The method of fundamental solutions and condition number analysis for inverse problems of Laplace equation

Young, D. L.; Tsai, Chia‐Cheng; Chen, C. W.; Fan, Chia-Ming

doi:10.1016/j.camwa.2007.05.015

Cited by 55 publications

(37 citation statements)

References 11 publications

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“…It was found, as in Ramachandran (2002), Young et al (2008) and Lockerby & Collyer (2016), that faster convergence to the analytic solutions was obtained when the singularities were furthest from the computational domain r s 1, but that this can lead to ill-conditioning. A compromise of r s = 0.1 was found to work well and was used in all that follows.…”

Section: Mfs Parametersmentioning

confidence: 94%

Fundamental solutions to the regularised 13-moment equations: efficient computation of three-dimensional kinetic effects

et al. 2017

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Fundamental solutions (Green's functions) are derived for the regularised 13-moment system (R13) of rarefied gas dynamics, for small departures from equilibrium; these solutions show the presence of Knudsen layers, associated with exponential decay terms, that do not feature in the solution of lower-order systems (e.g. the Navier-Stokes-Fourier equations). Incorporation of these new fundamental solutions into a numerical framework based on the method of fundamental solutions (MFS) allows for efficient computation of three-dimensional gas microflows at remarkably low computational cost. The R13-MFS approach accurately recovers analytic solutions for low-speed flow around a stationary sphere and heat transfer from a hot sphere (for which a new analytic solution has been derived), capturing non-equilibrium flow phenomena missing from lower-order solutions. To demonstrate the potential of the new approach, the influence of kinetic effects on the hydrodynamic interaction between approaching solid microparticles is calculated. Finally, a programme of future work based on the initial steps taken in this article is outlined.

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Section: Mfs Parametersmentioning

confidence: 94%

Fundamental solutions to the regularised 13-moment equations: efficient computation of three-dimensional kinetic effects

et al. 2017

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“…Young et al [13] used the double precision MFS to solve the temperature and heat flux distributions on the missing boundary. It showed that the results had good agreement with the exact solution when boundary conditions were imposed with noise level from 10 −5 to 10 −3 , and when larger noise 0.01 was used, regularization was necessary.…”

Section: Problem With Over-specified Boundary Conditionmentioning

confidence: 99%

“…is non-singular, the solution of Equation (10) can be solved by the Gaussian elimination method [16], the Linpack's Cholesky decomposition [13] or LU decomposition solver [17].…”

Section: The Rbcmmentioning

confidence: 99%

“…However, the MFS is a versatile boundary-type meshless numerical scheme and free from treatment of singularities since the source points are always placed outside the domain. It has been successfully applied to the inverse heat conduction problem [10], backward heat conduction problem [11], Cauchy problem associated with Helmholtz-type equation [12] and Cauchy problem of Laplace equation [13,14]. Another boundary-type meshfree technique is the BKM which does not require the artificial boundary and approximates the homogeneous equation's solution by a linear combination of non-singular general solutions of the differential operator instead of singular fundamental solutions in the MFS.…”

Section: Introductionmentioning

confidence: 99%

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Least‐square‐based radial basis collocation method for solving inverse problems of Laplace equation from noisy data

Mao

2010

Numerical Meth Engineering

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SUMMARYThe inverse problem of 2D Laplace equation involves an estimation of unknown boundary values or the locations of boundary shape from noisy observations on over-specified boundary or internal data points. The application of radial basis collocation method (RBCM), one of meshless and non-iterative numerical schemes, directly induces this inverse boundary value problem (IBVP) to a single-step solution of a system of linear algebraic equations in which the coefficients matrix is inherently ill-conditioned. In order to solve the unstable problem observed in the conventional RBCM, an effective procedure that builds an over-determined linear system and combines with least-square technique is proposed to restore the stability of the solution in this paper. The present work investigates three examples of IBVPs using over-specified boundary conditions or internal data with simulated noise and obtains stable and accurate results. It underlies that least-square-based radial basis collocation method (LS-RBCM) poses a significant advantage of good stability against large noise levels compared with the conventional RBCM.

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“…), have been used increasingly over the last decade for the numerical solution of inverse problems. For example, the Cauchy problem associated with the heat conduction equation [38][39][40][41][42][43][44][45][46][47][48], linear elasticity [49,50], steady-state heat conduction in functionally graded materials [51], Helmholtz-type equations [19][20][21]52], Stokes problems [53], the biharmonic equation [54], etc., have been successfully addressed by using the MFS.…”

Section: Introductionmentioning

confidence: 99%

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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The method of fundamental solutions and condition number analysis for inverse problems of Laplace equation

Cited by 55 publications

References 11 publications

Fundamental solutions to the regularised 13-moment equations: efficient computation of three-dimensional kinetic effects

Fundamental solutions to the regularised 13-moment equations: efficient computation of three-dimensional kinetic effects

Least‐square‐based radial basis collocation method for solving inverse problems of Laplace equation from noisy data

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

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