Graduate Thesis of Mario Muñoz
 | Program | PhD in Applied Sciences with mention in Mathematical Engineering, Universidad de Concepción |
|---|---|---|
| Enrollment Year | 2018 | |
| Senior Year | 2023 | |
| Thesis Title | Numerical Methods for two types of Stochastic Differential Equations with Nonglobally Lipschitz Coefficients | |
Thesis Summary:This doctoral thesis focuses on the numerical solution of Stochastic Differential Equations (SDEs) with non-globally Lipschitz coefficients. It involves two independent investigations that propose different procedures for the effective numerical simulation of these models. The first investigation centers on the numerical solution of the non-linear stochastic Schr¨odinger equation, which is a stochastic differential equation with locally Lipschitz con- tinuous coefficients commonly used to model quantum measurement processes. We analyze the rate of weak convergence of an exponential scheme that reproduces the norm of the de- sired solution by using a projection onto the unit sphere. In particular, we prove that the exponential scheme converges with weak-order one, and obtain the leading order term of its weak error expansion. This justifies using the Talay-Tubaro extrapolation procedure in the numerical simulation of open quantum systems. By employing this procedure, a second- order method for computing mean values of smooth functions of the solution is obtained. Furthermore, we prove that the exponential scheme under study has order of strong con- vergence 1/2, validating its application in the Multilevel Monte Carlo method. Numerical experiments involving a quantized electromagnetic field interacting with a reservoir showcase the effectiveness of the proposed methods. The second investigation introduces a new methodology for the effective pathwise nu- merical simulation of stochastic differential equations with non-globally Lipschitz continuous coefficients. Specifically, we focus on SDEs with linear multiplicative noise. We employ a suitable invertible continuous transformation to establish a connection between the orig- inal SDE and an auxiliary Random Differential Equation (RDE). This explicit conjugacy enables the development of new pathwise numerical schemes for the studied SDE, utilizing numerical approximations of the auxiliary RDE. In particular, we introduce two numerical methods: one based on an exponential scheme and the other based on the Heun scheme. In order to showcase the practical applicability of our approach, we implement it within a compartmental epidemic model, specifically the stochastic SVIR model. This SDE captures the dynamics of a continuous vaccination strategy in the presence of environmental noise effects. Through comparative analysis with commonly used numerical approximations, we validate the effectiveness of our proposed numerical methods for simulating epidemiological models. | ||
| Thesis Director(s) | Hugo de la Cruz Cancino, Carlos Mora | |
| Thesis Project Approval Date | 2021, January 27 | |
| Thesis Defense Date | 2023, December 06 | |
| Professional Monitoring | ||
| PDF Thesis | Download Thesis PDF  | |
ISI Publications from the ThesisCarlos M. MORA, Mario MUñOZ: On the rate of convergence of an exponential scheme for the non-linear stochastic Schrödinger equation with finite-dimensional state space. Physica Scripta, vol. 98, 6, article: 065226, (2023). |
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