Time domain decomposition for PDE-constrained optimization

Abstract

Despite decades of advances in parallel computing, the time dimension has remained largely unexploited for parallelization. At first glance, parallelization in time seems paradoxical, the solution of a time-dependent partial differential equation (PDE) at a future instant cannot be computed without knowledge of the past. This sequential behavior appears to preclude any temporal decomposition. However, in the setting of PDE-constrained optimization, the special structure of the first-order optimality system “weakens” this causality principle, and thus opens the door to a new and largely unexplored paradigm for time-parallel computation. In this talk, we will explore some non-overlapping domain decomposition algorithms to solve this forward-backward optimality system. We will introduce the idea of time domain decomposition and compare it with the space decomposition, the so-called waveform relaxation methods. Based on the forward-backward structure of the optimality system, we will then discuss some properties of classical domain decomposition methods in the time decomposition framework. Some tests will be shown to reveal numerical properties of these methods.

卢 柳 䃅

Postdoctoral Fellow and Oberwolfach Leibniz Fellow

My research interests include numerical analysis, scientific computing, mathematical modelling, optimization and control.