Felipe Vinicio Caro Gutiérrez will defend his thesis on 29 November

• The defense of the thesis will take place in the Salón de Grados of the Faculty of Science and Technology of the UPV/EHU in Leioa

Felipe Vinicio Caro Gutiérrez began his academic journey at the Universidad de Guadalajara, where he earned his bachelor's degree in mathematics in 2016. He then received a master's degree from the prestigious Centro de Investigación Científica y de Educación Superior de Ensenada in 2018.

In September 2019, Felipe joined the Basque Center for Applied Mathematics – BCAM. His research primarily focuses on the cutting-edge areas of physics-informed neural networks (PINNs) and the adaptive finite element method (AFEM). 

His thesis, "Easy-to-implement hp-adaptivity for non-elliptic goal-oriented problems," is under the expert guidance of Professor David Pardo and Elisabete Alberdi Celaya, both academics at the Universidad del País Vasco / Euskal Herriko Unibertsitatea. 

The defense will take place on Wednesday 29 November in the Salón de Grados of the Faculty of Science and Technology of the UPV/EHU (Leioa) at 12:00h.

On behalf of all BCAM members, we would like to wish Felipe the best of luck in his upcoming thesis defense.

Abstract

In engineering, precise error control in specific domains related to a particular Quantity of Interest (QoI) is often more critical than a generalized focus on the overall solution. This necessity has led to the development of goal-oriented adaptive (GOA) strategies.

In this dissertation, we develop automatic GO hp-adaptive algorithms tailored explicitly for non-elliptic problems. These algorithms are robust and straightforward to implement, making them highly suitable for various industrial applications. A key advantage of our methodologies is their independence from computing reference solutions on globally refined grids. However, it should be noted that our approach primarily utilizes anisotropic p-adaptations and isotropic h-adaptations, which may limit its applicability in specific scenarios.

Our validation process involves a series of rigorous tests. We examine the convergence behavior of our GO h- and p-adaptive algorithms by applying Helmholtz and convection-diffusion equations in one-dimensional settings. Using our GO hp-adaptive algorithms, we extend our testing for two-dimensional scenarios to include Poisson, Helmholtz, and convection-diffusion equations. In three-dimensional contexts, we employ a Helmholtz-like scenario to demonstrate the versatility and adaptability of our GO algorithms. Additionally, we have developed efficient methodologies for constructing extensive databases, ideally suited for training Deep Neural Networks (DNNs).

These methods leverage the capabilities of the hp Multi-Adaptive Goal-Oriented (MAGO) Finite Element Method (FEM). As a result, we can efficiently generate large-scale databases, potentially comprising hundreds of thousands of synthetic datasets or measurements, which are crucial for the practical training of DNN models.