Conjugate gradient-boundary element solution to the Cauchy problem for Helmholtz-type equations L. Marin, L. Elliott, P. J. Heggs, D. B. Ingham, D. Lesnic, X. Wen Abstract In this paper, an iterative algorithm based on the conjugate gradient method (CGM) in combination with the boundary element method (BEM) for obtaining stable approximate solutions to the Cauchy problem for Helm- holtz-type equations is analysed. An efficient regularising stopping criterion for CGM proposed by Nemirovskii [25] is employed. The numerical results obtained confirm that the CGM + BEM produces a convergent and stable numerical solution with respect to increasing the number of boundary elements and decreasing the amount of noise added into the input data. Keywords Inverse problem, Cauchy problem, Helmholtz-type equations, CGM, BEM 1 Introduction The Helmholtz equation arises naturally in many physical applications related to wave propagation and vibration phenomena. It is often used to describe the vibration of a structure [1], the acoustic cavity problem [2], the radiation wave [3] and the scattering of a wave [4]. Another im- portant application of the Helmholtz equation is the problem of heat conduction in fins, see e.g. Kern and Kraus [5] and Manzoor et al. [6]. The knowledge of the Dirichlet, Neumann or mixed boundary conditions on the entire boundary of the solu- tion domain gives rise to direct problems for the Helm- holtz equation which have been extensively studied in the literature. For example, Niwa et al. [8] have studied the solution to the Helmholtz equation using the complex valued boundary element method (BEM). De Mey [9] has proposed a simplified formulation which used the real part of the complex valued fundamental solution to construct the real part BEM for the Helmholtz equation. Hutchinson [10] has used the real part BEM in order to solve the vibration problems of a membrane. Later, other real-val- ued formulations have been developed, e.g. the multiple reciprocity method (MRBEM) [11, 12, 13] and the dual reciprocity method (DRBEM) [14, 15, 16]. The well-posedness of the direct problems of the Helmholtz equation via the removal of the eigenvalues of the Laplacian operator is well established, see e.g. Chen and Zhou [17]. Unfortunately, many engineering problems do not belong to this category. In particular, the boundary conditions are often incomplete, either in the form of underspecified and overspecified boundary conditions on different parts of the boundary or the solution is pre- scribed at some internal points in the domain. These are inverse problems, and it is well known that they are gen- erally ill-posed, i.e. the existence, uniqueness and stability of their solutions are not always guaranteed. There are important studies of the Cauchy problem for the Helmholtz equation in the literature. Unlike in direct problems, the uniqueness of the Cauchy problem is guaranteed without the necessity of removing the eigen- values for the Laplacian. However, the Cauchy problem suffers from the non-existence of the solution and con- tinuous dependence on the input data. A BEM-based acoustic holography technique using the singular value decomposition (SVD) for the reconstruction of sound fields generated by irregularly shaped sources has been developed by Bai [18]. The vibrational velocity, sound pressure and acoustic power on the vibrating boundary comprising an enclosed space have been reconstructed by Kim and Ih [19] who have used the SVD in order to obtain the inverse solution in the least-squares sense and to express the acoustic modal expansion between the measurement and source field. Wang and Wu [20] have developed a method employing the spherical wave expansion theory and a least-squares minimisation to reconstruct the acoustic pressure field from a vibrating object and their method has been extended to the recon- struction of acoustic pressure fields inside the cavity of a vibrating object by Wu and Yu [21]. Recently, DeLillo et al. [22] have detected the source of acoustical noise inside the cabin of a midsize aircraft from measurements of the acoustical pressure field inside the cabin by solving a linear Fredholm integral equation of the first kind. Computational Mechanics 31 (2003) 367–377 Ó Springer-Verlag 2003 DOI 10.1007/s00466-003-0439-y 367 Received: 5 November 2002 / Accepted: 5 March 2003 L. Marin (&), X. Wen School of the Environment, University of Leeds, Leeds LS2 9JT, UK E-mail: liviu@env.leeds.ac.uk L. Elliott, D. B. Ingham, D. Lesnic Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK P. J. Heggs Department of Chemical Engineering, UMIST, P.O. Box 88, Manchester M60 1QD, UK L. Marin would like to acknowledge the financial support received from the EPSRC. The authors would like to thank Professor Dinh Nho Ha `o and Dr. Thomas Johansson for some useful discussions and suggestions.