Weighted Proper Orthogonal Decomposition for High-Dimensional Optimization

While proper orthogonal decomposition (POD) is widely used for model reduction, its standard form does not take into account any parametric model structure. Extensions to POD have been proposed to address this, but these either require large amounts of solution data, lack online adaptivity, or have limited approximation accuracy. We circumvent these limitations by instead assigning weights to the snapshot matrix columns, and updating these whenever the model is evaluated at a new point in the parameter space. We derive an a posteriori error bound that depends on these snapshot weights, show how these weights can be chosen to tighten the error bound, and present an algorithm to compute the corresponding reduced basis efficiently. We show how this weighted POD approach can be used to naturally generalize the calculation of reduced basis derivatives to situations with multidimensional parameter spaces and snapshots at multiple locations in the parameter space. Lastly, we cover how these approaches can be implemented within an optimization algorithm, without the need for an offline training phase. The proposed weighted POD methods with and without reduced basis derivatives are applied to a gradient-based shell thickness optimization problem with 105 design parameters and a time-dependent partial differential equation. The numerical solutions obtained for this problem attain errors that are several orders of magnitude smaller when using weighted POD than those computed with regular POD and Grassmann manifold interpolation, while having comparable wall times per query and requiring fewer high-dimensional model snapshots to reach an optimal solution.