Wavelet compression of anisotropic integrodifferential operators on sparse tensor product spaces

For a class of anisotropic integrodifferential operators ${\cal B}$ 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 ${\cal B}$ u = f on [0,1]n with possibly large n. Under certain conditions on ${\cal B}$, the scheme is of essentially optimal and dimension independent complexity $\mathcal{O}$(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 ${\cal B}$ are not satisfied, the complexity can be bounded by $\mathcal{O}$(h-(1+ε)), where ε $\ll 1$ 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$(\cdot,\cdot)$ that can be singular on all secondary diagonals. Practical examples of such operators from Mathematical Finance are given and some numerical results are presented