c Pleiades Publishing, Ltd., 2008. ISSN 1995-4239, Numerical Analysis and Applications, 2008, Vol. 1, No. 1, pp. 34–45. c S.I. Kabanikhin, A. Hasanov, A.V. Penenko, 2008, published in Sibirskii Zhurnal Vychislitel’noi Matematiki, 2008, Vol. 11, No. 1, pp. 41–54. Original Russian Text
A Gradient Descent Method for Solving an Inverse Coefficient Heat Conduction Problem S. I. Kabanikhin1* , A. Hasanov2** , and A. V. Penenko1*** 1
S. L. Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, pr. Akad. Koptyuga 4, Novosibirsk, 630090 Russia 2
Applied Mathematical Sciences Research Center, Kocaeli University, bul. Ataturk, Izmit, Kocaeli, 41300 Turkey Received November 23, 2006; in final form, February 8, 2007
Abstract—An iterative gradient descent method is applied to solve an inverse coefficient heat conduction problem with overdetermined boundary conditions. Theoretical estimates are derived showing how the target functional varies with varying the coefficient. These estimates are used to construct an approximation for a target functional gradient. In numerical experiments, iteration convergence rates are compared for different descent parameters. DOI: 10.1134/S1995423908010047 Key words: coefficient identification, inverse heat conduction problem, gradient, adjoint problem, descent parameter.
1. INTRODUCTION Problems of identifying unknown coefficients in heat conduction equations with overdetermined boundary conditions are an important part of inverse problem theory and its applications. These problems arise in modeling processes such as heat conduction, diffusion, filtration, in evaluating risks in financial mathematics, etc. Direct measurements of the parameters presented by unknown coefficients either are altogether impossible or call for complex physical experiments. The objective of solving inverse problems is to replace an in situ experiment with a numerical one, which allows, via mathematical modeling, unobservables to be estimated in terms of measurable characteristics of the processes under examination. The problem of identifying coefficients in parabolic equations has been extensively studied in the literature (see, e.g., [1, 2] and references therein). Methods that are based on minimization of a target functional are quite abundant in use. Usually, this functional is taken to be the squared norm of the residual of the solution of an inverse problem. Minimization is effected via a gradient descent method. Such methods are described in [2, 3]. The main difficulties hindering application of the methods are associated with issues dealing in uniqueness of solutions and convergence of iterative processes. These handicaps have been lifted in some particular cases [4, 5]. In the present paper, we construct an approximation for a target functional gradient and look at various choice options for the descent parameter. *
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A GRADIENT DESCENT METHOD FOR SOLVING AN INVERSE
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2. STATEMENT OF THE INVERSE PROBLEM In the region ΩT = [0, X] × [0, T ], we consider the following initial-boundary value problem: ut (x, t) − (k(x)ux (x, t))x = 0, t ∈ [0, T ], x ∈ [0, X], k(0)ux (0, t) = αL (t), t ∈ [0, T ], k(X)ux (X, t) = αR (t), t ∈ [0, T ], u(x, 0) = ϕ(x), x ∈ [0, X].
(1) (2) (3) (4)
A direct problem is one that determines the function u(x, t) given (1)–(4) and known functions αL (t), αR (t), ϕ(x), and k(x). An inverse problem is formulated as follows: find a coefficient k(x) so that u(x, t) satisfies system (1)–(4) and the conditions (5) u(0, t) = fL (t), t ∈ [0, T ], (6) u(X, t) = fR (t), t ∈ [0, T ]. Functions αL (t), αR (t) and ϕ(x), fL (t), and fR (t) are assumed known. Hereinafter, · := ·L2 and ·, · := ·, ·L2 . Definition 3. We define the set of admissible coefficients to be K = {k(x) ∈ L∞ (0, X) | 0 < kmin k(x) kmax }. Definition 4. A generalized solution u(x, t; k) of problem (1)–(4) in a space H 1,1 (ΩT ) is a function u(x, t) ∈ H 1,1 (ΩT ) satisfying the relations t
t αR (τ )η(X, τ )dτ −
[ut (x, τ )η(x, τ ) + k(x)ux (x, τ )ηx (x, τ )]dxdτ = 0
Ωt
αL (τ )η(0, τ )dτ , 0
u(x, 0) = ϕ(x) for any t ∈ (0, T ] and any η(x, t) ∈ H 1,0 (ΩT ). Theorem 1 [3]. Let k(x) ∈ K, ϕ(x) ∈ H 1 (0, X), αL (t) ∈ H 1 (0, T ), and αR (t) ∈ H 1 (0, T ). Then there exists a unique generalized solution u(x, t; k) to problem (1) − (4) in H 1,1 (ΩT ) and the following estimate holds true: ux (·, ·; k) + u(·, ·; k) + u(0, ·; k) + u(X, ·; k) cd (ϕ(·)H 1 (0,X) + αL (·)H 1 (0,T ) + αR (·)H 1 (0,T ) ),
(7)
where cd does not depend on u(x, t; k). Along with (1)–(4), we consider its adjoint problem ψt (x, t) + (k(x)ψx (x, t))x = 0, t ∈ [0, T ], x ∈ [0, X], k(0)ψx (0, t) = p(t), t ∈ [0, T ], k(X)ψx (X, t) = r(t), t ∈ [0, T ], ψ(x, T ) = 0, x ∈ [0, X].
(8) (9) (10) (11)
Definition 5. A generalized solution ψ(x, t; k) of problem (8) − (11) in a space V 1,0 (ΩT ) is a function ψ(x, t) ∈ V 1,0 (ΩT ) satisfying the relations
[ψ(x, τ )ut (x, τ ) + k(x)ψx (x, τ )ux (x, τ )]dxdτ =
T
r(τ )u(X, τ )dt −
0
ΩT
(12)
for all u(x, t) ∈ H 1,1 (ΩT ) and all u(x, 0) = 0. No. 1
p(τ )u(0, τ )dt,
0
ψ(·, T ) = 0
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Theorem 2 [6]. Let k(x) ∈ K, p(t) ∈ L2 (0, T ), and r(t) ∈ L2 (0, T ). Then there exists a unique generalized solution ψ(x, t; k) of system (8) − (11) in V 1,0 (ΩT ) and the following estimate holds true: sup ψ(·, t; k)2 + ψx (·, ·; k)2 + ψ(0, ·; k)2 + ψ(X, ·; k)2 cc 2 (p(·)2 + r(·)2 ),
(13)
0
