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In this thesis we explore uncertainty quantification of forward and inverse problems involving differential equations. Differential equations are widely employed for modeling natural and social phenomena, with applications in engineering, chemistry, meteor ...
EPFL2021
Mathematical models involving multiple scales are essential for the description of physical systems. In particular, these models are important for the simulation of time-dependent phenomena, such as the heat flow, where the Laplacian contains mixed and ind ...
We establish a sharp estimate on the negative moments of the smallest eigenvalue of the Malliavin matrix gamma z of Z := (u(s, y), u(t , x) - u(s, y)), where u is the solution to a system of d non-linear stochastic heat equations in spatial dimension k >= ...
We study the problem of learning unknown parameters of stochastic dynamical models from data. Often, these models are high dimensional and contain several scales and complex structures. One is then interested in obtaining a reduced, coarse-grained descript ...
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 ...
We propose a data-driven Model Order Reduction (MOR) technique, based on Artificial Neural Networks (ANNs), applicable to dynamical systems arising from Ordinary Differential Equations (ODEs) or time-dependent Partial Differential Equations (PDEs). Unlike ...
2019
In a recent work, Bourgain gave a fine description of the expectation of solutions of discrete linear elliptic equations on Zd with random coefficients in a perturbative regime using tools from harmonic analysis. This result is surprising for it goes beyon ...
We consider sample path properties of the solution to the stochastic heat equation, in Rd or bounded domains of Rd, driven by a Levy space-time white noise. When viewed as a stochastic process in time with values in an infinite-dimensional space, the solut ...
This paper proposes an algorithm to upper-bound maximal quantile statistics of a state function over the course of a Stochastic Differential Equation (SDE) system execution. This chance-peak problem is posed as a nonconvex program aiming to maximize the Va ...
Biological oscillators are pervasive in biology, covering all aspects of life from enzyme kinetics reactions to population dynamics. Although their behaviour has been intensively studied in the last decades, the recent advances of high-throughput experimen ...
