Nonlinear Optimization, Quadrature, and Interpolation

Authors:
H. Cheng;V. Rokhlin;N. Yarvin
Affiliations:
-;-;-
Venue:
SIAM Journal on Optimization
Year:
1999

Citing 0
Cited 10

Accurate numerical methods for micromagnetics simulations with general geometries

Journal of Computational Physics
Prolate spheroidal wavefunctions as an alternative to Chebyshev and Legendre polynomials for spectral element and pseudospectral algorithms

Journal of Computational Physics
A wideband fast multipole method for the Helmholtz equation in three dimensions

Journal of Computational Physics
High performance BLAS formulation of the multipole-to-local operator in the fast multipole method

Journal of Computational Physics
Corner singularities for elliptic problems: Integral equations, graded meshes, quadrature, and compressed inverse preconditioning

Journal of Computational Physics
A numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions

Computers & Mathematics with Applications
Efficient discretization of Laplace boundary integral equations on polygonal domains

Journal of Computational Physics
A Nonlinear Optimization Procedure for Generalized Gaussian Quadratures

SIAM Journal on Scientific Computing
A new Fast Multipole formulation for the elastodynamic half-space Green's tensor

Journal of Computational Physics
High-order accurate methods for Nyström discretization of integral equations on smooth curves in the plane

Advances in Computational Mathematics

Quantified Score

Hi-index	0.03

Visualization

Abstract

We present a nonlinear optimization procedure for the design of generalized Gaussian quadratures for a fairly broad class of functions. While some of the components of the algorithm have been published previously, we introduce an improved procedure for the determination of an acceptable initial point for the continuation scheme that stabilizes the Newton-type process used to find the quadratures. The resulting procedure never failed when applied to Chebyshev systems (for which the existence and uniqueness of generalized Gaussian quadratures are well known); it also worked for many non-Chebyshev systems, for which the generalized Gaussian quadratures are not guaranteed to exist. The performance of the algorithm is illustrated with several numerical examples; some of the presented quadratures integrate efficiently large classes of singular functions.