In this minisymposium, we explore the symbiotic relationship between computational statistics and computational dynamics. The interaction of the two fields have long been established. Efficiently computing statistics of dynamical quantities is of interest in science and engineering, and cleverly constructed dynamical systems are used to sample from high-dimensional probability distributions. We will highlight recent advances in numerical methods that utilize tools in one field to solve problems in the other in a novel fashion. We will exhibit new algorithms for sensitivity analysis, efficient sampling methods, and inference.
14:00
Couplings-based sensitivity estimates for stochastic dynamics
Kevin Lin | University of Arizona | United States
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Kevin Lin | University of Arizona | United States
Jonathan Mattingly | Duke University | United States
A common task in computational studies of stochastic dynamical systems is to estimate (local) sensitivities, i.e., how stationary expectations of observables vary with model parameters. Sensitivity estimates can be computationally expensive, and numerical methods based on couplings are often used to speed up such computations. Unfortunately, the simplest such methods are often ineffective for chaotic dynamics. In this talk, I will report on coupling-based sensitivity estimators applicable to chaotic systems, and examine their performance under different dynamical scenarios.
14:30
Computing statistical response to small perturbations in chaotic systems
Nisha Chandramoorthy | Massachusetts Institute of Technology | United States
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Nisha Chandramoorthy | Massachusetts Institute of Technology | United States
Qiqi Wang | Massachusetts Institute of Technology | United States
Gradient information computed from numerical simulations is used for design and optimization and uncertainty quantification in many engineering disciplines. Today’s high-fidelity simulations are able to capture complex physics including chaotic dynamics but sensitivity analysis in them is still nascent. The reason is that an infinitesimal perturbation applied to a chaotic system grows unbounded in time, rendering linearized perturbation solutions meaningless. Yet, the average response of the system, or the sensitivity of statistics to perturbations in system inputs is bounded. Computing this statistical response to infinitesimal changes in parameters without the unstable long-time evolution of instantaneous derivatives is the objective of the perturbation space-split sensitivity (S3) algorithm. An efficient computation of the statistical response would enable uncertainty quantification, mesh adaptation, parameter estimation and other gradient-based multidisciplinary design optimization techniques that are still nascent in chaotic systems. Based on the fact that perturbations that lie in the stable subspace decay intime exponentially, the contribution to the overall sensitivity from them is computed similar to in non-chaotic systems. The unstable contribution is reduced by an integration-by-parts procedure to a variance-reduced Monte-Carlo sampling. This method is provably convergent. The S3 algorithm will be demonstrated on low-dimensional uniformly hyperbolic systems.
15:00
Machine learning non-local closures for turbulent anisotropic multiphase fluid flows
Alexis-Tzianni Charalampopoulos | Massachusetts Institute of Technology | United States
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Alexis-Tzianni Charalampopoulos | Massachusetts Institute of Technology | United States
Themistoklis P. Sapsis | Massachusetts Institute of Technology | United States
We examine the capabilities of deep recurrent neural networks to act as nonlocal closure models for turbulent 2D multiphase flows. We employ a coarse-discretization of the governing equations together with the neural net closures to evolve in time the fluid flow. This approach allows us to resolve for the statistics of inertial tracers advected by the fluid flow. Utilizing recurrent and convolutional layers we capture nonlocal patio-temporal effects, induced by the large scale dynamics of the flow. We avoid the use of fully-connected layers due to their large computational overhead, tendency to over-fit on training data and non-physical implications. Numerical investigation of turbulent anisotropic flows is carried out with the aim to model the fluid flow and the transport of bubbles. Finally, we validate the derived closure in configurations close to the training flows but also assess its limits for configurations that are radically different from the training ones.
15:30
Numerical approximation for invariant measures of the 2D Navier-Stokes equations
Cecilia Mondaini | Drexel University | United States
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Cecilia Mondaini | Drexel University | United States
Nathan Glatt-Holtz | Tulane University | United States
We consider the problem of approximating statistically steady states of the 2D stochastic Navier-Stokes equations (SNSE) via an approximating sequence of measures generated from a space-time discretization of the 2D SNSE. More specifically, we consider a spectral Galerkin spatial discretization and a semi-implicit Euler time scheme. We show that successive iterations of the Markov semigroup associated to the discretized system, starting from any initial probability distribution, converge to the invariant measure of the continuous system. The proof is obtained with two main steps: a spectral gap result for the discretized system which is independent of the discretization parameters, and finite time L2-convergence of the discretized system towards the continuous one. Most importantly, this approach allows us to obtain explicit rates of convergence with respect to the number of iterations in the Markov chain, up to (also explicit) numerical discretization error.