Commun. Comput. Phys., 2 (2007), pp. 1139-1173.

High Performance Algorithms Based on a New Wavelet Expansion for Time Dependent Acoustic Obstacle Scattering

Lorella Fatone 1, Giuseppe Rao 2, Maria Cristina Recchioni 3, Francesco Zirilli 4*

1 Dipartimento di Matematica Pura e Applicata, Universit`a di Modena e Reggio Emilia, Via Campi 213/b, 41100 Modena, Italy.
2 Dipartimento di Matematica e Applicazioni, Universit`a di Palermo, Via Archirafi 34, 90123 Palermo, Italy.
3 Dipartimento di Scienze Sociali "D. Serrani", Universit`a Politecnica delle Marche, Piazza Martelli 8, 60121 Ancona, Italy.
4 Dipartimento di Matematica "G. Castelnuovo", Universit`a di Roma "La Sapienza", Piazzale Aldo Moro 2, 00185 Roma, Italy.

Received 15 December 2006; Accepted (in revised version) 20 March 2007
Communicated by Gang Bao
Available online 15 June 2007


This paper presents a highly parallelizable numerical method to solve time dependent acoustic obstacle scattering problems. The method proposed is a generalization of the ``operator expansion method" developed by Recchioni and Zirilli [SIAM J.~Sci.~Comput., 25 (2003), 1158-1186]. The numerical method proposed reduces, via a perturbative approach, the solution of the scattering problem to the solution of a sequence of systems of first kind integral equations. The numerical solution of these systems of integral equations is challenging when scattering problems involving realistic obstacles and small wavelengths are solved. A computational method has been developed to solve these challenging problems with affordable computing resources. To this aim a new way of using the wavelet transform and new bases of wavelets are introduced, and a version of the operator expansion method is developed that constructs directly element by element in a fully parallelizable way. Several numerical experiments involving realistic obstacles and ``small" wavelengths are proposed and high dimensional vector spaces are used in the numerical experiments. To evaluate the performance of the proposed algorithm on parallel computing facilities, appropriate speed up factors are introduced and evaluated.

AMS subject classifications: 34L25, 01-08

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Key words: Time dependent acoustic scattering, Helmholtz equation, integral equation methods, wavelet bases, sparse linear systems.

*Corresponding author.
Email: (F. Zirilli)

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