Cellular Probabilistic Automata - A Novel Method for Uncertainty Propagation

We propose a novel density based numerical method for uncertainty propagation under certain partial differential equation dynamics. The main idea is to translate them into objects that we call cellular probabilistic automata and to evolve the latter. The translation is achieved by state discretization as in set oriented numerics and the use of the locality concept from cellular automata theory. We develop the method at the example of initial value uncertainties under deterministic dynamics and prove a consistency result. As an application we discuss arsenate transportation and adsorption in drinking water pipes and compare our results to Monte Carlo computations

Stochastic inflation in the Constant Roll regime

Màster Oficial d'Astrofísica, Física de Partícules i Cosmologia, Facultat de Física, Universitat de Barcelona. Curs: 2022-2023. Tutor: Cristiano GermaniWe investigate the inhomogeneities generated during the inflationary epoch from the point of view of the stochastic formalism, which attempts to transform a problem of quantum fluctuations into a statistical one. The formalism, that we derive in the text, is based on the use of the Arnowitt-Deser-Misner (ADM) equations, which are convenient to describe inhomogeneities in the context of inflation, as well as gradient expansion, which works at zeroth order in spatial gradients but at all orders in the amplitudes of the fluctuations, and is therefore intended to capture non-perturbative effects. Finally, the perturbations are split into long- and short-wavelength modes, where the latter act as a stochastic noise for the former when crossing a certain scale. We demonstrate that the use of certain approximations in the derivation of this formalism, which are intended to make the system of stochastic differential equations (SDEs) Markovian and described with white noises, causes the method to become restricted to the reproduction of Linear Perturbation Theory (LPT). This framework, nonetheless, is still useful since it can be used as a test for the validity of the linear approximation, signalling the coming into play of non-perturbative effects. Specifically, we solve the system of SDEs numerically for the Constant Roll (CR) inflationary scenario, and show that this regime is in accordance with LPT

Runge-Kutta methods for rough differential equations

We study Runge-Kutta methods for rough differential equations which can be used to calculate solutions to stochastic differential equations driven by processes that are rougher than a Brownian motion. We use a Taylor series representation (B-series) for both the numerical scheme and the solution of the rough differential equation in order to determine conditions that guarantee the desired order of the local error for the underlying Runge-Kutta method. Subsequently, we prove the order of the global error given the local rate. In addition, we simplify the numerical approximation by introducing a Runge-Kutta scheme that is based on the increments of the driver of the rough differential equation. This simplified method can be easily implemented and is computational cheap since it is derivative-free. We provide a full characterization of this implementable Runge-Kutta method meaning that we provide necessary and sufficient algebraic conditions for an optimal order of convergence in case that the driver, e.g., is a fractional Brownian motion with Hurst index $\frac{1}{4} < H \leq \frac{1}{2}$. We conclude this paper by conducting numerical experiments verifying the theoretical rate of convergence