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Texas A&M University

Colloquium - Brendan Keith

Date: January 18, 2022

Time: 4:00PM - 5:00PM

Location: BLOC 117

Speaker: Brendan Keith, LLNL


Title: Adaptive numerical methods: From finite element analysis to stochastic programming
Abstract: 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.