• Residual Stress
• Ill-posedness
• Uncertainty Quantification
• Bias-variance Tradeoff
• Inverse Problems

This dissertation critically explores relaxation methods within the context of inverse problems, emphasizing the need for a nuanced understanding of their inherent properties. Its primary contributions include the recognition of their ill-posedness as an intrinsic mathematical feature, urging researchers and practitioners to adeptly navigate its distinctive effects. The notion of unregularized solutions is challenged, asserting the inevitability of regularization and highlighting the role of discretization as a form of regularization. While acknowledging the theoretical underpinnings of current methodologies, the work contends that achieving meaningful solutions without additional physical knowledge is simply impossible. A critical assessment of existing uncertainty quantification techniques is presented, with a caution against hidden errors that are invisible to the residual stress analyst. The proposal to shift from point-wise to average stress values is made to enable equitable result comparisons and achieve a well-posed problem that avoids a balance between an unknown bias and a computable variance. Additionally, a chapter delves into the hole-drilling method, underlining the significance of reference solutions for validation purposes and introducing a technique to address errors in the zero-depth datum. In essence, this work advocates a paradigm shift, encouraging a deeper comprehension of ill-posed problems and suggesting pragmatic alternatives for robust results in relaxation methods.