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Analysis of an adjoint problem approach to the identification of an unknown diffusion coefficient. (English) Zbl 1054.35125

The authors consider the nonlinear inverse problem of finding the diffusion coefficient \(D(u)\) depending on the state variable \(u\) in a quasilinear parabolic partial differential equation (locally 1-D diffusion equation) from time-dependent flux observations at the left end of the interval and state observations at the right end over some time interval. In a series of lemmas there are formulated and proven essential properties of the parameter-to-data mapping such as continuity, monotonicity and injectivity. For a given parameter function \(D\) this mapping (forward operator) is applied by solving the associated initial-boundary value problem of the parabolic equation. In this context, some integral identities are presented that provide explicit representations of the forward operator, where adjoint versions of the direct problem are used to derive equations for the interplay of changes in coefficients (input data) and measured functions (output data). Moreover, it is shown that the forward operator is explicitly invertible if the problem is dicretized by using polygonal coefficients. This assertion yields the basis to a numerical approach for computing the diffusion coefficient \(D\). Some error analysis concerning the accuracy of approximation and some numerical case studies illustrating the chances of the method are given.

MSC:

35R30 Inverse problems for PDEs
35K05 Heat equation
47J06 Nonlinear ill-posed problems
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