We provide an overview of recent progress in statistical inverse problems with random experimental design, covering both linear and nonlinear inverse problems. Different regularization schemes have been studied to produce robust and stable solutions. We discuss recent results in spectral regularization methods and regularization by projection, exploring both approaches within the context of Hilbert scales and presenting new insights particularly in regularization by projection. Additionally, we overview recent advancements in regularization using convex penalties. Convergence rates are analyzed in terms of the sample size in a probabilistic sense, yielding minimax rates in both expectation and probability. To achieve these results, the structure of reproducing kernel Hilbert spaces is leveraged to establish minimax rates in the statistical learning setting. We detail the assumptions underpinning these key elements of our proofs. Finally, we demonstrate the application of these concepts to nonlinear inverse problems in pharmacokinetic/pharmacodynamic (PK/PD) models, where the task is to predict changes in drug concentrations in patients.

翻译：本文综述了随机实验设计下统计逆问题的最新进展，涵盖线性和非线性逆问题。为获得稳健稳定的解，我们研究了多种正则化方案。讨论了谱正则化方法和投影正则化的近期成果，在Hilbert尺度背景下对两种方法进行了探索，特别针对投影正则化提出了新见解。此外，还综述了利用凸罚函数进行正则化的最新进展。从概率角度分析了样本规模下的收敛速度，获得了期望与概率意义上的极小化极大速率。为取得这些结果，我们利用再生核Hilbert空间的结构在统计学习框架下建立了极小化极大速率，详细阐述了支撑这些关键证明要素的假设条件。最后，将上述概念应用于药代动力学/药效学（PK/PD）模型中的非线性逆问题，该任务旨在预测患者体内药物浓度的变化。