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An adjoint problem approach and coarse-fine mesh method for identification of the diffusion coefficient in a linear parabolic equation. (English) Zbl 1135.35095

Summary: This article presents a mathematical and numerical analysis of the adjoint problem approach for inverse coefficient problems related to linear parabolic equations. Based on maximum principle a structure of the coefficient-to-data mapping is derived. The obtained integral identities permit one to prove the monotonicity and invertibility of the input-output mappings, as well as formulate the gradient of the cost functional via the solutions of the direct and adjoint problems. In the second part of the paper a numerical algorithm for determining the diffusion coefficient \(k=k(x)\) in the linear parabolic equation \(u_t=(k(x)u_x)_x\) from the measured output data is presented. The main distinguished feature of the proposed algorithm is the use of a fine mesh for the numerical solution of the well-posed forward and backward parabolic problems, and a coarse mesh for the interpolation of unknown coefficient \(k=k(x)\). The nodal values of the unknown coefficient on the coarse mesh are recovered sequentially, solving on each step the well-posed forward and the sequence of backward initial value problems. This guarantees a compromise between the accuracy and stability of the solution of the considered inverse problem. An efficiency and applicability of the method is demonstrated on various numerical examples with noisy free and noisy data.

MSC:

35R30 Inverse problems for PDEs
35K15 Initial value problems for second-order parabolic equations
65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs

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