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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Numerische Mathematik 86 (2000), S. 49-101 
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991): 65D15,65D32, 65Y20, 41A25, 46E35
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. This paper is concerned with the efficient evaluation of nonlinear expressions of wavelet expansions obtained through an adaptive process. In particular, evaluation covers here the computation of inner products of such expressions with wavelets which arise, for instance, in the context of Galerkin or Petrov Galerkin schemes for the solution of differential equations. The central objective is to develop schemes that facilitate such evaluations at a computational expense exceeding the complexity of the given expansion, i.e., the number of nonzero wavelet coefficients, as little as possible. The following issues are addressed. First, motivated by previous treatments of the subject, we discuss the type of regularity assumptions that are appropriate in this context and explain the relevance of Besov norms. The principal strategy is to relate the computation of inner products of wavelets with compositions to approximations of compositions in terms of possibly few dual wavelets. The analysis of these approximations finally leads to a concrete evaluation scheme which is shown to be in a certain sense asymptotically optimal. We conclude with a simple numerical example.
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    Integral equations and operator theory 10 (1987), S. 531-553 
    ISSN: 1420-8989
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
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  • 3
    Electronic Resource
    Electronic Resource
    Springer
    Integral equations and operator theory 13 (1990), S. 104-131 
    ISSN: 1420-8989
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Banach algebras of certain bounded operators acting on the half-spaceL p (R + n+1 ,x 0 α ) (1〈p〈∞, −1〈α〈p−1) are defined which contain for example Wiener-Hopf operators, defined by multidimensional singular convolution integral operators, as well as certain singular integral operators with fixed singularities. Moreover the symbol may be a positive homogeneous function only piecewise continuous on the unit sphere. Actually these multidimensional singular integral operators may be not Calderón-Zygmund operators but are built up by those in lower dimensions. This paper is a continuation of a joint paper of the author together with R.V. Duduchava [10]. The purpose is to investigate invertibility or Fredholm properties of these operators, while the continuity is given by definition. This is done in [10] forp=2 and −1〈α〈1, and in the present paper forL p (R + n+1 ,x 0 α ) with 1〈p〈∞ and −1〈α〈p−1.
    Type of Medium: Electronic Resource
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  • 4
    Publication Date: 2022-04-19
    Description: We consider a stochastic optimal exit time feedback control problem. The Bellman equation is solved approximatively via the Policy Iteration algorithm on a polynomial ansatz space by a sequence of linear equations. As high degree multi-polynomials are needed, the corresponding equations suffer from the curse of dimensionality even in moderate dimensions. We employ tensor-train methods to account for this problem. The approximation process within the Policy Iteration is done via a Least-Squares ansatz and the integration is done via Monte-Carlo methods. Numerical evidences are given for the (multi dimensional) double well potential and a three-hole potential.
    Language: English
    Type: article , doc-type:article
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