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Fast computation of volume potentials by approximate approximations

Record Type:	Language materials, printed : Monograph/item
Title/Author:	Fast computation of volume potentials by approximate approximations/ by Flavia Lanzara, Vladimir Maz'ya, Gunther Schmidt.
Author:	Lanzara, Flavia.
other author:	Maz'i︠a︡, V. G.
Published:	Cham :Springer Nature Switzerland : : 2025.,
Description:	x, 264 p. :ill. (some col.), digital ; : 24 cm.;
Contained By:	Springer Nature eBook
Subject:	Approximation theory. -
Online resource:	https://doi.org/10.1007/978-3-031-97442-7
ISBN:	9783031974427

Fast computation of volume potentials by approximate approximations
Lanzara, Flavia.

Fast computation of volume potentials by approximate approximations[electronic resource] /by Flavia Lanzara, Vladimir Maz'ya, Gunther Schmidt. - Cham :Springer Nature Switzerland :2025. - x, 264 p. :ill. (some col.), digital ;24 cm. - Lecture notes in mathematics,v. 23781617-9692 ;. - Lecture notes in mathematics ;1943..

Chapter 1. Introduction -- Chapter 2. Quasi-interpolation -- Chapter 3. Approximation of integral operators -- Chapter 4. Some other cubature problems -- Chapter 5. Approximate solution of non-stationary problems -- Chapter 6. Integral operators over hyper-rectangular domains.

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.

ISBN: 9783031974427

Standard No.: 10.1007/978-3-031-97442-7doiSubjects--Topical Terms:

527707
Approximation theory.

LC Class. No.: QA221

Dewey Class. No.: 511.4

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520 $a 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.
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