Convergence of an Alternating Method to Solve the Cauchy Problem for Poisson's Equation

Jourhmane, Mostafa; Nachaoui, Abdeljalil

doi:10.1080/0003681021000029819

Cited by 45 publications

(43 citation statements)

References 15 publications

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“…Consequently, an iterative procedure, which provides the selection of the optimal regularization parameter, occurs within each step of the iterative algorithm of Kozlov et al [12] and hence the computational cost of the iterative MFS-based algorithm is increased. To overcome this inconvenience and encouraged by the recent findings of Johansson and Marin [32], as well as similar results obtained for the Cauchy problem associated with the Poisson equation [55], the Laplace equation [56,57], and the Cauchy-Navier system of elasticity [58,59], we decided to employ the two relaxation procedures, as proposed and investigated using the BEM in [32], for the iterative MFS-based algorithm implemented by Marin [52] and study the influence of the relaxation parameter upon the rate of convergence of the modified method. The efficiency of these relaxation procedures is tested for Cauchy problems associated with the two-dimensional modified Helmholtz operator in simply and doubly connected domains with smooth or piecewise smooth boundaries.…”

Section: Introductionmentioning

confidence: 75%

A relaxation method of an alternating iterative MFS algorithm for the Cauchy problem associated with the two‐dimensional modified Helmholtz equation

Marin

2011

Numerical Methods Partial

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We propose two algorithms involving the relaxation of either the given Dirichlet data or the prescribed Neumann data on the over-specified boundary in the case of the alternating iterative algorithm of Kozlov et al. (USSR Comput Math Math Phys 31 (1991), 45-52) applied to the Cauchy problem for the two-dimensional modified Helmholtz equation. The two mixed, well-posed and direct problems corresponding to every iteration of the numerical procedure are solved using the method of fundamental solutions (MFS), in conjunction with the Tikhonov regularization method. For each direct problem considered, the optimal value of the regularization parameter is selected according to the generalized cross-validation criterion. The iterative MFS algorithms with relaxation are tested for Cauchy problems associated with the modified Helmholtz equation in two-dimensional geometries to confirm the numerical convergence, stability, accuracy and computational efficiency of the method.

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Section: Introductionmentioning

confidence: 75%

A relaxation method of an alternating iterative MFS algorithm for the Cauchy problem associated with the two‐dimensional modified Helmholtz equation

Marin

2011

Numerical Methods Partial

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“…Let B DD and G DD be defined by (16) and (17), respectively. Then the second version of the alternating method can be written in the form…”

Section: Recursion In the Second Version Of The Alternating Methodsmentioning

confidence: 99%

Fast reconstruction of harmonic functions from Cauchy data using integral equation techniques

Helsing

Johansson

2010

Inverse Problems in Science and Engineering

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We consider the problem of stable determination of a harmonic function from knowledge of the solution and its normal derivative on a part of the boundary of the (bounded) solution domain. The alternating method is a procedure to generate an approximation to the harmonic function from such Cauchy data and we investigate a numerical implementation of this procedure based on Fredholm integral equations and Nyström discretization schemes, which makes it possible to perform a large number of iterations (millions) with minor computational cost (seconds) and high accuracy. Moreover, the original problem is rewritten as a fixed point equation on the boundary, and various other direct regularization techniques are discussed to solve that equation. We also discuss how knowledge of the smoothness of the data can be used to further improve the accuracy. Numerical examples are presented showing that accurate approximations of both the solution and its normal derivative can be obtained with much less computational time than in previous works.Mathematics Subject Classification (2000): 35R25.

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“…There are two types approaches to the solution of inverse Cauchy problems. When the first type methods bring the problem into a class of well posed problem in the Tikhonov's sense [22][23][24], the second type methods are iterations [25][26][27][28][29].…”

Section: Introductionmentioning

confidence: 99%

A non-typical Lie-group integrator to solve nonlinear inverse Cauchy problem in an arbitrary doubly-connected domain

Liu

2015

Applied Mathematical Modelling

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a b s t r a c tThe inverse Cauchy problem for a nonlinear elliptic equation defined in an arbitrary doubly-connected plane domain is solved numerically. When the overspecified boundary data are imposed on the outer boundary, we seek the unknown data on the inner boundary, by using a mixed group preserving scheme (MGPS) to integrate the nonlinear inverse Cauchy problem as an initial value problem. We also reverse the order of the above inverse Cauchy problem by giving overspecified boundary data on the inner boundary and seeking the unknown data on the outer boundary. Several numerical examples are examined to show that the MGPS can overcome the highly ill-posed behavior of nonlinear inverse Cauchy problem defined in arbitrary doubly-connected plane domain. The proposed algorithm is robust against large noise and very time saving without needing of any iteration.

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Convergence of an Alternating Method to Solve the Cauchy Problem for Poisson's Equation

Cited by 45 publications

References 15 publications

A relaxation method of an alternating iterative MFS algorithm for the Cauchy problem associated with the two‐dimensional modified Helmholtz equation

A relaxation method of an alternating iterative MFS algorithm for the Cauchy problem associated with the two‐dimensional modified Helmholtz equation

Fast reconstruction of harmonic functions from Cauchy data using integral equation techniques

A non-typical Lie-group integrator to solve nonlinear inverse Cauchy problem in an arbitrary doubly-connected domain

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