Seminario de Matemáticas Aplicadas y Computacionales (SeMACs)

2021-04-28
13:00hrs.

Nicolás García Trillos. Department of Statistics, University of Wisconsin Madison
Adversarial classification, optimal transport, and geometric flows
https://zoom.us/j/93450667974
Abstract:
The purpose of this talk is to provide an explicit link between the three topics that form the talk's title, and to introduce a new perspective (more dynamic and geometric) to understand robust classification problems.  For concreteness, we will discuss a version of adversarial classification where an adversary is empowered to corrupt data inputs up to some distance \epsilon. We will first describe necessary conditions associated with the optimal classifier subject to such an adversary. Then, using the necessary conditions we derive a geometric evolution equation which can be used to track the change in classification boundaries as \veps varies. This evolution equation may be described as an uncoupled system of differential equations in one dimension, or as a mean curvature type equation in higher dimension. In one dimension we rigorously prove that one can use the initial value problem starting from \veps=0, which is simply the Bayes classifier, to solve for the global minimizer of the adversarial problem. Global optimality is certified using a duality principle between the original adversarial problem and an optimal transport problem. Several open questions and directions for further research will be discussed.

2021-04-07
13:00hrs.

Ngoc Cuong Nguyen. Massachusetts Institute of Technology
Discontinuous Galerkin Methods for Continuum Mechanics
https://zoom.us/j/93450667974
Abstract:
In this talk, we present the recent development of discontinuous Galerkin methods for solving  partial differential equations (PDEs) in continuum mechanics. The essential ingredients are a local Galerkin projection of the underlying PDEs at the element level onto spaces of polynomials of degree k; a judicious choice of the numerical flux to provide stability and consistency; and a jump condition that enforces the continuity of the numerical flux to arrive at a global weak formulation in terms of the numerical trace.  The present DG methods are fully implicit, high-order accurate and endowed with several unique features. First, they reduce the globally coupled unknowns to the approximate trace of the solution on element boundaries, thereby leading to a significant reduction in the degrees of freedom. Second, they provide, for smooth viscous-dominated problems, approximations of all the variables which converge with the optimal order of k + 1 in the L2-norm. Third, they possess some superconvergence properties that allow us to define inexpensive element-by-element postprocessing procedures to compute a new approximate solution which may converge with higher order than the original solution. And fourth, they allow for a novel and systematic way for imposing boundary conditions for the total stress, viscous stress, vorticity and pressure which are not naturally associated with the weak formulation of the methods. In addition, they possess other interesting properties for specific problems. Their approximate solution can be postprocessed to yield an exactly divergence-free and H(div)-conforming velocity field for incompressible flows. They do not exhibit volumetric locking for nearly incompressible solids. We provide extensive numerical results to illustrate their distinct characteristics and demonstrate their performance for a wide range of physical problems in solid mechanics, fluid mechanics, and electromagnetics. Finally, we discuss  an open-source software (https://github.com/exapde/Exasim) for generating discontinuous Galerkin codes to numerically solve parametrized partial differential equations on different computing platforms with distributed memory.

2021-03-31
13:00hrs.

Raphaël Pestourie. Department of Mathematics, Massachusetts Institute of Technology
Efficient inverse design for extreme applications in optics
https://zoom.us/j/93450667974
Abstract:
Optical metasurfaces are thin large-area structures with aperiodic subwavelength patterns, designed for focusing light and a variety of other wave transformations. Because of their irregularity and large scale, they are one of the most challenging tasks for computational design. This talk will present ways to harness the full computational power of modern large-scale optimization in order to design metasurfaces with thousands or millions of free parameters. To that end, we exploit domain-decomposition approximations and “surrogate” models based on Chebyshev interpolation or new active-learning neural networks. We will also present some recent experimental results on lenses with extended depth of field. Finally, we will discuss recent progress towards holistic "end-to-end" optimization that combines optical design with lensless image processing.

2020-12-02
13:00hrs.

Keaton Burns. Department of Mathematics, Massachusetts Institute of Technology
Flexible spectral methods and high-level programming for PDEs
https://zoom.us/j/91782510486
Abstract:
The large-scale numerical solution of partial differential equations (PDEs) is an essential part of scientific research. Decades of work have been put into developing fast numerical schemes for specific equations, but computational research in many fields is still largely software-limited. Here I will discuss how algorithmic flexibility and composability can enable new science, as illustrated by the Dedalus Project. Dedalus is an open-source Python framework that automates the solution of general PDEs using spectral methods. High-level abstractions allow users to symbolically specify equations, parallelize and scale their solvers to thousands of cores, and perform arbitrary analysis with the computed solutions. I will provide an overview the code’s design and the underlying sparse spectral algorithms, and show how they are enabling novel simulations of diverse hydrodynamical systems.  I will include astrophysical and geophysical applications using new bases for tensor-valued equations in spherical domains, immersed boundary methods for multiphase flows, and multi-domain simulations interfacing Dedalus with other PDE and integral equation solvers.

2020-11-25
13:00hrs.

Marc Bonnet. Cnrs-Inria-Ensta
Volume integral equations for scattering by inhomogeneities. Application to small-defect asymptotics and identification using topological derivative
https://zoom.us/j/91782510486
Abstract:
This talk addresses volume integral equation (VIE) formulation for the scattering of acoustic (or elastic) waves by material inhomogeneities that affect the leading-order term of the governing differential operator, and their use for the derivation and justification of the small-inclusion solution asymptotics and the topological derivatives (TDs) of objective functionals. In particular, we show how a simple reformulation of the zero-frequency VIE allows to establish its well-posedness by means of a simple Neumann series argument, for any inhomogeneity contrast. This in turn yields a well-posedness result for the frequency-domain VIE. We then show how the relevant VIEs provide (upon coordinate rescaling) a convenient and systematic foundation for both the derivation of asymptotic models and their justification. Finally, we explain the instrumental role played by the previously-mentioned reformulation of the zero-frequency VIE in the mathematical justification of qualitative inverse scattering methods based on the TD concept when the strength of the sought scatterers satisfies a limitation expressed in terms of the operator norm of a certain volume integral operator. We will close with numerical examples involving TD-based qualitative inverse scattering.
http://uma.ensta.fr/~mbonnet

2020-11-18
13:00hrs.

Digvijay Boob. Southern Methodist University (Engineering Management, Information and Systems)
First-order methods for some structured nonconvex function constrained optimization problems
https://zoom.us/j/91782510486
Abstract:
First-order (stochastic) methods have become popular for solving various problems ranging from composite optimization to saddle point problems. Recently, function constrained optimization became popular because it gives more modeling power and does not make assumption on the simplicity of constraint set. In particular, nonconvex function constrained optimization is very new area of algorithmic research. In this talk, I will present a first-order method for solving a structured nonconvex function constrained optimization where the objective can be deterministic or stochastic, smooth or nonsmooth, convex or nonconvex whereas the constraint will be a structured nonconvex function. Using the structure of the nonconvex constraint, we can show existence of KKT multiplier under a well-known Mangasarian-Fromovitz Constraint Qualification (MFCQ). Moreover, we establish convergence complexity for finding $\eps$-KKT point. Notably, the complexity is on par with gradient descent for unconstrained nonconvex unconstrained optimization up to some Lipschitz constants of constraint function while ensuring the feasibility of the solution. As an application, we consider the nonconvex sparsity inducing function constraints, e.g., SCAD, MCP, etc. We fit this problem into our framework and provide some numerical experiments to show effectiveness of the proposed method.

2020-11-11
13:00hrs.

Wernher Brevis. Escuela de Ingeniería UC
ESTELAS CONFINADAS DESARROLLADAS POR OBSTÁCULOS POROSOS DE MULTI-ESCALA EN FLUJOS EN CANALES
https://zoom.us/j/91782510486
http://imc.uc.cl/index.php/actividades/seminarios

2020-11-04
13:00hrs.

Nicolas Figueroa. Instituto de Economía, Pontificia Universidad Católica de Chile
Admisión Escolar y Universitaria: Algoritmos de Selección
https://zoom.us/j/91782510486
Abstract:
Presentamos una visión unificada de los sistemas de admisión escolar y universitarios en Chile, dado que ambos usan variantes del sistema de "Aceptación Diferida". Para el caso de la admisión universitaria, mostramos teórica y empíricamente como  la introducción de reglas especiales por la PUC y la UCH llevan a importantes pérdidas de bienestar en los estudiantes. Para el sistema escolar, mostramos como el nuevo sistema disminuye la brecha socioeconómica en la calidad académica de los colegios a los que son aceptados los estudiantes. Además, mostramos que la no-participación en el sistema es una razón importante de la desigualdad, y está concentrada en los alumnos más pobres. Finalmente, usando técnicas de bigdata, generamos un algoritmo para sugerir colegios a las familias, con resultados potencialmente importantes en la asignación final.

2020-10-28
13:00hrs.

Andres Abeliuk. University of Southern California / Universidad de Chile
El impacto de los sesgos algorítmicos en la sociedad
https://zoom.us/j/91782510486
Abstract:
Los algoritmos son los nuevos mediadores en la toma de decisiones en cada vez más áreas, desde ver una película hasta buscar trabajo o una cita. Sin embargo, al igual que el cerebro humano, la inteligencia artificial está sujeta a sesgos cognitivos, i.e., heurísticas mentales que pueden llevar a toma de decisiones y razonamientos incorrectos. En estos sistemas sociales, la interacción entre las decisiones individuales y los algorítmicos es capaz de reforzar sesgos y causar impactos negativos en la sociedad. En esta charla, se presentarán una serie de estudios realizados por Andrés Abeliuk, donde se analizará el impacto de los algoritmos en diversos sistemas sociales.

2020-10-21
13:00hrs.

Emmanuel Garza. University of Southern California, Department of Electrical and Computer Engineering
Boundary integral methods for simulation and optimization of photonic devices
https://zoom.us/j/91782510486
Abstract:
Devices capable of manipulating electromagnetic waves are a cornerstone of modern society. From radar sensors used for air traffic control, to the optical fibers that enable today’s high-speed Internet, the understanding of electromagnetic fields has revolutionized the world. In particular, nanophotonic devices are an emerging class of components which are capable of wielding light at wavelength scales. These devices promise to provide the next generational leap in data transfers by integrating with traditional electronic circuits. In this talk, we present boundary integral methods for the efficient simulation and optimization of nanophotonic devices. First, we show how the windowed Green function method can be used to simulate infinitely long nonuniform waveguide structures. In this approach, the boundaries of the waveguide are smoothly terminated by a smooth window function, while incident guided modes are accurately incorporated using Green’s representation formula to recast certain relevant slowly-decaying integrals into exponentially decaying integrals. In the second part of the talk, we present a framework for optimizing nanophotonic devices using boundary integral methods. In this framework, the gradient with respect to design parameters is computed efficiently using an adjoint formulation which requires only two simulations plus some sparse operations. We then demonstrate this technique by applying it to the design of waveguide splitters, tapers, gratings and metasurfaces.

2020-10-14
13:00hrs.

David Shirokoff. Department of Mathematical Sciences, New Jersey Institute of Technology
Unconditional stability for multistep ImEx schemes
https://zoom.us/j/91782510486
Abstract:
In this talk we devise unconditionally stable multistep ImEx schemes in problems where both the implicit, and explicit terms are stiff. Unconditional stability is a desirable property for a numerical scheme as it implies the absence of a (stiff) time step restriction.  One particular application where such an approach may be advantageous is in nonlinear problems, where a (simple) implicit term is taken to be a constant coefficient operator, and the stiff nonlinear terms are treated explicitly.  This then bypasses the need for nonlinear solvers.   We first use the new stability theory to explain the fundamental stability restrictions of the well-known semi-implicit backward differentiation formulas (SBDF). We then show, using the new theory, how to overcome the limitations of SBDF to obtain higher order schemes. Using this insight, rigorous, unconditionally stable schemes are devised for the linear variable coefficient diffusion problem.  We will then use the linear results to show that they can be used to avoid the implicit treatment of nonlinear terms in some nonlinear diffusion problems.  Numerical examples will be presented.

2020-10-07
13:00hrs.

Rajesh Jayaram. Carnegie Mellon University
Learning Two Layer Rectified Neural Networks in Polynomial Time
https://zoom.us/j/91782510486
Abstract:
Given pairs (x_i,y_i) of samples x_i and (vector-valued) labels y_i drawn from some distribution, a fundamental problems in machine learning is to train a neural network to correctly classify the data. This problem can be framed as a learning problem. Namely, given pairs (x_i,y_i) with the promise that the samples are classified by some ground-truth neural network M(x), one can attempt to learn the weights of this network. In this talk, we focus on two-layer networks M(x) with a single hidden layer containing rectified (e.g. ReLU) activation units f( ). Such a network is specified by weight matrices U,V, so that M(x) = U f(V x), and f is applied coordinate-wise. More generally, the observations y_i may be corrupted by noise e_i, and instead we observe M(x_i) + e_i, and the goal is to learn the matrices U,V. In this talk, we discuss state of the art learning algorithms and hardness results under varying assumptions on the input and noise. Our central result is a polynomial time algorithm for Gaussian inputs and Sub-gaussian noise. In addition, we discuss a poly-time exact recovery algorithm for the noiseless case, and fixed-parameter tractable algorithms for more general noise distributions.

2020-10-01
11:30hrs.

Mircea Petrache. Facultad de Matemáticas y Iimc, PUC
Transporte óptimo y moléculas
https://zoom.us/j/99363985515?pwd=Mk9ySWZwMGxXdGlzRTRrejNzNFM0dz09
Abstract:
El transporte óptimo es un problema de optimización en el cual se requiere enviar una cantidad de masa en otra, modeladas por dos medidas positivas, minimizando un "costo de transporte". Miraremos una aplicación sorprendente, para el cálculo de las formas de moléculas en mecánica cuántica computacional, en lo que se llama "Density Functional Theory" (DFT). La nube de electrones de una molécula, está descrita en mecánica cuántica por una densidad de probabilidad en 3N dimensiones con N el número de electrones de la molécula. Es imposible calcular esta probabilidad numéricamente con precisión desde la ecuación de Schrodinger, debido a la "explosión dimensional" del problema: Walter Kohn, el inventor del "DFT", obtuvo el premio Nobel en química en 1998 por una primera simplificación del problema. En la charla veremos como una versión "exótica" del problema de transporte óptimo ayuda a controlar que hace el problema de Kohn en el límite de N largo. Encontraremos nuevas cotas precisas para varios términos de error, lo que requiere armar nuevas herramientas mezclando ideas de análisis armónico y optimizacion.
http://escueladoc.mat.uc.cl/programa.php

2020-09-30
11:30hrs.

Eduardo Cerpa. Instituto de Ingeniería Matemática y Computacional, PUC
Análisis Funcional y Control
https://zoom.us/j/99363985515?pwd=Mk9ySWZwMGxXdGlzRTRrejNzNFM0dz09
Abstract:
En esta charla expondremos sobre un tema de investigación llamado Control de Ecuaciones en Derivadas Parciales. Una de las principales preguntas en este tema es la de la controlabilidad, que es la propiedad de poder llevar un sistema dinámico desde una condición inicial a una condición final mediante la elección adecuada de algún elemento del sistema dinámico que llamamos control. Cuando el sistema dinámico está descrito por una ecuación en derivadas parciales, lo que llamamos control puede ser típicamente un término fuente o una condición de borde. Veremos que esta pregunta se puede plantear y estudiar mediante el uso de herramientas avanzadas de Análisis Funcional lo que hace de este tema un punto de intersección de áreas como Análisis, Ecuaciones Diferenciales y Optimización, con interesantes aplicaciones en otras ciencias.
http://escueladoc.mat.uc.cl/programa.php

2020-09-29
15:30hrs.

Thomas Führer. Facultad de Matemáticas y Iimc, PUC
Problema de Obstáculo y aproximaciones
https://zoom.us/j/99363985515?pwd=Mk9ySWZwMGxXdGlzRTRrejNzNFM0dz09
Abstract:
Problemas de obstáculos son un tipo de problemas muy importantes en aplicaciones y aparecen en varias formas y áreas distintas. En esta charla consideramos un problema clásico: Hallar la posición del equilibrio de una membrana elástica bajo una fuerza externa y forzada a estar sobre un obstáculo. Discutimos el problema de optimización correspondiente y las desigualdades variacionales que describen el problema. Luego revisamos métodos de elementos finitos para obtener aproximaciones. Finalmente presento resultados recientes que muestran cómo se puede aplicar métodos de cuadrados mínimos para resolver el problema de obstáculo.
http://escueladoc.mat.uc.cl/programa.php

2020-09-28
15:30hrs.

Niao He. Department of Industrial Systems Engineering and Coordinated Science Laboratory, University of Illinois At Urbana-Champaign
Conditional Stochastic Optimization: from Theory to Practice
https://zoom.us/j/99363985515?pwd=Mk9ySWZwMGxXdGlzRTRrejNzNFM0dz09
Abstract:
In this talk, we are going to introduce a class of compositional stochastic optimization involving conditional expectations, which finds a wide spectrum of applications in reinforcement learning, meta-learning, and many other decision-making problems under uncertainty. We introduce a modified Sample Average Approximation (SAA) and a family of biased Stochastic Approximation (SA) for solving such problems and establish their sample complexities under various structural assumptions, for both convex and nonconvex settings. We also provide information-theoretic lower bounds showing that some of these complexity bounds are tight. Furthermore, we demonstrate the efficiency of the proposed framework and algorithms in a number of machine learning applications. This talk is based on joint work with Yifan Hu and Xin Chen from UIUC.
http://escueladoc.mat.uc.cl/programa.php

2020-09-25
11:30hrs.

Fleurianne Bertrand. Humboldt-Universität Zu Berlin, Berlin, Germany
Stress-based finite element methods with application to solid mechanics
https://zoom.us/j/99363985515?pwd=Mk9ySWZwMGxXdGlzRTRrejNzNFM0dz09
Abstract:
Due to the fact that large local stresses are related to failure, accurate stress approximations are of interest in many applications in solid mechanics.The finite element method for elasticity usually consists in minimizing an energy depending on the displacement variable in an appropriate finite element space. This leads in general to discontinuous stresses, and the reconstruction of accurate stresses in a localizable post-processing step for elasticity is an ongoing research field. In the best case, this reconstruction can be built on each element or on vertex-patches, and involved constants depend only on the shape regularity. An alternative approach minimizes a dual energy under the constraints of momentum and leads to an approximation of the stress directly in a conforming space. This approach is of saddle-point type and the compatibility of the FE spaces has to be proven. In particular, the asymmetry of the stress tensor has to be controlled. To circumvent this restriction, the hyperelasticity problem can be considered directly in the deformed configuration. Here, parametric Raviart-Thomas elements are essential to deal with a domain with curved boundaries.
http://escueladoc.mat.uc.cl/programa.php

2020-09-22
15:30hrs.

Pablo Barceló. Instituto de Ingenieria Matemática y Computacional UC
The expressive power of modern neural networks architectures
https://zoom.us/j/99363985515?pwd=Mk9ySWZwMGxXdGlzRTRrejNzNFM0dz09
Abstract:
Applications of neural networks architectures are becoming impressively popular. Still, very little is known about their expressive power, i.e., which properties can these properties learn and recognize. This is not just an interesting theoretical problem, but can actually have relevant practical implications. For instance, this analysis might yield a better understanding of which parts of the architecture are superfluous, and thus can be removed consequently improving efficiency and effectiveness. I will show that techniques developed for decades in the theoretical computer science community can be used to provide a deep understanding of the expressiveness of modern neural network architectures. To do so I will provide two recent examples from my own research. First, I will show that Transformer networks, which are often used by Google in NLP tasks, can actually express any computable function. This provides a theoretical foundation to the claim that such architectures can learn arbitrary algorithms from example. Second, I will present recent characterizations of the expressive power of message-passing graph neural networks (GNNs) in terms of well-known algorithms for checking graph isomorphism and fragments of first-order logic. This shows concrete upper bounds on the properties that GNNs can learn.
http://escueladoc.mat.uc.cl/programa.php

2020-09-16
13:00hrs.

Álvaro Lorca. Escuela de Ingeniería UC
OPTIMIZACIÓN Y MODELACIÓN ESTOCÁSTICA EN PROBLEMAS DE PLANIFICACIÓN ENERGÉTICA
https://zoom.us/j/91782510486
Abstract:

http://imc.uc.cl/index.php/actividades/seminarios

2020-09-09
13:00hrs.

José Verschae. Instituto de Ingeniería Matemática y Computacional UC
Simetrías en Optimización Discreta: Grupos y Geometría para un Mejor Diseño de Algoritmos
https://zoom.us/j/91782510486
Abstract:
Las simetrías están presentes en una infinidad de objetos de nuestra vida. Los problemas de optimización no son la excepción. Más aún, su presencia juega un rol importante en su resolución práctica, especialmente en técnicas basadas en enumeración. En esta charla introduciré los conceptos básicos para lidiar con simetrías en problemas de optimización discreta. Nos enfocaremos en los llamados dominios fundamentales, que son regiones de R^n que buscan romper las simetrías de un problema. Lamentablemente, los dominios fundamentales en la literatura son computacionalmente difíciles de manejar, ya que tienen una cantidad exponencial (en la dimensión) de facetas y son NP-difíciles de separar. Mostraremos como conexiones entre geometría y teoría de grupos nos ayudan a definir mejores dominios fundamentales: regiones más simples (que se describen con menos desigualdades) y que rompen las simetrías de mejor manera. En particular, daremos un método general para generar dominios fundamentales que, entre otros, nos lleva a definir un dominio fundamental con a lo más n-1 facetas para cualquier grupo de permutación. Concluiremos con varios problemas abiertos.
http://imc.uc.cl/index.php/actividades/seminarios