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On Spectral Approximations with Nonstandard Weight Functions and Their Implementations to Generalized Chaos Expansions

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Abstract

In this manuscript, we analyze the expansions of functions in orthogonal polynomials associated with a general weight function in a multidimensional setting. Such orthogonal polynomials can be obtained, e.g, by Gram–Schmidt orthogonalization. However, in most cases, they are not eigenfunctions of some singular Sturm–Liouville problem, as is the case for known polynomials, such as the Jacobi polynomials. Therefore, the standard convergence theorems do not apply. Furthermore, since in general multidimensional cases the weight functions are not a tensor product of one-dimensional functions, the orthogonal polynomials are not a product of one-dimensional orthogonal polynomials, as well. This work provides a way of estimating the convergence rate using a comparison lemma. We also present a spectrally convergent, multidimensional, integration method. Numerical examples demonstrate the efficacy of the proposed method. We also show that the use of non-standard weight functions can allow for efficient integration of singular functions. We demonstrate the use of this method to uncertainty quantification problem using Generalized Polynomial Chaos Expansions in the case of dependent random variables, as well.

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  1. School of Mathematical Sciences, Tel Aviv University, 69978, Tel Aviv, Israel

    A. Ditkowski & R. Katz

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  1. A. Ditkowski
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  2. R. Katz
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Correspondence to A. Ditkowski.

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A Appendix: Local Errors for the L-Shaped Domain and Condition Numbers in a Triangular Domain.

A Appendix: Local Errors for the L-Shaped Domain and Condition Numbers in a Triangular Domain.

Table 1 The error in each subdomain, \(\Omega _j\) for different degree of non-standard polynomials expansions
Full size table
Table 2 The error in each subdomain, \(\Omega _j\) for different degree of Legendre-type polynomials expansions
Full size table
Table 3 Condition numbers in a triangular domain. The condition number grows less than linear with M (the slope of the linear least squares regression of \(\log (\text {the condition number}\,)\) versus \(\log (M)\) is 0.7086)
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Ditkowski, A., Katz, R. On Spectral Approximations with Nonstandard Weight Functions and Their Implementations to Generalized Chaos Expansions. J Sci Comput 79, 1981–2005 (2019). https://doi.org/10.1007/s10915-019-00922-5

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  • DOI: https://doi.org/10.1007/s10915-019-00922-5

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