Discretized Tikhonov regularization by reproducing kernel Hilbert space for backward heat conduction problem
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Discretized Tikhonov regularization by reproducing kernel Hilbert space for backward heat conduction problem Y. C. Hon · Tomoya Takeuchi
Received: 5 August 2009 / Accepted: 27 November 2009 / Published online: 8 June 2010 © Springer Science+Business Media, LLC 2010
Abstract In this paper we propose a numerical reconstruction method for solving a backward heat conduction problem. Based on the idea of reproducing kernel approximation, we reconstruct the unknown initial heat distribution from a finite set of scattered measurement of transient temperature at a fixed final time. Standard Tikhonov regularization technique using the norm of reproducing kernel is adopt to provide a stable solution when the measurement data contain noises. Numerical results indicate that the proposed method is stable, efficient, and accurate. Keywords Tikhonov regularization · Reproducing Hilbert kernel · Inverse problem Mathematics Subject Classifications (2010) 65J20 · 65M30
Communicated by Charles A. Micchelli. The work described in this paper was fully supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 101209). Y. C. Hon (B) Department of Mathematics, City University of Hong Kong, Hong Kong SAR, People’s Republic of China e-mail: [email protected] T. Takeuchi Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC, USA e-mail: [email protected]
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Y.C. Hon, T. Takeuchi
1 Introduction Let ⊂ Rd , d ∈ N be a bounded domain with sufficiently smooth boundary ∂. Consider the following initial boundary value problem for heat conduction equation: ⎧ ⎪ ⎨ ∂t u(t, x) = u(t, x), x ∈ , t ∈ (0, t f ), x ∈ ; u(0, x) = f0 (x), ⎪ ⎩ u(t, x) = 0, x ∈ ∂, t ∈ (0, t f ),
(1)
where t f > 0 is a fixed final time. The backward heat conduction problem (BHCP) is to recover the heat distribution at any earlier time 0 ≤ t < t f from the temperature distribution u(t f , ·). This is a well known highly ill-posed problem [3, 25] in the Hadamard sense: There exists no solution in general that satisfies the heat equation with final data and the boundary conditions. Even if the solution exists, any small change in the observation data may induce enormous change in the solution. Moreover, in practical situation, the data u(t f , ·) are collected only at a finite set of points {z1 , . . . , z M } ⊂ and are contaminated with measurement noises. Therefore, most conventional numerical methods often fail to give an acceptable approximation to the solution of the BHCP. There are quite a large number of works devoted to stable numerical methods for BHCP. The following is a partial list of articles which contain numerical tests: the method of fundamental solutions [22], boundary element method [10, 29], iterative boundary element method [21], inversion methods [18, 20], Tikhonov regularization by maximum entropy principle [23], operatorsplitting methods [14], lattice-free finite difference method [12], Fourier regularization [7, 8], quasi-reversibility
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