Approximate optimal projection for reduced-order models

Computer Methods in Applied Mechanics and Engineering, 2013

In this work, a domain decomposition strategy for non-linear hyper-reduced-order models is presented. The basic idea consists of restricting the reduced-order basis functions to the nodes belonging to each of the subdomains into which the physical domain is partitioned. An extension of the proposed domain decomposition strategy to a hybrid full-order / reduced-order model is then described. The general domain decomposition approach is particularized for the reduced-order finite element approximation of the incompressible Navier-Stokes equations with hyper-reduction. When solving the reduced incompressible Navier-Stokes equations, instabilities in the form of large gradients of the recovered reduced-order unknown at the subdomain interfaces may appear, which is the motivation for the design of additional stability terms giving rise to penalty matrices. Numerical examples illustrate the behavior of the proposed method for the simulation of the reduced-order systems, showing the capability of the approach to adapt to configurations which are not present in the original snapshot set.

The Galerkin projection method based on modes generated by the Proper Orthogonal Decomposition (POD) technique is very popular for the dimensional reduction of linearized Computational Fluid Dynamics (CFD) models, among many other typically high-dimensional models in computational engineering. This, despite the fact that it cannot guarantee neither the optimality nor the stability of the Reduced-Order Models (ROMs) it constructs. Short of proposing any variant of this model order reduction method that guarantees the stability of its outcome, this paper contributes a best practice to its application to the construction of linearized CFD ROMs. It begins by establishing that whereas the solution snapshots computed using the descriptor and non-descriptor forms of the discretized Euler or Navier-Stokes equations are identical, the ROMs obtained by reducing these two alternative forms of the governing equations of interest are different. Focusing next on compressible fluid-structure interaction problems associated with computational aeroelasticity, this paper shows numerically that the POD-based fluid ROMs originating from the non-descriptor form of the governing linearized CFD equations tend to be unstable, but their counterparts originating from the descriptor form of these equations are typically stable and reliable for aeroelastic applications. Therefore, this paper argues that whereas many computations are performed in CFD codes using the non-descriptor form of discretized Euler and/or Navier-Stokes equations, POD-based model reduction in these codes should be performed using the descriptor form of these equations.

In this paper we consider (hierarchical, Lagrange) reduced basis approximation and a posteriori error estimation for linear functional outputs of affinely parametrized elliptic coercive partial differential equations. The essential ingredients are (primal-dual) Galerkin projection onto a low-dimensional space associated with a smooth "parametric manifold"-dimension reduction; efficient and effective greedy sampling methods for identification of optimal and numerically stable approximations-rapid convergence; a posteriori error estimation procedures-rigorous and sharp bounds for the linear-functional outputs of interest; and Offline-Online computational decomposition strategies-minimum marginal cost for high performance in the real-time/embedded (e.g., parameter-estimation, con

2003

Modeling issues of infinite dimensional system is studied in this paper. Although the modeling problem has been solved to some extent, use of decomposition techniques still poses several difficulties. A prime one of this is the amount of data to be processed. Method of snapshots integrated with POD is a remedy. The second difficulty is the fact that the decomposition followed by a projection yields an autonomous set of finite dimensional ODEs that is not useful for developing a concise understanding of the input operator of the system. A numerical approach to handle this issue is presented in this paper. As the example, we study 2D heat flow problem. The results obtained confirm the theoretical claims of the paper and emphasize that the technique presented here is not only applicable to infinite dimensional linear systems but also to nonlinear ones.

Applied Numerical Mathematics, 2005

We present an application in multi-parametrized sub-domains based on a technique for the rapid and reliable prediction of linear-functional outputs of elliptic coercive partial differential equations with affine parameter dependence (reduced-basis methods). The main components are (i) rapidly convergent global reduced-basis approximations-Galerkin projection onto a space W N spanned by solutions of the governing equation at N selected points in parameter space (chosen by an adaptive procedure to minimize the estimated error and the effectivity; (ii) a posteriori error estimation-relaxations of the error-residual equation that provide inexpensive bounds for the error in the outputs of interest; and (iii) off-line/on-line computational procedures-methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage-in which, given a new parameter value, we calculate the output of interest and associated error bounddepends only on N (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control. The application is based on a heat transfer problem in a parametrized geometry in view of haemodynamics applications and biomechanical devices optimization, such as the bypass configuration problem.

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.

2020

The energy-conserving sampling and weighting (ECSW) method is a hyperreduction method originally developed for accelerating the performance of Galerkin projection-based reduced-order models (PROMs) associated with large-scale finite element models, when the underlying projected operators need to be frequently recomputed as in parametric and/or nonlinear problems. In this paper, this hyperreduction method is extended to Petrov-Galerkin PROMs where the underlying high-dimensional models can be associated with arbitrary finite element, finite volume, and finite difference semi-discretization methods. Its scope is also extended to cover local PROMs based on piecewise-affine approximation subspaces, such as those designed for mitigating the Kolmogorov $n$-width barrier issue associated with convection-dominated flow problems. The resulting ECSW method is shown in this paper to be robust and accurate. In particular, its offline phase is shown to be fast and parallelizable, and the potenti...

International Journal for Numerical Methods in Fluids, 2013

In this paper we present an explicit formulation for reduced order models of the stabilized finite element approximation of the incompressible Navier-Stokes equations. The basic idea is to build a reduced order model based on a proper orthogonal decomposition and a Galerkin projection and treat all the terms in an explicit way in the time integration scheme, including the pressure. This is possible because the reduced model snapshots do already fulfill the continuity equation. The pressure field is automatically recovered from the reduced order basis and solution coefficients. The main advantage of this explicit treatment of the incompressible Navier-Stokes equations is that it allows for the easy use of hyper-reduced order models, since only the right-hand side vector needs to be recovered by means of a gappy data reconstruction procedure. A method for choosing the optimal set of sampling points at the discrete level in the gappy procedure is also presented. Numerical examples show the performance of the proposed strategy.

ArXiv, 2019

We provide first the functional analysis background required for reduced order modeling and present the underlying concepts of reduced basis model reduction. The projection-based model reduction framework under affinity assumptions, offline-online decomposition and error estimation is introduced. Several tools for geometry parametrizations, such as free form deformation, radial basis function interpolation and inverse distance weighting interpolation are explained. The empirical interpolation method is introduced as a general tool to deal with non-affine parameter dependency and non-linear problems. The discrete and matrix versions of the empirical interpolation are considered as well. Active subspaces properties are discussed to reduce high-dimensional parameter spaces as a pre-processing step. Several examples illustrate the methodologies.

Applied Mathematics and Computation, 2019

Minimization of energy in gradient systems leads to formation of oscillatory and Turing patterns in reaction-diffusion systems. These patterns should be accurately computed using fine space and time meshes over long time horizons to reach the spatially inhomogeneous steady state. In this paper, a reduced order model (ROM) is developed which preserves the gradient dissipative structure. The coupled system of reaction-diffusion equations are discretized in space by the symmetric interior penalty discontinuous Galerkin (SIPG) method. The resulting system of ordinary differential equations (ODEs) are integrated in time by the average vector field (AVF) method, which preserves the energy dissipation of the gradient systems. The ROMs are constructed by the proper orthogonal decomposition (POD) with Galerkin projection. The nonlinear reaction terms are computed efficiently by discrete empirical interpolation method (DEIM). Preservation of the discrete energy of the FOMs and ROMs with POD-DEIM ensures the long term stability of the steady state solutions. Numerical simulations are performed for the gradient dissipative systems with two specific equations; real Ginzburg-Landau equation and Swift-Hohenberg equation. Numerical results demonstrate that the POD-DEIM reduced order solutions preserve well the energy dissipation over time and at the steady state.