Wavelet compression of anisotropic integrodifferential operators on sparse tensor product spaces
ESAIM: Modélisation mathématique et analyse numérique, Tome 44 (2010) no. 1, pp. 33-73.

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 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 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.

DOI : 10.1051/m2an/2009039
Classification : 47A20, 65F50, 65N12, 65Y20, 68Q25, 45K05, 65N30
Mots clés : wavelet compression, sparse grids, anisotropic integrodifferential operators, norm equivalences
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Reich, Nils. Wavelet compression of anisotropic integrodifferential operators on sparse tensor product spaces. ESAIM: Modélisation mathématique et analyse numérique, Tome 44 (2010) no. 1, pp. 33-73. doi : 10.1051/m2an/2009039. http://archive.numdam.org/articles/10.1051/m2an/2009039/

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