The non-linear stochastic wave equation in high dimensions : existence, Hölder-continuity and Itô-Taylor expansion

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We prove global in time well-posedness for perturbations of the 2D stochastic Navier-Stokes equations partial derivative( t)u + u center dot del u = Delta u - del p + sigma + xi, u(0, center dot ) = u(0),div (u) = 0, driven by additive space-time white noi ...

We present a combination technique based on mixed differences of both spatial approximations and quadrature formulae for the stochastic variables to solve efficiently a class of optimal control problems (OCPs) constrained by random partial differential equ ...

Consider the wave equation with heterogeneous coefficients in the homogenization regime. At large times, the wave interacts in a nontrivial way with the heterogeneities, giving rise to effective dispersive effects. The main achievement of the present wor ...

We consider a class of nonlocal conservation laws with exponential kernel and prove that quantities involving the nonlocal term W := 1(-infinity,0](center dot)exp(center dot) * rho satisfy an Oleinik-type entropy condition. More precisely, under different ...

We establish shape holomorphy results for general weakly- and hyper-singular boundary integral operators arising from second-order partial differential equations in unbounded two-dimensional domains with multiple finite-length open arcs. After recasting th ...

We propose a local, non -intrusive model order reduction technique to accurately approximate the solution of coupled multi -component parametrized systems governed by partial differential equations. Our approach is based on the approximation of the boundar ...

In this work, we analyze space-time reduced basis methods for the efficient numerical simulation of haemodynamics in arteries. The classical formulation of the reduced basis (RB) method features dimensionality reduction in space, while finite difference sc ...

We consider on the torus the scaling limit of stochastic 2D (inviscid) fluid dynamics equations with transport noise to deterministic viscous equations. Quantitative estimates on the convergence rates are provided by combining analytic and probabilistic ar ...

In this thesis we will present two results on global existence for nonlinear dispersive equations with data at or below the scaling regularity. In chapter 1 we take a probabilistic perspective to study the energy-critical nonlinear Schrödinger equation in ...

Predicting the evolution of systems with spatio-temporal dynamics in response to external stimuli is essential for scientific progress. Traditional equations-based approaches leverage first principles through the numerical approximation of differential equ ...

Despite the widespread empirical success of ResNet, the generalization properties of deep ResNet are rarely explored beyond the lazy training regime. In this work, we investigate scaled ResNet in the limit of infinitely deep and wide neural networks, of wh ...

We study the hitting probabilities of the solution to a system of d stochastic heat equations with additive noise subject to Dirichlet boundary conditions. We show that for any bounded Borel set with positive (d-6)\documentclass[12pt]{minimal} \usepackage{ ...

The thesis is dedicated to the study of two main partial differential equations (PDEs) in fluid dynamics: the Navier-Stokes equations, which describe the motion of incompressible fluids, and the transport equation with divergence-free velocity fields, whic ...

In algorithms for solving optimization problems constrained to a smooth manifold, retractions are a well-established tool to ensure that the iterates stay on the manifold. More recently, it has been demonstrated that retractions are a useful concept for ot ...

There is a bias in the inference pipeline of most diffusion models. This bias arises from a signal leak whose distribution deviates from the noise distribution, creating a discrepancy between training and inference processes. We demonstrate that this signa ...

Population equations for infinitely large networks of spiking neurons have a long tradition in theoret-ical neuroscience. In this work, we analyze a recent generalization of these equations to populations of finite size, which takes the form of a nonlinear ...

We consider the problem of nonparametric estimation of the drift and diffusion coefficients of a Stochastic Differential Equation (SDE), based on n independent replicates {Xi(t) : t is an element of [0 , 1]}13 d B(t), where alpha is an element of {0 , 1} a ...

Functional data are typically modeled as sample paths of smooth stochastic processes in order to mitigate the fact that they are often observed discretely and noisily, occasionally irregularly and sparsely. The smoothness assumption is imposed to allow for ...

Correct prediction of particle transport by surface waves is crucial in many practical applications such as search and rescue or salvage operations and pollution tracking and clean-up efforts. Recent results by Deike et al. (J. Fluid Mech., vol. 829, 2017, ...

We introduce the elliptical Ornstein-Uhlenbeck (OU) process, which is a generalisation of the well-known univariate OU process to bivariate time series. This process maps out elliptical stochastic oscillations over time in the complex plane, which are obse ...

The non-linear stochastic wave equation in high dimensions : existence, Hölder-continuity and Itô-Taylor expansion

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Global existence for perturbations of the 2D stochastic Navier-Stokes equations with space-time white noise

A Combination Technique for Optimal Control Problems Constrained by Random PDEs

A Spectral Ansatz for The Long-Time Homogenization of The Wave Equation

Oleinik-type estimates for nonlocal conservation laws and applications to the nonlocal-to-local limit

Shape Holomorphy of Boundary Integral Operators on Multiple Open Arcs

Model reduction of coupled systems based on non-intrusive approximations of the boundary response maps

SPACE-TIME REDUCED BASIS METHODS FOR PARAMETRIZED UNSTEADY STOKES EQUATIONS

Quantitative convergence rates for scaling limit of SPDEs with transport noise

Probabilistic and Deterministic Wellposedness for Low Regularity Dispersive Equations

Learning the intrinsic dynamics of spatio-temporal processes through Latent Dynamics Networks

Generalization of Scaled Deep ResNets in the Mean-Field Regime

Hitting with Probability One for Stochastic Heat Equations with Additive Noise

Anomalous dissipation and other non-smooth phenomena in fluids

From low-rank retractions to dynamical low-rank approximation and back

Exploiting the Signal-Leak Bias in Diffusion Models

On a Finite-Size Neuronal Population Equation

Nonparametric estimation for SDE with sparsely sampled paths: An FDA perspective

Functional data analysis with rough sample paths?

Stochastic particle transport by deep-water irregular breaking waves

The elliptical Ornstein-Uhlenbeck process