Ill-posedness and the bias-variance tradeoff in residual stress
measurement inverse solutions

## Abstract

**Background:** Relaxation methods determine residual stresses by
measuring the deformations produced by incremental removal of a
subdomain of the specimen.

Measured strains at any given increment, determined by the cumulative
effect of the relieved stresses, appear as an integral equation, which
must be inverted to obtain residual stresses. In practice, stress
distributions are discretized by a finite-dimensional basis, to
transform the integral equations into a linear system of equations,
which is often ill-conditioned.

**Objective:** This article demonstrates that the problem is
actually ill-posed and comes with an inherent bias-variance tradeoff.

**Methods:** The hole drilling method is used as an example
application, and the practical effects of ill-posedness are illustrated.

**Results:** Traditional regularization of the solution by limiting
the resolution of the discretization reduces solution variance (noise)
at the expense of increased bias

and often results in the ultimately harmful practice of taking fewer
data points. A careful analysis including the alternate Tikhonov
regularization approach shows

that the highest number of measurements should always be taken to reduce
the variance for a given regularization scheme. Unfortunately, the
variability of a regularized solution cannot be used to build a valid
confidence interval, since an unknown bias term is always present in the
true overall error.

**Conclusions:** The mathematical theory of ill-posed problems
provides tools to manage the bias-variance tradeoff on a reasonable
statistical basis, especially

when the statistical properties of measurement errors are known. In the
long run, physical arguments that provide constraints on the true
solution would be of utmost importance, as they could regularize the
problem

without introducing an otherwise unknown bias. Constraining the minimum
length scale to some physically meaningful value is one promising
possibility.