Global polynomial-time estimation in statistical nonlinear inverse problems via generalized stability

TL;DR

This work develops a class of generalized M-estimators for nonlinear statistical inverse problems governed by PDEs, replacing exact PDE constraints with weak, PDE-penalized terms to achieve tractable, globally computable optimization. It introduces PDE-penalized and plug-in M-estimators, grounded by generalized stability estimates that connect parameter error to PDE residuals and solution deviations, enabling minimax-optimal rates for forward and inverse problems in elliptic PDEs such as Darcy flow and Schrödinger equations. A key methodological advance is a sub-quadratic, polynomial-time scheme based on a high-dimensional wavelet discretization that attains the best known rates, plus adaptive procedures that handle unknown smoothness; the framework also yields polynomial-time Bayesian computation with warm-start guarantees for MCMC. Collectively, these results offer a principled, scalable approach to non-linear PDE-based inverse problems with rigorous statistical guarantees and practical computational feasibility, and they suggest broad applicability to non-linear operators beyond the elliptic setting.

Abstract

Non-linear statistical inverse problems pose major challenges both for statistical analysis and computation. Likelihood-based estimators typically lead to non-convex and possibly multimodal optimization landscapes, and Markov chain Monte Carlo (MCMC) methods may mix exponentially slowly. We propose a class of computationally tractable estimators--plug-in and PDE-penalized M-estimators--for inverse problems defined through operator equations of the form $L_f u = g$, where $f$ is the unknown parameter and $u$ is the observed solution. The key idea is to replace the exact PDE constraint by a weakly enforced relaxation, yielding conditionally convex and, in many PDE examples, nested quadratic optimization problems that avoid evaluating the forward map $G(f)$ and do not require PDE solvers. For prototypical non-linear inverse problems arising from elliptic PDEs, including the Darcy flow model $L_f u = \nabla\!\cdot(f\nabla u)$ and a steady-state Schrödinger model, we prove that these estimators attain the best currently known statistical convergence rates while being globally computable in polynomial time. In the Darcy model, we obtain an explicit sub-quadratic $o(N^2)$ arithmetic runtime bound for estimating $f$ from $N$ noisy samples. Our analysis is based on new generalized stability estimates, extending classical stability beyond the range of the forward operator, combined with tools from nonparametric M-estimation. We also derive adaptive rates for the Darcy problem, providing a blueprint for designing provably polynomial-time statistical algorithms for a broad class of non-linear inverse problems. Our estimators also provide principled warm-start initializations for polynomial-time Bayesian computation.

Figures (2)

• Figure 1: Numerical illustration of the proposed plug-in estimation procedure for the Darcy flow inverse problem. Left to right: ground truth solution $u_0=\mathcal{G}(f_0)$ and coefficient $f_0$, preliminary estimator $\hat{u}_N$ obtained by a smoothing regression step, and plug-in estimator $\hat{f}_N$, based on $N=3000$ noisy measurements.
• Figure 2: Schematic representation of the log-posterior landscape. Often (e.g. in the Darcy flow and Schrödinger models) the negative log-posterior is convex with high probability NW22nickl2023bayesian within some region $\mathcal{I}$ containing $f_0$, while it may be multimodal in $\mathcal{M}$. Cold-start MCMC algorithms may thus suffer from computational difficulties arising from multimodality or 'free energy barriers' BMNW23.

Theorems & Definitions (5)

• proof
• proof : Proof of Lemma \ref{['darcy-generalized-stability']}
• proof : Proof of Lemma \ref{['schroed-generalized-stability']}
• proof
• proof