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A shape-optimization approach for inverse diffusion problems using a single boundary measurement (2503.14764v2)

Published 18 Mar 2025 in math.NA, cs.NA, and math.OC

Abstract: This paper explores the reconstruction of a space-dependent parameter in inverse diffusion problems, proposing a shape-optimization-based approach. The main objective is to recover the absorption coefficient from a single boundary measurement. While conventional gradient-based methods rely on the Fr\'{e}chet derivative of a cost functional with respect to the unknown parameter, we also utilize its shape derivative with respect to the unknown boundary interface for recovery. This non-conventional approach addresses the problem of parameter recovery from a single measurement, which represents the key innovation of this work. Numerical experiments confirm the effectiveness of the proposed method, even for intricate and non-convex boundary interfaces.

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