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ON THE COMPUTATION OF HIGH-DIMENSIONAL POTENTIALS OF ADVECTION–DIFFUSION OPERATORS

Part of: Numerical approximation and computational geometry

Published online by Cambridge University Press:  24 February 2015

Flavia Lanzara  and
Gunther Schmidt
Flavia Lanzara
Affiliation:
Department of Mathematics, Sapienza University of Rome, Piazzale Aldo Moro 2, 00185 Rome, Italy email lanzara@mat.uniroma1.it
Gunther Schmidt
Affiliation:
Weierstrass Institute forApplied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin, Germany email schmidt@wias-berlin.de
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Abstract

We study a fast method for computing potentials of advection–diffusion operators $-{\rm\Delta}+2\mathbf{b}\boldsymbol{\cdot }{\rm\nabla}+c$ with $\mathbf{b}\in \mathbb{C}^{n}$ and $c\in \mathbb{C}$ over rectangular boxes in $\mathbb{R}^{n}$. By combining high-order cubature formulas with modern methods of structured tensor product approximations, we derive an approximation of the potentials which is accurate and provides approximation formulas of high order. The cubature formulas have been obtained by using the basis functions introduced in the theory of approximate approximations. The action of volume potentials on the basis functions allows one-dimensional integral representations with separable integrands, i.e. a product of functions depending on only one of the variables. Then a separated representation of the density, combined with a suitable quadrature rule, leads to a tensor product representation of the integral operator. Since only one-dimensional operations are used, the resulting method is effective also in the high-dimensional case.

MSC classification

Primary: 65D32: Quadrature and cubature formulas
Type
Research Article
Information
Mathematika , Volume 61 , Issue 2 , May 2015 , pp. 309 - 327
Copyright
Copyright © University College London 2015 

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