Energy-Conservative Data-Driven Modelling for the two-scale Navier-Stokes Equations

This thesis presents an energy-conservative data-driven approach in modelling the closure terms of the Navier-Stokes equations casted through the Variational Multiscale (VMS) framework. For context, the VMS framework is applied in designing stabilised finite element methods for multiscale phenomena in which stability is not guaranteed. Under...

Stochastic FitzHugh-Nagumo Equations

The stochastic FitzHugh-Nagumo equations are a system of stochastic partial differential equations that describes the propagation of action potentials along nerve axons. In the present work we obtain well-posedness and regularisation results for the FitzHugh-Nagumo equations with domain R^d. We begin by considering the weak critical variational...

Application of Deep Neural Networks to the Operator Space of Nonlinear PDE for Physics-based Proxy Modeling

Compositional simulation is computationally intensive for high-fidelity models due to thermodynamic equilibrium relations and the coupling of flow, transport and mass transfer. In this report, two methods for accelerated compositional simulation are outlined and demonstrated for a gas vaporization problem. The first method uses a proxy model...

Multiple Markov properties for fractional parabolic SPDEs

Spatiotemporal stochastic processes have applications in various fields, but they can be difficult to numerically approximate in a reasonable time, in particular, in the context of statistical inference for large datasets. <br/>Recently, a new approach for efficient spatiotemporal statistical modeling has been proposed, where the space-time...

Stabilizing convection-dominated flow problems using neural networks based on flux-limiting techniques

Convection-dominated flow problems are well-known to have non-physical oscillations near steep gradients or discontinuities in the solution when solved with standard numerical methods, such as finite elements or finite difference methods. To overcome this limitation, algebraic flux correction (AFC) can be used, which is a stabilization method....

A Stochastic Parametrically-Forced NLS Equation

In this thesis, a variation on the nonlinear Schrödinger (NLS) equation with multiplicative noise is studied. In particular, we consider a stochastic version of the parametrically-forced nonlinear Schrödinger equation (PFNLS), which models the effect of linear loss and the compensation thereof by phase-sensitive amplification in pulse...

Magnetic Behaviour of a Steel Ellipsoid

Reliable and efficient modelling of magnetic hysteresis in inhomogeneous and aniso-tropic media is an important step in developing a state-of-the-art closed-loop degaussing system for naval ships and submarines, to be developed by TNO and to be used by the Royal Netherlands Navy in an updated generation of naval vessels and submarines. Different...

Multi-Level Inversion Based On Mesh Decoupling

Understanding the permeability of the subsurface is a crucial step to simulate fluid flow in the subsurface. A parameter estimation problem for the flow equations can be solved to find the permeability. The robust identification of material parameters remains a significant challenge. In classical approaches, the non-linear least-squares problem...

Solving Partial Differential Equations with Neural Networks

Recent works have shown that neural networks can be employed to solve partial differential equations, bringing rise to the framework of physics informed neural networks.The aim of this project is to gain a deeper understanding of these novel methods, and to use these insights to further improve them. We show that solving a partial differential...

Optimal boundary point control for linear elliptic equations

This master thesis is concerned with optimal boundary point and function control problems for linear elliptic equations subject to control constraints. The elliptic partial differential equation with Robin boundary condition is considered. The control is chosen as a linear combination of the Dirac delta functions in the point control problem....

A nonlinear Schrödinger equation in L² with multiplicative white noise

In this thesis existence and uniqueness of the solution of a nonlinear Schrödinger equation in L² with multiplicative white noise is proven under some assumptions. Furthermore, pathwise L²-norm conservation of the solution is studied.