A survey of applications of the MFS to inverse problems

DOI10.1080/17415977.2011.551830zbMath1220.65157OpenAlexW2015137881MaRDI QIDQ3017891

Liviu Marin, Daniel Lesnic, Andreas Karageorghis

Publication date: 20 July 2011

Published in: Inverse Problems in Science and Engineering (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1080/17415977.2011.551830

zbMATH Keywords

heat equationregularizationinverse problemsLaplace equationHelmholtz equationPoisson equationmethod of fundamental solutionsStokes equationsurvey article

Mathematics Subject Classification ID

Heat equation (35K05) Inverse problems for PDEs (35R30) Navier-Stokes equations (35Q30) Research exposition (monographs, survey articles) pertaining to numerical analysis (65-02) Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation (35J05) Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs (65M32) Numerical methods for inverse problems for boundary value problems involving PDEs (65N21) Fundamental solutions, Green's function methods, etc. for boundary value problems involving PDEs (65N80) Fundamental solutions, Green's function methods, etc. for initial value and initial-boundary value problems involving PDEs (65M80)

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Uses Software

• NAG
• HYBRJ
• minpack

Cites Work

• On the choice of source points in the method of fundamental solutions
• A method of fundamental solutions for transient heat conduction in layered materials
• Solution of potential flow problems by the modified method of fundamental solutions: formulations with the single layer and the double layer fundamental solutions
• The time-marching method of fundamental solutions for wave equations
• A meshless method for the stable solution of singular inverse problems for two-dimensional Helmholtz-type equations
• Some remarks concerning the shape of the source contour with application of the method of fundamental solutions to elastic torsion of prismatic rods
• The method of fundamental solutions for the inverse heat source problem
• An iterative MFS approach for the detection of immersed obstacles
• A method of fundamental solutions for transient heat conduction
• An inverse boundary problem for one-dimensional heat equation with a multilayer domain
• The method of fundamental solutions for free surface Stefan problems
• The method of fundamental solutions for the Cauchy problem in two-dimensional linear elasticity
• An adaptive greedy technique for inverse boundary determination problem
• Boundary reconstruction in two-dimensional steady state anisotropic heat conduction using a regularized meshless method
• Convergence analysis for the Cauchy problem of Laplace's equation by a regularized method of fundamental solutions
• Regularized method of fundamental solutions for boundary identification in two-dimensional isotropic linear elasticity
• Relaxation procedures for an iterative MFS algorithm for the stable reconstruction of elastic fields from Cauchy data in two-dimensional isotropic linear elasticity
• A relaxation method of an alternating iterative algorithm for the Cauchy problem in linear isotropic elasticity
• The method of fundamental solutions for problems in potential flow
• A meshless method for solving the Cauchy problem in three-dimensional elastostatics
• The method of fundamental solutions for inverse boundary value problems associated with the two-dimensional biharmonic equation
• The method of fundamental solutions and condition number analysis for inverse problems of Laplace equation
• An alternating iterative MFS algorithm for the Cauchy problem for the modified Helmholtz equation
• Two regularization methods for the Cauchy problems of the Helmholtz equation
• A meshless method for solving an inverse spacewise-dependent heat source problem
• Method of fundamental solutions with optimal regularization techniques for the Cauchy problem of the Laplace equation with singular points
• Reconstruction of the corrosion boundary for the Laplace equation by using a boundary collocation method
• The method of fundamental solutions for a biharmonic inverse boundary determination problem
• The method of fundamental solutions for the solution of steady-state free boundary problems
• The method of fundamental solutions for elliptic boundary value problems
• The method of fundamental solutions for scattering and radiation problems.
• A numerical computation for inverse boundary determination problem
• A meshless, integration-free, and boundary-only RBF technique
• Uniqueness and convergence of numerical solution of the Cauchy problem for the Laplace equation by a charge simulation method
• A new method of fundamental solutions applied to nonhomogeneous elliptic problems
• A meshless method for the numerical solution of the Cauchy problem associated with three-dimensional Helmholtz-type equations
• Optimality of the method of fundamental solutions
• The MFS for numerical boundary identification in two-dimensional harmonic problems
• Relaxation procedures for an iterative MFS algorithm for two-dimensional steady-state isotropic heat conduction Cauchy problems
• Application of the MFS to inverse obstacle scattering problems
• Solution of a two-dimensional bubble shape in potential flow by the method of fundamental solutions
• A method of fundamental solutions for inverse heat conduction problems in an anisotropic medium
• Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators
• Full identification of acoustic sources with multiple frequencies and boundary measurements
• Boundary knot method for the Cauchy problem associated with the inhomogeneous Helmholtz equation
• A meshless method for some inverse problems associated with the Helmholtz equation
• Numerical solution of the Cauchy problem for steady-state heat transfer in two-dimensional functionally graded materials
• The direct method of fundamental solutions and the inverse Kirsch-Kress method for the reconstruction of elastic inclusions or cavities
• A fundamental solution method for inverse heat conduction problem
• A three-dimensional boundary determination problem in potential corrosion damage
• Density results using Stokeslets and a method of fundamental solutions for the Stokes equations
• The method of fundamental solutions for inverse 2D Stokes problems
• The method of fundamental solutions for the inverse space-dependent heat source problem
• Conditional stability estimation for an inverse boundary problem with non-smooth boundary in $\mathcal {R}^3$
• THE METHOD OF FUNDAMENTAL SOLUTIONS FOR AN INVERSE INTERNAL BOUNDARY VALUE PROBLEM FOR THE BIHARMONIC EQUATION
• Inverse problems: A Bayesian perspective
• Applicability and applications of the method of fundamental solutions
• Detection of cavities using the method of fundamental solutions
• Boundary knot method for some inverse problems associated with the Helmholtz equation
• The method of fundamental solutions for inverse boundary value problems associated with the steady-state heat conduction in anisotropic media
• Estimation of the heat transfer coefficient by means of the method of fundamental solutions
• The method of fundamental solutions for the inverse conductivity problem
• The method of fundamental solutions for inverse source problems associated with the steady-state heat conduction
• The method of fundamental solutions for the backward heat conduction problem
• Recovering the source term in a linear diffusion problem by the method of fundamental solutions
• A computational method for inverse free boundary determination problem
• Reconstruction of a moving boundary from Cauchy data in one-dimensional heat equation
• Reconstruction of part of a boundary for the Laplace equation by using a regularized method of fundamental solutions
• Fundamental Solutions Method for Elliptic Boundary Value Problems
• The Approximate Solution of Elliptic Boundary-Value Problems by Fundamental Solutions
• Practical Approximate Solutions to Linear Operator Equations When the Data are Noisy
• The simple layer potential method of fundamental solutions for certain biharmonic problems
• An inverse Robin problem for Laplace's equation: theoretical results and numerical methods
• The Method of Fundamental Solutions for Stokes flows with a free surface
• Stable Determination of Boundaries from Cauchy Data
• The method of fundamental solutions for axisymmetric potential problems
• The Use of the L-Curve in the Regularization of Discrete Ill-Posed Problems
• The method of fundamental solutions for Signorini problems
• New Insights in Boundary-only and Domain-type RBF Methods
• Hierarchical Bayesian models for inverse problems in heat conduction
• The method of fundamental solutions for solving a Cauchy problem of Laplace's equation in a multi-connected domain
• A DISCREPANCY PRINCIPLE FOR THE SOURCE POINTS LOCATION IN USING THE MFS FOR SOLVING THE BHCP
• Discrete Inverse Problems
• The method of functional equations for the approximate solution of certain boundary value problems
• A method for the approximate solution of limiting problems in mathematical physics
• Some comments on the use of radial basis functions in the dual reciprocity method
• The numerical solution of three-dimensional Signorini problems with the method of fundamental solutions
• Boundary element method for the Cauchy problem in linear elasticity.
• Trefftz method: An overview