Avanços e aplicações de problemas inversos em mecânica computacional

Abstract

Discrete inverse problems represent a class of numerical challenges that aim to compute missing information from mathematical-physical models, given a system response often contaminated by measuring errors and noise. The applications of these problems span various scientific fields, having been used in medical imaging to enhance the resolution of brain MRI images, in biomedical science to reconstruct blood perfusion coefficients, in thermal science to obtain the heat transfer coefficients in coiled tubes, and in materials science to identify elastic and viscoelastic material properties. As these problems are often ill-posed and include operators with large condition numbers, this work presents a comprehensive overview and numerical comparisons between classical and novel regularization schemes that can address these challenges. Specifically, the effectiveness of these regularization techniques is evaluated and validated by applying them to established test problems from the literature, as well as computing the shear correction factor for a Timoshenko beam through a forward-inverse finite element framework. Overall, this work advances the discrete inverse problems area by thoroughly analyzing various regularization techniques and their effectiveness in solving ill-posed problems and proposing a novel regularization method for finding the optimal Tikhonov regularization parameter.