Session: Nonsmooth PDE Constrained Optimization: Algorithms, Analysis and Applications Part 2
Chair: Robert Baraldi
Cluster: PDE-constrained Optimization
Talk 1: An Inexact Trust-Region Algorithm for Nonsmooth Risk-Averse Optimization
Speaker: Drew Kouri
Abstract: Many practical problems require the optimization of systems (e.g., differential equations) with uncertain inputs such as noisy problem data, unknown operating conditions, and unverifiable modeling assumptions. In this talk, we formulate these problems as infinite-dimensional, risk-averse stochastic programs for which we minimize a quantification of risk associated with the system performance. For many popular risk measures, the resulting risk-averse objective function is not differentiable, significantly complicating the numerical solution of the optimization problem. Unfortunately, traditional methods for nonsmooth optimization converge slowly (e.g., sublinearly) and consequently are often intractable for problems in which the objective function and any derivative information is expensive to evaluate. To address this challenge, we introduce a novel trust-region algorithm for solving large-scale nonsmooth risk-averse optimization problems. This algorithm is motivated by the primal-dual risk minimization algorithm and employs smooth approximate risk measures at each iteration. In addition, this algorithm permits and rigorously controls inexact objective function value and derivative (when available) computations, enabling the use of inexpensive approximations such as adaptive discretizations. We discuss convergence of the algorithm under mild assumptions and demonstrate its efficiency on various examples from PDE-constrained optimization.
Talk 2: Variational problems with gradient constraints: A priori and a posteriori error identities
Speaker: Rohit Khandelwal
Abstract: Nonsmooth variational problems are ubiquitous in science and engineering, for e.g., fracture modeling and contact mechanics. This talk presents a generic primal-dual framework to tackle these types of nonsmooth problems. Special attention is given to variational problems with gradient constraints. The key challenge here is how to project onto the constraint set both at the continuous and discrete levels. In fact, both a priori and a posteriori error analysis for such nonsmooth problems has remained open. In this talk, on the basis of a (Fenchel) duality theory at the continuous level, an a posteriori error identity for arbitrary conforming approximations of primal-dual formulations is derived. In addition, on the basis of a (Fenchel) duality theory at the discrete level, an a priori error identity for primal (Crouzeix–Raviart) and dual (Raviart–Thomas) formulations is established. The talk concludes by deriving the optimal a priori error decay rates.
Talk 3: Optimal Insulation: Numerical Analysis and Spacecraft
Speaker: Keegan Kirk
Abstract: Given a fixed amount of insulating material, how should one coat a heat-conducting body to optimize its insulating properties? A rigorous asymptotic analysis reveals this problem can be cast as a convex variational problem with a non-smooth boundary term. As this boundary term is difficult to treat numerically, we consider an equivalent (Fenchel) dual variational formulation more amenable to discretization. We propose a numerical scheme to solve this dual formulation on the basis of a discrete duality theory inherited by the Raviart-Thomas and Crouzeix-Raviart finite elements and show that the solution of the original primal problem can be reconstructed locally from the discrete dual solution. We discuss the a posteriori and a priori error analysis of our scheme, derive a posteriori estimators based on convex optimality conditions, and present numerical examples to verify theory. As an application, we consider the design of an optimally insulated spacecraft heat shield.
