# A Geometric Approach to Dynamical Model-Order Reduction

## TL;DR

This paper introduces a geometric framework for model-order reduction of stochastic PDEs, analyzing the manifold of fixed rank matrices and deriving dynamical systems that optimize low-rank approximations.

## Contribution

It provides a detailed geometric analysis of the fixed rank matrix manifold and develops explicit dynamical systems for low-rank approximation and model reduction.

## Key findings

- The curvature of the fixed rank matrix manifold is characterized.
- Error bounds for the DO approximation are established based on singular value gaps.
- Algorithms for adaptive low-rank matrix approximation are proposed.

## Abstract

Any model order reduced dynamical system that evolves a modal decomposition to approximate the discretized solution of a stochastic PDE can be related to a vector field tangent to the manifold of fixed rank matrices. The Dynamically Orthogonal (DO) approximation is the canonical reduced order model for which the corresponding vector field is the orthogonal projection of the original system dynamics onto the tangent spaces of this manifold. The embedded geometry of the fixed rank matrix manifold is thoroughly analyzed. The curvature of the manifold is characterized and related to the smallest singular value through the study of the Weingarten map. Differentiability results for the orthogonal projection onto embedded manifolds are reviewed and used to derive an explicit dynamical system for tracking the truncated Singular Value Decomposition (SVD) of a time-dependent matrix. It is demonstrated that the error made by the DO approximation remains controlled under the minimal condition that the original solution stays close to the low rank manifold, which translates into an explicit dependence of this error on the gap between singular values. The DO approximation is also justified as the dynamical system that applies instantaneously the SVD truncation to optimally constrain the rank of the reduced solution. Riemannian matrix optimization is investigated in this extrinsic framework to provide algorithms that adaptively update the best low rank approximation of a smoothly varying matrix. The related gradient flow provides a dynamical system that converges to the truncated SVD of an input matrix for almost every initial data.