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2018 | OriginalPaper | Chapter

Using the Collage Method to Solve Inverse Problems for Vector-Valued Variational Problems on a Perforated Domain in Reflexive Banach Spaces

Authors : Herb Kunze, Davide La Torre, Manuel Ruiz Galán

Published in: Recent Advances in Mathematical and Statistical Methods

Publisher: Springer International Publishing

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Abstract

Recent work establishes that the solution of the parameter estimation on a perforated domain can be approximated by instead solving the inverse problem on the much easier to work with associated solid domain. In this work, we consider vector-valued variational problems on a perforated domain and show that the inverse problems on the perforated and associated solid domains can be similarly connected. The approach relies on a “generalized collage theorem” built from the vector-valued Lax-Milgram Theorem in reflexive Banach spaces. The method will be demonstrated on a numerical example.

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previous chapter Inverse Problems and Total Variation Minimization for Iterated Function Systems on Maps
next chapter Inverse Problems Using Iterated Function Systems with Place-Dependent Probabilities
Literature
1.
Kunze, H., Mendivil, F., La Torre, D., Vrscay, E.: Fractal-Based Methods in Analysis. Springer-Verlag (2012). ISBN 1461418909
2.
Kunze, H., La Torre, D., Levere, K., Ruiz Galán, M.: Inverse Problems via the “Generalized Collage Theorem” for Vector-valued Lax-Milgram-Based Variational Problems. Mathematical Problems in Engineering (2015)
3.
Kunze, H., La Torre, D.: Collage-type approach to inverse problems for elliptic PDEs on perforated domains. Electron. J. Differen. Equat. 48, 1–11 (2015)MathSciNetMATH
4.
Kunze, H., La Torre, D.: An inverse problem for a system of steady-state reaction-diffusion equations acting on a perforated domain. AIP Conference Proc. 1798, 020089 (2017)CrossRef
5.
Marchenko, V.A., Khruslov, E.Y.: Homogenization of Partial Differential Equations. Birkhauser, Boston (2006)MATH
6.
Berenguer, M.I., Kunze, H., La Torre, D., Ruiz, Galán M.: Galerkin schemes and inverse boundary value problems in reflexive banach spaces. J. Comput. Appl. Math. 275, 1–8 (2015)MathSciNetCrossRef
7.
Kunze, H., La Torre, D., Vrscay, E.R.: A generalized collage method based upon the Lax-Milgram functional for solving boundary value inverse problems. Nonlinear Anal. 71(12), 1337–1343 (2009)MathSciNetCrossRef
Metadata
Title
Using the Collage Method to Solve Inverse Problems for Vector-Valued Variational Problems on a Perforated Domain in Reflexive Banach Spaces
Authors
Herb Kunze
Davide La Torre
Manuel Ruiz Galán
Copyright Year
2018
Book
Recent Advances in Mathematical and Statistical Methods

Print ISBN: 978-3-319-99718-6

Electronic ISBN: 978-3-319-99719-3

Copyright Year: 2018

DOI
https://doi.org/10.1007/978-3-319-99719-3_10

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