When confidence intervals are too confident: how inverse methods fool us and how to fix them

The identification of residual stresses through inverse methods is a mathematically ill-posed problem. To achieve a solution with a manageable signal-to-noise ratio, a certain degree of theoretically uncomputable bias must be introduced—a well-known phenomenon referred to as the bias-variance tradeoff. This raises a fundamental question: how can solution uncertainty be quantified if part of it remains inherently inaccessible?

While additional physical knowledge could, in theory, help characterize bias, this is rarely feasible in practical applications. A review of biases in established methods reveals that eliminating them would require information that is never available in practice. Instead, an alternative approach is proposed: considering average stresses over a distance, which leads to a well-posed problem. A numerical and experimental example illustrates this concept. Since determining average stresses avoids the bias-variance tradeoff, result uncertainties can be estimated using standard methods, allowing for exact confidence intervals.

More broadly, residual stresses and relaxation methods highlight the limitations of pointwise stress values, which are usually a good model when a natural unstressed state can be assumed, as in classical continuum mechanics. This suggests shifting the focus from evaluating stress at specific points to understanding its impact on material performance—often governed by stresses that are inherently averaged in space.