Wavelet compression of anisotropic integrodifferential operators on sparse tensor product spaces

ETH Zurich, Seminar for Applied Mathematics, 8092 Zurich, Switzerland. reich@math.ethz.ch

Received: 6 August 2008

Abstract

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.