Stochastic asymptotical regularization for nonlinear ill-posed problems

TL;DR

This work extends Stochastic Asymptotical Regularization (SAR) from linear to nonlinear ill-posed inverse problems, combining regularization theory with stochastic analysis to achieve mean-square convergence guarantees. The SAR framework evolves via a stochastic differential equation driven by a Q-Wiener process, with a carefully chosen time-dependent noise term and an a posteriori stopping rule to realize regularization. The authors establish convergence and rate results under canonical source conditions and tangential cone nonlinearity, and demonstrate SAR’s practical advantages: uncertainty quantification, potential to escape local minima, and the ability to identify multiple solutions through sampling and clustering. Numerical experiments on PDE parameter identification and a nonlinear autoconvolution model illustrate improved accuracy, robustness to noise, and the capability to reveal multiple plausible solutions, highlighting SAR as a valuable tool for nonlinear inverse problems with deterministic forward models.

Abstract

Recently, the stochastic asymptotical regularization (SAR) has been developed in (\emph{Inverse Problems}, 39: 015007, 2023) for the uncertainty quantification of the stable approximate solution of linear ill-posed inverse problems. In this paper, we extend the regularization theory of SAR for nonlinear inverse problems. By combining techniques from classical regularization theory and stochastic analysis, we prove the regularizing properties of SAR with regard to mean-square convergence. The convergence rate results under the canonical sourcewise condition are also studied. Several numerical examples are used to show the accuracy and advantages of SAR: compared with the conventional deterministic regularization approaches for deterministic inverse problems, SAR can quantify the uncertainty in error estimates for ill-posed problems, improve accuracy by selecting the optimal path, escape local minima for nonlinear problems, and identify multiple solutions by clustering samples of obtained approximate solutions.

Figures (12)

• Figure 1: Results for the 1D case, with $\delta = 2\%$, $N=500$, $\theta=0.5$. (a) Relative errors; (b) Semi-log plot of residual errors; (c) Approximate solution by Landweber; (d) Approximate solution by SAR.
• Figure 2: Estimation of $P(\|x_n - x^\dag\|\geq \delta_i)$ using $N=500$ independent runs for 1D case with $\theta = 0.5$ and different noise levels $\delta_1= 1 \%, \delta_2=3\%, \delta_3=5\%$. The numerical results are derived from 500 independent simulations, with the starting point uniformly sampled using the formula $0.01 + 4 \cdot \text{rand}$.
• Figure 3: (a) The expectation of SAR and the 75$\%$ confidence interval for problem \ref{['elliptic equation']} with noise level $\delta = 1\%$; (b) The expectation of SAR and the 99$\%$ confidence interval for problem \ref{['elliptic equation']} with noise level $\delta = 2\%$. Other parameters in both (a) and (b): sample size N = 800, $\tau$ = 1.1, $\theta=0.1$.
• Figure 4: Reconstructions for the 2D case, with $90\%$ confidence interval and $\delta = 1\%$, $\theta=0.02$.
• Figure 5: Reconstructions for the 2D case, with $85\%$ confidence interval and $\delta = 1\%$, $N=200$.

Theorems & Definitions (37)

• Proposition 2.1
• Theorem 2.2
• Theorem 2.3
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Proposition 3.3
• proof
• Example 3.4