Wavelet compression of anisotropic integrodifferential operators on sparse tensor product spaces

Reich, Nils

ESAIM: Modélisation mathématique et analyse numérique, Tome 44 (2010) no. 1, pp. 33-73

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 $ℬ$ u = 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 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$(·,·)$ that can be singular on all secondary diagonals. Practical examples of such operators from Mathematical Finance are given and some numerical results are presented.

Classification : 47A20, 65F50, 65N12, 65Y20, 68Q25, 45K05, 65N30
Keywords: wavelet compression, sparse grids, anisotropic integrodifferential operators, norm equivalences

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