In this short course, I will discuss connections between mean-field games (MFG) systems and modern machine-learning (ML) techniques and problems. In the first part of the course, roughly the first two lectures, I will present how various ML techniques can be applied to solve high-dimensional MFG systems that are far out of reach for traditional methods. In the second part of the course, I will discuss the reverse relation, namely, how the MFG framework can be useful for solving specific ML problems.

In this course, we will consider the so-called Lagrangian approach to mean-field games. We will introduce the problem by recalling some basic results for nonatomic games in a static framework. Next, based on recent results in collaboration with Markus Fischer (University of Padua), I will introduce the analogous setting in the deterministic and dynamic framework. In the second part of the course, I will present the details of a convergence result, obtained in collaboration with Saeed Hadikhanloo, of equilibria of a suitable sequence of discrete-time and finite mean-field games, as introduced by Gomes, Mohr, and Souza in 2010, to an equilibrium of the first-order mean-field game system. The convergence proof relies importantly on the Lagrangian formulation of equilibria.

In this course, we will consider the so-called Lagrangian approach to mean-field games. We will introduce the problem by recalling some basic results for nonatomic games in a static framework. Next, based on recent results in collaboration with Markus Fischer (University of Padua), I will introduce the analogous setting in the deterministic and dynamic framework. In the second part of the course, I will present the details of a convergence result, obtained in collaboration with Saeed Hadikhanloo, of equilibria of a suitable sequence of discrete-time and finite mean-field games, as introduced by Gomes, Mohr, and Souza in 2010, to an equilibrium of the first-order mean-field game system. The convergence proof relies importantly on the Lagrangian formulation of equilibria.

In biochemically reactive systems with small copy numbers of one or more reactant molecules, stochastic effects dominate the dynamics. In the first part of this thesis, we design novel efficient simulation techniques, based on multilevel Monte Carlo methods and importance sampling, for a reliable and fast estimation of various statistical quantities for stochastic biological and chemical systems under the framework of Stochastic Reaction Networks (SRNs). In the second part of this thesis, we design novel numerical methods for pricing financial derivatives. Option pricing is usually challenging due to a combination of two complications: 1) The high dimensionality of the input space, and 2) The low regularity of the integrand on the input parameters. We address these challenges by using different techniques for smoothing the integrand to uncover the available regularity and improve quadrature methods' convergence behavior. We develop different ways of smoothing that depend on the characteristics of the problem at hand. Then, we approximate the resulting integrals using hierarchical quadrature methods combined with Brownian bridge construction and Richardson extrapolation.

In this short course, I will address some regularity aspects in the theory of Mean-Field Games systems, with special emphasis on stationary and uniformly elliptic problems. I will first describe some regularity results for linear uniformly elliptic PDEs and semi-linear PDEs of Hamilton-Jacobi type. Then, I will show how to use these tools to prove the existence (and in some cases uniqueness) of solutions to MFG systems.

Mean-field games (MFGs) study the behavior of rational and indistinguishable agents in a large population. Agents seek to minimize their cost based upon statistical information on the population's distribution. In this dissertation, we study the homogenization of a stationary first-order MFG and seek to find a numerical method to solve the homogenized problem. More precisely, we characterize the asymptotic behavior of a first-order stationary MFG with a periodically oscillating potential. Our main tool is the two-scale convergence. Using this convergence, we rigorously derive the two-scale homogenized and the homogenized MFG problems. Moreover, we prove the existence and uniqueness of the solution to these limit problems. Next, we notice that the homogenized problem resembles the problem involving effective Hamiltonians and Mather measures, which arise in several problems, including homogenization of Hamilton--Jacobi equations, nonlinear control systems, and Aubry--Mather theory. Thus, we develop algorithms to solve the homogenized problem, effective Hamiltonians, and Mather measures.

In this short course, I will address some regularity aspects in the theory of Mean-Field Games systems, with special emphasis on stationary and uniformly elliptic problems. I will first describe some regularity results for linear uniformly elliptic PDEs and semi-linear PDEs of Hamilton-Jacobi type. Then, I will show how to use these tools to prove the existence (and in some cases uniqueness) of solutions to MFG systems.

We introduce several PDE tools which are useful in the study of mean field game systems with local couplings. Due to the lack of regularity of solutions, refined compactness and renormalization arguments are needed for a general approach leading to existence and uniqueness results. If time is enough, congestion models will be treated by similar techniques.

We introduce several PDE tools which are useful in the study of mean field game systems with local couplings. Due to the lack of regularity of solutions, refined compactness and renormalization arguments are needed for a general approach leading to existence and uniqueness results. If time is enough, congestion models will be treated by similar techniques.

In this work, we develop a new framework of trajectory planning for AUVs in realistic ocean scenarios. We divide this work into three parts. In the first part, we provide a new approach for deterministic trajectory planning in steady current, described using Ocean General Circulation Model (OGCM) data. The latter are used to specify both the ocean current and the bathymetry. We apply a NLP to the optimal-time trajectory planning problem. To demonstrate the effectivity of the resulting model, we consider the optimal time trajectory planning of an AUV operating in the Red Sea and the Gulf of Aden. In the second part, we generalize our 3D trajectory planning framework to time-dependent ocean currents. We also extend the framework to accommodate multi-objective criteria, focusing specifically on the Pareto front curve between time and energy. The scheme is demonstrated for time-energy trajectory planning problems in the Gulf of Aden. In the last part, we address uncertainty in the ocean current field. The uncertainty in the current is described in terms of a finite ensemble of flow realizations. The proposed approach is based on a non-linear stochastic programming methodology that uses a risk-aware objective function, accounting for the full variability of the flow ensemble. Advanced visualization tools are used to amplify simulation results.

In many problems in statistical signal processing, regularization is employed to deal with uncertainty, ill-posedness, and insufficiency of training data. It is possible to tune these regularizers optimally asymptotically, i.e. when the dimension of the problem becomes very large, by using tools from random matrix theory and Gauss Process Theory. In this talk, we demonstrate the optimal turning of regularization for three problems : i) Regularized least squares for solving ill-posed and/or uncertain linear systems, 2) Regularized least squares for signal detection in multiple antenna communication systems and 3) Regularized linear and quadratic discriminant binary classifiers.

No summary is available.

Transcription factors are an important family of proteins that control the transcription rate from DNAs to messenger RNAs through the binding to specific DNA sequences. Transcription factor regulation is thus fundamental to understanding not only the system-level behaviors of gene regulatory networks, but also the molecular mechanisms underpinning endogenous gene regulation. In this talk, I will introduce our efforts on developing novel optimization and deep learning methods to quantitatively understanding transcription factor regulation at network- and molecular-levels. Specifically, I will talk about how we estimate the kinetic parameters from sparse time-series readout of gene circuit models, and how we model the relationship between the transcription factor binding sites and their binding affinities.

An important stream of research in computational design aims at digital tools which support users in realizing their design intent in a simple and intuitive way, while simultaneously taking care of key aspects of function and fabrication. Such tools are expected to shorten the product development cycle through a reduction of costly feedback loops between design, engineering and fabrication. The strong coupling between shape generation, function and fabrication is a rich source for the development of new geometric concepts, with an impact to the original applications as well as to geometric theory. This will be illustrated at hand of applications in architecture and fabrication with a mathematical focus on discrete differential geometry and geometric optimization problems.

This talk presents a new classification method for functional data. We consider the case where different groups of functions have similar means so that it is difficult to classify them based on only the mean function. To overcome this limitation, we propose the second moment based functional classifier (SMFC). Here, we demonstrate that the new method is sensitive to divergence in the second moment structure and thus produces lower rate of misclassification compared to other competitor methods. Our method uses the Hilbert-Schmidt norm to measure the divergence of second moment structure. One important innovation of our classification procedure lies in the dimension reduction step. The method data-adaptively discovers the basis functions that best capture the discrepancy between the second moment structures of the groups, rather than uses the functional principal component of each individual group, and good performance can be achieved as unnecessary variability is removed so that the classification accuracy is improved. Consistency properties of the classification procedure and the relevant estimators are established. Simulation study and real data analysis on phoneme and rat brain activity trajectories empirically validate the superiority of the proposed method.

Functional data analysis is a very active research area due to the overwhelming existence of functional data. In the first part of this talk, I will introduce how functional data depth is used to carry out exploratory data analysis and explain recently-developed depth techniques. In the second part, I will discuss spatio-temporal statistical modeling. It is challenging to build realistic space-time models and assess the validity of the model, especially when datasets are large. I will present a set of visualization tools we developed using functional data analysis techniques for visualizing covariance structures of univariate and multivariate spatio-temporal processes. I will illustrate the performance of the proposed methods in the exploratory data analysis of spatio-temporal data.

In the lecture we present a three dimensional mdoel for the simulation of signal processing in neurons. To handle problems of this complexity, new mathematical methods and software tools are required. In recent years, new approaches such as parallel adaptive multigrid methods and corresponding software tools have been developed allowing to treat problems of huge complexity. Part of this approach is a method to reconstruct the geometric structure of neurons from data measured by 2-photon microscopy. Being able to reconstruct neural geometries and network connectivities from measured data is the basis of understanding coding of motoric perceptions and long term plasticity which is one of the main topics of neuroscience. Other issues are compartment models and upscaling.

Approximating the solution to an evolutionary partial differential equation by a set of "moving particles" has several advantages. It validates the use of a continuity equation in an "individuals-based" modeling setting, it provides a link between Lagrangian and Eulerian description, and it defines a "natural" numerical approach to those equations. I will describe recent rigorous results in that context. The main one deals with one-dimensional scalar conservation laws with nonnegative initial data, for which we prove that the a suitably designed "follow-the-leader" particle scheme approximates entropy solutions in the sense of Kruzkov in the many particle limit. Said result represents a new way to solve scalar conservation laws with bounded and integrable initial data. The same method applies to second order traffic flow models, to nonlocal transport equations, and to the Hughes model for pedestrian movements.

In this talk I will discuss statistical models which incorporate temperature response to the radiative forcing components. The models can be used to estimate important climate sensitivity measures and give temperature forecasts. Bayesian inference is obtained using the methodology of integrated nested Laplace approximation and Monte Carlo simulations. The resulting approach will be demonstrated in analyzing instrumental data and Earth system model ensembles.

Secondary quantities such as energy and entropy can be very important for numerical methods. Firstly, preserving these quantities can ensure that non-physical behavior is excluded. Secondly, preserving such quantities can result in stability estimates. Finally, preserving the correct energy/entropy evolution in time can result in additional desirable properties such as lower numerical errors. In this talk, a brief overview of some recent advances concerning energy and entropy preserving numerical methods for ordinary and partial differential equations will be given, together with an outlook on future research directions and applications.

Abstract

The demand for increasingly multidisciplinary reliable simulations, for both analysis and

In this talk we consider the Cauchy problem for the 2D Euler equations for incompressible inviscid fluids. It is well-known that given a smooth initial datum, the Cauchy problem is  well-posed and in particular the energy is conserved and the vorticity is transported by the flow of the velocity. When we consider weak solutions this might not be the case anymore. We will review some recent results obtained in collaboration with Gianluca Crippa and Gennaro Ciampa where we extend those properties to the case of irregular vorticities. In particular, under low integrability assumptions on the vorticity we show that certain approximations important from a physical and a numerical point of view converge to solutions satisfying those properties.

​Author of more than 290 journal and conference publications, Professor Stenchikov's research interests are in multi-scale modeling of environmental processes and numerical methods; global climate change, climate downscaling, atmospheric convection; assessment of anthropogenic impacts and geoengineering; air-sea interaction, evaluating environmental consequences of catastrophic events like volcanic eruptions, nuclear explosions, forest and urban fires; and air pollution, transport of aerosols, chemically and optically active atmospheric tracers, their radiative forcing and effect on climate.

In this talk we present the derivation of a new Boussinesq-type system to describe the propagation of long waves of small amplitude in a basin with mildly varying bottom topography. We prove the existence and uniqueness of weak solutions for maximal times that do not depend on the amplitude of the waves. We then present the numerical solution of the new system using Galerkin finite element methods and prove the convergence of the semidiscrete solution to the exact solution. The system appears to describe well water waves even in benchmark experiments that involve also general bathymetries.

In this talk, we review some recent advances in the analysis and design of algebraic flux correction (AFC) schemes for hyperbolic problems. In contrast to most variational stabilization techniques, AFC approaches modify the standard Galerkin discretization in a way which provably guarantees the validity of discrete maximum principles for scalar conservation laws and invariant domain preservation for hyperbolic systems. The corresponding inequality constraints are enforced by adding diffusive fluxes, and bound-preserving antidiffusive corrections are performed to obtain nonlinear high-order approximations. After introducing the AFC methodology and the underlying theoretical framework in the context of continuous piecewise-linear finite element discretizations, we present some of the limiting techniques that we use in high-resolution AFC schemes. This presentation is based on joint work with Dr. Manuel Quezada de Luna (KAUST) and other collaborators.