Fundamental solution neural networks for solving inverse Cauchy problems for the Laplace and biharmonic equations

Localized MFS for the inverse Cauchy problems of two-dimensional Laplace and biharmonic equations

The pre/post equilibrated conditioning methods to solve Cauchy problems

We consider the inverse Cauchy problems for Laplace equation in simply and doubly connected plane domains by recoverning the unknown bound- ary value on an inaccessible part of a noncircular contour from overspecified data. A modified Trefftz method is used directly to solve those problems with a simple collocation technique to determine unknown coefficients, which is named a mod- ified collocation Trefftz method (MCTM). Because the condition number is small for the MCTM, we can apply it to numerically solve the inverse Cauchy problems without needing of an extra regularization, as that used in the solutions of direct problems for Laplace equation. So, the computational cost of MCTM is very sav- ing. Numerical examples show the effectiveness of the new method in providing an excellent estimate of unknown boundary data, even by subjecting the given data to a large noise. Keyword: Inverse Cauchy problem, Modified Trefftz method, Laplace equation, Modified collocation Trefftz method (MCTM)

Previous article Next article The Numerical Solution of Some Biharmonic Problems by Mathematical Programming TechniquesJ. R. Cannon and Maria M. CecchiJ. R. Cannon and Maria M. Cecchihttps://doi.org/10.1137/0703039PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Heinrich Behnke and , Friedrich Sommer, Theorie der analytischen Funktionen einer komplexen Veränderlichen., Zweite veränderte Auflage. Die Grundlehren der mathematischen Wissenschaften, Bd. 77, Springer-Verlag, Berlin, 1962, 128– MR0147622 0101.29502 CrossrefGoogle Scholar[2] J. R. Cannon, The numerical solution of the Dirichlet problem for Laplace's equation by linear programming, J. Soc. Indust. Appl. Math., 12 (1964), 233–237 10.1137/0112022 MR0164453 0221.65186 LinkISIGoogle Scholar[3] J. R. Cannon and , Keith Miller, Some problems in numerical analytic continuation, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal., 2 (1965), 87–98 MR0179908 0214.14805 LinkGoogle Scholar[4] T. Carleman, Fonctions Quasi Analytiques, Gauthier-Villars, Paris, 1926, 3–5 Google Scholar[5] Jim Douglas, Jr., R. E. Langer, A numerical method for analytic continuation, Boundary problems in differential equations, Univ. of Wisconsin Press, Madison, 1960, 179–189 MR0117866 0100.12405 Google Scholar[6] Saul I. Gass, Linear programming: methods and applications, McGraw-Hill Book Co., Inc., New York, 1958xii+223 MR0096554 0081.36702 Google Scholar[7] Fritz John, Continuous dependence on data for solutions of partial differential equations with a presribed bound, Comm. Pure Appl. Math., 13 (1960), 551–585 MR0130456 0097.08101 CrossrefISIGoogle Scholar[8] G. Lauricella, Integrazione dell' equazione $\Delta^{2}(\Delta^{2}u)=0$ in un campo di forma circolare, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., 31 (1895–1896), 610–618 Google Scholar[9] O. L. Mangasarian, Numerical solution of the first biharmonic problem by linear programming, Internat. J. Engrg. 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L. Schryer14 July 2006 | SIAM Journal on Numerical Analysis, Vol. 9, No. 4AbstractPDF (2179 KB)Determination of an unknown forcing function in a hyperbolic equation from overspecified dataAnnali di Matematica Pura ed Applicata, Vol. 85, No. 1 Cross Ref Applications of Linear Programming to Numerical AnalysisPhilip Rabinowitz18 July 2006 | SIAM Review, Vol. 10, No. 2AbstractPDF (3637 KB)Numerical Experiments on the Solution of Some Biharmonic Problems by Mathematical Programming TechniquesJ. R. Cannon and Maria M. Cecchi14 July 2006 | SIAM Journal on Numerical Analysis, Vol. 4, No. 2AbstractPDF (575 KB) Volume 3, Issue 3| 1966SIAM Journal on Numerical Analysis History Submitted:01 December 1965Published online:14 July 2006 InformationCopyright © 1966 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0703039Article page range:pp. 451-466ISSN (print):0036-1429ISSN (online):1095-7170Publisher:Society for Industrial and Applied Mathematics

The MFS and MAFS for solving Laplace and biharmonic equations

We solve the Calderon inverse conductivity problem (Calderon (1980, 2006)), for an elliptic type equation in a rectangular plane domain, to recover an unknown conductivity function inside the domain, from the over-specified Cauchy data on the bottom of the rectangle. The Calderon inverse problem exhibits three- fold simultaneous difficulties: ill-posedness of the inverse Cauchy problem, ill- posedness of the parameter identification, and no information inside the domain being available on the impedance function. In order to solve this problem, we discretize the whole domain into many sub-domains of finite strips, each with a small height. Thus the Calderon inverse problem is reduced to an inverse Cauchy problem and a parameter identification problem in each finite strip. An effective combination of the Lie-group adaptive method (LGAM), together with a finite- strip method is developed, where the Lie-group equation can adaptively solve the semi-discretized ODEs to find the unknown conductivity coefficients through it- erations. The success of the present method hinges on a rationale that the local ODEs and the global Lie-group equation have to be self-adaptive during the itera- tion process. Thus, we have a computationally inexpensive mathematical algorithm to solve the Calderon inverse problem. The feasibility, accuracy and efficiency of present method are evaluated by comparing the estimated results for the unknown impedance function in the domain, in the Calderon inverse problem, with some postulated exact solutions. It may be concluded that the iterative and adaptive Lie- group method presented in this paper, may provide a simple and effective means of solving the Calderon inverse problem in general domains.

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Determination of an Unknown Heat Source from Overspecified Boundary Data

In Chapter 10, we examined finite element methods for the numerical solution of Laplace's equation. In this chapter, we propose an alternative approach. We introduce the idea of reformulating Laplace's equation as a boundary integral equation (BIE), and then we consider the numerical solution of Laplace's equation by numerically solving its reformulation as a BIE. Some of the most important boundary value problems for elliptic partial differential equations have been studied and solved numerically by this means; and depending on the requirements of the problem, the use of BIE reformulations may be the most efficient means of solving these problems. Examples of other equations solved by use of BIE reformulations are the Helmholtz equation (Δυ + λυ = 0) and the biharmonic equation (Δ2υ = 0). We consider here the use of boundary integral equations in solving only planar problems for Laplace's equation. For the domain D for the equation, we restrict it or its complement to be a simply-connected set with a smooth boundary S. Most of the results and methods given here will generalize to other equations (e.g. Helmholtz's equation).

Localized method of fundamental solutions for solving two-dimensional Laplace and biharmonic equations

In this paper, a localized boundary knot method is proposed, based on the local concept in the localized method of fundamental solutions. The localized boundary knot method is formed by combining the classical boundary knot method and the localization approach. The localized boundary knot method is truly free from mesh and numerical quadrature, so it has great potential for solving complicated engineering applications, such as multiply connected problems. In the proposed localized boundary knot method, both of the boundary nodes and interior nodes are required, and the algebraic equations at each node represent the satisfaction of the boundary condition or governing equation, which can be derived by using the boundary knot method at every subdomain. A sparse system of linear algebraic equations can be yielded using the proposed localized boundary knot method, which can greatly reduce the computer time and memory required in computer calculations. In this paper, several cases of simply connected domains and multi-connected domains of the Laplace equation and bi-harmonic equation are demonstrated to evidently verify the accuracy, convergence and stability of this proposed meshless method.

The boundary element method (BEM) is applied to discretise numerically a Cauchy problem for the biharmonic equation which involves over- and underspecified boundary portions of the solution domain. The resulting ill-conditioned system of linear equations is solved using the regularization method. It is shown that the regularization method performs better than the minimal energy method in the case of the biharmonic equation, unlike the Laplace equation where the minimal energy method is more efficient. Moreover, the stability of the numerical solution obtained by the regularization method is also investigated.

The inverse Cauchy problems for elliptic equations, such as the Laplace equation, the Poisson equation, the Helmholtz equation and the modified Helmholtz equation, defined in annular domains are investigated. The outer boundary of the annulus is imposed by overspecified boundary data, and we seek unknown data on the inner boundary through the numerical solution by a spring-damping regularization method and its Lie-group shooting method (LGSM). Several numerical examples are examined to show that the LGSM can overcome the ill-posed behavior of inverse Cauchy problem against the disturbance from random noise, and the computational cost is very cheap.

and obtained some partial results. These enabled us to obtain estimates on the rates of convergence of the two-line iterative methods of the Laplace and biharmonic difference equations in rectangular domains(2). In the case of Laplace's equation we obtained an exact asymptotic result. However, in the case of the biharmonic equation we obtained only a one-sided estimate. The purpose of this report is two-fold. In ??2, 3, and 4 we extend the results of Kac, Murdoch and Szego, and Widom. We will make very strong use of Widom's results and technique. In ?5 we discuss the application of the preceding results to the general problem of the extreme eigenvalues of block Toeplitz matrices. These include the matrices of elliptic difference equations Presented to the Society, January 26, 1961; received by the editors October 18, 1960. (') Some of these results were obtained while the author was at the Brookhaven National Laboratory, summer 1959. (2) The two-line iterative methods for the Laplace and biharmonic difference equations were studied by R. S. Varga [18] at the same time. His approach is totally different from the one we investigated in [14]. His approach to the solution of the iteration equations is more general and probably preferable. Varga also estimated the rate of convergence in the Laplace case using the theory of non-negative matrices. That theory does not apply to the biharmonic case.

Numerical solutions of two-dimensional Laplace and biharmonic equations by the localized Trefftz method