Approximation of radiating waves in the near-field: error estimates and application to a class of inverse problems. (English) Zbl 07403087

Summary: Numerical approximation of radiating waves, with a priori truncation parameter and error estimates, is crucial for efficient simulation of forward and inverse scattering models. Convergence of a series ansatz for the wave field using the classical radiating wave functions is known only when the field is evaluated exterior to a ball circumscribing the configuration. If the configuration comprises non-convex and/or elongated scatterers, evaluation of the scattered field in the interior region of the ball is important for applications including for the far-field-data based inverse problem of identifying the scatterer boundary. In this article we develop a new error estimate for the series ansatz that facilitates identification of the truncation-parameter dependent interior convergence region. This in turn facilitates an estimate-based approach for solving the boundary identification inverse problem. We demonstrate, through numerical experiments, excellent agreement of the theoretical error estimate with respect to the truncation parameter, and the efficiency of the approach to identify scatterer shapes.


65Nxx Numerical methods for partial differential equations, boundary value problems
78Axx General topics in optics and electromagnetic theory
35Jxx Elliptic equations and elliptic systems


TMATROM; MieSolver
Full Text: DOI


[1] Colton, D.; Kress, R., Inverse Acoustic and Electromagnetic Scattering Theory (2012), Springer
[2] Mishchenko, M. I.; Travis, L. D.; Lacis, A. A., Multiple Scattering of Light by Particles: Radiative Transfer and Coherent Backscattering (2006), Cambridge University Press
[3] van de Hulst, H. C., Light Scattering by Small Particles (1981), Dover Publications Inc.
[4] Rayleigh, H. C., On the electromagnetic theory of light, Philos. Mag. S. 5, 12, 81-101 (1881)
[5] Hawkins, S. C., Algorithm 1009: MieSolver—an object-oriented Mie Series software for wave scattering by cylinders, ACM Trans. Math. Software, 46, 19:1-19:28 (2020) · Zbl 1484.65328
[6] Ganesh, M.; Hawkins, S. C., Algorithm 975: TMATROM—A T-matrix reduced order model software, ACM Trans. Math. Software, 44, 9:1-9:18 (2017) · Zbl 1484.78006
[7] Ganesh, M.; Hawkins, S. C., An efficient \(\mathcal{O} ( N )\) algorithm for computing \(\mathcal{O} ( N^2 )\) acoustic wave interactions in large \(N\)-obstacle three dimensional configurations, BIT, 55, 117-139 (2015) · Zbl 1311.65156
[8] Ganesh, M.; Hawkins, S. C., A fast high order algorithm for multiple scattering from large sound-hard configurations in three dimensions, J. Comput. Appl. Math., 362, 324-340 (2019) · Zbl 1433.78025
[9] Mishchenko, M. I., Comprehensive T-matrix reference database: a 2017-2019 update, J. Quant. Spectrosc. Radiat. Transfer, 242, Article 106692 pp. (2020)
[10] Mishchenko, M. I.; Travis, L. D.; Mackowski, D. W., T-matrix computations of light scattering by nonspherical particles: a review, J. Quant. Spectrosc. Radiat. Transfer, 55, 535-575 (1996)
[11] Angell, T. S.; Kleinman, R. E.; Roach, G. F., An inverse transmission problem for the Helmholtz equation, Inverse Problems, 3, 149-180 (1987) · Zbl 0658.35093
[12] Angell, T. S.; Kleinman, R. E.; Kok, B.; Roach, G. F., A constructive method for identification of an impenetrable scatterer, Wave Motion, 11, 185-200 (1989) · Zbl 0686.65087
[13] Ganesh, M.; Hawkins, S. C., Three dimensional electromagnetic scattering T-matrix computations, J. Comput. Appl. Math., 234, 1702-1709 (2010) · Zbl 1195.78016
[14] Ganesh, M.; Hawkins, S. C., A far-field based T-matrix method for two dimensional obstacle scattering, ANZIAM J., 51, C215-C230 (2010) · Zbl 1386.76146
[15] Ganesh, M.; Hawkins, S. C.; Hiptmair, R., Convergence analysis with parameter estimates for a reduced basis acoustic scattering T-matrix method, IMA J. Numer. Anal., 32, 1348-1374 (2012) · Zbl 1275.65074
[16] Kirsch, A., The denseness of the far field patterns for the transmission problem, IMA J. Appl. Math., 37, 213-225 (1986) · Zbl 0652.35104
[17] Ganesh, M.; Hawkins, S. C., A far-field based T-matrix method for three dimensional acoustic scattering, ANZIAM J., C121-C136 (2008) · Zbl 1359.76253
[18] Ganesh, M.; Hawkins, S. C., An efficient algorithm for simulating scattering by a large number of two dimensional particles, ANZIAM J., 52, C139-C155 (2011) · Zbl 1390.65155
[19] Sloan, I. H., Polynomial interpolation and hyperinterpolation over general regions, J. Approx. Theory, 83, 238-254 (1995) · Zbl 0839.41006
[20] Sloan, I. H.; Womersley, R. S., Constructive polynomial approximation on the sphere, J. Approx. Theory, 103, 91-118 (2000) · Zbl 0946.41007
[21] Barnett, A. H.; Betcke, T., An exponentially convergent nonpolynomial finite element method for time-harmonic scattering from polygons, SIAM J. Sci. Comput., 32, 1417-1441 (2010) · Zbl 1216.65151
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.