Elliptic PDEs are ubiquitous both in Mathematics and in applications of Mathematics. Regularity of generalized solutions is a fundamental issue necessary to handle in a proper way if one wants to obtain qualitative information about solutions. My goal is to introduce the audience to the topic of regularity for elliptic PDEs under assumptions on the coefficients that are of minimal requirements.

This thesis studies novel and efficient computational sampling methods for applications in three types of stochastic inversion problems: seismic waveform inversion, filtering problems, and static parameter estimation.

Backward error analysis is a tool that allows us to understand the effect of errors without knowing exact solutions.

Filtering (or data assimilation) is the process of combining noisy observational data with an output of a numerical model to produce an optimal estimate of the evolving state of the system.

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.

Many alternatives to the probabilistic modelling of information have been proposed since the birth of modern Statistics; yet, few have been successfully applied to the complex inference problems that modern Statisticians are faced with.

Fast and accurate hourly forecasts of wind speed and power are crucial in quantifying and planning the energy budget in the electric grid.

Abstract

We present double enriched finite volume spaces for the simulation of free particles in a

In the classical theory of the finite element approximation of elliptic partial differential equations, based on standard Galerkin schemes, the energy norm of the error decays with the same rate of convergence as the best finite element approximation, without any additional requirements on the involved spaces.

Bayesian inference for nonlinear diffusions, observed at discrete times,is a challenging task that has prompted the development of a number of algorithms, mainly within the computational statistics community. We propose a new direction, and accompanying methodology - borrowing ideas from statistical physics and computational chemistry - for inferring the posterior distribution of latent diffusion paths and model parameters, given observations of the process

We consider two sets of new priors for Bayesian inversion and machine learning: The first one is based on mixture of experts models with Gaussian processes. The target is to estimate the number of experts and their parameters, and to make state estimation. For sampling, we use SMC^2. For non-Gaussian priors, we discuss Cauchy priors and the generalisation to high-order Cauchy fields and further generalisation to alpha-stable fields. For sampling, we use a selection of modern MCMC tools. Finally, we apply some of the methods and models to an industrial tomography problem on estimating log internal structure, measured at sawmills, based on X-ray, RGB camera and laser scanning.

Constructing functional representations of the key quantities of interest (QoIs), the ignition delay time (_ign), of an uncertain ignition reaction in high dimension is our main goal. First, attention is focused on the ignition delay time of an iso-octane air mixture, using a detailed chemical mechanism with 3,811 elementary reactions. Uncertainty in all reaction rates is directly accounted for using associated uncertainty factors, assuming independent log uniform priors. A Latin hypercube sample (LHS) of the ignition delay times was first generated, and the resulting database was then exploited to assess the possibility of constructing polynomial chaos (PC) representations in terms of the canonical random variables parameterizing the uncertain rates.

his talk reviews some fundamental and practical issues related to the formulation and analysis of joint models of mixed types of outcomes with latent variables, with particular emphasis on both several case-studies in applied statistics and their computational implementation

Fluid dynamics models are ubiquitous in a multitude of applications. One of the most important applications of fluid dynamics models is numerical weather prediction. Modern numerical weather prediction combines sophisticated nonlinear fluid dynamics models with increasingly accurate high-dimensional data.  This process is called data assimilation and it is performed every day at all major operational weather centers across the world. Data assimilation  (DA) requires massive computing capabilities as realistic atmosphere-ocean models typically have billions of degrees of freedom. I will give a short overview of the ongoing research that aims to drastically decrease the required DA computational effort by reducing the dimension of the models involved and using stochastic perturbations to account for the unresolved scales. The incorporation of observation data is done by using particle approximations suitably adapted to solve high-dimensional problems.

Partial differential equations (PDEs) are used to describe multi-dimensional physical phenomena. However, some of these phenomena are described by a more general class of systems called fractional systems (FS). 

Semi-implicit schemes for evolutionary partial differential equations. Topic 3 - construction of more general schemes for evolutionary partial differential equations, in which the stiffness may be of a different type than the one previously considered. Several examples will be given illustrating the general procedure.

The qualitative study of PDEs often relies on integral identities and inequalities. For example, for time-dependent  PDEs, conserved integral quantities or quantities that are dissipated play an important role. In particular, if these integral quantities have a definite sign, they are of great interest as they may provide control on the solutions to establish well-posedness.

Implicit-Explicit schemes for hyperbolic systems with stiff relaxation. Topic 2 - hyperbolic relaxation models and to the methods for their numerical solution. After introduction of hyperbolic-hyperbolic and hyperbolic-parabolic type relaxation problem, conservative finite difference space discretization will be introduced.

Construction of high order finite volume and finite difference shock-capturing schemes for conservation laws. Topic 1 - illustrating how to construct shock capturing schemes for conservation laws. We focus on semi-discrete schemes based on the method of lines.

The numerical solution of multi-physics problems relying on the Navier-Stokes equations has kept multiple generations of supercomputers busy. For fundamental problems, computational fluid dynamic aims to resolve all relevant time and length scales, which is then known as direct numerical simulation (DNS).

During my Ph.D. program, we have studied mean-field games (MFGs). MFGs model games with large populations of rational agents. The agents search for their optimal strategies and trajectories to minimize an individual cost, which depends on the statistical distribution of the population. Although it is quite hard to consider the systems of large populations in the numerical analysis, we can expect to consider the average effect given by the populations because the influence of each agent should be small.

Singular and degenerate partial differential equations are unavoidable in the modelling of several phenomena, from phase transitions to flows in porous media or chemotaxis. They encompass a crucial issue in the analysis of pdes, namely wether we can still derive analytical estimates when the crucial algebraic assumption of ellipticity collapses.

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.

Assessing the effectiveness of cancer treatments in clinical trials raises multiple methodological challenges that need to be properly addressed in order to produce a reliable estimate of treatment effects.

We consider the problem of parameter estimation for a McKean stochastic differential equation, and the associated system of weakly interacting particles. The problem is motivated by many applications in areas such as neuroscience, social sciences (opinion dynamics, cooperative behaviours), financial mathematics, statistical physics. We will first survey some model properties related to propagation of chaos and ergodicity and then move on to discuss the problem of parameter estimation both in offline and on-line settings. In the on-line case, we propose an online estimator, which evolves according to a continuous-time stochastic gradient descent algorithm on the asymptotic log-likelihood of the interacting particle system. The talk will present our convergence results and then show some numerical results for two examples, a linear mean field model and a stochastic opinion dynamics model. This is joint work with Louis Sharrock, Panos Parpas and Greg Pavliotis. Preprint: https://arxiv.org/abs/2106.13751