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Reconstruction algorithm for unknown cavities via Feynman–Kac type formula

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  • Hajime Kawakami

Abstract

In this paper, we consider an inverse problem of identifying one or more unknown cavities in a heat conductor. We propose an algorithm for reconstructing the position and shape of the unknown cavities using the measured surface temperature of the heat conductor. The heat conductor is discretized into small components, and we attempt to determine the components of the unknown cavities. Each of the components of the cavities is encoded as $$0,$$ 0 , and each of the other components is encoded as $$1.$$ 1 . Thus, the inverse problem is translated into a binary optimization problem. Our algorithm is based on a discrete probabilistic representation of a solution of an initial boundary value problem for the heat equation, which we call a discrete Feynman–Kac type formula. It uses a set of sample paths generated by the Monte-Carlo method. We can use this formula to naturally transform the binary optimization problem to an optimization problem with continuous variables. This continuous approach is used by the algorithm. Although the algorithm comprises some iterations, each iteration step can use a common set of sample paths. Thus, we only need one round of the Monte-Carlo-based simulation to obtain the set of sample paths. Our experiments suggest that the algorithm has an acceptable performance when there are one or two cavities. Copyright Springer Science+Business Media New York 2015

Suggested Citation

  • Hajime Kawakami, 2015. "Reconstruction algorithm for unknown cavities via Feynman–Kac type formula," Computational Optimization and Applications, Springer, vol. 61(1), pages 101-133, May.
    • Handle: RePEc:spr:coopap:v:61:y:2015:i:1:p:101-133
    DOI: 10.1007/s10589-014-9706-4
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    References listed on IDEAS

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    1. Bayer Christian & Szepessy Anders & Tempone Raúl, 2010. "Adaptive weak approximation of reflected and stopped diffusions," Monte Carlo Methods and Applications, De Gruyter, vol. 16(1), pages 1-67, January.
    2. Walter Murray & Kien-Ming Ng, 2010. "An algorithm for nonlinear optimization problems with binary variables," Computational Optimization and Applications, Springer, vol. 47(2), pages 257-288, October.
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