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Moments, Time-Inversion and Source Identification for the Heat Equation

Kang Liu, Enrique Zuazua

TL;DR

This work addresses the ill-posed problem of identifying the initial heat source from terminal data for the heat equation by introducing a moment-based framework. Finite-order moments of the solution evolve according to a linear ODE with a nilpotent generator, allowing backward propagation to recover initial moments, which are then used in a convex total-variation minimization under moment constraints to recover a sparse initial measure. The authors prove the existence of atomic solutions and provide explicit Kantorovich-distance error bounds that scale polynomially with the terminal time and with observation noise, offering a clear advantage over classical exponential ill-posedness. They develop practical numerical schemes, including Gauss--Hermite quadrature for moment observation and a discretize-then-optimize LP solved by simplex, demonstrating robust performance in 1D and 2D experiments up to sizeable times $T$. Overall, the method offers a stable and quantifiable alternative to regularization-based approaches and provides guidance on moment order, domain choice, and observation accuracy.

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 ODE 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.

Moments, Time-Inversion and Source Identification for the Heat Equation

TL;DR

This work addresses the ill-posed problem of identifying the initial heat source from terminal data for the heat equation by introducing a moment-based framework. Finite-order moments of the solution evolve according to a linear ODE with a nilpotent generator, allowing backward propagation to recover initial moments, which are then used in a convex total-variation minimization under moment constraints to recover a sparse initial measure. The authors prove the existence of atomic solutions and provide explicit Kantorovich-distance error bounds that scale polynomially with the terminal time and with observation noise, offering a clear advantage over classical exponential ill-posedness. They develop practical numerical schemes, including Gauss--Hermite quadrature for moment observation and a discretize-then-optimize LP solved by simplex, demonstrating robust performance in 1D and 2D experiments up to sizeable times . Overall, the method offers a stable and quantifiable alternative to regularization-based approaches and provides guidance on moment order, domain choice, and observation accuracy.

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 ODE 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.

Paper Structure

This paper contains 41 sections, 14 theorems, 140 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Problem statement and motivation
  3. Ill-posedness.
  4. Moment method
  5. Inverting the dynamics of the moments.
  6. Recovering initial distribution from its moments.
  7. Overall formulation.
  8. Main results
  9. Main contributions.
  10. Techniques used in the proof.
  11. Numerical implementation
  12. Related work
  13. Tikhonov regularization methods.
  14. Algorithms for the discretized problem.
  15. Organization and notations
  16. ...and 26 more sections

Key Result

Theorem 2.1

Assume that $\Omega$ is compact and has a non-empty interior. Then, for any $k \in \mathbb{Z}_+$, any $T>0$ and any vector $\mathbf{y} = (y_{\alpha})_{\|\alpha\|_1 \leq k} \in \mathbb{R}^{\binom{k + d}{d}}$, the following statements hold:

Figures (3)

  • Figure 1: Initial conditions and terminal solutions ($T=10$) of \ref{['eq:heat']}.
  • Figure 2: Moment errors obtained with the quadrature methods. (a) Varying sensor count $n \in \{1,\dots,100\}$ at fixed $T = 10$. (b) Varying terminal time $T \in [1, 1000]$ at fixed $n = 100$. (c) Joint influence of moment order $k$ and terminal time for the Gauss--Hermite scheme.
  • Figure 3: Numerical results for the 2D case: (A) Values of $W_1(u_0^k,u_0^*)$ for $k\in[0,10]$ and $T=100$. (B) Recovered initial distributions $u_0^k$ for $k=5,\ldots,9$, versus the true $u_0^*$.

Theorems & Definitions (32)

  • Definition 1.1: Kantorovich Norm hanin1999kantorovichpiccoli2016properties
  • Theorem 2.1: Existence
  • proof
  • Theorem 2.2: Error estimate
  • proof
  • Remark 2.3: Error estimate analysis
  • Remark 2.4: Optimal choice of the moment order $k$
  • Lemma 2.5: The ODE system of moments
  • proof
  • Lemma 2.6: Growth rate
  • ...and 22 more