Past Event: Oden Institute Seminar

Finite Element Approximation of Constrained and Ill-Posed Problems

Mats Larson, Umea University, Sweden

Abstract

We review recent advances in applying the augmented Lagrangian method as a general approach for generating stabilized methods with and without explicit multipliers. The augmented Lagrangian method enhances the standard Lagrange multiplier approach by incorporating a penalty term that enforces the constraint equations. Traditionally known as the basis for iterative algorithms in constrained optimization, its application as a stabilization method in computational mechanics has recently gained significant interest. We examine both equality and inequality constraints, showcasing the framework through various examples in computational mechanics, including ill-posed problems. Finally, we introduce machine learning tools aimed at accelerating the solution of ill-posed problems and integrating statistical observations into the results.

Biography

Dr. Larson received his Ph.D. in 1997 with a thesis on Adaptive Finite Element Methods under the supervision of Prof. Claes Johnson. Then, he was a postdoc fellow at Stanford University, working with Prof. Tom Hughes. He was appointed Professor of Applied Mathematics at Umea University in 2004.

Dr. Larson's research encompasses various aspects of finite element methods, including adaptivity, discontinuous Galerkin methods, and immersed methods. He has also explored the integration of machine learning tools into computational mechanics. 

In addition to his research, Dr. Larson holds leadership positions in professional organizations. He serves as Chairman of the Nordic Association of Computational Mechanics and the ECCOMAS Technical Committee for Computational and Applied Mathematics. In 2024, he was elected as a fellow of the International Association for Computational Mechanics (IACM), recognizing his significant contributions to the field.