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A nonlinear optimization procedure for generalized Gaussian quadratures. (English) Zbl 1215.65045

Summary: We present a new nonlinear optimization procedure for the computation of generalized Gaussian quadratures for a broad class of square integrable functions on intervals. While some of the components of this algorithm have been previously published, we present a simple and robust scheme for the determination of a sparse solution to an underdetermined nonlinear optimization problem which replaces the continuation scheme of the previously published works. The new algorithm successfully computes generalized Gaussian quadratures in a number of instances in which the previous algorithms fail.

Four applications of our scheme to computational physics are presented: the construction of discrete plane wave expansions for the Helmholtz Green’s function, the design of linear array antennae, the computation of a quadrature for the discretization of Laplace boundary integral equations on certain domains with corners, and the construction of quadratures for the discretization of Laplace and Helmholtz boundary integral equations on smooth surfaces.

MSC:
65D32Quadrature and cubature formulas (numerical methods)
41A55Approximate quadratures
65N38Boundary element methods (BVP of PDE)
35J05Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation