Probabilistic and Bayesian methods for uncertainty quantification of deterministic and stochastic differential equations

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, meteorology, and economics. Mathematical models of complex systems in these fields require numerical methods, which introduce uncertainties in the outcome. Moreover, there has recently been a steep rise in the availability of data, which also come with an uncertainty. Therefore, blending mathematical models, data, and their respective uncertainties is nowadays of the utmost importance. The first part of this thesis is dedicated to two novel methods for multiscale inverse problems. We first consider an elliptic partial differential equation (PDE), with a diffusion tensor oscillating at a small scale. Given noisy observations of the solution, we consider the problem of inferring a slow-scale parametrization of the multiscale tensor. For this purpose, we combine numerical homogenization, which yields a single-scale surrogate of the full model, and the ensemble Kalman filter. The scheme we propose is accurate in the homogenized limit, and outperforms existing methods in terms of computational cost. We then study the error due to the mismatch between the full and the homogenized models, and show how to combine statistical techniques for model misspecification and our scheme. We then move to multiscale diffusion processes, and consider the problem of inferring effective dynamics from multiscale observations. A homogenized single-scale equation reproducing the full model exists also in this case. The effective model, though, is the subject of the inference procedure, and not only a computational tool. The resulting issue of model misspecification is usually bypassed by subsampling at an appropriate rate, which is non-trivial to choose, and which may give misleading results. We avoid subsampling by designing a novel technique based on filtered data, and show how to modify classical estimators and obtain an effective equation consistently with homogenization. Our technique is robust and can be employed as a black-box tool for inferring effective surrogates of complex stochastic models.In the second part we present two novel schemes belonging to the field of probabilistic numerics, whose purpose is to provide a statistical description of the uncertainty due to numerical discretization. We first consider ordinary differential equations (ODEs), and introduce a probabilistic integrator based on random time steps and Runge-Kutta methods (RTS-RK). Tuning the distribution of the time steps, we generate a probability measure on the solution which allows for a consistent uncertainty quantification of numerical errors. Unlike previous probabilistic methods in literature, our scheme inherits the geometric properties of the underlying deterministic integrators. In particular, we show long-time energy conservation when the RTS-RK is applied to Hamiltonian ODEs. We employ the idea of randomizing the discretization to propose a random mesh finite element method (RM-FEM) for elliptic PDEs. We prove that the measure induced by the RM-FEM on the solution can be employed to derive a posteriori error estimators. Hence, the RM-FEM provides a consistent statistical characterization of numerical errors. For both our novel schemes, we demonstrate the usefulness of the probabilistic approach in Bayesian inverse problems.