# Wavelet compression of anisotropic integrodifferential operators on sparse tensor product spaces

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 44, Issue: 1, page 33-73
- ISSN: 0764-583X

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topReich, Nils. "Wavelet compression of anisotropic integrodifferential operators on sparse tensor product spaces." ESAIM: Mathematical Modelling and Numerical Analysis 44.1 (2010): 33-73. <http://eudml.org/doc/250788>.

@article{Reich2010,

abstract = {
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.
},

author = {Reich, Nils},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Wavelet compression; sparse grids; anisotropic integrodifferential operators; norm equivalences; wavelet compression; convergence; partial integrodifferential equations; financial mathematics; complexity},

language = {eng},

month = {3},

number = {1},

pages = {33-73},

publisher = {EDP Sciences},

title = {Wavelet compression of anisotropic integrodifferential operators on sparse tensor product spaces},

url = {http://eudml.org/doc/250788},

volume = {44},

year = {2010},

}

TY - JOUR

AU - Reich, Nils

TI - Wavelet compression of anisotropic integrodifferential operators on sparse tensor product spaces

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/3//

PB - EDP Sciences

VL - 44

IS - 1

SP - 33

EP - 73

AB -
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.

LA - eng

KW - Wavelet compression; sparse grids; anisotropic integrodifferential operators; norm equivalences; wavelet compression; convergence; partial integrodifferential equations; financial mathematics; complexity

UR - http://eudml.org/doc/250788

ER -

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