Fast Low‐Rank Empirical Cross Gramians

Abstract

The cross Gramian matrix encodes the input-output coherence of linear control systems and is used in projection-based model reduction. The empirical cross Gramian is a data-driven variant of the cross Gramian which also extends to nonlinear systems. A drawback of the empirical cross Gramian for large-scale systems is its full order and dense structure; yet, it may be computed column-wise. Using the hierarchical approximate proper orthogonal decomposition (HAPOD), this partial computability can be exploited to obtain a truncated projection for model order reduction without assembling a full cross Gramian. Model ReductionThe simulation of large-scale input-output systems is a computationally challenging task, even though, typically, a lowdimensional input is mapped to a low-dimensional output. Yet, this mapping involves the solution of a, possibly highdimensional, differential equation system. Model order reduction (MOR) accelerates such simulations by computing reduced order models that approximate the full order model's output [3]. In this work we summarize the computation of the low-rank empirical-cross-Gramian-based model reduction technique [4]. Generally, this presented MOR method is applicable to nonlinear systems, but for the sake of exposition, we focus on the reduction of linear time-invariant systems with M := dim(u(t)) inputs, N := dim(x(t)) and Q := dim(y(t)) outputs:The reduced order model has a lower dimension in the differential equation system dim(x r (t)) dim(x(t)) =: N and exhibits a small output error y − y r 1. Cross GramianCross-Gramian-based model reduction is a balancing approach closely related to balanced truncation. Balancing refers to the transformation of the underlying system into a coordinate system in which the system-theoretic attributes controllability and observability are equalized. For square systems, M = Q, a cross Gramian matrix, which jointly encodes these properties, can be computed and a truncated singular value decomposition (tSVD) of this cross Gramian results in an approximate balancing Galerkin projection U 1 that enables the projection-based model reduction of the original system: Empirical Cross GramianThe cross Gramian can be computed in a data-driven manner by the empirical cross Gramian W X , which is equal to W X for linear systems up to numerical error [5]:for a system with M inputs and outputs, state trajectories x m (t) with a perturbed m-th input component, output trajectories y j (t) with a perturbed j-th initial state component and respective temporal averagesx m ,ȳ j . Note that the computation does not require the linear system components A, B, C.

Cross-Gramian-based dominant subspaces

“…3.1.3] is used, as in-memory storage of the Gramian(s) is possible. For settings, where only parts of the cross Gramian can be kept in memory, the low-rank empirical cross Gramian [19] for the left singular vectors, and the low-rank empirical cross Gramian of the adjoint system for the right singular vectors, can be utilized, since the cross Gramian of the adjoint system is equal to the system's transposed cross Gramian:…”

“…For large-scale systems, the computation of dense system Gramians, which are of dimension N × N , may be infeasible or at least inefficient. To this end, lowrank representations of the Gramians can be computed, for the cross Gramian, in example by the implicitly restarted Arnoldi algorithm [40], the factorized iteration [3], a factored ADI [5] or a low-rank empirical cross Gramian [19]. Overall, the cross-Gramian-based dominant subspace algorithm is summarized by:…”

Emgr - Empirical Gramian Framework (5.4)

“…hence this distributed empirical cross Gramian [37,Sec. 4.2] is computable in parallel and communication-free on a distributed memory computer system, or sequentially in a memory-economical manner as a low-rank empirical cross Gramian [38] on a unified memory computer system. This column-wise computability translates also to the empirical joint Gramian and the non-symmetric variants of the empirical cross and joint Gramian.…”

emgr—The Empirical Gramian Framework

“…hence this distributed empirical cross Gramian [72] (Section 4.2) is computable in parallel and communication-free on a distributed memory computer system, or sequentially in a memory-economical manner as a low-rank empirical cross Gramian [73] on a unified memory computer system. This column-wise computability translates also to the empirical joint Gramian and the non-symmetric variants of the empirical cross and joint Gramian.…”

“…Together with a partitioned singular value decomposition, such as the HAPOD [72], an empirical-cross-Gramian-based Galerkin projection is computable with minimal communication in parallel on a distributed memory system, or sequentially on a shared memory system [73].…”