An accelerated alternating procedure for the Cauchy problem for the Helmholtz equation

Berntsson, Fredrik; Kozlov, Vladimir; Mpinganzima, Lydie; Turesson, Bengt-Ove

doi:10.1016/j.camwa.2014.05.002

Cited by 28 publications

(18 citation statements)

References 21 publications

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“…In [2] we noted that, if the modified alternating algorithm produces a convergent sequence, the bilinear form associated with the Helmholtz equation, the interior boundary γ and the parameter µ is coercive. This means that we can introduce a scalar product natural to the problem.…”

Section: Discussionmentioning

confidence: 99%

“…If k = 0 then the bilinear form corresponding to (1.2) may change sign. In [3,2] we have suggested to use instead,…”

Section: Introductionmentioning

confidence: 99%

“…However, since the quadratic form associated with the problem is positive, we can introduce an inner product, as discussed above, that gives us a natural framwork that can be used to implement more sophisticated iterative methods. This was done in [2], where we showed that the conjugate gradient method can used for solving the Cauchy problem efficiently. Since the conjugate gradient iterations have a regularizing effect (see [8]) we could also solve the problem with noisy data.…”

Section: Introductionmentioning

confidence: 99%

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Iterative Tikhonov regularization for the Cauchy problem for the Helmholtz equation

Berntsson

Kozlov

Mpinganzima

et al. 2017

Computers & Mathematics with Applications

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The Cauchy problem for the Helmholtz equation appears in various applications. The problem is severely ill-posed and regualrization is needed to obtain accurate solutions. We start from a formulation of this problem as an operator equation on the boundary of the domain and consider the equation in (H 1/2) * spaces. By introducing an artificial boundary in the interior of the domain we obtain an inner product for this Hilbert space in terms of a quadratic form associated with the Helmholtz equation; perturbed by an integral over the artificial boundary. The perturbation guarantees positivity property of the quadratic form. This inner product allows an efficient evaluation of the adjoint operator in terms of solution of a well-posed boundary value problem for the Helmholtz equation with transmission boundary conditions on the artificial boundary. In an earlier paper we showed how to take advantage of this framework to implement the conjugate gradient method for solving the Cauchy problem. In this work we instead use the Conjugate gradient method for minimizing a Tikhonov functional. The added penalty term regularize the problem and gives us a regularization parameter that can be used to easily control the stability of the numerical solution with respect to measurement errors in the data. Numerical tests show that the proposed algorithm works well.

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Section: Discussionmentioning

confidence: 99%

“…If k = 0 then the bilinear form corresponding to (1.2) may change sign. In [3,2] we have suggested to use instead,…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Iterative Tikhonov regularization for the Cauchy problem for the Helmholtz equation

Berntsson

Kozlov

Mpinganzima

et al. 2017

Computers & Mathematics with Applications

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“…It has been noted that the Dirichlet-Neumann algorithm does not always work even if L is the Helmholtz operator + k 2 . Thus, several variants of the alternating iterative algorithm have been proposed, see, for instance, [2,7,18,19], and also [3,4] where an artificial interior boundary was introduced in such a way that convergence was restored. Also, it has been suggested that replacing the Neumann conditions by Robin conditions can improve the convergence [6].…”

Section: Introductionmentioning

confidence: 99%

Analysis of Dirichlet–Robin Iterations for Solving the Cauchy Problem for Elliptic Equations

Achieng

Berntsson

Chepkorir

et al. 2020

Bull. Iran. Math. Soc.

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The Cauchy problem for general elliptic equations of second order is considered. In a previous paper (Berntsson et al. in Inverse Probl Sci Eng 26(7):1062–1078, 2018), it was suggested that the alternating iterative algorithm suggested by Kozlov and Maz’ya can be convergent, even for large wavenumbers $$k^2$$ k 2 , in the Helmholtz equation, if the Neumann boundary conditions are replaced by Robin conditions. In this paper, we provide a proof that shows that the Dirichlet–Robin alternating algorithm is indeed convergent for general elliptic operators provided that the parameters in the Robin conditions are chosen appropriately. We also give numerical experiments intended to investigate the precise behaviour of the algorithm for different values of $$k^2$$ k 2 in the Helmholtz equation. In particular, we show how the speed of the convergence depends on the choice of Robin parameters.

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“…For large wave numbers, refining the mesh to maintain this requirement may become prohibitively expensive. We advise the reader to see [3][4][5][6][7][8][9][10][11][12][13][14] for some new works about the Helmholtz equation.…”

Section: Introductionmentioning

confidence: 99%

A new wavelet method for solving the Helmholtz equation with complex solution

Heydari

Hooshmandasl

Ghaini

et al. 2015

Numer. Methods Partial Differential Eq.

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The Helmholtz equation which is very important in a variety of applications, such as acoustic cavity and radiation wave, has been greatly considered in recent years. In this article, we propose a new efficient computational method based on the Legendre wavelets (LWs) expansion together with their operational matrices of integration and differentiation to solve this equation with complex solution. Because of the fact that both of the operational matrices of integration and differentiation are used in the proposed method, the boundary conditions are taken into account automatically. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifies the problems. As an applied example, "propagation of plane waves" is investigated to demonstrate the validity and applicability of the presented method.

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An accelerated alternating procedure for the Cauchy problem for the Helmholtz equation

Cited by 28 publications

References 21 publications

Iterative Tikhonov regularization for the Cauchy problem for the Helmholtz equation

Iterative Tikhonov regularization for the Cauchy problem for the Helmholtz equation

Analysis of Dirichlet–Robin Iterations for Solving the Cauchy Problem for Elliptic Equations

A new wavelet method for solving the Helmholtz equation with complex solution

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