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Polynomial approximation of anisotropic analytic functions of several variables. (English) Zbl 1462.41003

Solving PDEs, especially when the underlying space dimensions are large, is an important and not-easy task in numerical analysis. In the present paper, the authors study an approach using approximations with multivariate polynomials and have in mind to limit the computational effort that is required to compute these approximations. In particular, it is desirable to meet a “budget” of computational cost while achieving a pre-chosen accurary. To this end, the authors define multivariable polynomials on lower sets of exponents that have a certain maximal size (cardinality of the set), approximate solutions of a PDE with those and study the optimal error with respect to these approximations. Lower sets are also known for rendering multivariate polynomials available for polynomial interpolation in more than one dimension.
The approximations are available in any dimension and the goal is to overcome the curse of dimensionality in multivariable approximation that prohibits normally the use of finite elements, say, or finite differences, for the PDE approximations. (The particular problems are not only the constructions of the finite elements if the dimension \(d\) is large, but also the deterioration of the minimal error with increasing space dimension.) But here, any number, including in theory infinitely many, of paramateres to the solution \(u\) of the PDE is admitted. The paper takes not only the usual features of \(u\) (e.g. smoothness) into account but in particular the fact that it stems from solving a partial differential equation.
The errors are normally measured for instance in Chebyshev norms but it is found in this article that a so-called surrogate norms is easier to handle. The latter is defined via the sums of norms of Taylor expansion coefficients of an analytic function rather than using the function itself. The error estimates are not only asymptotic estimates with respect to the cardinality \(n\) of the sets of exponents (the lower sets) but also certified, i.e., explicit with respect to \(n\) on an one-by-one basis and not only for \(n\to\infty\). An advantage of this approach is that, even with these relatively strong requirements, the suitable lower sets are easy to find via certain simplices that are explicitly derived in the paper. The errors \(E_n(K)\) are upper bounds to the Kolmogorov \(n\)-widths, but they are defined in a closely related way, namely the minimal surrogate error over all lowers sets of a cardinality \(n\), where the error is the maximum error of the best approximation over a model set \(K\) of \(u\)s of solutions by multivariate polynomials with those particular lower sets \(\Lambda\).
Then the authors study the behaviour of this number \(E_n\) with respect to \(n\), using the particular fact about \(u\) that it is a solution of a PDE, give optimal bounds on \(E_n\) and provide means of finding the best lower sets \(\Lambda=\Lambda_n\) such that the \(E_\Lambda\) is close to \(E_n(K)\). The search for those is made efficient by defining them by simplices which are well-understood by means of results from number theory.

MSC:

41A10 Approximation by polynomials
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
41A63 Multidimensional problems
65N15 Error bounds for boundary value problems involving PDEs
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