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Wavelet compression of anisotropic integrodifferential operators on sparse tensor product spaces

by N. Reich

(Report number 2008-26)

Abstract
For a class of anisotropic integrodifferential operators B arising as semi- group 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 pos- sibly large n. Under certain conditions on B, the scheme is of essentially optimal and dimension independent complexity O(h-1(ogh)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+e), where e<<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 k(.,.) that can be sin- gular on all secondary diagonals. Practical examples of such operators from Mathematical Finance are given and some numerical results are presented.

Keywords: Parabolic differential equations, wavelets, adaptivity, optimal computational complexity, best N-term approximation, matrix compression

@Techreport{R08_390,
  author = {N. Reich},
  title = {Wavelet compression of anisotropic integrodifferential operators on sparse tensor product spaces},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2008-26},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2008/2008-26.pdf },
  year = {2008}
}

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