engleski

Relaxation of inverse problems by homogenisation method

We study the inverse problem of identifying the arrangement of two given isotropic materials such that the measurements at the boundary, or at the final time for evolution problems, are achieved. The proposed approach can be used for stationary diffusion equation, as well as for the wave or the heat equation, but with coefficients depending only on space variables. We interpret the inverse problem as an optimal design problem and use the homogenisation method as the relaxation tool, which enables us to use standard variational techniques for obtaining necessary conditions of optimality. These conditions are used as the basis for the optimality criteria method, which is the most used numerical method for optimal design problems. The update of design parameters in this iterative method is based on the recent results on multiple state optimal design problems.

inverse problems; homogenisation; optimality criteria method

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