Sequential Structure-Sensitive Residual Diagnostics for PDE Inverse Problems

2026-07-02

AI Summary

This work addresses the limitation of traditional residual norm diagnostics in inverse problems for partial differential equations (PDEs), which often fail to detect structured model errors and may erroneously accept biased estimates. The study introduces e-processes—a novel sequential diagnostic framework—into PDE inversion by constructing a dictionary of spatial residual patterns to form an expert mixture, coupled with a likelihood-ratio-based wealth update mechanism. This approach guarantees strict Type-I error control at any stopping time. Demonstrated on elliptic diffusion, Stokes flow, and ice-sheet (icepack) inversion problems, the method detects and attributes model misspecification earlier and more accurately than conventional techniques, exhibiting both structural sensitivity and interpretability, thereby effectively guiding model refinement.

Abstract

Computational models in science and engineering are often assessed by checking whether the residual norm is consistent with the assumed noise level. This can be misleading in smoothing inverse problems: structured model errors may be attenuated in observation space, leaving residual magnitudes below practitioner discrepancy thresholds while coherent residual patterns remain. As a result, residual-norm diagnostics can accept fitted models that still give biased parameters, predictions, or quantities of interest. We propose a structure-sensitive sequential diagnostic based on e-processes. The method uses a portfolio of spatial residual-pattern experts, updates their likelihood-ratio wealth as observations are processed, and rejects the fitted model when the aggregate wealth crosses a prescribed threshold, giving anytime-valid type-I error control for a fixed fitted model. We compare the method with Morozov discrepancy checks, fixed-sample residual tests, and batch projection tests. Across three inverse problems (elliptic diffusion, two-dimensional Stokes flow, and a glaciological ice-stream inversion implemented in the community finite-element model icepack) we demonstrate how standard discrepancy checks accept misspecified fits that produce materially wrong quantities of interest. Structure-sensitive batch tests detect these failures using the full dataset, while the e-process detects them earlier from a fraction of the observations. After rejection, the expert wealth attributes the evidence to residual patterns in the chosen dictionary and provides a basis for exploratory model correction.

Problem

• inverse problems
• residual diagnostics
• model misspecification
• structured errors
• PDE

Innovation

• e-processes
• structure-sensitive diagnostics
• sequential testing
• inverse problems
• residual pattern detection