# Solution of Integral and Differential Equations using Algebraic<br/><br/>Splines

Saturday, September 16, 2006 - 10:30am - 11:20am

EE/CS 3-180

Chandrajit Bajaj (The University of Texas at Austin)

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.

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.