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arXiv · 2608.16133

An adjoint-free integral feedback method for a parabolic inverse source problem with conditional stability

Sedar Ngoma
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Abstract

We study an inverse source problem for a linear non-autonomous parabolic equation with additive source term $f(t)+ζ(t,x)$, where the unknown component depends only on time and is recovered from an integral observation of the solution. After reducing the problem to an equivalent linear inverse problem, we establish existence and uniqueness of the Tikhonov-regularized solution and derive a first-order optimality condition. We show that the forward operator admits a Volterra representation in time, yielding a weak-norm stability estimate in $H^{-1}(0,T)$ and, under an a priori $H^r(0,T)$ bound on the source, a conditional Hölder stability estimate in $L^2(0,T)$. Motivated by this Volterra structure, we introduce an adjoint-free integral feedback method that reconstructs the source using only forward solves. We analyze the feedback iteration by establishing its well-definedness and fixed-point properties, convergence for exact data, and finite-iteration stability with respect to noisy data. We further show that, with an appropriate noise-dependent stopping rule, the method constitutes an iterative regularization scheme. Numerical experiments for smooth and piecewise constant sources, supplemented by temporal regularization and automatic parameter selection, demonstrate accurate and stable reconstructions in the presence of noise.

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Sedar Ngoma. 2026-08-17. An adjoint-free integral feedback method for a parabolic inverse source problem with conditional stability. https://arxiv.org/abs/2608.16133

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