In this talk we will give some results regarding the existence and regularity of densities for stochastic differential equations (sde's) with irregular coefficients starting with Holder coefficients and then bounded and measurable. If time allows we will also discuss issues related to the simulation schemes for such sde's.

In many problems in science and engineering, one would like to perform inference on parametric fields, of space and/or time, in order to reduce uncertainties and improve predictive capabilities, for example, the computation of quantities of interest such as outputs of the model or various expectations thereof.  Examples of such fields include the initial condition of fluid dynamical equations in the context of numerical weather prediction and oceanography, the permeability and porosity of multiphase subsurface flow in the context of oil exploration, or the forcing of a stochastic dynamical system in the context of molecular dynamics.

In this talk, we focus on two sample applications: traffic networks and smart energy systems. We will explain how optimal control tools and user incentive designs can be used to better manage traffic congestion and address demand response in medium and large-scale systems.

The analysis and performance of numerical computations for optimal control problems is complicated by the fact that they are ill-posed. It is for example often the case that optimal solutionsdepend discontinuously on data. Moreover, the optimal control, if it exists, may be a highly non-regular function, with many points of discontinuity etc. On the other hand optimal control problems are well-posed in the sense that the associated value function is well-behaved, with such properties as continuous dependence on data.

Stochastic partial differential equations (SPDEs) are used in many areas of applied science and engineering. Examples of systems that are modeled by SPDEs arise in applications as diverse as satellite guidance, tumor detection and financial markets. In many instances analytic solutions to these equations are unavailable and approximations with a high order of accuracy are difficult to obtain.

The marriage of mathematics and epidemics has a long and distinguished history with a plethora of successes that go back to the work of Daniel Bernoulli (1700 – 1782) and Nobel Laureate and physician Sir Ronald Ross (1911) and associates. These individuals, mostly physicians, created the field of mathematical epidemiology in their efforts to meet their commitment to diminish health disparities, the consequences of poverty and the lack of access to health services.

Exact Sampling and Inference for Non-Linear Stochastic Differential Equations. I will present a surprisingly simple algorithm we developed some years ago for the exact simulation of non-linear diffusion processes. Exact sampling of such processes was considered impossible for such models. The presence of stochasticity allows for developing a rejection sampling algorithm by proposing candidate Brownian paths; the accept/reject rule then involves events of an appropriate Poisson process. We have thus termed the method the Weiner-Poisson factorization of a non-linear SDE. The developed sampling method provides also a natural approach for carrying out maximum-likelihood-based inference for unknown parameters of the SDE.

Sequential Monte Carlo (SMC) methods are nowadays routinely applied in a variety of complex applications: hidden Markov models, dynamical systems, target tracking, control problems, just to name a few. Whereas SMC methods have been greatly refined in the last decades and are now much better understood, they are still known to suffer from the curse of dimensionality: algorithms can sometimes break down exponentially fast with the dimension of the state space. As a consequence, practitioners in high-dimensional Data Assimilation applications in atmospheric sciences, oceanography and elsewhere will typically use 3D-Var or Kalman-filter-type approximations that will provide biased estimates in the presence of non-linear model dynamics.

This is a joint work with Professor Jens Frehse and Professor Phillip Yam. We give a unified presentation of two research domains which have been very popular in the recent years, but considered independently. We provide a Dynamic Programming as well as a Stochastic Maximum Principle approach. We discuss many extensions and the issue of time consistency. Extended slides will be provided.

Game Theory has been a staple of economics research since 1950, when John Nash who is the subject of the movie A Beautiful Mind, published the seminal paper that would win him the Nobel Prize in economics. As game theory has matured, it’s become even more central to the field of economics and social sciences.

Discrete Fracture Network (DFN) models are widely used in the simulation of subsurface flows; they describe a geological reservoir as a system of many intersecting planar polygons representing the underground network of fractures. Among the different approaches to DFN simulations, recently (Berrone et al, 2013) a numerical model has been formulated as a PDE-constrained optimization problem, in which neither fracture/fracture nor fracture/trace mesh conformity is required.

In this talk, I will explain what is uncertainty quantification and how to model uncertainties via random variables/random fields. I will give several examples of stochastic partial differential equations with uncertain coefficients, uncertain computations domain, uncertain right-hand side, or boundary conditions.

This course will introduce the audience to some of the essential ingredients of learning in games under uncertainty (random matrix games), particularly reinforcement learning, cost-of-learning, Q-learning, mean-field learning, combined learning, heterogeneous learning and hybrid learning.

The Monte Carlo forward Euler method with uniform time stepping is the standard technique to compute an approximation of the expected payoff of a solution of an Itô SDE. For a given accuracy requirement TOL, the complexity of this technique for well-behaved problems, that is the amount of computational work to solve the problem, is O(TOL-3).

One of the simplest model problems in the kinetic theory of plasma--physics is the Vlasov-Poisson system with periodic boundary conditions. Such system describes the evolution of a plasma of charged particles (electrons and ions) under the effects of the transport and self-consistent electric field. In this talk, we present a family of discontinuous Galerkin (DG) methods for the approximation of the Vlasov-Poisson system.

I will provide a quick glance into the research of my group, giving particular emphasis on the activities of my KAUST students. This will hopefully give a connection between our research and our educational activities.

The course treats completely observed control problems with a state equation of Itô type and with a performance function which is the expected value of a functional of the controlled process. As is well known,  Bellman's dynamic programming leading to the celebrated Hamilton-Jacobi-Bellman equation and Pontryagin's maximum principle are the most commonly used approaches to solving the control problem.

The student will be introduced to Bayesian modeling in selected, but relevant, stochastic processes and their applications: Markov chains, Poisson processes, reliability and queues. The use of real examples will be helpful in understanding why and how perform a Bayesian analysis. Students will be asked to analyze real data, from the elicitation of priors and modeling to (numerical) computation of estimates and forecasts and interpretation of findings.

Many problems in the physical sciences require the determination of an unknown field from a finite set of indirect measurements. Examples include oceanography, oil recovery, water resource management and weather forecasting. This may be formulated as a least squares problem to match the model output to the data. I will demonstrate that ideas from the Ensemble Kalman Filter can be adapted to solve such problems: by running multiple interacting copies of the model, and exposing their output to the (suitably randomized) data, a derivative-free minimization tool is constructed.

Molecular dynamics (MD) is widely employed in both industrial and academic environments, playing a key role in the study of a large variety of systems, ranging fromliquids to solids, as well as biomolecules, such as proteins and nucleic acids (DNA, RNA). MD simulations provide a suitable and convenient method to explore dynamical properties of a system at the atomic level which, in general, aresignificantly difficult and expensive to investigate in experimental settings.

Unstable dynamical systems can be stabilized, and hence the solution recovered from noisy data, provided two conditions hold. First, observe enough of the system: the unstable modes. Second, weight the observed data sufficiently over the model. In this talk I will illustrate this for the 3DVAR filter applied to three dissipative dynamical systems of increasing dimension: the Lorenz 1963 model, the Lorenz 1996 model, and the 2D Navier-Stokes equation.