(P1) Métodos analítico-numéricos para sistemas de EDPs débilmente acoplados
Silvia Jerez Galiano, Centro de Investigación en Matemáticas, A.C.
Resumen: En esta charla se presentan algoritmos convergentes para la resolución de sistemas del tipo advección-difusión-reacción no lineales. La idea clave de estos algoritmos es la discretización de ciertas técnicas analíticas usadas en el estudio de la existencia y unicidad de solución para problemas parabólicos débilmente acoplados. De este modo, la aproximación numérica mantiene propiedades cualitativas que verifica la solución exacta (positividad, acotación, monotonía, etc.) y se asegura su convergencia para problemas no lineales. Se presentan algunos ejemplos de aplicación.
(P2) Exponential, Non-Exponential Splitting Methods and Their Applications for Solving Singular Reaction-Diffusion Equations
Dr. Qin Sheng, Department of Mathematics, Center for Astrophysics, Space Physics and Engineering Research (CASPER), Baylor University, Texas, USA.
Abstract: This talk consists of two interactive components. First, we will pay an attention to optimized classical splitting methods, such as the non-exponential ADI and exponential LOD methods, and explore their modernizations. Then we will focus at interesting issues involving the design and analysis of highly-effective and highly-efficient finite difference methods for solving singular reaction-diffusion equations which are fundamental in numerical combustion simulatyions. We will outline the physical background of the quenching phenomena anticipated. Adaptive splitting approaches will be introduced. Numerical analysis on their monotonicity, convergence and stability will be discussed. We will also present ideas of the latest exponential evolving grid development inspired by moving grid strategies which can be extended for solving multiphysics equations with similar singularities from studies of biomathematics, oil pipeline decay detections, cancer treatments, and laser-materials interactions. Certain stochastic inferences will be mentioned. Potentials of further investigations and collaborations will be discussed.
(P3) Relaxation Techniques in Optimization and Control
Dr. Vadim Azhmyako, Universidad de Medellin
ABSTRACT: “Relaxing the initial problem” has various meanings in Applied Mathematics, depending on the areas where it is defined, depending also on what one relaxes (a functional, the underlying space, etc.). In the context of an Optimal Control Problem, when dealing with the minimization of an objective functional, the most common way of looking at relaxation is to consider the lower-semicontinuous hull of this functional determined on a convexification of the set of admissible controls. The concept of relaxed controls was introduced by L.C. Young in 1937 under the name of generalized curves and surfaces. It has been used extensively in the professional literature for the study of diverse Optimal Control problems. It is common knowledge that a real-world Optimal Control problem does not always have a (mathematical) solution. On the other hand, the corresponding relaxed problem has an optimal solution under some mild assumptions. In practice, this solution can be considered as a suitable approximation for the sophisticated initial problem. In the absence of the so-called “relaxation gap”, the generalized problem is of prime interest for the initial Optimal Control Problem. In this case, the minimal value of the objective functional in the initial problem coincides with the minimum of the objective functional in the relaxed problem. Therefore, in that situation, an adequately relaxed problem can be used as a theoretic fundament for adequate numerical solution algorithms for the initial problem. When solving Optimal Control Problems with ordinary differential equations, we deal with functions and systems which, except in very special cases, are to be replaced by numerically tractable approximations. In contrast to the conventional Optimal Control problems an effective implementation of adequate computational schemes for Hybrid / Switched systems optimization is predominantly based on the relaxed controls. Therefore, our aim is to consider the relaxations of the Hybrid and Switched Optimal Control problems in a close methodological relationship to the corresponding numerical methods and possible engineering applications.
Recall that various types of Hybrid and Switched control systems and the related Optimal Control problems have been comprehensively studied in the past several years due to their important engineering applications. Let us mention here some real-world applications from the mobile robot technology, intelligent automotive control, modern telecommunications, process control and data science. We first give an extensive overview of the existing (conventional and newly developed) relaxation techniques associated with the “conventional” systems described by ordinary differential equations. Next we construct a self-contained relaxation theory for Optimal Control processes governed by various types (sub-classes) of general Hybrid and Switched Systems. Note that due to the extreme complexity of Hybrid / Switched dynamic systems this “construction” is a challenging analytic and computational problem and cannot be considered as a simple “theory / facts transfers” from the conventional Optimal Control to hybrid and switched cases. Let us also note that the book we propose contains all mathematical tools that are necessary for an adequate understanding and using of the sophisticated relaxation techniques. All in all, this manuscript follows the “engineering” and “numerical” concepts. However, it can also be considered as a mathematical “compendium” that contains all the necessary formal results and some important algorithms related to the modern relaxation theory. This fact makes it possible to use this book in systems engineering (specifically in electrical- aerospace- and financial engineering) and in practical systems optimization.