The time-dependent PDE system for mean field games is a coupled pair of parabolic equations, one forward in time and the other backward in time. The lecturer will demonstrate two techniques for proving existence and uniqueness of solutions for this system. The first of these techniques is inspired by work in fluid dynamics, as a similar forward-backward structure for vortex sheets was discovered by Duchon and Robert in the 1980s. Adapting the ideas of Duchon and Robert gives existence and uniqueness of solutions for the time-dependent mean field games system in function spaces based on the Wiener algebra. The second technique to be demonstrated by the speaker uses Sobolev spaces, and is an adaptation of the energy method to the forward-backward setting.

This session will help you to improve the efficiency of your literature searches. You will learn basic search techniques – using Boolean operators, keyword search, concept search, truncation, etc. Two major interdisciplinary scientific databases – Web of Science and Scopus, as well as CEMSE specific resources in KAUST library collections will be presented.

The time-dependent PDE system for mean field games is a coupled pair of parabolic equations, one forward in time and the other backward in time. The lecturer will demonstrate two techniques for proving existence and uniqueness of solutions for this system. The first of these techniques is inspired by work in fluid dynamics, as a similar forward-backward structure for vortex sheets was discovered by Duchon and Robert in the 1980s. Adapting the ideas of Duchon and Robert gives existence and uniqueness of solutions for the time-dependent mean field games system in function spaces based on the Wiener algebra. The second technique to be demonstrated by the speaker uses Sobolev spaces, and is an adaptation of the energy method to the forward-backward setting.

State-space models are widely used for the analysis of time-series data but full Bayesian inference is still elusive. I will review some applications of state-space models and recent endeavours towards efficient Monte Carlo sampling, in particular using Particle filtering and more recent Particle Markov Chain Monte Carlo methods. I will discuss the theoretical scalability of these methods with respect to the length of the observed time-series. Our theoretical results on their efficiency align well with many documented instances of their effectiveness based on extensive numerical studies. The talk will conclude with some open challenges to be pursued.

In this course, we provide a brief introduction to fractional calculus with a view to applying it to the study of time fractional partial differential equations. We will introduce the definitions and main properties of  fractional integrals and derivatives, including those of Riemann-Liouville, Caputo and Grunwald-Letnikov. The previous results will serve as the main modeling tools for partial differential equations related to a class of non-Markovian stochastic processes, called subdiffusions. Then we will examine some results regarding time-fractional linear partial differential equations and conclude with a brief introduction to control problems and Mean Field Games for subdiffusion processes.

Computational mathematics has a millennium long history but its modern incarnation started after the advent of electronic computers about 80 years ago. Scientifically, it lies in the intersection between mathematics, a subject with a long history, and computer sciences, a relatively new discipline. Its motivations, approaches and practitioners have derived from different fields, and it has also had to evolve and adapt to new tools and opportunities. My own scientific career overlaps quite a bit with the field’s modern evolution and in this talk, I’ll give a personal, as well as a “historical” view of the field.

In this course, we provide a brief introduction to fractional calculus with a view to applying it to the study of time fractional partial differential equations. We will introduce the definitions and main properties of fractional integrals and derivatives, including those of Riemann-Liouville, Caputo and Grunwald-Letnikov. The previous results will serve as the main modeling tools for partial differential equations related to a class of non-Markovian stochastic processes, called subdiffusions. Then we will examine some results regarding time-fractional linear partial differential equations and conclude with a brief introduction to control problems and Mean Field Games for subdiffusion processes.

Elliptic PDE are ubiquitous both in Mathematics and in the applications of Mathematics. The regularity of the generalized solutions is a very important issue that it is necessary to handle in proper way if one want to obtain useful information. The goal of my lectures is to introduce the audience to the topic of regularity for elliptic PDE under assumptions on the coefficients that are of minimal requirements.

Elliptic PDE are ubiquitous both in Mathematics and in the applications of Mathematics. The regularity of the generalized solutions is a very important issue that it is necessary to handle in proper way if one want to obtain useful information. The goal of my lectures is to introduce the audience to the topic of regularity for elliptic PDE under assumptions on the coefficients that are of minimal requirements.

In this talk we prove that suitable deep neural network approximations do indeed overcome the curse of dimensionality in the case of a general class of semilinear parabolic PDEs and we thereby prove, for the first time, that a general semilinear parabolic PDE can be solved approximatively without the curse of dimensionality.

In this talk, several numerical methods will be presented and illustrated on examples. Borrowing tools from stochastic analysis, optimization, partial differential equations and machine learning, these methods enable us to solve mean field games with possibly complex sources of noise or high dimensional state variables.

Geospatial health data are essential to inform public health and policy. These data can be used to quantify disease burden, understand geographic and temporal patterns, identify risk factors, and measure inequalities. In this talk, I will give an overview of my research which focuses on the development of geospatial methods and interactive visualization applications for health surveillance. I will present disease risk models where environmental, demographic and climatic data are used to predict the risk and identify targets for intervention of lymphatic filariasis in sub-Saharan Africa, and leptospirosis in a Brazilian urban slum. I will also show the R packages epiflows for risk assessment of travel-related spread of disease, and SpatialEpiApp for disease mapping and the detection of clusters. Finally, I will describe my future research and how it can inform better surveillance and improve population health globally.

The theory of Mean Field Games (MFG) has been developed in the last two decades by economists, engineers, and mathematicians in order to study decision making in very large populations of “small" interacting agents. This short course will be focused on deterministic MFG, which are associated with a first order PDE system. We will address the problem assuming that agents are subject to state constraints, when classical PDE techniques are of little help. First, we will show how to prove the existence of solutions by the so-called Lagrangian approach, which interprets equilibria as certain measures on the space of paths that each agent can choose. Then, we will address regularity issues for such generalized solutions, deriving point-wise properties that allow to recover the typical MFG system. Finally, we will study the asymptotic behavior of solutions to the constrained MFG system as time goes to infinity, borrowing ideas from weak KAM theory.

Biological systems are distinguished by their enormous complexity and variability. That is why mathematical modeling and computational simulation of those systems is very difficult, in particular thinking of detailed models which are based on first principles. The difficulties start with geometric modeling which needs to extract basic structures from highly complex and variable phenotypes, on the other hand also has to take the statistic variability into account. Moreover, the models of the processes running on these geometries are not yet well established, since these are equally complex and often couple many scales in space and time. Thus, simulating such systems always means to put the whole frame to test, from modelling to the numerical methods and software tools used for simulation. These need to be advanced in connection with validating simulation results by comparing them to experiments.

The theory of Mean Field Games (MFG) has been developed in the last two decades by economists, engineers, and mathematicians in order to study decision making in very large populations of “small" interacting agents. This short course will be focused on deterministic MFG, which are associated with a first order PDE system. We will address the problem assuming that agents are subject to state constraints, when classical PDE techniques are of little help. First, we will show how to prove the existence of solutions by the so-called Lagrangian approach, which interprets equilibria as certain measures on the space of paths that each agent can choose. Then, we will address regularity issues for such generalized solutions, deriving point-wise properties that allow to recover the typical MFG system. Finally, we will study the asymptotic behavior of solutions to the constrained MFG system as time goes to infinity, borrowing ideas from weak KAM theory.

The overarching goal of Prof. Michels' Computational Sciences Group within KAUST's Visual Computing Center is enabling accurate and efficient simulations for applications in Scientific and Visual Computing. Towards this goal, the group develops new principled computational methods based on solid theoretical foundations. This talk covers a selection of previous and current work presenting a broad spectrum of research highlights ranging from simulating stiff phenomena such as the dynamics of fibers and textiles, over liquids containing magnetic particles, to the development of complex ecosystems and weather phenomena. Moreover, connection points to the growing field of machine learning are addressed and an outlook is provided with respect to selected technology transfer activities.

The main goal of this mini course is to discuss the state-of-the-art in understanding the phenomena of long time asympotitcs and phase transitions for a range of nonlinear Fokker-Planck equations with linear and nonlinear diffusion. They appear as natural macroscopic PDE descriptions of the collective behavior of particles such as Cucker-Smale models for consensus, the Keller Segel model for chemotaxis, and the Kuramoto model for synchronization. We will discuss the existence of phase transitions in a variety of these models using the natural free energy of the system and their interpretation as natural gradient flow structure with respect to the Wasserstein distance in probability measures. We will discuss both theoretical aspects as well as numerical schemes and simulations keeping those properties at the discrete level.

The main goal of this mini course is to discuss the state-of-the-art in understanding the phenomena of long time asympotitcs and phase transitions for a range of nonlinear Fokker-Planck equations with linear and nonlinear diffusion. They appear as natural macroscopic PDE descriptions of the collective behavior of particles such as Cucker-Smale models for consensus, the Keller Segel model for chemotaxis, and the Kuramoto model for synchronization. We will discuss the existence of phase transitions in a variety of these models using the natural free energy of the system and their interpretation as natural gradient flow structure with respect to the Wasserstein distance in probability measures. We will discuss both theoretical aspects as well as numerical schemes and simulations keeping those properties at the discrete level.

Modular robotics deals with robots that are an assemblage of smaller sized and often identical robots. The benefits of modular robots are many, chief among them being how easily they can be transported from one location to another. Moreover, their size can be adjusted according to the task at hand without requiring extensive redesign or specialization, therefore making them the object of significant research efforts.

The main goal of this mini course is to discuss the state-of-the-art in understanding the phenomena of long time asympotitcs and phase transitions for a range of nonlinear Fokker-Planck equations with linear and nonlinear diffusion. They appear as natural macroscopic PDE descriptions of the collective behavior of particles such as Cucker-Smale models for consensus, the Keller Segel model for chemotaxis, and the Kuramoto model for synchronization. We will discuss the existence of phase transitions in a variety of these models using the natural free energy of the system and their interpretation as natural gradient flow structure with respect to the Wasserstein distance in probability measures. We will discuss both theoretical aspects as well as numerical schemes and simulations keeping those properties at the discrete level.

In this talk we consider the problem of estimating the score function (or gradient of the log-likelihood) associated to a class of partially observed diffusion processes, with discretely observed, fixed length, data and finite dimensional parameters. We construct an estimator that is unbiased with no time-discretization bias. Using a simple Girsanov change of measure method to represent the score function, our methodology can be used for a wide class of diffusion processes and requires only access to a time-discretization method such as Euler-Maruyama. Our approach is based upon a novel adaptation of the randomization schemes developed by Glynn and co-authors along with a new coupled Markov chain simulation scheme. The latter methodology is an original type of coupling of the coupled conditional particle filter. We prove that our estimator is unbiased and of finite variance. We then illustrate our methodology on several challenging statistical examples. This is a joint work with Jeremy Heng (ESSEC, Singapore) and Jeremie Houssineau (Warwick, UK)

We propose a new optimization formulation for training federated learning models. The standard formulation has the form of an empirical risk minimization problem constructed to find a single global model trained from the private data stored across all participating devices. In contrast, our formulation seeks an explicit trade-off between this traditional global model and the local models, which can be learned by each device from its own private data without any communication. Further, we develop several efficient variants of SGD (with and without partial participation and with and without variance reduction) for solving the new formulation and prove communication complexity guarantees.

The mini-course is an introduction to intrinsic scaling, a powerful method in the analysis of degenerate and singular parabolic PDEs. The local Hölder continuity of bounded weak solutions will be derived from scratch for the model case of the degenerate p-Laplace equation. Our approach is entirely self-contained and focused on the essence of the method, leaving aside technical refinements needed to deal with more general equations.

In this talk, we start by introducing optimization and interesting optimization applications. We review some optimization formulations and focus on applications studied in our research, such as energy systems, and trajectory planning of autonomous underwater vehicles. After the introduction, we address the self-scheduling and market involvement of a virtual power plant using adaptive robust optimization under uncertainty in the wind speed and electricity prices.