Wavelet compression of anisotropic integrodifferential operators on sparse tensor product spaces

Nils Reich

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 44, Issue: 1, page 33-73
  • ISSN: 0764-583X

Abstract

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For a class of anisotropic integrodifferential operators ℬ arising as semigroup generators of Markov processes, we present a sparse tensor product wavelet compression scheme for the Galerkin finite element discretization of the corresponding integrodifferential equations ℬ u = f on [0,1]n with possibly large n. Under certain conditions on ℬ , the scheme is of essentially optimal and dimension independent complexity 𝒪 (h-1| log h |2(n-1)) without corrupting the convergence or smoothness requirements of the original sparse tensor finite element scheme. If the conditions on ℬ are not satisfied, the complexity can be bounded by 𝒪 (h-(1+ε)), where ε ≪ 1 tends to zero with increasing number of the wavelets' vanishing moments. Here h denotes the width of the corresponding finite element mesh. The operators under consideration are assumed to be of non-negative (anisotropic) order and admit a non-standard kernel κ ( · , · ) that can be singular on all secondary diagonals. Practical examples of such operators from Mathematical Finance are given and some numerical results are presented.

How to cite

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Reich, Nils. "Wavelet compression of anisotropic integrodifferential operators on sparse tensor product spaces." ESAIM: Mathematical Modelling and Numerical Analysis 44.1 (2010): 33-73. <http://eudml.org/doc/250788>.

@article{Reich2010,
abstract = { For a class of anisotropic integrodifferential operators $\{\cal B\}$ arising as semigroup generators of Markov processes, we present a sparse tensor product wavelet compression scheme for the Galerkin finite element discretization of the corresponding integrodifferential equations $\{\cal B\}$ u = f on [0,1]n with possibly large n. Under certain conditions on $\{\cal B\}$, the scheme is of essentially optimal and dimension independent complexity $\mathcal\{O\}$(h-1| log h |2(n-1)) without corrupting the convergence or smoothness requirements of the original sparse tensor finite element scheme. If the conditions on $\{\cal B\}$ are not satisfied, the complexity can be bounded by $\mathcal\{O\}$(h-(1+ε)), where ε$\ll 1$ tends to zero with increasing number of the wavelets' vanishing moments. Here h denotes the width of the corresponding finite element mesh. The operators under consideration are assumed to be of non-negative (anisotropic) order and admit a non-standard kernel κ$(\cdot,\cdot)$ that can be singular on all secondary diagonals. Practical examples of such operators from Mathematical Finance are given and some numerical results are presented. },
author = {Reich, Nils},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Wavelet compression; sparse grids; anisotropic integrodifferential operators; norm equivalences; wavelet compression; convergence; partial integrodifferential equations; financial mathematics; complexity},
language = {eng},
month = {3},
number = {1},
pages = {33-73},
publisher = {EDP Sciences},
title = {Wavelet compression of anisotropic integrodifferential operators on sparse tensor product spaces},
url = {http://eudml.org/doc/250788},
volume = {44},
year = {2010},
}

TY - JOUR
AU - Reich, Nils
TI - Wavelet compression of anisotropic integrodifferential operators on sparse tensor product spaces
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 44
IS - 1
SP - 33
EP - 73
AB - For a class of anisotropic integrodifferential operators ${\cal B}$ arising as semigroup generators of Markov processes, we present a sparse tensor product wavelet compression scheme for the Galerkin finite element discretization of the corresponding integrodifferential equations ${\cal B}$ u = f on [0,1]n with possibly large n. Under certain conditions on ${\cal B}$, the scheme is of essentially optimal and dimension independent complexity $\mathcal{O}$(h-1| log h |2(n-1)) without corrupting the convergence or smoothness requirements of the original sparse tensor finite element scheme. If the conditions on ${\cal B}$ are not satisfied, the complexity can be bounded by $\mathcal{O}$(h-(1+ε)), where ε$\ll 1$ tends to zero with increasing number of the wavelets' vanishing moments. Here h denotes the width of the corresponding finite element mesh. The operators under consideration are assumed to be of non-negative (anisotropic) order and admit a non-standard kernel κ$(\cdot,\cdot)$ that can be singular on all secondary diagonals. Practical examples of such operators from Mathematical Finance are given and some numerical results are presented.
LA - eng
KW - Wavelet compression; sparse grids; anisotropic integrodifferential operators; norm equivalences; wavelet compression; convergence; partial integrodifferential equations; financial mathematics; complexity
UR - http://eudml.org/doc/250788
ER -

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