# Seminario de Ingeniería Matemática y Computacional

El seminario de Ingeniería Matemática y Computacional reune a investigadores y alumnos del área homónima de la PUC cada miércoles durante el semestre. En un ambiente interdisciplinario, abarca diversos tópicos en el área incluyendo Optimización, Análisis Numérico, Cuantificación de Incertidumbre, Ciencias de Datos y Teoría de la Computación, con una fuerte inclinación a distintas aplicaciones en las más diversas áreas.

2021-06-30
13:00hrs.
Ignacio Madrid. École Polytechnique, Centre de Mathématiques Appliquées (Cmap), France
Modelo estocástico de crecimiento bacteriano bajo la acción de un antibiótico: de células individuales a poblaciones
https://zoom.us/j/93450667974
Abstract:
Se propone un modelo estocástico individuo-centrado para el crecimiento de bacterias E. coli bajo la acción de un antibiótico anti-ADN que perturba la dinámica de crecimiento y provoca la emergencia de una subpoblación de células de tamaño anormal. Nos interesa saber si las observaciones realizadas a la escala de células individuales son suficientes para inferir el comportamiento de una población, y en particular, la distribución de los tamaños.
El modelo se formaliza como un Proceso Estocástico a Valor Medida (PEVM), que definirá un Problema de Martingala a partir del cual derivaremos una EDP para la densidad de número de bacterias. A partir de esta EDP estudiaremos el comportamiento en tiempo largo  y mostraremos la existencia de una distribución estacionaria para el tamaño de las células.
Finalmente veremos cómo simular este tipo de procesos y comentaremos algunos resultados numéricos. Para producir simulaciones realistas propondremos además un modelo estadístico que permite ajustar los parámetros del PEVM a partir de datos single-cell obtenidos en el microscopio. Para concluir comentaremos algunas simulaciones un poco más complejas que pueden arrojar luz sobre mecanismos más realistas de emergencia de resistencia contra los antibióticos.
2021-06-16
13:00hrs.
Nicolas Barnafi. Dipartimento Di Matematica, Università Degli Studi Di Milano, Italia
Modelos y métodos matemáticos para la perfusión cardíaca
https://zoom.us/j/93450667974
Abstract:
El desarrollo de modelos matemáticos para el corazón es un desafío intrínsecamente interdisciplinario, que involucra médicos, biólogos, matemáticos, físicos e ingenieros. Su relevancia va más allá de la clínica, y actualmente presenta una riquísima fuente de problemas de gran interés. En este seminario, haré una breve introducción a los modelos usados para la perfusión cardíaca,  que es el flujo de sangre hacia el interior del corazón. Haré énfasis en los desafíos fisico/matemáticos que surgen, que abarcan la termodinámica de medios porosos, ecuaciones diferenciales parciales y el análisis numérico de los métodos usados para aproximar numéricamente  dichos modelos.
2021-06-09
13:00hrs.
Bernardo Cockburn. Department of Mathematics, University of Minnesota
Static condensation, hybridization and the devising of the HDG methods
https://zoom.us/j/93450667974
Abstract:

The hybridizable discontinuous Galerkin (HDG) methods were introduced in the framework of second-order diffusion problems by hybridization and static condensation. We show that the exact solution can be characterized as the solution of local Dirichlet problems (hybridization) which can then be patched together by the transmission conditions (static condensation). Our goal is to show that the HDG methods are nothing but a discrete version of this characterization. To do so, we show that this is also the case for the well-known continuous Galerkin and the mixed methods. We end by sketching how to define HDG methods for general PDEs.

2021-06-02
13:00hrs.
Pedro J. Saenz. Department of Mathematics, University of North Carolina At Chapel Hill
Hydrodynamic Spin Lattices
https://zoom.us/j/93450667974
2021-05-26
13:00hrs.
Eduardo Corona. Department of Applied Mathematics, University of Colorado-Boulder
Un 'Crash Course' en métodos de ecuaciones integrales y su aplicación a simulación de suspensiones densas en fluido Stokesiano
https://zoom.us/j/93450667974
Abstract:
El método de ecuaciones integrales consiste en formular la solución de un problema de valores en la frontera en términos de operadores integrales. Si esta fórmula se elige con cuidado, este método permite reducir la dimensionalidad del problema, y su discretización produce sistemas lineales bien condicionados. En esta charla, seguiremos un mini curso intensivo ("crash course"), explicando paso a paso la aplicación de este método a problemas de suspensiones de objetos rígidos en fluido viscoso (Stokes), con énfasis en la necesidad de métodos numéricos rápidos.

Los materiales granulares y las suspensiones de partículas en fluido son dos modelos indispensables de lo que se conoce como "materia suave" (soft matter). Son cruciales para el estudio de áreas como mecánica de suelos, auto-ensamblaje de materiales biológicos e ingeniería de materiales. En esta plática presento una plataforma general de modelado y simulación de suspensiones Stokesianas, y en particular, aplicaciones a suspensiones de partículas de Jano (de caracter dual) de tres tipos distintos: anfifílicas, bipolares y foréticas.
2021-05-19
13:00hrs.
Leonardo Zepeda. Department of Mathematics, University of Wisconsin-Madison
Wide-Band Butterfly Networks: Sub-Wavelength Imaging via Multi-Frequency Neural Networks
https://zoom.us/j/93450667974
Abstract:

For most wave-based inverse problems the resolution of the reconstruction is usually limited by the so-called diffraction limit, i.e., the smallest features to be reconstructed cannot be smaller than the smallest wavelength of available data. If one properly restricts the class of features to, for example, point-scatterers, the seminal work of Donoho in the early ’90s demonstrates that the recovery of these sub-wavelength features is tractable. However, algorithms to recover a more general class of structured scatterers containing features below the diffraction limit in the presence of noise remain an open question.

In this talk, we aim to surpass the diffraction limit using deep learning techniques coupled with computational harmonic analysis tools. In particular, I will introduce a new neural network architecture for inverting wide-band data to recover acoustic scatterers at resolutions finer than the classical limit. The architecture incorporates insights from the butterfly factorization and the Cooley-Tukey algorithm to explicitly account for the physics of wave propagation. The dimensions of the network seamlessly adapt to the desired image resolution, resulting in a number of trainable weights that scale quasilinearly with the image resolution and the data bandwidth. In addition, the data is optimally assimilated across frequencies thus enhancing the stability of the training stage. I will provide the rationale for such construction and showcase its properties for several classes of scatterers with sub-Nyquist features embedded in a known background media.

(Joint work with Matthew Li and Laurent Demanet).

2021-05-05
13:00hrs.
Raimundo Elicer Coopman. Danish School of Education, Aarhus University, Denmark
https://zoom.us/j/93450667974
Abstract:

El objetivo de esta charla es sentar bases generales para una conversación sobre educación matemática en tres campos de problemas: justificaciones, posibilidades e implementación. Para concretizar, hago uso de lo estudiado en mi tesis doctoral, anclada en el contexto chileno de probabilidad y estadística en enseñanza media, orientado a la elusiva noción de ciudadanía crítica.

Investigar justificaciones implica indagar en “por qué” enseñar un contenido en cierto nivel educacional. En particular, trazas del llamado argumento de la competencia crítica están evidenciados en el currículo chileno de probabilidad y estadística.

Estudiar posibilidades significa hacerse preguntas fundamentales sobre “qué” es el saber matemático y cómo se evidencia. En general, me refiero a la epistemología crítica de Skovsmose, y cómo se conecta con marcos teóricos de alfabetización probabilística y estadística.

Los problemas de implementación buscan responder al “cómo” de la educación matemática. En mi tesis, delineo tres principios generales para el diseño de ambientes de aprendizaje: ejemplaridad, enfoque de indagación y pragmatismo.

Finalmente, a modo de recapitulación y comparación, comento sobre mi actual proyecto de investigación acerca de las interconexiones entre competencias matemáticas y pensamiento computacional.

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
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:
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.