A comparative study on applying the method of fundamental solutions to the backward heat conduction problem

Johansson, Björn; Lesnic, D.; Reeve, Thomas

doi:10.1016/j.mcm.2011.02.030

Cited by 18 publications

(7 citation statements)

References 34 publications

(87 reference statements)

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“…The method of fundamental solutions is a powerful, accurate, easy to implement and low cost numerical meshless method and has been applied for solving a wide class of stationary and time dependent equations . To present a numerical study, we apply this method for solving IP1–IP3 and investigate its advantages or probable drawbacks in comparison with the Ritz–Galerkin method.…”

Section: Approximation Based On Fundamental Solutionsmentioning

confidence: 99%

Efficient numerical methods for boundary data and right‐hand side reconstructions in elliptic partial differential equations

Rashedi

Adibi

Dehghan

2015

Numerical Methods Partial

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In this article, we discuss the application of two important numerical methods, Ritz-Galerkin and Method of Fundamental Solutions (MFS), for solving some inverse problems, arising in the context of two-dimensional elliptic equations. The main incentive for studying the considered problems is their wide applications in engineering fields. In the previous literature, the use of these methods, particularly MFS for right hand side reconstruction has been limited, partly due to stability concerns. We demonstrate that these diculties may be surmounted if the aforementioned methods are combined with techniques such as dual reciprocity method(DRM). Moreover, we incorporate some iterative regularization techniques. This fact is especially veried by taking into account the noisy data with boundary conditions. In addition, parts of this article are dedicated to the problem of boundary data approximation and the issue of numerical stability, ending with a general discussion on the advantages and disadvantages of various methods.

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Section: Approximation Based On Fundamental Solutionsmentioning

confidence: 99%

Efficient numerical methods for boundary data and right‐hand side reconstructions in elliptic partial differential equations

Rashedi

Adibi

Dehghan

2015

Numerical Methods Partial

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“…There has been growing interest in recent years in the development of numerical methods for solving parabolic equations backward in time. In , various useful methods are analyzed and illustrated with interesting test computations. Currently, the two most significant areas of application of backward parabolic equations are hydrologic inversion and image deblurring .…”

Section: Introductionmentioning

confidence: 99%

Reconstructing the past from imprecise knowledge of the present: Effective non‐uniqueness in solving parabolic equations backward in time

Carasso

2012

Math Methods in App Sciences

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Identifying sources of ground water pollution and deblurring astronomical galaxy images are two important applications generating growing interest in the numerical computation of parabolic equations backward in time. However, while backward uniqueness typically prevails in parabolic equations, the precise data needed for the existence of a particular backward solution is seldom available. This paper discusses previously unexplored non‐uniqueness issues, originating from trying to reconstruct a particular solution from imprecise data. Explicit 1D examples of linear and nonlinear parabolic equations are presented, in which there is strong computational evidence for the existence of distinct solutions wred(x,t) and wgreen(x,t), on 0 ≤ t ≤ 1. These solutions have the property that the traces wred(x,1) and wgreen(x,1) at time t = 1 are close enough to be visually indistinguishable, while the corresponding initial values wred(x,0) and wgreen(x,0) are vastly different, well‐behaved, physically plausible functions, with comparable L2 norms. This implies effective non‐uniqueness in the recovery of wred(x,0) from approximate data for wred(x,1). In all these examples, the Van Cittert iterative procedure is used as a tool to discover unsuspected, valid, additional solutions wgreen(x,0). This methodology can generate numerous other examples and indicates that multidimensional problems are likely to be a rich source of striking non‐uniqueness phenomena. Published 2012. This article is a US Government work and is in the public domain in the USA.

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“…Recently, however, investigations into the application, accuracy, and the placement of source points have been carried out for time-dependent problems, see, for example, [3,9,11,17,21,24]. The method has been applied to direct problems, as well as to inverse problems, for example, heat conduction in one-dimensional layered materials [12], the free surface Stefan problem [4], heat conduction in two-dimensional domains [15], the inverse Stefan problem [14], the inverse Cauchy-Stefan problem [16] and the backward heat conduction problem [13].…”

Section: Introductionmentioning

confidence: 99%

A method of fundamental solutions for the radially symmetric inverse heat conduction problem

Johansson

Lesnic

Reeve

2012

International Communications in Heat and Mass Transfer

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A comparative study on applying the method of fundamental solutions to the backward heat conduction problem

Cited by 18 publications

References 34 publications

Efficient numerical methods for boundary data and right‐hand side reconstructions in elliptic partial differential equations

Efficient numerical methods for boundary data and right‐hand side reconstructions in elliptic partial differential equations

Reconstructing the past from imprecise knowledge of the present: Effective non‐uniqueness in solving parabolic equations backward in time

A method of fundamental solutions for the radially symmetric inverse heat conduction problem

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