Physics, P-DO
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Stochastic Closure for Multiscale Simulations
D.M. Schmidt, S.C. Jun (P-21), B. Nadiga, D. Livescu (CCS-2), D. Higdon (D-1), D. Ranken (CCN-12) IntroductionIn nonlinear systems, small-scale phenomena affect large-scale behavior. Computer simulations of large, nonlinear systems, therefore, require very large grids in order to both cover the domain of the problem and to resolve the finest relevant small-scale phenomenon. Existing computing capabilities are often inadequate to compute on grids that fully resolve the small-scale phenomena resulting in the use of coarser grids. In such cases, the equations governing the dynamics of the system to be simulated must be modified to try to replicate the effects of the now missing, subgrid-scale phenomena. This problem of how to modify the equations to replicate the subgrid phenomena is known as the problem of closure, and any particular modified set of partial differential equations (PDEs) is called a particular choice of closure relation. In practice, generating closure relations is a very problem-specific, time-consuming endeavor, often involving the generation and fine-tuning of models to incorporate phenomenological information and/or theoretical insights about the particular system to be simulated.1,2 Yet another ubiquitous problem associated with simulating complex systems is uncertainty quantification. Input parameters, boundary conditions, and physics-model parameters are often unknown and affect the outcome of the simulation. In order to quantify the uncertainty of the output of the simulation, given that of the input parameters, the simulation must be run many times with different values of the parameters.3,4 The problems of closure and of uncertainty quantification are correlated for a number of reasons. First, closure must be accomplished on a sufficiently coarse grid to allow for multiple runs of the simulation. Second, the act of up-scaling a system of equations to a coarse grid and choosing a closure relation is a source of uncertainty; different closure relations can be used, producing different results. Our Approach to ClosureWe are working to develop, validate, and apply a new probabilistic approach to the problems of closure and uncertainty quantification in multiscale simulations through stochastic closure (SC). Our SC approach uses a probability distribution of closure relations as the solution of the closure problem. This probability distribution represents information about the unresolved phenomena that may be obtained from the output of a simulation on a fine grid that resolves the small-scale phenomena over a smaller domain, as well as from relevant experimental results. Statistical methods for density estimation can be used to generate a distribution that encodes this information. Once the probability distribution is obtained, individual closure relations are drawn at random and used throughout the evolution of the simulation. Uncertainties in the choice of closure relations are clearly defined and better sampled in this approach. Moreover, the effects of unresolved phenomena tend to be better represented with this SC approach than with the use of a single closure relation. Finally, by addressing the closure issue with generalized probabilistic methods, the tools and technologies that we are developing will be applicable to a range of multiscale systems. RationaleThe rationale for SC derives from the following considerations. Any coarsely sampled or gridded field could have resulted from one of a large number of different nonsampled, or continuous fields. The different continuous fields generate different dynamics in the nonlinear terms of the dynamical equations, resulting in a large number of different closure relations that could be used. Constructing a probability distribution of closure relations allows one to incorporate information about the likelihood of any particular closure relation based on prior information about the subgrid-scale phenomenon for that dynamical system. This information is contained within the original dynamical equations and may be found by simulating the system over a small domain with a fine enough grid to resolve the small-scale phenomenon. If successfully utilized, a number of potential benefits may result from these ideas. By treating closure as a probabilistic problem where a probability distribution is generated based on the output of a fine-scale simulation, the statistical techniques developed to generate such a distribution can be applied to a wide range of problems and in a nearly automated fashion. This diminishes the historical problem of generating closure relations as being a problem-specific, time-consuming endeavor. By drawing from a distribution of closure relations throughout multiple simulations, the uncertainty associated with up-scaling to a coarse grid are inherently included and in a manner that allows for other uncertainties to be included as well. This significantly aids the uncertainty quantification goal. Moreover, because multiple closure relations are being used, the time average of multiple moments of fields is well reconstructed, not just one or a few moments. Though others have used SC terms5, they have been problem specific and have not been formulated in the generalized statistical terms we propose here, nor did they address the important problem of uncertainty quantification. Example
To construct a distribution of closure relations for the low-resolution simulation, we used the results of the high-resolution simulation. The dynamical system involved a single nonlinear term in the PDEs that was a Jacobian of two fields, J(p, q). To construct the SC distribution, we examined the local spatial average (to represent the act of down-sampling to a coarser grid) of the Jacobian versus the Jacobian of the spatially averaged fields. Specifically, we looked at 〈J(p,q)〉versus J(〈p,q〉), where 〈 〉 denotes local spatial average. A scatter plot of these quantities from the output of the fine-scale simulation for a local region in the two-dimensional spatial domain is shown in Figure 2. When integrating the PDEs on a coarse grid we are given J(〈p,q〉) but would like to know 〈J(p,q)〉. Clearly, no single choice exists for this but rather a range of choices that we approximate by constructing a distribution (Figure 2). The low-resolution simulation was rerun, but modified to include a SC term to model the subgrid turbulence that was randomly sampled from the previously found distribution, assuming a temporally homogeneous model for the subgrid phenomena temporal correlation. Adding this sampling step did not significantly add to the run time of the code so that we could run multiple stochastic simulations in much less time than it took for one run of the high-resolution simulation. The middle panel of Figure 1 shows the average over multiple stochastic simulations of the time-averaged stream function. The outer pair of gyres now re-emerges and the magnitude of circulation in the wind-driven pair is improved as well. We also calculated and compared the temporal variability (standard deviation over time) of the stream function for each type of simulation (Figure 3). The low-resolution run completely failed to reproduce the temporal variability found in the high-resolution run, but the low-resolution stochastic run did a good job in reproducing this variability. These results indicate that multiple moments of the fields are reproduced well with this SC approach. Moreover, the variance or uncertainty of any quantity may be estimated by simply calculating the variance of that quantity across the multiple stochastic closure runs, as shown in Figure 4. |
We have used a simple barotropic model of the subtropical/subpolar gyre (rotating flow) circulation in the ocean as a prototypical multiscale problem to illustrate the SC approach. The time-averaged stream function from a high-resolution, 200 × 100 grid simulation shows a pair of inner wind-driven, counter-rotating gyres and a pair of outer gyres (left panel of Figure 1) that are driven by turbulent eddy fluxes.6 In a low-resolution 50 × 25 grid simulation, using the same dynamical equations and identical setting, the time-averaged stream function does not contain the outer pair of eddy-driven gyres and the magnitude of circulation is lower in the inner pair of wind-driven gyres (right panel of Figure 1). 
