Using B-spline frames to approximate solutions of acoustic scattering problems

TL;DR

This paper explores the use of B-spline based Gabor frames to efficiently approximate solutions to 2D acoustic scattering problems, highlighting implementation strategies and comparing coefficient determination methods.

Contribution

It introduces a novel application of B-spline Gabor frames for acoustic scattering, including practical implementation and coefficient calculation techniques.

Findings

Coefficients from a modified orthogonal matching pursuit algorithm yield the best accuracy-to-sparsity ratio.

Frame representations are effective in approximating solutions with fewer coefficients.

Numerical experiments compare efficiency of different coefficient determination methods.

Abstract

Although frames, which are a generalization of bases, are important tools used in signal processing, their potential in other fields of engineering and applied mathematics (e.g. acoustics) has not been fully explored yet. Gabor frames, that are specific type of frames, are very well adapted to oscillating functions, and therefore have a great potential to efficiently represent functions in connection with the Helmholtz operator. In this paper representations of the solution of a scattering problem in 2D using Gabor frames based on B-splines as building blocks are investigated. Practical issues concerning the implementation of frames like the restriction of the frame elements to a finite interval and methods to determine the unknown coefficients for the representation, i.e. by using dual frames or by solving a least squares problem, are discussed. Numerical experiments are made comparing the different ways to determine the unknown expansion coefficients and the representations are compared in terms of their efficiency, i.e. the number of used frame coefficients versus the accuracy of the approximation. In all cases the coefficients calculated with a slightly modified orthogonal matching pursuit algorithm provide the best accuracy versus sparsity ratio.