Higher Order Mori–Zwanzig Models for the Euler Equations

Stinis, Panos

doi:10.1137/06066504x

Cited by 38 publications

(47 citation statements)

References 28 publications

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“…However, the construction of reduced models based on the original MZ formalism can be quite expensive to obtain and implement. In addition, the reduced models may be unstable [17]. The generalized MZ models which were briefly presented in this article can alleviate these difficulties.…”

Section: Resultsmentioning

confidence: 99%

“…This assumption may be true for small t but it does not have to hold for larger t. In other words, the Taylor expansion of the operator e −tPL has, in general, only a finite radius of convergence. Insisting on using the Taylor expansion of the operator e −tPL as is for later times is dangerous and can lead to the instability of the reduced model [17]. In fact, the breakdown of the Taylor expansion of the operator e −tPL is related to the onset of underresolution on the part of the full system.…”

Section: The Mori-zwanzig Formalismmentioning

confidence: 99%

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Renormalized Mori–Zwanzig-reduced models for systems without scale separation

Stinis

2015

Proc. R. Soc. A.

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Model reduction for complex systems is a rather active area of research. For many real-world systems, constructing an accurate reduced model is prohibitively expensive. The main difficulty stems from the tremendous range of spatial and temporal scales present in the solution of such systems. This leads to the need to develop reduced models where, inevitably, the resolved variables do not exhibit (spatial and/or temporal) scale separation from the unresolved ones. We present a brief survey of recent results on the construction of Mori-Zwanzigreduced models for such systems. The construction is inspired by the concepts of scale dependence and renormalization which first appeared in the context of high-energy and statistical physics.

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Section: Resultsmentioning

confidence: 99%

Section: The Mori-zwanzig Formalismmentioning

confidence: 99%

Renormalized Mori–Zwanzig-reduced models for systems without scale separation

Stinis

2015

Proc. R. Soc. A.

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“…Applications of the Mori-Zwanzig approach to fluid dynamics can be found in Stinis [2007], Chandy and Frankel [2009], Hald and Stinis [2007], and Hou [2007]. A simple approximation to Mori-Zwanzig has been applied to jet formation on a beta plane in Tobias and Marston [2013].…”

Section: Projection Operator Techniquesmentioning

confidence: 99%

Mathematical and physical ideas for climate science

et al. 2014

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The climate is a forced and dissipative nonlinear system featuring nontrivial dynamics on a vast range of spatial and temporal scales. The understanding of the climate's structural and multiscale properties is crucial for the provision of a unifying picture of its dynamics and for the implementation of accurate and efficient numerical models. We present some recent developments at the intersection between climate science, mathematics, and physics, which may prove fruitful in the direction of constructing a more comprehensive account of climate dynamics. We describe the Nambu formulation of fluid dynamics and the potential of such a theory for constructing sophisticated numerical models of geophysical fluids. Then, we focus on the statistical mechanics of quasi-equilibrium flows in a rotating environment, which seems crucial for constructing a robust theory of geophysical turbulence. We then discuss ideas and methods suited for approaching directly the nonequilibrium nature of the climate system. First, we describe some recent findings on the thermodynamics of climate, characterize its energy and entropy budgets, and discuss related methods for intercomparing climate models and for studying tipping points. These ideas can also create a common ground between geophysics and astrophysics by suggesting general tools for studying exoplanetary atmospheres. We conclude by focusing on nonequilibrium statistical mechanics, which allows for a unified framing of problems as different as the climate response to forcings, the effect of altering the boundary conditions or the coupling between geophysical flows, and the derivation of parametrizations for numerical models.

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“…A range of research examining the MZ formalism as a multiscale modeling tool exists in the community. Most notably, Stinis and co-workers [44,45,46,47,48,49] have developed several models for approximating the memory, including finite memory and renormalized models, and examined their application to the semi-discrete systems emerging from Fourier-Galerkin and Polynomial Chaos Expansions of Burgers' equation and the Euler equations. Application of MZ-based techniques to the classic POD-ROM approach has not been undertaken.This manuscript leverages work that the authors have performed on the use of the MZ formalism to develop closure models of partial differential equations [50,51,52,53,54].…”

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confidence: 99%

The Adjoint Petrov–Galerkin method for non-linear model reduction

Parish

Wentland

Duraisamy

2020

Computer Methods in Applied Mechanics and Engineering

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We formulate a new projection-based reduced-ordered modeling technique for non-linear dynamical systems. The proposed technique, which we refer to as the Adjoint Petrov-Galerkin (APG) method, is derived by decomposing the generalized coordinates of a dynamical system into a resolved coarse-scale set and an unresolved fine-scale set. A Markovian finite memory assumption within the Mori-Zwanzig formalism is then used to develop a reduced-order representation of the coarse-scales. This procedure leads to a closed reduced-order model that displays commonalities with the adjoint stabilization method used in finite elements. The formulation is shown to be equivalent to a Petrov-Galerkin method with a non-linear, time-varying test basis, thus sharing some similarities with the Least-Squares Petrov-Galerkin method. Theoretical analysis examining a priori error bounds and computational cost is presented. Numerical experiments on the compressible Navier-Stokes equations demonstrate that the proposed method can lead to improvements in numerical accuracy, robustness, and computational efficiency over the Galerkin method on problems of practical interest. Improvements in numerical accuracy and computational efficiency over the Least-Squares Petrov-Galerkin method are observed in most cases.(G ROM) has been used successfully in a variety of problems. When applied to general non-self-adjoint and non-linear problems, however, theoretical analysis and numerical experiments have shown that Galerkin ROM lacks a priori guarantees of stability, accuracy, and convergence [9]. This last issue is particularly challenging as it demonstrates that enriching a ROM basis does not necessarily improve the solution [10]. The development of stable and accurate reduced-order modeling techniques for complex non-linear systems is the motivation for the current work.A significant body of research aimed at producing accurate and stable ROMs for complex non-linear problems exists in the literature. These efforts include, but are not limited to, "energy-based" inner products [9,11], symmetry transformations [12], basis adaptation [13,14], L 1 -norm minimization [15], projection subspace rotations [16], and least-squares residual minimization approaches [17,18,19,20,21,22,23,24]. The Least-Squares Petrov-Galerkin (LSPG) [22] method comprises a particularly popular leastsquares residual minimization approach and has been proven to be an effective tool for non-linear model reduction. Defined at the fully-discrete level (i.e., after spatial and temporal discretization), LSPG relies on least-squares minimization of the FOM residual at each time-step. While the method lacks a priori stability guarantees for general non-linear systems, it has been shown to be effective for complex problems of interest [24,23,25]. Additionally, as it is formulated as a minimization problem, physical constraints such as conservation can be naturally incorporated into the ROM formulation [26]. At the fully-discrete level, LSPG is sensitive to both the time integration scheme as ...

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Higher Order Mori–Zwanzig Models for the Euler Equations

Cited by 38 publications

References 28 publications

Renormalized Mori–Zwanzig-reduced models for systems without scale separation

Renormalized Mori–Zwanzig-reduced models for systems without scale separation

Mathematical and physical ideas for climate science

The Adjoint Petrov–Galerkin method for non-linear model reduction

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