Certified Randomized Model Order Reduction Methods for High-Dimensional Approximation

Abstract

In this talk, we present randomized methods that provide high-probability guarantees for the accuracy of reduced-order approximations of parametric partial differential equations (PDEs) with high-dimensional parameter sets. The underlying philosophy is to combine classical reduced-basis and greedy approximation ideas with concentration phenomena and data-dependent sampling to obtain certified approximations in high dimensions.

We first present non-asymptotic error bounds for Proper Orthogonal Decomposition (POD) under the sole assumption that the parameter-to-solution map is uniformly bounded for almost all parameter values. In contrast to existing results, the leading term in our bounds is governed by the sum of the neglected eigenvalues and scales inversely with the number of samples, thereby allowing one to exploit rapid eigenvalue decay. The resulting estimates are independent of the dimension of the parameter space. Consequently, even a modest number of samples can be sufficient for the empirical POD approximation to perform comparably to the ideal POD constructed from the full parameter distribution, including in infinite-dimensional parameter settings.

While the POD result holds for arbitrary sampling strategies, the quality of randomized model reduction can be significantly improved by exploiting problem-dependent sampling. We therefore turn to provably near-optimal sampling strategies. Sampling from Christoffel-type measures provides a near-optimal strategy for identifying large function values from finitely many samples. Building on this idea, we design a randomized greedy algorithm that guarantees, with high probability, the desired approximation accuracy for almost every parameter value in the parameter set, thereby eliminating the need for costly online error assessment. We conclude by showing that, with high probability, the resulting randomized greedy algorithm achieves quasi-optimal convergence rates.

Kathrin Smetana

Kathrin Smetana is a tenure-track assistant professor in the Department of Mathematical Sciences at the Stevens Institute of Technology. Before joining Stevens in January 2021, she was an assistant professor at the University of Twente in the Netherlands. She previously held postdoctoral positions with Mario Ohlberger at the University of Münster and Anthony T. Patera at the Massachusetts Institute of Technology.

She received the Professor De Winter Award in 2018 for her work on randomized local model order reduction methods and a National Science Foundation (NSF) CAREER Award in Computational Mathematics in 2022.

Her research focuses on the development and analysis of randomized methods for approximating solutions to heterogeneous and parametric partial differential equations, with applications in multiscale modeling, inverse problems, uncertainty quantification, and scientific machine learning.