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arXiv · 2603.27729

Global Convergence and Uniqueness for an Inverse Problem Posed by Gelfand

Michael V. Klibanov·Jingzhi Li·Tian Niu·Vladimir G. Romanov
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Abstract

The first globally convergent numerical method is developed for a coefficient inverse problem (CIP) for the $n-$d, $n\geq 2$ wave equation with the unknown potential in the most challenging case when the $\delta -$ function is present in the initial condition with a single location of the point source. In fact, an approximate mathematical model for that CIP is derived. That globally convergent numerical method is developed for this model. This is a new version of the so-called convexification numerical method. Uniqueness theorem is proven as well within the framework of that approximate mathematical model. The question about uniqueness of this CIP was first posed by a famous mathematician I. M. Gelfand in 1954 as an $n-$d ($n=2,3$) extension of the fundamental theorem of V.A. Marchenko in the 1-d case (1950). Based on a Carleman estimate, convergence analysis is carried out. This analysis ensures the global convergence of the proposed numerical method, i.e. it is not necessary to have a good first guess for the solution. Exhaustive computational experiments with noisy data demonstrate a high reconstruction accuracy of complicated structures. In particular, this accuracy points towards a high adequacy of that approximate mathematical model.

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Michael V. Klibanov, Jingzhi Li, Tian Niu, Vladimir G. Romanov. 2026-03-29. Global Convergence and Uniqueness for an Inverse Problem Posed by Gelfand. https://arxiv.org/abs/2603.27729

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