Sequential Structure-Sensitive Residual Diagnostics for PDE Inverse Problems

• URL: http://arxiv.org/abs/2607.02101v1
• Date: Thu, 02 Jul 2026 12:36:48 GMT
• Title: Sequential Structure-Sensitive Residual Diagnostics for PDE Inverse Problems
• Abstract summary: We propose a structure-sensitive sequential diagnostic based on e-processes.<n>We demonstrate how standard discrepancy checks accept misspecified fits that produce materially wrong quantities of interest.
• Score: 0.9778150054416824
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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