Dual reciprocity boundary element method solution of the Cauchy problem for Helmholtz-type equations with variable coefficients

Marin, Liviu; Elliott, L.; Heggs, Peter J.; Ingham, D.B.; Lesnic, D.; Wen, Xin

doi:10.1016/j.jsv.2006.03.045

Cited by 38 publications

(34 citation statements)

References 20 publications

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“…In the previous corollary, (19) is an estimate that is typical for well-posed problem with an error of the order O(α + δ). The rest, (20), (21) are estimates typical for ill-posed equations.…”

Section: Error Estimates For Tikhonov Regularizationmentioning

confidence: 99%

“…It appears in several applications such as electromagnetics [24] or acoustics [6]. Computing approximate solutions by various regularizations have been suggested by several authors, e.g., by an initial value approach [22], backpropagation [21], frequency space cut-off [24], iterative methods [19] or Tikhonov regularization [16,27,20,28,23].…”

Section: Problem Statement and Backgroundmentioning

confidence: 99%

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Subspaces of stability in the Cauchy Problem for the Helmholtz equation

Isakov

Kindermann

2011

Methods and Applications of Analysis

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Abstract. We study the stability in the Cauchy Problem for the Helmholtz equation in dependence of the wave number k. For simple geometries, we show analytically that this problem is getting more stable with increasing k. In more detail, there is a subspace of the data space on which the Cauchy Problem is well posed, and this subspace grows with larger k. We call this a subspace of stability. For more general geometries, we study the ill-posedness by computing the singular values of some operators associated with the corresponding well-posed (direct) boundary value problems. Numerical computations of the singular values show that an increasing subspace of stability exists for more general domains and boundary data. Assuming the existence of such subspaces we develop a theory of (Tikhonov type) regularization for the Cauchy Problem, giving convergence rates and regularization parameter choice which improve with growing k.

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Section: Error Estimates For Tikhonov Regularizationmentioning

confidence: 99%

Section: Problem Statement and Backgroundmentioning

confidence: 99%

Subspaces of stability in the Cauchy Problem for the Helmholtz equation

Isakov

Kindermann

2011

Methods and Applications of Analysis

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“…In the direct problem formulation, the acoustic pressureū(x) is prescribed on part ∂ D Ω of the boundary ∂Ω and the normal velocityt(x) on the remaining ∂ N Ω part of ∂Ω, see [2,3,25] (Lu)(…”

Section: Reduction Of the Helmholtz Equation With Variable Coefficienmentioning

confidence: 99%

Numerical solution of the two-dimensional Helmholtz equation with variable coefficients by the radial integration boundary integral and integro-differential equation methods

AL-Jawary¹,

Wrobel²

2012

International Journal of Computer Mathematics

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Abstract. This paper presents new formulations of the boundary-domain integral equation (BDIE) and the boundary-domain integro-differential equation (BDIDE) methods for the numerical solution of the two-dimensional Helmholtz equation with variable coefficients. When the material parameters are variable (with constant or variable wave number), a parametrix is adopted to reduce the Helmholtz equation to a BDIE or BDIDE. However, when material parameters are constant (with variable wave number), the standard fundamental solution for the Laplace equation is used in the formulation. The radial integration method is then employed to convert the domain integrals arising in both BDIE and BDIDE methods into equivalent boundary integrals. The resulting formulations lead to pure boundary integral and integrodifferential equations with no domain integrals. Numerical examples are presented for several simple problems, for which exact solutions are available, to demonstrate the efficiency of the proposed methods.

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“…Traditional numerical methods, in conjunction with an appropriately chosen regularization/stabilization method, have been employed to solve inverse problems associated with Helmholtz-type equations, such as the finite-difference method (FDM) [4,5], the finite element method (FEM) [25,26] and the boundary element method (BEM) [39,40], respectively. Both the FDM and the FEM require the discretization of the domain of interest which is time consuming and tedious, especially for complicated geometries.…”

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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Dual reciprocity boundary element method solution of the Cauchy problem for Helmholtz-type equations with variable coefficients

Cited by 38 publications

References 20 publications

Subspaces of stability in the Cauchy Problem for the Helmholtz equation

Subspaces of stability in the Cauchy Problem for the Helmholtz equation

Numerical solution of the two-dimensional Helmholtz equation with variable coefficients by the radial integration boundary integral and integro-differential equation methods

The Plane Waves Method for Numerical Boundary Identification

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