Wavelet compression of anisotropic integrodifferential operators on sparse tensor product spaces
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 1, p. 33-73

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 $\ll 1$ 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$\left(·,·\right)$ that can be singular on all secondary diagonals. Practical examples of such operators from Mathematical Finance are given and some numerical results are presented.

DOI : https://doi.org/10.1051/m2an/2009039
Classification:  47A20,  65F50,  65N12,  65Y20,  68Q25,  45K05,  65N30
Keywords: wavelet compression, sparse grids, anisotropic integrodifferential operators, norm equivalences
@article{M2AN_2010__44_1_33_0,
author = {Reich, Nils},
title = {Wavelet compression of anisotropic integrodifferential operators on sparse tensor product spaces},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {44},
number = {1},
year = {2010},
pages = {33-73},
doi = {10.1051/m2an/2009039},
zbl = {1189.65311},
mrnumber = {2647753},
language = {en},
url = {http://www.numdam.org/item/M2AN_2010__44_1_33_0}
}

Reich, Nils. Wavelet compression of anisotropic integrodifferential operators on sparse tensor product spaces. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 1, pp. 33-73. doi : 10.1051/m2an/2009039. http://www.numdam.org/item/M2AN_2010__44_1_33_0/

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