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.

翻译：非线性统计反问题在统计分析与计算方面均构成重大挑战。基于似然的估计器通常导致非凸且可能多模态的优化曲面，而马尔可夫链蒙特卡洛（MCMC）方法可能呈现指数级缓慢的混合速度。针对通过算子方程 $L_f u = g$ 定义的反问题（其中 $f$ 为未知参数，$u$ 为观测解），我们提出一类计算可行的估计器——插件型与PDE惩罚型M估计器。其核心思想是通过弱强制松弛替代精确的PDE约束，从而产生条件凸的优化问题；在众多PDE示例中，该问题可转化为嵌套二次优化形式，既无需计算前向映射 $G(f)$，也不依赖PDE求解器。对于椭圆型PDE衍生的典型非线性反问题（包括达西流模型 $L_f u = \nabla\!\cdot(f\nabla u)$ 与稳态薛定谔模型），我们证明这些估计器在保持全局多项式时间可计算性的同时，达到了当前已知的最优统计收敛速率。在达西模型中，我们获得了从 $N$ 个含噪样本估计 $f$ 的显式次二次 $o(N^2)$ 算术运行时间上界。本研究的分析基于新型广义稳定性估计——将经典稳定性理论拓展至前向算子值域之外，并结合非参数M估计的理论工具。我们还推导了达西问题的自适应收敛速率，为设计广泛非线性反问题类中可证明多项式时间的统计算法提供了理论框架。所提出的估计器同时为多项式时间贝叶斯计算提供了理论启发的预热初始化方案。