On Spectral Approximations With Nonstandard Weight Functions and Their Implementations to Generalized Chaos Expansions
For researchers in spectral methods and uncertainty quantification, this work extends convergence theory and integration techniques to nonstandard weight functions, enabling efficient handling of singular functions and dependent random variables.
The paper analyzes expansions in orthogonal polynomials with nonstandard weight functions, proving convergence rates via a comparison lemma and presenting a spectrally convergent multidimensional integration method. Numerical examples show efficient integration of singular functions and application to generalized polynomial chaos expansions for dependent random variables.
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 by Gram-Schmidt orthogonalization. However, in most cases, they are not eigenfunctions of some singular Sturm-Liouville problem, as is the case for classical polynomials. Therefore, standard results regarding convergence cannot be applied. Furthermore, since in general, the weight functions are not a tensor product of one-dimensional functions, the orthogonal polynomials are not a tensor product of one-dimensional orthogonal polynomials, as well. In this work, we determine 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 show that the use of nonstandard weight functions can allow for efficient integration of singular functions. We also apply this method to Generalized Polynomial Chaos Expansions in the case of dependent random variables.