Past Event: Oden Institute Seminar

High-dimensional approximation of Hamilton-Jacobi-Bellman PDEs in deterministic optimal control: architectures, algorithms, and applications

Dante Kalise, Senior Lecturer, Computational Optimization and Control, Department of Mathematics, Imperial College London

Abstract

Optimal feedback synthesis for nonlinear dynamics -a fundamental problem in optimal control- is enabled by solving fully nonlinear Hamilton-Jacobi-Bellman type PDEs arising in dynamic programming. While our theoretical understanding of dynamic programming and HJB PDEs has seen a remarkable development over the last decades, the numerical approximation of HJB-based feedback laws has remained largely an open problem due to the curse of dimensionality. More precisely, the associated HJB PDE must be solved over the state space of the dynamics, which is extremely high-dimensional in applications such as distributed parameter systems or agent-based models.

In this talk we will review recent approaches regarding the effective numerical approximation of very high-dimensional HJB PDEs. We will explore modern scientific computing methods based on tensor decompositions of the value function of the control problem, and the
construction of data-driven schemes in supervised and semi-supervised learning environments. We will highlight some novel research
directions at the intersection of control theory, scientific computing, and statistical machine learning.

Biography

Dante Kalise is Senior Lecturer in Computational Optimisation and Control at the Department of Mathematics, Imperial College London, since 2021. He received B.Sc. and M.Sc. degrees (2008) from the Federico Santa María Technical University in Valparaíso, Chile, and a Ph.D. (2012) from the University of Bergen, Norway. Before joining Imperial, he was Assistant Professor in Applied Mathematics at the University of Nottingham, and held research positions at RICAM Linz, and at La Sapienza University of Rome. He serves as Associate Editor of Mathematics of Control, Signals, and Systems, and of Advances in Continuous and Discrete Models. Dr. Kalise's research interests lie at the interface between scientific computing, optimal control, and PDEs. His current research is centered around the analysis and design of computational methods for the solution of high-dimensional Hamilton-Jacobi-Bellman PDEs and applications in nonlinear feedback control for PDE dynamics and agent-based models across scales.