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 my thesis, I consider the system of thermoelasticity endowed with polyconvex energy. After presenting the equations in their mathematical and physical context, I embed the equations of polyconvex thermoviscoelasticity into an augmented, symmetrizable, hyperbolic system which possesses a convex entropy. This allows to prove many important stability results, such as convergence from thermoviscoelasticity (with Newtonian viscosity and Fourier heat conduction) to smooth solutions of the system of adiabatic thermoelasticity, and convergence from thermoviscoelasticity to smooth solutions of thermoelasticity in the zero-viscosity limit. In addition, I establish a weak-strong uniqueness result in the class of entropy weak solutions and in a suitable class of measure-valued solutions, defined by means of generalized Young measures that describe both oscillatory and concentration effects. Also, I construct a variational scheme for isentropic processes of adiabatic polyconvex thermoelasticity: I establish existence of minimizers which converge to a measure-valued solution that dissipates the total energy, while the scheme converges when the limiting solution is smooth.

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.

Advances in imaging technology have given neuroscientists unprecedented access to examine various facets of how the brain “works”. Brain activity is complex. A full understanding of brain activity requires careful study of its multi-scale spatial-temporal organization (from neurons to regions of interest; and from transient events to long-term temporal dynamics). Motivated by these challenges, we will explore some characterizations of dependence between components of a multivariate time series and then apply these to the study of brain functional connectivity.

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.

Eigenvalue problems associated with partial differential equations are key ingredients for the modeling and simulation of a variety of real world applications, ranging from fluid-dynamics, structural mechanics, acoustics, to electromagnetism and medical problems. We review some properties related to the approximation of eigenvalue problems. Starting from matrix algebraic problems, we present a series of examples and counterexamples showing that even extremely simple situations can produce unexpected results.

Our suggested criteria are more useful for the determination of tuning parameters for sophisticated approximation methods of spatial model fitting. To illustrate this, we investigate the trade-off between the execution time, estimation accuracy, and prediction efficiency for the TLR method with intensive simulation studies and suggest proper settings of the TLR tuning parameters.

The mini-course is an introduction to the analysis of infinity-harmonic functions. We detail the proof of the equivalence between enjoying comparison with cones and solving the infinity-Laplace equation in the viscosity sense, thus making a seamless connection with the previous mini-course. Further material includes the existence of infinity-harmonic functions in the case of an unbounded domain and an easy and self-contained proof, due to Armstrong and Smart, of the celebrated uniqueness theorem of Jensen.

Compartmental epidemiological models are one of the simplest models for the spread of a disease.  They are based on statistical models of interactions in large populations and can be effective in the appropriate circumstances.  Their application historically and in the present pandemic has sometimes been successful and sometimes spectacularly wrong.  In this talk I will review some of these models and their application.  I will also discuss the behavior of the corresponding dynamical systems, and discuss how the theory of optimal control can be applied to them.  I will describe some of the challenges in using such a theory to make decisions about public policy.

This dissertation presents our efforts to build an operational ensemble forecasting system for the Red Sea, based on the Data Research Testbed (DART) package for ensemble data assimilation and the Massachusetts Institute of Technology general circulation ocean model (MITgcm) for forecasting. The Red Sea DART-MITgcm system efficiently integrates all the ensemble members in parallel, while accommodating different ensemble assimilation schemes. The promising ensemble adjustment Kalman filter (EAKF), designed to avoid manipulating the gigantic covariance matrices involved in the ensemble assimilation process, possesses relevant features required for an operational setting. We developed new schemes aiming at lowering the computational burden while preserving reliable assimilation results. The ensemble data assimilation system is implemented and tested on Shaheen, our world-class supercomputer, and will form the basis of the first ever operational Red Sea forecasting system that is currently being implemented to support Saudi Aramco operations in this basin.

The mini-course is an introduction to the analysis of infinity-harmonic functions. We detail the proof of the equivalence between enjoying comparison with cones and solving the infinity-Laplace equation in the viscosity sense, thus making a seamless connection with the previous mini-course. Further material includes the existence of infinity-harmonic functions in the case of an unbounded domain and an easy and self-contained proof, due to Armstrong and Smart, of the celebrated uniqueness theorem of Jensen.

The mini-course is an introduction to the analysis of infinity-harmonic functions. We detail the proof of the equivalence between enjoying comparison with cones and solving the infinity-Laplace equation in the viscosity sense, thus making a seamless connection with the previous mini-course. Further material includes the existence of infinity-harmonic functions in the case of an unbounded domain and an easy and self-contained proof, due to Armstrong and Smart, of the celebrated uniqueness theorem of Jensen.

We present Exascale GeoStatistics (ExaGeoStat) software, a high-performance library implemented on a wide variety of contemporary hybrid distributed-shared supercomputers whose primary target is climate and environmental prediction applications.

The mini-course is an introduction to the analysis of infinity-harmonic functions. We detail the proof of the equivalence between enjoying comparison with cones and solving the infinity-Laplace equation in the viscosity sense, thus making a seamless connection with the previous mini-course. Further material includes the existence of infinity-harmonic functions in the case of an unbounded domain and an easy and self-contained proof, due to Armstrong and Smart, of the celebrated uniqueness theorem of Jensen.

Individual-based models of collective behavior represent a very active research field with applications in physics (spontaneous magnetization), biology (flocking and swarming) and social sciences (opinion formation). They are also a hot topic engineering (swarm robotics). A particularly interesting aspect of the dynamics of multi-agent systems is the emergence of global self-organized patterns, while individuals typically interact only on short scales. In this talk I shall discuss the impact of delay on asymptotic consensus formation in Hegselmann-Krause-type models, where agents adapt their „opinions“ (in broad sense) to the ones of their close neighbors. We shall understand the two principial types/sources of delay - information propagation and processing - and explain their qualitatively different impacts on the consensus dynamics. We then discuss various mathematical methods that provide asymptotic consensus results in the respective settings: Lyapunov functional-type approach, direct estimates, convexity arguments and forward-backward estimates.

The study of waves in fluids is one of the most significant branches of fluid mechanics. Part of this study is the theory of nonlinear and dispersive waves which has recently emerged and is still under development. Nonlinear and dispersive waves appear in fluids of any form and have significant role in the fields of oceanic waves (surface and internal), atmospheric modelling, electromagnetism, nonlinear optics, ultra-cold matter and even in blood flow problems. In this presentation we will review relevant applications, such as tsunami waves, the El Nino southern oscillation, blood flow in arteries and solitons propagating in optical fibres. Mathematical modelling techniques for deriving equations that describe such phenomena will be introduced in the context of surface water waves. We will also review the minimum required theoretical background in order to proceed with safe numerical simulations. Finally, we will discuss the numerical modelling of such problems where methods such as standard and mixed Galerkin / Finite element methods are of central focus. We close this presentation by showcasing a topic of much current interest, namely, the development of modern mathematical models for nonlinear and dispersive waves by combining machine learning techniques with classical methodologies.

We will discuss the Lipschitz extension problem, its solution via MacShane-Whitney extensions, and its several drawbacks, leading to the notion of AMLE (Absolutely Minimizing Lipschitz Extension). We then present a rigorous and detailed analysis of the equivalence between being absolutely minimizing Lipschitz and enjoying comparison with cones. Finally, we explore some consequences of this geometric notion, chiefly the derivation of a Harnack inequality.

In this work, we estimate extreme sea surface temperature (SST) hotspots, i.e., high threshold exceedance regions, for the Red Sea, a vital region of high biodiversity. We analyze high-resolution satellite-derived SST data comprising daily measurements at 16703 grid cells across the Red Sea over the period 1985–2015. We propose a semiparametric Bayesian spatial mixed-effects linear model with a flexible mean structure to capture spatially-varying trend and seasonality, while the residual spatial variability is modeled through a Dirichlet process mixture (DPM) of low-rank spatial Student-t processes (LTPs). By specifying cluster-specific parameters for each LTP mixture component, the bulk of the SST residuals influence tail inference and hotspot estimation only moderately. Our proposed model has a nonstationary mean, covariance and tail dependence, and posterior inference can be drawn efficiently through Gibbs sampling. In our application, we show that the proposed method outperforms some natural parametric and semiparametric alternatives.

We will discuss the Lipschitz extension problem, its solution via MacShane-Whitney extensions, and its several drawbacks, leading to the notion of AMLE (Absolutely Minimizing Lipschitz Extension). We then present a rigorous and detailed analysis of the equivalence between being absolutely minimizing Lipschitz and enjoying comparison with cones. Finally, we explore some consequences of this geometric notion, chiefly the derivation of a Harnack inequality.

We will discuss the Lipschitz extension problem, its solution via MacShane-Whitney extensions, and its several drawbacks, leading to the notion of AMLE (Absolutely Minimizing Lipschitz Extension). We then present a rigorous and detailed analysis of the equivalence between being absolutely minimizing Lipschitz and enjoying comparison with cones. Finally, we explore some consequences of this geometric notion, chiefly the derivation of a Harnack inequality.

When constructing high-order schemes for solving hyperbolic conservation laws with multi-dimensional finite volume schemes, the corresponding high-order reconstructions are commonly performed in characteristic spaces to eliminate spurious oscillations as much as possible. For multi-dimensional finite volume schemes, we need to perform the characteristic decomposition several times in different normal directions of the target cell, which is very time-consuming. We propose a rotated characteristic decomposition technique that requires only one-time decomposition for multi-dimensional reconstructions. This technique not only reduces the computational cost remarkably, but also controls spurious oscillations effectively. We take a third-order weighted essentially non-oscillatory finite volume scheme for solving the Euler equations as an example to demonstrate the efficiency of the proposed technique. We apply the new methodology to the simulation of instabilities in direct initiation of gaseous detonations in free space.

We will discuss the Lipschitz extension problem, its solution via MacShane-Whitney extensions, and its several drawbacks, leading to the notion of AMLE (Absolutely Minimizing Lipschitz Extension). We then present a rigorous and detailed analysis of the equivalence between being absolutely minimizing Lipschitz and enjoying comparison with cones. Finally, we explore some consequences of this geometric notion, chiefly the derivation of a Harnack inequality.

Discussing the concept of correlation and how to interpret it alone (marginally) or within a more complex environment (conditionally). This rather simple observation is the key observation behind a lot of exciting developments and connections in statistics that can be leveraged for improved computations and better motivated statistical models.

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 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.