Moments, time-inversion and source identification for the heat equation
Liu K, Zuazua Iriondo E (2026)
Publication Type: Journal article
Publication year: 2026
Journal
Book Volume: 42
Article Number: 015009
Journal Issue: 1
Abstract
We address the initial source identification problem for the heat equation, a notably ill-posed inverse problem characterized by exponential instability. Departing from classical Tikhonov regularization, we propose a novel approach based on moment analysis of the heat flow, transforming the problem into a more stable inverse moment formulation. By evolving the measured terminal time moments backward through their governing ordinary differential equation system, we recover the moments of the initial distribution. We then reconstruct the source by solving a convex optimization problem that minimizes the total variation of a measure subject to these moment constraints. This formulation naturally promotes sparsity, yielding atomic solutions that are sums of Dirac measures. Compared to existing methods, our moment-based approach reduces exponential error growth to polynomial growth with respect to the terminal time. We provide explicit error estimates on the recovered initial distributions in terms of moment order, terminal time, and measurement errors. In addition, we develop efficient numerical discretization schemes and demonstrate significant stability improvements of our approach through comprehensive numerical experiments.
Authors with CRIS profile
Enrique Zuazua Iriondo
Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur)
Involved external institutions
France (FR)How to cite
APA:
Liu, K., & Zuazua Iriondo, E. (2026). Moments, time-inversion and source identification for the heat equation. Inverse Problems, 42(1). https://doi.org/10.1088/1361-6420/ae3797
MLA:
Liu, Kang, and Enrique Zuazua Iriondo. "Moments, time-inversion and source identification for the heat equation." Inverse Problems 42.1 (2026).
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