Boundary knot method for some inverse problems associated with the Helmholtz equation

Inverse and ill-posed problems which consist of reconstructing the unknown support of a source from a single pair of exterior boundary Cauchy data are investigated. The underlying dependent variable, e.g. potential, temperature or pressure, may satisfy the Laplace, Poisson, Helmholtz or modified Helmholtz partial differential equations (PDEs). For constant coefficients, the solutions of these elliptic PDEs are sought as linear combinations of explicitly available fundamental solutions (free-space Greens functions), as in the method of fundamental solutions (MFS). Prior to this application of the MFS, the free-term inhomogeneity represented by the intensity of the source is removed by the method of particular solutions. The resulting transmission problem then recasts as that of determining the interface between composite materials. In order to ensure a unique solution, the unknown source domain is assumed to be starshaped. This in turn enables its boundary to be parametrised by the radial coordinate, as a function of the polar or, spherical angles. The problem is nonlinear and the numerical solution which minimizes the gap between the measured and the computed data is achieved using the Matlab toolbox routine lsqnonlin which is designed to minimize a sum of squares starting from an initial guess and with no gradient required to be supplied by the user. Simple bounds on the variables can also be prescribed. Since the inverse problem is still ill-posed with respect to small errors in the data and possibly additional ill-conditioning introduced by the spectral feature of the MFS approximation, the least-squares functional which is minimized needs to be augmented with regularizing penalty terms on the MFS coefficients and on the radial function for a stable estimation of these couple of unknowns. Thorough numerical investigations are undertaken for retrieving regular and irregular shapes of the source support from both exact and noisy input data.

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“…and the inhomogeneous part is just a particular solution of equation (10) (in the whole R d ) given by, see for example Jin and Zheng (2005),…”

Section: The Methods Of Fundamental Solutions (Mfs)mentioning

confidence: 99%

Reconstruction of a source domain from boundary measurements

Bin‐Mohsin

Lesnic

2017

Applied Mathematical Modelling

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“…It is noted that the truncated singular value decomposition (TSVD) is clearly superior to Gaussian elimination for noisy boundary conditions [14]. On the other hand, the TSVD is also employed in the BKM solution of inverse problems [15,16]. All these studies, however, have mainly focused on the solution accuracy rather than on the solution INVESTIGATION OF REGULARIZED TECHNIQUES FOR BKM 1869 stability without a detailed investigation on the convergence behaviors.…”

Section: Introductionmentioning

confidence: 96%

Investigation of regularized techniques for boundary knot method

Wang

Chen

Jiang

2009

Numer Methods Biomed Eng

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SUMMARYThis study investigates regularization techniques for the boundary knot method (BKM). We consider three regularization methods and two approaches for the determination of the regularization parameter. Our numerical experiments show that Tikhonov regularization in conjunction with generalized cross-validation approach outperforms the other regularization techniques in the BKM solution of Helmholtz and modified Helmholtz problems.

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“…The advantages of meshless methods are the ease with which they can be implemented, in particular for problems in complex geometries, their low computational cost and the fact that, in general, they are exempted from integrations that may become cumbersome, especially in three dimensions. Such methods include the boundary particle method (BPM) [13], the singular boundary method (SBM) [14], the method of fundamental solutions (MFS) [16], the boundary knot method (BKM) [23], Kansa's method [28], etc.…”

Section: Introductionmentioning

confidence: 99%

The Plane Waves Method for Numerical Boundary Identification

Karageorghis

Lesnic

Marin

2017

Adv. Appl. Math. Mech.

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Abstract. We study the numerical identification of an unknown portion of the boundary on which either the Dirichlet or the Neumann condition is provided from the knowledge of Cauchy data on the remaining, accessible and known part of the boundary of a two-dimensional domain, for problems governed by Helmholtz-type equations. This inverse geometric problem is solved using the plane waves method (PWM) in conjunction with the Tikhonov regularization method. The value for the regularization parameter is chosen according to Hansen's L-curve criterion. The stability, convergence, accuracy and efficiency of the proposed method are investigated by considering several examples.

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Boundary knot method for some inverse problems associated with the Helmholtz equation

Cited by 54 publications

References 26 publications

Reconstruction of a source domain from boundary measurements

Reconstruction of a source domain from boundary measurements

Investigation of regularized techniques for boundary knot method

The Plane Waves Method for Numerical Boundary Identification

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