A Spatial Clustering Algorithm for Constructing Local Reduced-order Bases for Nonlinear Model Reduction

Cristina White, Stanford University

Projection-based model order reduction (PMOR) techniques rely on the precomputation of an approximation subspace that, despite having a dimension much smaller than that of its underlying high-dimensional model, exhibits the ability to capture its dominant features. Compressing a matrix of solution snapshots using a singular value decomposition (SVD) is a widely used method for constructing such a subspace. However, for highly nonlinear problems characterized by distinct physical features and/or scales, a global application of this method cannot be expected to produce a reduced-order basis (ROB) with the smallest possible dimension. Therefore, first an existing approach is used to address this issue, called local ROBs. There, the solution space is first partitioned into subregions using a clustering algorithm applied to the columns of the snapshot matrix. Local ROBs are then constructed and assigned to the various subregions. Although this approach has demonstrated a significant potential for achieving large speedups while maintaining good accuracy, it has room for improvement, particularly for problems characterized by spatially varying features and/or scales. To this end, a complementary method for constructing local ROBs is presented here. The solution space is partitioned into subregions by clustering not only the columns of the snapshot matrix, but also the rows of each column-wise snapshot cluster. Local ROBs are then constructed and assigned to the various subregions. Row-wise clustering allows PMOR to leverage information regarding spatial structures for parallel computation that may be encoded in the snapshot matrix. The proposed method is amenable to existing hyper-reduction techniques.

Abstract Author(s): Tina White, Phil Avery, Charbel Farhat