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Stochastic Convergence Analysis of Inverse Potential Problem

Bangti Jin, Qimeng Quan, Wenlong Zhang

TL;DR

This paper addresses the inverse problem of recovering a potential $q$ in the elliptic equation $- abla^2 u + q u=f$ from noisy point observations of the state. It proposes a variational regularization framework with an $H^1(Ω)$ penalty and solves the forward problem and regularized inverse problem using Galerkin FEM, establishing high-probability $L^2(Ω)$ error bounds for both the continuous and discrete problems and providing an adaptive strategy for choosing the regularization parameter $γ$. The results rigorously quantify how the error depends on $γ$, the number of observations $n$, and the mesh size $h$, and they connect conditional stability with stochastic error control to yield explicit convergence rates under low regularity ($q∈H^1(Ω)$). The work also offers a self-contained adaptive γ algorithm with monotone convergence guarantees and supports the theory with numerical experiments demonstrating accurate reconstruction of $q$ and the state, including boundary-layer effects. Overall, the paper advances deterministic-regularization techniques for inverse potentials by incorporating stochastic observations and discretization effects, yielding practical guidance for parameter tuning and robust reconstruction in PDE-inverse problems.

Abstract

In this work, we investigate the inverse problem of recovering a potential coefficient in an elliptic partial differential equation from the observations at deterministic sampling points in the domain subject to random noise. We employ a least squares formulation with an $H^1(Ω)$ penalty on the potential in order to obtain a numerical reconstruction, and the Galerkin finite element method for the spatial discretization. Under mild regularity assumptions on the problem data, we provide a stochastic $L^2(Ω)$ convergence analysis on the regularized solution and the finite element approximation in a high probability sense. The obtained error bounds depend explicitly on the regularization parameter $γ$, the number $n$ of observation points and the mesh size $h$. These estimates provide a useful guideline for choosing relevant algorithmic parameters. Furthermore, we develop a monotonically convergent adaptive algorithm for determining a suitable regularization parameter in the absence of \textit{a priori} knowledge. Numerical experiments are also provided to complement the theoretical results.

Stochastic Convergence Analysis of Inverse Potential Problem

TL;DR

This paper addresses the inverse problem of recovering a potential in the elliptic equation from noisy point observations of the state. It proposes a variational regularization framework with an penalty and solves the forward problem and regularized inverse problem using Galerkin FEM, establishing high-probability error bounds for both the continuous and discrete problems and providing an adaptive strategy for choosing the regularization parameter . The results rigorously quantify how the error depends on , the number of observations , and the mesh size , and they connect conditional stability with stochastic error control to yield explicit convergence rates under low regularity (). The work also offers a self-contained adaptive γ algorithm with monotone convergence guarantees and supports the theory with numerical experiments demonstrating accurate reconstruction of and the state, including boundary-layer effects. Overall, the paper advances deterministic-regularization techniques for inverse potentials by incorporating stochastic observations and discretization effects, yielding practical guidance for parameter tuning and robust reconstruction in PDE-inverse problems.

Abstract

In this work, we investigate the inverse problem of recovering a potential coefficient in an elliptic partial differential equation from the observations at deterministic sampling points in the domain subject to random noise. We employ a least squares formulation with an penalty on the potential in order to obtain a numerical reconstruction, and the Galerkin finite element method for the spatial discretization. Under mild regularity assumptions on the problem data, we provide a stochastic convergence analysis on the regularized solution and the finite element approximation in a high probability sense. The obtained error bounds depend explicitly on the regularization parameter , the number of observation points and the mesh size . These estimates provide a useful guideline for choosing relevant algorithmic parameters. Furthermore, we develop a monotonically convergent adaptive algorithm for determining a suitable regularization parameter in the absence of \textit{a priori} knowledge. Numerical experiments are also provided to complement the theoretical results.

Paper Structure

This paper contains 19 sections, 21 theorems, 181 equations, 4 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Numerical procedure, main results and discussions
  3. Regularized reconstruction and its discretization
  4. Main results and discussions
  5. Review on existing studies
  6. Deterministic variational regularization
  7. Statistical analysis
  8. Preliminaries
  9. Basic theory of sub-Gaussian random variables
  10. Discrete Sobolev semi-norm
  11. Proof of Theorem \ref{['thm:con-err-potential']}
  12. Proof of Theorem \ref{['thm:err-potential']}
  13. Numerical experiments and discussion
  14. Adaptive parameter choice
  15. Numerical results and discussions
  16. ...and 4 more sections

Key Result

Theorem 2.2

\newlabelthm:con-err-potential0 Let Assumptions ass:quasi-points and ass:con-reg be fulfilled, and let $q^*\in\mathcal{A}\cap H^1(\Omega)$ be a minimizer to problem eqn:conti-optim-prob-potential-eqn:conti-weak-potential. Fix $\tau\in(0,\frac{1}{4})$, Then with probability at least $1-2\tau$, with $\ell_\tau = \sqrt{\log\frac{2}{\tau}}$, there holds where the constant $c$ is independent of $q^*

Figures (4)

  • Figure 1: A schematic illustration of Assumption \ref{['ass:quasi-points']} on scattered sampling points $(x_i)_{i=1}^n$: (a) uniform, (b) quasi-uniform and (c) nonquasi-uniform.
  • Figure 1: The numerical results on the optimal regularization parameter $\gamma^*$ at two noise levels: the error $e_q$ versus the parameter $\gamma$ (left); the error $e_u$ versus the parameter $\gamma$ (right).
  • Figure 2: The numerical results for Example \ref{['exam:adap']} with $\sigma=5.0\%$. The top row shows the exact potential $q^\dag$ (left), recovered potential $q_{h}^*$ from Algorithm \ref{['alg:adpa-gamma']} (middle), and the pointwise error $e:=q^\dag - q_{h}^*$ (right). The bottom row shows the value of $\gamma$ (left), the error $e_q$ (middle) and the error $e_u$ (right), all versus the iteration index $k$.
  • Figure 3: The reconstruction $q_{h}^*$ for Example \ref{['exam:rate']} (a)-(d): (a) and (b) at the first two rows; (c) and (d) at the last two rows.

Theorems & Definitions (51)

  • Theorem 2.2
  • Remark 2.1
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.3
  • Definition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Definition 3.4
  • Definition 3.5
  • ...and 41 more