Past Event: Oden Institute Seminar

Adaptive numerical methods: From finite element analysis to stochastic programming

Brendan Keith, Postdoctoral Researcher, Center for Applied Scientific Computing, Lawrence Livermore National Laboratory

Abstract

** This seminar will be presented LIVE in person in POB 6.304 and streamed live via Zoom. **

Intelligent resource allocation is fundamental to balancing solution accuracy and computational cost. Adaptive numerical methods can provide a rigorous mechanism to keep this balance, so it comes as no surprise that they also play a prominent role in many of the most efficient codes.

In the first half of this talk, we revisit the marking decisions made in the prototypical adaptive finite element method (AFEM). We will show that a naive marking policy leads to inefficient use of adaptive mesh refinement (AMR). To address this issue, we choose to recast AMR as a partially-observed Markov decision process that can be optimized using methods from reinforcement learning. This recasting delivers a tractable optimization framework which eliminates the need for parameter tuning by expert users. We use the Poisson equation to showcase our framework in three representative AFEM applications inspired by the literature: (1) $h$-refinement and (2) $hp$-refinement in non-convex polyhedra and (3) dynamic $h$-refinement and derefinement for a transient source. Our experiments indicate that superior marking policies remain undiscovered for many canonical AFEM applications.

In the second half of this talk, we consider adaptivity in parameter space. More specifically, the focus is adaptive sampling as a mechanism for more efficient stochastic optimization algorithms. We will introduce and analyze new adaptive sampling methods for risk-averse stochastic programs with deterministic and, if time allows, probabilistic constraints. In particular, we propose a variant of the stochastic projected gradient method where the sample size used to approximate the reduced gradient is determined a posteriori and updated adaptively. We also propose an SQP-type method based on similar adaptive sampling principles. Both methods lead to a significant reduction in cost. PDE-constrained optimization problems from engineering will be used to illustrate the performance and efficacy of the presented algorithms.

Biography

Brendan Keith, Postdoctoral Researcher at Lawrence Livermore National Laboratory, is a Canadian mathematician and computational scientist. He received his Ph.D. from the Oden Institute for Computational Engineering and Sciences in August 2018, under the supervision of Leszek Demkowicz. His main research contributions are in the development and analysis of finite element methods, especially the DPG method. More recently, he has contributed to the fields of stochastic optimization, uncertainty quantification, fractional PDEs, and scientific machine learning.