Preserving general physical properties in model reduction of dynamical systems via constrained‐optimization projection

Schein, Alexander; Carlberg, Kevin; Zahr, Matthew J.

doi:10.1002/nme.6667

Cited by 11 publications

(7 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…General constrained-optimization formulations for projection-based model reduction were proposed in Schein, Carlberg and Zahr (2021) to enforce the resulting reduced models to satisfy specific physical properties such as conservation of invariants.…”

Section: Methods Based On Constraintsmentioning

confidence: 99%

Reduced basis methods for time-dependent problems

Hesthaven¹,

Pagliantini²,

Rozza³

2022

Acta Numerica

View full text Add to dashboard Cite

Numerical simulation of parametrized differential equations is of crucial importance in the study of real-world phenomena in applied science and engineering. Computational methods for real-time and many-query simulation of such problems often require prohibitively high computational costs to achieve sufficiently accurate numerical solutions. During the last few decades, model order reduction has proved successful in providing low-complexity high-fidelity surrogate models that allow rapid and accurate simulations under parameter variation, thus enabling the numerical simulation of increasingly complex problems. However, many challenges remain to secure the robustness and efficiency needed for the numerical simulation of nonlinear time-dependent problems. The purpose of this article is to survey the state of the art of reduced basis methods for time-dependent problems and draw together recent advances in three main directions. First, we discuss structure-preserving reduced order models designed to retain key physical properties of the continuous problem. Second, we survey localized and adaptive methods based on nonlinear approximations of the solution space. Finally, we consider data-driven techniques based on non-intrusive reduced order models in which an approximation of the map between parameter space and coefficients of the reduced basis is learned. Within each class of methods, we describe different approaches and provide a comparative discussion that lends insights to advantages, disadvantages and potential open questions.

show abstract

Section: Methods Based On Constraintsmentioning

confidence: 99%

Reduced basis methods for time-dependent problems

Hesthaven¹,

Pagliantini²,

Rozza³

2022

Acta Numerica

View full text Add to dashboard Cite

show abstract

“…is the left Cauchy-Green deformation, tensor, where F M denotes the mechanical part of the deformation gradient. For a pure mechanics problem, F M = F. The mechanical equations (39) are solved for the displacements, the primary unknowns in these equations.…”

Section: Mechanical Formulationmentioning

confidence: 99%

“…11,22,25,[32][33][34] We note that the LSPG method has been extended in the time domain, 35,36 to allow for nonlinear trial manifolds rather than linear trial subspaces, 37 to operate in a domain-decomposition setting, 38 and to account for physics-based constraints. 12,14,39 The aim of the present article is to develop a methodology for improving the accuracy of ROMs constructed using the LSPG projection method for a wide range of applications through the introduction of preconditioning. As shown in Reference 11, LSPG errors are subject to a stability constant that is dictated by the residual.…”

Section: Introductionmentioning

confidence: 99%

Preconditioned least‐squares Petrov–Galerkin reduced order models

Lindsay

Fike

Tezaur

et al. 2022

Numerical Meth Engineering

View full text Add to dashboard Cite

In this article, we introduce a methodology for improving the accuracy and efficiency of reduced order models (ROMs) constructed using the least‐squares Petrov–Galerkin (LSPG) projection method through the introduction of preconditioning. Unlike prior related work, which focuses on preconditioning the linear systems arising within the ROM numerical solution procedure to improve linear solver performance, our approach leverages a preconditioning matrix directly within the minimization problem underlying the LSPG formulation. Applying preconditioning in this way has the potential to improve ROM accuracy for several reasons. First, preconditioning the LSPG formulation changes the norm defining the residual minimization, which can improve the residual‐based stability constant bounding the ROM solution's error. The incorporation of a preconditioner into the LSPG formulation can have the additional effect of scaling the components of the residual being minimized to make them roughly of the same magnitude, which can be beneficial when applying the LSPG method to problems with disparate scales (e.g., dimensional equations, multi‐physics problems). Importantly, we demonstrate that an “ideal preconditioned” LSPG ROM (a ROM in which the preconditioner is the inverse of the Jacobian of its corresponding full order model) emulates projection of the full order model solution increment onto the reduced basis. This quantity defines a lower bound on the error of a ROM solution for a given reduced basis. By designing preconditioners that approximate the Jacobian inverse—as is common in designing preconditioners for solving linear systems—it is possible to obtain a ROM whose error approaches this lower bound. The proposed approach is evaluated on several mechanical and thermo‐mechanical problems implemented within the Albany HPC code and run in the predictive regime, with prediction across material parameter space. We demonstrate numerically that the introduction of simple Jacobi, Gauss‐Seidel, and ILU preconditioners into the proper orthogonal decomposition/LSPG formulation reduces significantly the ROM solution error, the reduced Jacobian condition number, the number of nonlinear iterations required to reach convergence, and the wall time (thereby improving efficiency). Moreover, our numerical results reveal that the introduction of preconditioning can deliver a robust and accurate solution for test cases in which the unpreconditioned LSPG method fails to converge.

show abstract

“…into both model reduction [48,90] and system identification [19,49,91,92]. Precisely in this way, the energy-preserving constraint in Eq.…”

Section: Constraints In Model Reduction and System Identificationmentioning

confidence: 99%

Promoting global stability in data-driven models of quadratic nonlinear dynamics

Kaptanoglu,

Callaham,

Hansen

et al. 2021

Preprint

View full text Add to dashboard Cite

Modeling realistic fluid and plasma flows is computationally intensive, motivating the use of reducedorder models for a variety of scientific and engineering tasks. However, it is challenging to characterize, much less guarantee, the global stability (i.e., long-time boundedness) of these models. The seminal work of Schlegel and Noack [1] provided a theorem outlining necessary and sufficient conditions to ensure global stability in systems with energy-preserving, quadratic nonlinearities, with the goal of evaluating the stability of projection-based models. In this work, we incorporate this theorem into modern data-driven models obtained via machine learning. First, we propose that this theorem should be a standard diagnostic for the stability of projection-based and data-driven models, examining the conditions under which it holds. Second, we illustrate how to modify the objective function in machine learning algorithms to promote globally stable models, with implications for the modeling of fluid and plasma flows. Specifically, we introduce a modified "trapping SINDy" algorithm based on the sparse identification of nonlinear dynamics (SINDy) method. This method enables the identification of models that, by construction, only produce bounded trajectories. The effectiveness and accuracy of this approach are demonstrated on a broad set of examples of varying model complexity and physical origin, including the vortex shedding in the wake of a circular cylinder.

show abstract

Preserving general physical properties in model reduction of dynamical systems via constrained‐optimization projection

Cited by 11 publications

References 32 publications

Reduced basis methods for time-dependent problems

Reduced basis methods for time-dependent problems

Preconditioned least‐squares Petrov–Galerkin reduced order models

Promoting global stability in data-driven models of quadratic nonlinear dynamics

Contact Info

Product

Resources

About