POD/DEIM reduced-order strategies for efficient four dimensional variational data assimilation

Ştefanescu, Răzvan; Sandu, Adrian; Navon, I. M.

doi:10.1016/j.jcp.2015.04.030

Cited by 87 publications

(67 citation statements)

References 88 publications

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“…Higher‐order integrators and automatic tuning of parameters should be considered when theses algorithms are applied to more complicated, for example, when

scriptℋ

is nonlinear or when the Gaussian prior assumption is relaxed. The reduced basis V is constructed using initial trajectories of the high‐fidelity forward and adjoint models as well as the associated gradient of the full cost function to allow for consistent reduced Karush–Kuhn–Tucker system. Later on, this basis is updated using the current proposal and the corresponding trajectories.…”

Section: Numerical Resultsmentioning

confidence: 99%

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The reduced‐order hybrid Monte Carlo sampling smoother

Attia

Ştefanescu

Sandu

2016

Numerical Methods in Fluids

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SUMMARYHybrid Monte Carlo sampling smoother is a fully non-Gaussian four-dimensional data assimilation algorithm that works by directly sampling the posterior distribution formulated in the Bayesian framework. The smoother in its original formulation is computationally expensive owing to the intrinsic requirement of running the forward and adjoint models repeatedly. Here we present computationally efficient versions of the hybrid Monte Carlo sampling smoother based on reduced-order approximations of the underlying model dynamics. The schemes developed herein are tested numerically using the shallow-water equations model on Cartesian coordinates. The results reveal that the reduced-order versions of the smoother are capable of accurately capturing the posterior probability density, while being significantly faster than the original full-order formulation.

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“…Higher‐order integrators and automatic tuning of parameters should be considered when theses algorithms are applied to more complicated, for example, when

scriptℋ

Section: Numerical Resultsmentioning

confidence: 99%

“…The associated reduced Karush–Kuhn–Tucker conditions are:

\begin{matrix} ARRA reduced forward model: \\ {\tilde{x}}_{k + 1} = {\tilde{ℳ}}_{t_{k} \to t_{k + 1}} ({\tilde{x}}_{k}), k = 0, \dots, N_{NOBS}, \end{matrix}

\begin{matrix} ARRA reduced adjoint model : \\ {\tilde{λ}}_{NOBS} = V^{T} {\hat{H}}_{NOBS}^{T} R_{NOBS}^{- 1} (y_{NOBS} - H_{NOBS} (V {\tilde{x}}_{NOBS})), \\ {\tilde{λ}}_{k} = V^{T} {\hat{M}}_{t_{k} \to t_{k + 1}}^{T} V {\tilde{λ}}_{k + 1} + V^{T} {\hat{H}}_{k}^{T} R_{k}^{- 1} (y_{k} - H_{k} (V x) \end{matrix}

…”

Section: Four‐dimensional Variational Data Assimilation With Reduced‐mentioning

confidence: 99%

The reduced‐order hybrid Monte Carlo sampling smoother

Attia

Ştefanescu

Sandu

2016

Numerical Methods in Fluids

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“…Reduced order models (ROMs) are popular and powerful techniques for circumventing the intensive computational burden in large complex numerical simulations in engineering and science, for example, ocean modelling, weather prediction, uncertainty quantification, sensitive analysis, data assimilation, sensor placement optimization, porous media, structural problem, convection diffusion reaction equations, molecular dynamics simulation and optimal control [1,15,22,24,42,45,30,16,10,26,46,11,9,2,17,21]. The basic idea of reduced order modelling is to find an approximate solution by a linear combination of a set of basis functions.…”

Section: Introductionmentioning

confidence: 99%

A parameterized non-intrusive reduced order model and error analysis for general time-dependent nonlinear partial differential equations and its applications

Xiao¹,

Fang²,

Pain³

et al. 2017

Computer Methods in Applied Mechanics and Engineering

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A novel parameterized non-intrusive reduced order model (P-NIROM) based on proper orthogonal decomposition (POD) has been developed. This P-NIROM is a generic and efficient approach for model reduction of parameterized partial differential equations (P-PDEs). Over existing parameterized reduced order models (P-ROM) (most of them are based on the reduced basis method), it is non-intrusive and independent on partial differential equations and computational codes. During the training process, the Smolyak sparse grid method is used to select a set of parameters over a specific parameterized space (Ω p ∈ R P ). For each selected parameter, the reduced basis functions are generated from the snapshots derived from a run of the high fidelity model. More generally, the snapshots and basis function sets for any parameters over Ω p can be obtained using an interpolation method. The P-NIROM can then be constructed by using our recently developed technique [50,53] where either the Smolyak or radial basis function (RBF) methods are used to generate a set of hyper-surfaces representing the underlying dynamical system over the reduced space.The new P-NIROM technique has been applied to parameterized Navier-Stokes equations and implemented with an unstructured mesh finite element model. The capability of this P-NIROM has been illustrated numerically by two test cases: flow past a cylinder and lock exchange case. The prediction capabilities of the P-NIROM have been evaluated by varying the viscosity, initial and boundary conditions. The results show that this P-NIROM has captured the quasi-totality of the details of the flow with CPU speedup of three orders of magnitude. An error analysis for the P-NIROM has been carried out.

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“…ROMs have been successfully used in numerous applications, such as fluid flow optimization and control, cardiovascular flows, or geophysical flows (see, e.g., the surveys in [1][2][3][4][5][6]). The proper orthogonal decomposition (POD) is one of the most successful methods for ROM development.…”

Section: Introductionmentioning

confidence: 99%

Approximate deconvolution reduced order modeling

Xie

Wells

Wang

et al. 2017

Computer Methods in Applied Mechanics and Engineering

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Highlights• A new large eddy simulation reduced order model (LES-ROM) framework.• A new approximate deconvolution ROM (AD-ROM) not of eddy viscosity type.• Successful application of AD-ROM to 3D flow past cylinder at Re = 1000. AbstractThis paper proposes a large eddy simulation reduced order model (LES-ROM) framework for the numerical simulation of convection-dominated flows. In this LES-ROM framework, the proper orthogonal decomposition (POD) is used to define the ROM basis and a ROM differential filter is used to define the large ROM structures. An approximate deconvolution (AD) approach is used to solve the ROM closure problem and develop a new AD-ROM. This AD-ROM is tested in the numerical simulation of the three-dimensional flow past a circular cylinder at a Reynolds number Re = 1000.

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POD/DEIM reduced-order strategies for efficient four dimensional variational data assimilation

Cited by 87 publications

References 88 publications

The reduced‐order hybrid Monte Carlo sampling smoother

The reduced‐order hybrid Monte Carlo sampling smoother

A parameterized non-intrusive reduced order model and error analysis for general time-dependent nonlinear partial differential equations and its applications

Approximate deconvolution reduced order modeling

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