Uncertainty quantification for complex systems with very high dimensional response using Grassmann manifold variations

TL;DR

This paper introduces an adaptive stochastic simulation method that uses Grassmann manifold variations to efficiently quantify uncertainty in high-dimensional, full-field responses of complex systems, enabling targeted refinement and solution approximation.

Contribution

It presents a novel adaptive sampling approach leveraging Grassmann manifold metrics for uncertainty quantification in high-dimensional responses, avoiding surrogate models.

Findings

Effectively identifies regions with significant response changes.

Provides accurate probability estimates for shear band formation.

Reduces computational cost by targeted sampling.

Abstract

This paper addresses uncertainty quantification (UQ) for problems where scalar (or low-dimensional vector) response quantities are insufficient and, instead, full-field (very high-dimensional) responses are of interest. To do so, an adaptive stochastic simulation-based methodology is introduced that refines the probability space based on Grassmann manifold variations. The proposed method has a multi-element character discretizing the probability space into simplex elements using a Delaunay triangulation. For every simplex, the high-dimensional solutions corresponding to its vertices (sample points) are projected onto the Grassmann manifold. The pairwise distances between these points are calculated using appropriately defined metrics and the elements with large total distance are sub-sampled and refined. As a result, regions of the probability space that produce significant changes in the full-field solution are accurately resolved. An added benefit is that an approximation of the solution within each element can be obtained by interpolation on the Grassmann manifold without the need to develop a mathematical surrogate model. The method is applied to study the probability of shear band formation in a bulk metallic glass using the shear transformation zone theory.