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For a class of anisotropic integrodifferential operators 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 u = f on [0,1]n with possibly large n. Under certain conditions on , the scheme is of essentially optimal and dimension independent complexity (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 are not satisfied, the complexity can be bounded by (h-(1+ε)), where ε 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.
@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 },
pages = {33--73},
publisher = {EDP-Sciences},
volume = {44},
number = {1},
year = {2010},
doi = {10.1051/m2an/2009039},
mrnumber = {2647753},
zbl = {1189.65311},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2009039/}
}
TY - JOUR AU - Reich, Nils TI - Wavelet compression of anisotropic integrodifferential operators on sparse tensor product spaces JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2010 SP - 33 EP - 73 VL - 44 IS - 1 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2009039/ DO - 10.1051/m2an/2009039 LA - en ID - M2AN_2010__44_1_33_0 ER -
%0 Journal Article %A Reich, Nils %T Wavelet compression of anisotropic integrodifferential operators on sparse tensor product spaces %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2010 %P 33-73 %V 44 %N 1 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2009039/ %R 10.1051/m2an/2009039 %G en %F 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 , Tome 44 (2010) no. 1, pp. 33-73. doi: 10.1051/m2an/2009039
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