On an Inverse Problem for the Heat Equation

On an Inverse Problem for the Heat Equation: Uniqueness and Regularization

Adriano De Cezaro Federal University of Rio Grande adrianocezaro@furg.br

and

B. Tomas Johansson Link¨ oping University tomas.johansson@liu.se

In this talk we will investigate uniqueness and regularization for the inverse problem of recon- structing simultaneously a spacewise conductivitya(x) and a heat sourcef(x) in the parabolic heat equation

u_t− ∇ ·(a(x)∇u) = f(x) in Ω×(0, T)

u(x, t) = 0 on (x, t)∈∂Ω×(0, T) (1) u(x,0) = g(x) forx∈Ω,

for a given initial temperature g and given the information that the temperature u is zero on ∂Ω.

We assume that we have additional information from a supplementary temperature measurement at a given single instant of time [5,6], i.e.,

u(x, T) = u_T(x) for x∈Ω, T >0. (2) The given data and the solution domain are assumed sufficiently smooth such that the required norms and derivatives of the conductivity, source and solution of the parabolic heat equation exist and are continuous throughout the solution domain. We use Carleman estimates for parabolic equations [7, 8] to obtain a uniqueness result for the inverse problem.

In the second part of this talk we will investigate level set regularization methods [1, 2, 3, 4] for the identification of a piecewise constant and spacewise dependent heat conductivity and heat source. We prove continuity of the parameter-to-solution map in suitable L^p-norms, which implies convergence and stability of the regularized solution obtained by level set type approaches.

The numerical examples are implemented using first order optimality conditions of the proposed Tikhonov functional with total variation T V −H¹ penalization.

References

[1] A. De Cezaro, A. Leit˜ao, and X.-C. Tai,On multiple level-set regularization methods for inverse problems, Inverse Problems 25 (2009), 035004.

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[2] , On piecewise constant level-set (pcls) methods for the identication of discontinuous parameters in ill-posed problems, submitted (2012), 1–25.

[3] A. De Cezaro and A. Leit˜ao, Level-set of L²type for recovering shape and contrast in inverse problems, Inverse Problems in Science and Enginnering 20 (2010), 571–587.

[4] F. Fr¨uhauf, O. Scherzer, and A. Leit˜ao, Analysis of regularization methods for the solution of ill-posed problems involving discontinuous operators, SIAM J. Numer. Anal.43(2005), 767–786.

[5] B. T. Johansson and D. Lesnic, A procedure for determining a spacewise dependent heat source and the initial temperature, Appl. Anal. 87 (2008), no. 3, 265–276.

[6] T. Johansson and D. Lesnic, Determination of a spacewise dependent heat source, J. Comput.

Appl. Math. 209 (2007), no. 1, 66–80.

[7] M. V. Klibanov and A. Timonov, Carleman estimates for coefficient inverse problems and numerical applications, Inverse and Ill-posed Problems Series, VSP, Utrecht, 2004.

[8] M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems 25 (2009), no. 12, 123013, 75.

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