In this talk we consider the problem of computing statistics of the solution of a partial differential equation with random data, where the random coefficient is parametrized by means of a finite or countable sequence of terms in a suitable expansion. We describe and analyze a Multi-Index Monte Carlo (MIMC) and a Multi-Index Stochastic Collocation method (MISC). the former is both a stochastic version of the combination technique introduced by Zenger, Griebel and collaborators and an extension of the Multilevel Monte Carlo (MLMC) method first described by Heinrich and Giles. Instead of using firstorder differences as in MLMC, MIMC uses mixed differences to reduce the variance of the hierarchical differences dramatically. This in turn yields new and improved complexity results, which are natural generalizations of Giles s MLMC analysis, and which increase the domain of problem parameters for which we achieve the optimal convergence, O(TOL-2). On the same vein, MISC is a deterministic combination technique based on mixed differences of spatial approximations and quadratures over the space of random data. Provided enough mixed regularity, MISC can achieve better complexity than MIMC. Moreover, we show that in the optimal case the convergence rate of MISC is only dictated by the convergence of the deterministic solver applied to a one-dimensional spatial problem. We propose optimization procedures to select the most effective mixed differences to include in MIMC and MISC. Such optimization is a crucial step that allows us to make MIMC and MISC computationally effective. We finally show the effectiveness of MIMC and MISC with some computational tests, including tests with a infinite countable number of random parameters.
We present an overview of mean field games formulation. A comparative analysis of the optimality for a stochastic McKean-Vlasov process with time-dependent probability is presented. Then we examine mean-field games for social interactions and we show that optimizing the long-term well-being through effort and social feeling state distribution (mean-field) will help to stabilize couple (marriage). However , if the cost of effort is very high, the couple fluctuates in a bad feeling state or the marriage breaks down. We then examine the influence of society on a couple using mean field sentimental games. We show that, in mean-field equilibrium, the optimal effort is always higher than the one-shot optimal effort. Finally we introduce the Wiener chaos expansion for the construction of solution of stochastic differential equations of Mckean-Vlasov type. The method is based on the Cameron-Martin version of the Wiener Chaos expansion and allow to quantify the uncertainty in the optimality system.
Darcy s law is a widely used model and the limit of its validity is fairly well known. When the flow is sufficiently slow and the porosity relatively homogeneous and low, Darcy s law is the homogenized equation arising from the Stokes and Navier- Stokes equations and depends on a single effective parameter (the absolute permeability). However when the model is extended to multiphase flows, the assumptions are much more restrictive and less realistic. Therefore it is often used in conjunction with empirical models (such as relative permeability and capillary pressure curves), derived usually from phenomenological speculations and experimental data fitting. In this work, we present the results of a Bayesian calibration of a two-phase flow model, using high-fidelity DNS numerical simulation (at the pore-scale) in a realistic porous medium. These reference results have been obtained from a Navier-Stokes solver coupled with an explicit interphase-tracking scheme. The Bayesian inversion is performed on a simplified 1D model in Matlab by using adaptive spectral method. Several data sets are generated and considered to assess the validity of this 1D model.
Here, we consider numerical methods for stationary mean-field games (MFG) and investigate two classes of algorithms. The first one is a gradient flow method based on the variational characterization of certain MFG. The second one uses monotonicity properties of MFG. We illustrate our methods with various examples, including one-dimensional periodic MFG, congestion problems, and higher-dimensional models.
The Monte Carlo (and Multi-level Monte Carlo) finite element method can be used to approximate observables of solutions to diffusion equations with lognormal distributed diffusion coefficients, e.g. modeling ground water flow. Typical models use lognormal diffusion coefficients with H´ older regularity of order up to 1/2 a.s. This low regularity implies that the high frequency finite element approximation error (i.e. the error from frequencies larger than the mesh frequency) is not negligible and can be larger than the computable low frequency error. We address how the total error can be estimated by the computable error.
As the KAUST Strategic Research Initiative for Uncertainty Quantification completes its fourth year of existence we recall several results produced during its exciting journey of discovery. These include, among others, contributions on Multi-level and Multi-index sampling techniques that address both direct and inverse problems. We may discuss also several techniques for Bayesian Inverse Problems and Optimal Experimental Design.
A study of Monte Carlo methods for weak approximations of stochastic particle systems in the mean-field?
I discuss using single level and multilevel Monte Carlo methods to compute quantities of interests of a stochastic particle system in the mean-field. In this context, the stochastic particles follow a coupled system of Ito stochastic differential equations (SDEs). Moreover, this stochastic particle system converges to a stochastic mean-field limit as the number of particles tends to infinity. I start by recalling the results of applying different versions of Multilevel Monte Carlo (MLMC) for particle systems, both with respect to time steps and the number of particles and using a partitioning estimator. Next, I expand on these results by proposing the use of our recent Multi-index Monte Carlo method to obtain improved convergence rates.
The ensemble Kalman filter (EnKF) is a sequential filtering method that uses an ensemble of particle paths to estimate the means and covariances required by the Kalman filter by the use of sample moments, i.e., the Monte Carlo method. EnKF is often both robust and efficient, but its performance may suffer in settings where the computational cost of accurate simulations of particles is high. The multilevel Monte Carlo method (MLMC) is an extension of classical Monte Carlo methods which by sampling stochastic realizations on a hierarchy of resolutions may reduce the computational cost of moment approximations by orders of magnitude. In this work we have combined the ideas of MLMC and EnKF to construct the multilevel ensemble Kalman filter (MLEnKF) for the setting of finite dimensional state and observation spaces. The main ideas of this method is to compute particle paths on a hierarchy of resolutions and to apply multilevel estimators on the ensemble hierarchy of particles to compute Kalman filter means and covariances. Theoretical results and a numerical study of the performance gains of MLEnKF over EnKF will be presented. Some ideas on the extension of MLEnKF to settings with infinite dimensional state spaces will also be presented.
PDE with stochastic data usually lead to very high-dimensional algebraic problems which easily become unfeasible for numerical computations because of the dense coupling structure of the discretised stochastic operator. Recently, an adaptive stochastic Galerkin FEM based on a residual a posteriori error estimator was presented and the convergence of the adaptive algorithm was shown. While this approach leads to a drastic reduction of the complexity of the problem due to the iterative discovery of the sparsity of the solution, the problem size and structure is still rather limited. To allow for larger and more general problems, we exploit the tensor structure of the parametric problem by representing operator and solution iterates in the tensor train (TT) format. The (successive) compression carried out with these representations can be seen as a generalisation of some other model reduction techniques, e.g. the reduced basis method. We show that this approach facilitates the efficient computation of different error indicators related to the computational mesh, the active polynomial chaos index set, and the TT rank. In particular, the curse of dimension is avoided.