An Algorithm for Fast Calculation of Flow Ensembles

Jiang, Nan; Layton, William

doi:10.1615/int.j.uncertaintyquantification.2014007691

Cited by 80 publications

(91 citation statements)

References 24 publications

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“…In such applications, spatial resolution is often sacrificed to reduce the total computational time. For these reasons the development of efficient methods that allow for fast calculation of flow ensembles at a sufficiently fine spatial resolution is of great practical interest and significance.Only recently, a first step was taken in [22,23] where a new algorithm was proposed for computing an ensemble of solutions of the time-dependent Navier-Stokes equations (NSE) with different initial condition and/or body forces. At each time step, the new method employs the same coefficient matrix for all ensemble members.…”

mentioning

confidence: 99%

An Ensemble-Proper Orthogonal Decomposition Method for the Nonstationary Navier--Stokes Equations

Gunzburger¹,

Jiang²,

Schneier³

2017

SIAM J. Numer. Anal.

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Abstract. The definition of partial differential equation (PDE) models usually involves a set of parameters whose values may vary over a wide range. The solution of even a single set of parameter values may be quite expensive. In many cases, e.g., optimization, control, uncertainty quantification, and other settings, solutions are needed for many sets of parameter values. We consider the case of the time-dependent Navier-Stokes equations for which a recently developed ensemble-based method allows for the efficient determination of the multiple solutions corresponding to many parameter sets. The method uses the average of the multiple solutions at any time step to define a linear set of equations that determines the solutions at the next time step. To significantly further reduce the costs of determining multiple solutions of the Navier-Stokes equations, we incorporate a proper orthogonal decomposition (POD) reduced-order model into the ensemble-based method. The stability and convergence results for the ensemble-based method are extended to the ensemble-POD approach. Numerical experiments are provided that illustrate the accuracy and efficiency of computations determined using the new approach.Key words. Ensemble methods, proper orthogonal decomposition, reduced-order models, Navier-Stokes equations.1. Introduction. Computing an ensemble of solutions of fluid flow equations for a set of parameters or initial/boundary conditions for, e.g., quantifying uncertainty or sensitivity analyses or to make predictions, is a common procedure in many engineering and geophysical applications. One common problem faced in these calculations is the excessive cost in terms of both storage and computing time. Thanks to recent rapid advances in parallel computing as well as intensive research in ensemble-based data assimilation, it is now possible, in certain settings, to obtain reliable ensemble predictions using only a small set of realizations. Successful methods that are currently used to generate perturbations in initial conditions include the Bred-vector method, [30], the singular vector method, [5], and the ensemble transform Kalman filter, [4]. Despite all these efforts, the current level of available computing power is still insufficient to perform high-accuracy ensemble computations for applications that deal with large spatial scales such as numerical weather prediction. In such applications, spatial resolution is often sacrificed to reduce the total computational time. For these reasons the development of efficient methods that allow for fast calculation of flow ensembles at a sufficiently fine spatial resolution is of great practical interest and significance.Only recently, a first step was taken in [22,23] where a new algorithm was proposed for computing an ensemble of solutions of the time-dependent Navier-Stokes equations (NSE) with different initial condition and/or body forces. At each time step, the new method employs the same coefficient matrix for all ensemble members. This reduces the problem of solving multi...

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confidence: 99%

An Ensemble-Proper Orthogonal Decomposition Method for the Nonstationary Navier--Stokes Equations

Gunzburger¹,

Jiang²,

Schneier³

2017

SIAM J. Numer. Anal.

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“…Stability tests. Next, we check the stability of our algorithm by considering the problem of a flow between two offset circles [11,12,14,15]. The domain is a disk with a smaller off-center obstacle inside.…”

Section: Convergence Testmentioning

confidence: 99%

“…The ensemble algorithm we use was first developed in [11] to find a set of J solutions of the Navier-Stokes equations (NSE) subject to different initial condition and forcing functions. The main idea is that, based on the introduction of an ensemble average and a special semi-implicit time discretization, the discrete systems for the multiple flow simulations share a common coefficient matrix.…”

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confidence: 99%

A Second-Order Time-Stepping Scheme for Simulating Ensembles of Parameterized Flow Problems

Gunzburger

Jiang

Wang

2017

Computational Methods in Applied Mathematics

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Abstract. We consider settings for which one needs to perform multiple flow simulations based on the Navier-Stokes equations, each having different values for the physical parameters and/or different initial condition data, boundary conditions data, and/or forcing functions. For such settings, we propose a second-order time accurate ensemble-based method that to simulate the whole set of solutions, requires, at each time step, the solution of only a single linear system with multiple righthand-side vectors. Rigorous analyses are given proving the conditional stability and error estimates for the proposed algorithm. Numerical experiments are provided that illustrate the analyses.Key words. Navier-Stokes equations, parameterized flow, ensemble method MSC (2010). 65M60, 76D051. Introduction. Many computational fluid dynamics applications require multiple simulations of a flow under different input conditions. For example, the ensemble Kalman filter approach used in data assimilation first simulates a forward model a large number of times by perturbing either the initial condition data, boundary condition data, or uncertain parameters, then corrects the model based on the model forecasts and observational data. A second example is the construction of low-dimensional surrogates for solutions of partial differential equation (PDE) such as sparse-grid interpolants or proper orthogonal decomposition approximations for which one has to first obtain expensive approximations of solutions corresponding to several parameter samples. Another example is sensitivity analyses of solutions for which one often has to determine approximate solutions for a number of perturbed inputs such as the values of certain physical parameters. In this paper, we consider such applications and develop a second-order time-stepping scheme for efficiently simulating an ensemble of flows. In particular, we consider the setting in which one wishes to determine the PDE solutions for several different values of the physical parameters and with several different choices of initial condition and boundary condition data and forcing functions appearing in the PDE model.The ensemble algorithm we use was first developed in [11] to find a set of J solutions of the Navier-Stokes equations (NSE) subject to different initial condition and forcing functions. The main idea is that, based on the introduction of an ensemble average and a special semi-implicit time discretization, the discrete systems for the multiple flow simulations share a common coefficient matrix. Thus, instead of solving J linear system with J right-hand sides (RHS), one only need solve one linear system with J RHS. This leads to great computational saving in linear solvers when either the LU factorization (for small-scale systems) or a block iterative algorithm (for large-

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“…A first order accurate, ensemble algorithm was proposed in [20]. The algorithm results in J linear systems with the same coefficient matrix instead of J linear systems with J different coefficient matrices at each time step, which allows the use of block iterative methods, e.g., [10][11][12]31], to reduce the computing time and required memory substantially.…”

Section: Introductionmentioning

confidence: 99%

“…The algorithm results in J linear systems with the same coefficient matrix instead of J linear systems with J different coefficient matrices at each time step, which allows the use of block iterative methods, e.g., [10][11][12]31], to reduce the computing time and required memory substantially. While efficient, the method of [20] is only first order accurate. In applications such as the climate and ocean forecasts, which involve both turbulent flows and long time integration, higher order methods incorporating turbulence models are indispensable.…”

Section: Introductionmentioning

confidence: 99%

A Higher Order Ensemble Simulation Algorithm for Fluid Flows

Jiang

2014

J Sci Comput

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This report presents an efficient, higher order method for fast calculation of an ensemble of solutions of the Navier-Stokes equations. We give a complete stability and convergence analysis of the method for laminar flows and an extension to turbulent flows. For high Reynolds number flows, we propose and analyze an eddy viscosity model with a recent reparameterization of the mixing length. This turbulence model depends on an ensemble mean compatible with the higher order method. We show the turbulence model has superior stability, also demonstrated in numerical tests. We also give tests showing the potential of the new method for exploring flow problems to compute turbulence intensities, effective Lyapunov exponents, windows of predictability and to verify the selective decay principle.

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An Algorithm for Fast Calculation of Flow Ensembles

Cited by 80 publications

References 24 publications

An Ensemble-Proper Orthogonal Decomposition Method for the Nonstationary Navier--Stokes Equations

An Ensemble-Proper Orthogonal Decomposition Method for the Nonstationary Navier--Stokes Equations

A Second-Order Time-Stepping Scheme for Simulating Ensembles of Parameterized Flow Problems

A Higher Order Ensemble Simulation Algorithm for Fluid Flows

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