An efficient, globally convergent method for optimization under uncertainty using adaptive model reduction and sparse grids

Understand

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.