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

Abstract

SUMMARYA 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.

Low-dimensional modelling of high-Reynolds-number shear flows incorporating constraints from the Navier–Stokes equation

“…This is particularly true in the proposed approach since the tensor,Q must be evaluated at each iteration. Over the years, various approximations to tensor products have been derived (Carlberg et al 2011;Wang & Ahuja 2008). In this paper, an alternative method to alleviate tensor computations is presented.…”

Group-wise construction of reduced models for understanding and characterization of pulmonary blood flows from medical images

“…It is important to remind that, even when the number of degrees of freedom is drastically reduced (from about 10 5 to a few dozen in our case), the nonlinearities may compromise the reduction of computational time if they are not correctly handled. A number of works have addressed this issue, see, e.g., (Carlberg et al, 2011;Grepl et al, 2007;Baiges et al, 2012;Wang et al, 2012;Yano et al, 2012;Grinberg et al, 2009;Grinberg, 2012). With the current implementation of our method, we observed a reduction of 20% to 30% of the CPU time with respect to the full order model.…”

A partitioned model order reduction approach to rationalise computational expenses in nonlinear fracture mechanics

“…In a first subset of these strategies, the nonlinear function is reconstructed by interpolation over an other POD basis ("gappy" technique) [5,34,51,36]. The expansion of the nonlinear term reads: Kerfriden …”