Model reduction for multi-scale transport problems using model-form preserving least-squares projections with variable transformation

Huang, Cheng; Wentland, Christopher R.; Duraisamy, Karthik; Merkle, Charles L.

doi:10.1016/j.jcp.2021.110742

Cited by 28 publications

(39 citation statements)

References 89 publications

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“…We additionally note that Refs. [23] and [5] provide a similar scaling for POD. While this scaling process results in dimensionallyconsistent ROMs, their resulting performance will be, of course, tied to the choice of scaling; such a drawback was pointed out in Rowley et al [42].…”

Section: Introductionmentioning

confidence: 90%

“…Ref. [23], for instance, performs POD on the snapshots where each state variable is scaled by its L 2 (Ω) norm on the training set. Ref.…”

Section: Non-dimensional Vector-valued L 2 Inner Product In Conserved...mentioning

confidence: 99%

“…We emphasize that non-dimensional inner products have been considered in the literature, see, e.g., Refs. [23,5], and typically have resulted in more robust ROMs than those obtained with direct L 2 (Ω) residual minimization. Despite being dimensionally consistent, this inner product is arguably not a "physics-based" norm as it simply adds scaled variables of different units.…”

Section: Non-dimensional Lmentioning

confidence: 99%

“…Projection-based model reduction of fluid flows based on proper orthogonal decomposition (POD) and, e.g., (Petrov-)Galerkin projection has been a research area of great interest for the past several decades. A wealth of literature exists examining applications to incompressible flows [4,9,47,22,29], and more recently POD-based reduced-order models (ROMs) have been examined within the context of compressible and multiphysics compressible flows (e.g., compressible flows closed with the Reynoldsaveraged Navier-Stokes (RANS) equations [13], reacting flows [23]). While these works have demonstrated the potential of POD-based ROMs for more complex configurations, a common trend that has emerged is a lack of robustness.…”

Section: Introductionmentioning

confidence: 99%

“…This has been increasingly realized in recent years. Washabaugh ans Huang et al [23], for example, formulate least-squares ROMs that minimizes a scaled residual. Provided that the scaling is done consistently, this process can result in a ROM formulation that is dimensionally consistent.…”

Section: Introductionmentioning

confidence: 99%

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On the impact of dimensionally-consistent and physics-based inner products for POD-Galerkin and least-squares model reduction of compressible flows

Parish¹,

Rizzi²

2022

Preprint

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Model reduction of the compressible Euler equations based on proper orthogonal decomposition (POD) and Galerkin orthogonality or least-squares residual minimization requires the selection of inner product spaces in which to perform projections and measure norms. The most popular choice is the vectorvalued L 2 (Ω) inner product space. This choice, however, yields dimensionally-inconsistent reduced-order model (ROM) formulations which often lack robustness. In this work, we try to address this weakness by studying a set of dimensionally-consistent inner products with application to the compressible Euler equations. First, we demonstrate that non-dimensional inner products have a positive impact on both POD and Galerkin/least-squares ROMs. Second, we further demonstrate that physics-based inner products based on entropy principles result in drastically more accurate and robust ROM formulations than those based on non-dimensional L 2 (Ω) inner products.As test cases, we consider the following well-known problems: the one-dimensional Sod shock tube, the two-dimensional Kelvin-Helmholtz instability and two-dimensional homogeneous isotropic turbulence.

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Section: Introductionmentioning

confidence: 90%

“…Ref. [23], for instance, performs POD on the snapshots where each state variable is scaled by its L 2 (Ω) norm on the training set. Ref.…”

Section: Non-dimensional Vector-valued L 2 Inner Product In Conserved...mentioning

confidence: 99%

Section: Non-dimensional Lmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On the impact of dimensionally-consistent and physics-based inner products for POD-Galerkin and least-squares model reduction of compressible flows

Parish¹,

Rizzi²

2022

Preprint

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Preconditioned least‐squares Petrov–Galerkin reduced order models

Lindsay

Fike

Tezaur

et al. 2022

Numerical Meth Engineering

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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.

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Reduced Models with Nonlinear Approximations of Latent Dynamics for Model Premixed Flame Problems

Uy,

Wentland,

Huang

et al. 2024

Lecture Notes in Computational Science and Engineering

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Model reduction for multi-scale transport problems using model-form preserving least-squares projections with variable transformation

Cited by 28 publications

References 89 publications

On the impact of dimensionally-consistent and physics-based inner products for POD-Galerkin and least-squares model reduction of compressible flows

On the impact of dimensionally-consistent and physics-based inner products for POD-Galerkin and least-squares model reduction of compressible flows

Preconditioned least‐squares Petrov–Galerkin reduced order models

Reduced Models with Nonlinear Approximations of Latent Dynamics for Model Premixed Flame Problems

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