Fast Computation of Volume Potentials by Approximate Approximations
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Beschreibung
This book introduces a new fast high-order method for approximating volume potentials and other integral operators with singular kernel. These operators arise naturally in many fields, including physics, chemistry, biology, and financial mathematics. A major impediment to solving real world problems is the so-called curse of dimensionality, where the cubature of these operators requires a computational complexity that grows exponentially in the physical dimension. The development of separated representations has overcome this curse, enabling the treatment of higher-dimensional numerical problems. The method of approximate approximations discussed here provides high-order semi-analytic cubature formulas for many important integral operators of mathematical physics. By using products of Gaussians and special polynomials as basis functions, the action of the integral operators can be written as one-dimensional integrals with a separable integrand. The approximation of a separated representation of the density combined with a suitable quadrature of the one-dimensional integrals leads to a separated approximation of the integral operator. This method is also effective in high-dimensional cases. The book is intended for graduate students and researchers interested in applied approximation theory and numerical methods for solving problems of mathematical physics.
This book introduces a new fast high-order method for approximating volume potentials and other integral operators with singular kernel. These operators arise naturally in many fields, including physics, chemistry, biology, and financial mathematics. A major impediment to solving real world problems is the so-called curse of dimensionality, where the cubature of these operators requires a computational complexity that grows exponentially in the physical dimension. The development of separated representations has overcome this curse, enabling the treatment of higher-dimensional numerical problems. The method of approximate approximations discussed here provides high-order semi-analytic cubature formulas for many important integral operators of mathematical physics. By using products of Gaussians and special polynomials as basis functions, the action of the integral operators can be written as one-dimensional integrals with a separable integrand. The approximation of a separated representation of the density combined with a suitable quadrature of the one-dimensional integrals leads to a separated approximation of the integral operator. This method is also effective in high-dimensional cases. The book is intended for graduate students and researchers interested in applied approximation theory and numerical methods for solving problems of mathematical physics.
Provides a new approach to solving high-dimensional partial differential problems of mathematical physics Obtains new one-dimensional integral representations for operators of mathematical physics Provides fast and effective high-order methods in very high dimensions for applied problems
Autor*in
Flavia Lanzara
Themen in »Fast Computation of Volume Potentials by Approximate Approximations«
Approximations of high-dimensional volume potentials Effective treatment of multivariate singular integral operators High-order semi-analytic cubature formulas Fast and accurate computation even for very high dimensions Basis functions introduced by Approximate Approximations One-dimensional integral representation with separable integrand Tensor product approximation Approximations via Gaussians and special polynomials Efficient computation to harmonic and biharmonic potentials Approximate solution of the Cauchy problem for the heat equation Computation of solutions to nonstationary Stokes system Computation of solutions to the Lamé system A fast solution method for time dependent Schroedinger equation 3-dimensional problems in elasticity and thermoelasticity Cubature of pseudo-differential operators