The Inverse Eigenvalue Problem with Finite Data for Partial Differential Equations

This work is concerned with the inverse eigenvalue problem for the partial differential equation $\nabla ^2 u + (\lambda - q(x,y))u = 0$ . We study the problem of reconstructing the coefficient function $q(x,y)$ (or at least a numerical approximation to it) using only a finite amount of spectral dat...

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Vydáno v:SIAM journal on mathematical analysis Ročník 26; číslo 3; s. 616 - 632
Hlavní autoři: Barnes, David C., Knobel, Roger
Médium: Journal Article
Jazyk:angličtina
Vydáno: Philadelphia Society for Industrial and Applied Mathematics 01.05.1995
Témata:
Applied mathematics
Approximation
Eigenvalues
Inverse problems
Mathematical functions
Norms
Partial differential equations
ISSN:0036-1410, 1095-7154
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Shrnutí:This work is concerned with the inverse eigenvalue problem for the partial differential equation $\nabla ^2 u + (\lambda - q(x,y))u = 0$ . We study the problem of reconstructing the coefficient function $q(x,y)$ (or at least a numerical approximation to it) using only a finite amount of spectral data, say, $\lambda _n (q)$ for $n = 1,2, \cdots ,N$. One of the essential tasks considered here is that of determining how much information about the unknown function can be contained in such a fixed and finite amount of spectral data. A numerical method, based on a constrained least squares procedure, is devised for extracting such information, and several examples are given. A proof of convergence for the numerical method is provided. We show that the main difficulty with the finite inverse problem is that the eigenvalues are continuous in some very weak topologies. This work is a higher-dimensional version of the problem considered by Barnes [SIAM J. Math Anal., 22 (1991), pp. 732-753] for ordinary differential equations.
Bibliografie:ObjectType-Article-1
SourceType-Scholarly Journals-1
content type line 14
ISSN:0036-1410
1095-7154
DOI:10.1137/S0036141093253133