms of Numerical
Algebraic Geometry. Foremost among these is the notion of a witness
set, a numerical approximation to a linear section of an algebraic
set. We will describe how witness sets are computed, how they are
used in finding numerical irreducible decompositions, and how the
witness set for the intersection of two algebraic sets, say A and B,
can be found from the witness sets for A and B, via a diagonal
homotopy. Some recent avenues of research, such as how to find the
real solutions inside a complex curve, will be mentioned briefly.
Suggested reading: A.J. Sommese and C.W. Wampler, The numerical
solution of systems of polynomials arising in engineering and science,
World Scientific, 2005.