Efficient non‐linear model reduction via a least‐squares Petrov–Galerkin projection and compressive tensor approximations

TLDR

The authors propose a Petrov–Galerkin projection method to reduce the dimensionality of discrete nonlinear static or dynamic models for real‑time processing. They construct an invariant right basis via proper orthogonal decomposition and an iteration‑dependent left basis that minimizes the residual norm at each Newton step, approximating the reduced operators as a product of a large invariant matrix and a smaller iteration‑dependent matrix to enable least‑squares reconstruction and a globally convergent Gauss–Newton scheme. Tests on turbulent flow and nonlinear structural dynamics problems show that the method satisfies the proposed consistency requirements and can dramatically lower computational cost while preserving accuracy. © 2010 John Wiley & Sons, Ltd.

Abstract

Abstract A Petrov–Galerkin projection method is proposed for reducing the dimension of a discrete non‐linear static or dynamic computational model in view of enabling its processing in real time. The right reduced‐order basis is chosen to be invariant and is constructed using the Proper Orthogonal Decomposition method. The left reduced‐order basis is selected to minimize the two‐norm of the residual arising at each Newton iteration. Thus, this basis is iteration‐dependent, enables capturing of non‐linearities, and leads to the globally convergent Gauss–Newton method. To avoid the significant computational cost of assembling the reduced‐order operators, the residual and action of the Jacobian on the right reduced‐order basis are each approximated by the product of an invariant, large‐scale matrix, and an iteration‐dependent, smaller one. The invariant matrix is computed using a data compression procedure that meets proposed consistency requirements. The iteration‐dependent matrix is computed to enable the least‐squares reconstruction of some entries of the approximated quantities. The results obtained for the solution of a turbulent flow problem and several non‐linear structural dynamics problems highlight the merit of the proposed consistency requirements. They also demonstrate the potential of this method to significantly reduce the computational cost associated with high‐dimensional non‐linear models while retaining their accuracy. Copyright © 2010 John Wiley & Sons, Ltd.