Structure-Aware Tensorial Model Reduction

TL;DR

This paper introduces a two-stage tensorial model reduction method for parameterized PDEs, enhancing efficiency and accuracy in nonlinear and data-limited scenarios through novel basis construction and interpolation techniques.

Contribution

It proposes a new tensorial ROM approach with extensions for structured PDEs and sparse sampling, improving on existing methods in nonlinear and practical applications.

Findings

Effective reduction of nonlinear PDE models demonstrated on Maxwell's equations in 3D.

The nonlinear basis ROM mitigates linear restrictions on Kolmogorov n-width.

Error estimates validate the proposed modifications.

Abstract

This work investigates a two-stage method for constructing projection-based reduced-order models (ROMs) of parameterized partial differential equations (PDEs). Based on established tensorial ROM methodology, the proposed approach reduces dimensionality offline by encoding solution snapshots using a multi-linear Tucker factorization, so that a reduced basis which varies nonlinearly with PDE parameters can be rapidly constructed online and used in a Galerkin ROM. Two novel extensions of this strategy, tailored to the cases of structured PDEs and sparse parameter sampling, are presented: the construction of reduced bases orthonormalized with respect to a general discrete inner product, and the interpolation of encoded states via radial basis functions. Basic representation and ROM error estimates are presented demonstrating the validity of these modifications, and the approach is challenged on examples where monolithic-basis ROMs are known to struggle, including a realistic instance of Maxwell's equations in 3D. Results suggest that the proposed nonlinear basis ROM can effectively mitigate linear restrictions on Kolmogorov $n$-width while improving upon previous tensorial ROM technology, particularly in the highly nonlinear and data-limited regimes characteristic of practical use cases.