An Efficient, Globally Convergent Method for Optimization Under Uncertainty Using Adaptive Model Reduction and Sparse Grids

Abstract

This work introduces a new method to efficiently solve optimization problems constrained by partial differential equations (PDEs) with uncertain coefficients. The method leverages two sources of inexactness that trade accuracy for speed: (1) stochastic collocation based on dimension-adaptive sparse grids (SGs), which approximates the stochastic objective function with a limited number of quadrature nodes, and (2) projection-based reduced-order models (ROMs), which generate efficient approximations to PDE solutions. These two sources of inexactness lead to inexact objective function and gradient evaluations, which are managed by a trust-region method that guarantees global convergence by adaptively refining the sparse grid and reduced-order model until a proposed error indicator drops below a tolerance specified by trust-region convergence theory. A key feature of the proposed method is that the error indicator-which accounts for errors incurred by both the sparse grid and reduced-order model-must be only an asymptotic error bound, i.e., a bound that holds up to an arbitrary constant that need not be computed. This enables the method to be applicable to a wide range of problems, including those where sharp, computable error bounds are not available; this distinguishes the proposed method from previous works. Numerical experiments performed on a model problem from optimal flow control under uncertainty verify global convergence of the method and demonstrate the method's ability to outperform previously proposed alternatives.

Time-series machine-learning error models for approximate solutions to parameterized dynamical systems

“…Note that the approximate solution incurs no error if the approximate solutions exactly satisfy the FOM O∆E (2) such that the residual is zero at all time instances. This also illustrates why the residual norm is often viewed as a useful error indicator for guiding greedy methods for snapshot collection [7,6,19,48,49] or trust-region optimization algorithms [52,50,51].…”

Fast Neural Network Predictions from Constrained Aerodynamics Datasets

“…ROMs use knowledge gained from previous simulations to construct the reduced-order basis, and can begin to approximate solutions well starting from a handful of examples-hence, they are often useful for industry problems where getting extremely large CFD datasets for training is infeasible. [18,19]. For training, ROMs utilize an adaptive sampling method known as a "greedy" procedure for sampling the parameter space and generating high-dimensional training solution snapshots, which allows them to minimize the training time as much as possible.…”

Time-series learning of latent-space dynamics for reduced-order model closure

“…To address this issue, there has recently been a significant quantity of research into the reducedorder modeling (ROM) of such systems to reduce the degrees of freedom of the forward problem to manageable magnitudes [1,2,3,4,5,6,7]. As such, this field finds extensive application in control [8], multi-fidelity optimization [9] and uncertainty quantification [10,11] among others. However, ROMs are limited in how they handle nonlinear dependence and perform poorly for complex physical phenomena which are inherently multiscale in space and time [12,13,14,15].…”