Wavelet Compression of Anisotropic Integrodifferential Operators on Sparse Tensor Product Spaces

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 Bu = f on [0, 1] 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. Mathematics Subject Classification. 47A20, 65F50, 65N12, 65Y20, 68Q25, 45K05, 65N30. Received August 6, 2008. Published online October 9, 2009.