WAVELET COMPRESSION OF ANISOTROPIC INTEGRODIFFERENTIAL OPERATORS ON SPARSE TENSOR PRODUCT SPACES

摘要

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]~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 ε1 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.