Session: Optimization with Approximate or Uncertain Models
Chair: Matthias Heinkenschloss
Cluster: PDE-constrained Optimization
Talk 1: Preconditioned Pseudo-time Continuation for Parameterized Inverse Problems
Speaker: Bart van Bloemen Waanders
Abstract: In this presentation we discuss a continuation approach to overcome linearization limitations associated with evaluating the post-optimality sensitivity of uncertainties with respect to optimization solutions. The post-optimality sensitivities (sensitivities of the first order optimality condition) arise from the Implicit Function Theorem and depend on second-order derivatives. If the magnitude of uncertainty is large, the post-optimality sensitivities are insufficient to predict the effects of perturbed uncertainty parameters on the optimization solution. To address this issue, we introduce a continuation process that uses a pseudo time-stepping scheme to evolve the sensitivities. A combination of specialized time-discretization and preconditioning helps to accelerate convergence. A key computational challenge is the calculation of the inverse Hessian as part of the post-optimality sensitivity evaluation at each iteration of the continuation process. To that end, we use a preconditioned Conjugate Gradient (PCG) solution strategy in which two novel Quasi-Newton update schemes are implemented that exploit the pseudo-time continuation structure. Our first update scheme introduces a secant equation to captures the uncertainty variations. The second is an adaption of the block BFGS methods that leverages the PCG iteration history. We demonstrate our approach on an insightful yet simple Poisson PDE with nonlinear boundary conditions and a nonlinear forcing term that in turn embeds uncertainty. We invert for a spatially distributed diffusion coefficient and demonstrate the efficacy of our time-stepping and preconditioning algorithms.
Talk 2: Shape and topology optimization under uncertainty by robust approaches with application to electric machines
Speaker: Stefan Ulbrich
Abstract: We consider shape and topology optimization for PDE-constrained problems, where parameters in the PDE (e.g. coefficients) as well as the design itself (e.g. manufacturing tolerances) are uncertain. We propose a robust optimization approach, where the usually nonsmooth maximum value functions of constraints and objective function on the uncertainty sets are used in the robust counterpart. We discuss the efficient calculation of generalized derivatives of the robustified objective function and constraints. In particular, we introduce a novel robust topological derivative that can be used for robust topology optimization. We apply the methodology to shape and topology optimization of electric machines.
Talk 3: Adaptive Surrogate Modeling for Trajectory Optimization with Model Inexactness
Speaker: Matthias Heinkenschloss
Abstract: In many applications, one must compute optimal trajectories from imperfect knowledge of the dynamics. For example, solving trajectory optimization problems for hypersonic vehicles requires computing lift and drag coefficients at many flight configurations. Determining these coefficients over the entire state space would require expensive high-fidelity computations using detailed representations of the hypersonic vehicle at prohibitively many samples. This talk proposes using computationally inexpensive adaptive kernel regression models constructed from high-fidelity samples to approximate the components of the dynamics that are expensive to evaluate. To reduce the effect of model errors on the optimal trajectory, the current kernel regression model is updated as needed at the cost of evaluating the components of the dynamics at a small number of additional sample points. First, the optimal control problem is solved using the current kernel model to represent the dynamics. Next, a new optimality sensitivity analysis is combined with error estimates of the kernel model to determine whether the kernel regression model needs to be updated and, if so, at which samples the dynamics should be evaluated to update it. This talk outlines our current model refinement procedure and demonstrates its performance on a trajectory optimization problem for a hypersonic vehicle with lift and drag models that are known but expensive to evaluate.
