Actividades

Nishant Mehta, Department of Computer Science, University of Victoria.

Miércoles 10 de agosto de 2022, 13 hrs. (Vía Zoom, enlace a través del correo Esta dirección de correo electrónico está siendo protegida contra los robots de spam. Necesita tener JavaScript habilitado para poder verlo.)

ABSTRACT

I will begin by introducing an online learning problem motivated by group fairness considerations. It is standard in online learning to prove sublinear upper bounds on the regret, a key performance measure in online learning and online convex optimization. An alternative concept is a best-case lower bound — the largest improvement an algorithm can obtain relative to the single best action in hindsight. Best-case lower bounds have connections to fairness: it is known that best-case lower bounds are instrumental in obtaining algorithms for the popular decision-theoretic online learning (DTOL) setting that satisfy a notion of group fairness. A parallel motivation of this work is to better understand the adaptivity of a learning algorithm; while some algorithms provably exhibit certain types of adaptivity, we show that they are provably prohibited from obtaining another desirable form of adaptivity (related to what is known as the shifting regret). Our contributions are a general method to provide best-case lower bounds in Follow the Regularized Leader (FTRL) algorithms with time-varying regularizers, which we use to show that best-case lower bounds are often of the same order as existing upper regret bounds: this includes situations with a fixed learning rate, decreasing learning rates, and adaptive gradient methods. We also show that the popular linearized version of FTRL can attain negative linear regret and hence does not admit non-trivial best-case lower bounds in general.

This is joint work with Cristóbal Guzmán and Ali Mortazavi.

Cristián Escauriaza, Departamento de Ingeniería Hidráulica y Ambiental, Pontificia Universidad Católica de Chile.

Miércoles 1 de junio de 2022, 13 hrs. (Presencial en Auditorio Edificio San Agustín.)

ABSTRACT

Las zonas de almacenamiento superficial en ambientes fluviales y costeros se caracterizan por grandes regiones laterales de recirculación, dominadas por múltiples estructuras coherentes turbulentas que interactúan entre sí y con los bordes. Estos flujos que poseen velocidades más bajas, juegan un papel fundamental en el transporte de contaminantes y de sedimentos, y en la absorción de nutrientes en ríos y en la costa. Sin embargo, la dinámica de las estructuras coherentes en estas zonas es altamente compleja, con múltiples escalas espaciales y temporales. Modelos numéricos de alta resolución que capturan estos flujos a altos números de Reynolds proporcionan información sobre los mecanismos de transporte y los factores que influyen a escalas espaciales más grandes. En este trabajo estudiamos los procesos físicos utilizando simulaciones numéricas de las ecuaciones filtradas de Navier-Stokes junto con ecuaciones de transporte. Implementamos un modelo Lagrangiano de partículas para estudiar tiempos de residencia y realizar análisis estadísticos de trayectorias que permiten comprender los impactos a mayor escala, y sus implicancias en parametrizaciones de transporte.

David M. Hernandez, Institute for Theory and Computation, Harvard-Smithsonian Center for Astrophysics.

Miércoles 18 de mayo de 2022, 13 hrs. (Presencial en Auditorio San Agustín)

ABSTRACT

I describe new mathematical tools I've built to solve different problems in gravitational dynamics. I first describe maps that solve the gravitational system of ordinary differential equations describing asteroids in the Solar System.  Enforcing that these maps be time-reversible and symplectic can significantly improve the reliability of the long-term dynamics of these bodies.

I then tackle the problem of the stability of the Solar System.  Although great progress has been made in the last decades towards an understanding of chaos and stability of the Solar System due to the development of modern computers, I show that important studies are affected by numerical chaos, which causes artificial Solar System chaos and instability.  This numerical instability arises from resonances between the time step and physical Solar System frequencies, and is an inherent property of symplectic maps.  I discuss our current work to calculate Solar System stability, and in particular Mercury's future trajectory, without the effects of numerical chaos.

I next describe a suite of tools, including powerful new Kepler solvers and new symplectic integrators and their tangent equations that are designed to solve for the orbits of planets in exoplanetary systems.  Unlike other popular methods, we can solve planetary systems with arbitrary geometries and orbits including moons.  We have implemented these tools to solve the transit timing variation problem, and derive the properties and possible compositions of TRAPPIST-1 planets.  Some of this work has been incorporated in the popular Rebound code.

 Benjamín Palacios, Departamento de Matemáticas, Pontificia Universidad Católica de Chile.

Miércoles 15 de junio de 2022, 13 hrs. (Presencial en Auditorio Edificio San Agustín.)

ABSTRACT

Photoacoustic Tomography (PAT) is a promising hybrid medical imaging modality that is able to generate high-resolution and high-contrast images by exploiting the coupling of electromagnetic pulses (in the visible region) and ultrasound waves via de photoacoustic effect. The mathematical problem splits into two steps, one involving the inversion of boundary acoustic data to determine the initial source of waves, and the second step uses this internal information to retrieve optical properties of the medium and it is commonly known as Quantitative PAT.

In this talk, I will introduce the modality and focus on the ultrasound propagation component which is mathematically modeled as an inverse initial source problem for the wave equation. I will then discuss mathematical aspects of this inverse problem and present some recent theoretical results. The last part of the presentation will be devoted to addressing some open questions related to reconstruction methods and numerical implementations.