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

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.

References in corpus (5)

• The generalization of Latin hypercube sampling
• Shear-transformation-zone theory of plastic deformation near the glass transition
• Strain localization in a shear transformation zone model for amorphous solids
• Rate dependent shear bands in a shear transformation zone model of amorphous solids
• Stochastic collocation approach with adaptive mesh refinement for parametric uncertainty analysis

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• A physics and data co-driven surrogate modeling method for high-dimensional rare event simulation
• Data-driven Uncertainty Quantification in Computational Human Head Models
• Dimensionality reduction can be used as a surrogate model for high-dimensional forward uncertainty quantification
• Grassmannian diffusion maps based surrogate modeling via geometric harmonics
• Polynomial Chaos Expansions on Principal Geodesic Grassmannian Submanifolds for Surrogate Modeling and Uncertainty Quantification