Past Computational Mathematics and Applications Seminar

13 February 2014
14:00
Dr Leonardo Figueroa
Abstract
We report on a numerical implementation of a quasi-static model of rock detachment based on Allaire, Jouve and Van Goethem's implementation of Francfort and Marigo's model of damage in brittle solids, As such, local minimizers of a cost functional involving both stored elastic energy and a damage penalization term are sought by using a procedure which alternates between approximately solving a linear elasticity system and advancing a transport equation for a level set function describing the loci of still-attached rock. We pay special attention to the mixed finite element method used in the approximation of the linear elasticity system.
  • Computational Mathematics and Applications Seminar
6 February 2014
14:00
Professor Grady Wright
Abstract
Radial basis function (RBF) methods are becoming increasingly popular for numerically solving partial differential equations (PDEs) because they are geometrically flexible, algorithmically accessible, and can be highly accurate. There have been many successful applications of these techniques to various types of PDEs defined on planar regions in two and higher dimensions, and to PDEs defined on the surface of a sphere. Originally, these methods were based on global approximations and their computational cost was quite high. Recent efforts have focused on reducing the computational cost by using ``local’’ techniques, such as RBF generated finite differences (RBF-FD). In this talk, we first describe our recent work on developing a new, high-order, global RBF method for numerically solving PDEs on relatively general surfaces, with a specific focus on reaction-diffusion equations. The method is quite flexible, only requiring a set of ``scattered’’ nodes on the surface and the corresponding normal vectors to the surface at these nodes. We next present a new scalable local method based on the RBF-FD approach with this same flexibility. This is the first application of the RBF-FD method to general surfaces. We conclude with applications of these methods to some biologically relevant problems. This talk represents joint work with Edward Fuselier (High Point University), Aaron Fogelson, Mike Kirby, and Varun Shankar (all at the University of Utah).
  • Computational Mathematics and Applications Seminar
23 January 2014
14:00
Professor Luis Nunes Vicente
Abstract
Direct-search methods are a class of popular derivative-free algorithms characterized by evaluating the objective function using a step size and a number of (polling) directions. When applied to the minimization of smooth functions, the polling directions are typically taken from positive spanning sets which in turn must have at least n+1 vectors in an n-dimensional variable space. In addition, to ensure the global convergence of these algorithms, the positive spanning sets used throughout the iterations must be uniformly non-degenerate in the sense of having a positive (cosine) measure bounded away from zero. \\ \\ However, recent numerical results indicated that randomly generating the polling directions without imposing the positive spanning property can improve the performance of these methods, especially when the number of directions is chosen considerably less than n+1. \\ \\ In this talk, we analyze direct-search algorithms when the polling directions are probabilistic descent, meaning that with a certain probability at least one of them is of descent type. Such a framework enjoys almost-sure global convergence. More interestingly, we will show a global decaying rate of $1/\sqrt{k}$ for the gradient size, with overwhelmingly high probability, matching the corresponding rate for the deterministic versions of the gradient method or of direct search. Our analysis helps to understand numerical behavior and the choice of the number of polling directions. \\ \\ This is joint work with Clément Royer, Serge Gratton, and Zaikun Zhang.
  • Computational Mathematics and Applications Seminar
5 December 2013
14:00
Dr Gabriel Barrenechea
Abstract

We propose a strategy which allows computing eigenvalue enclosures for the Maxwell operator by means of the finite element method. The origins of this strategy can be traced back to over 20 years ago. One of its main features lies in the fact that it can be implemented on any type of regular mesh (structured or otherwise) and any type of elements (nodal or otherwise). In the first part of the talk we formulate a general framework which is free from spectral pollution and allows estimation of eigenfunctions.

We then prove the convergence of the method, which implies precise convergence rates for nodal finite elements. Various numerical experiments on benchmark geometries, with and without symmetries, are reported.

  • Computational Mathematics and Applications Seminar
Dr Amal Khabou
Abstract
We present a block LU factorization with panel rank revealing pivoting (block LU_PRRP), an algorithm based on strong rank revealing QR for the panel factorization. Block LU_PRRP is more stable than Gaussian elimination with partial pivoting (GEPP), with a theoretical upper bound of the growth factor of $(1+ \tau b)^{(n/ b)-1}$, where $b$ is the size of the panel used during the block factorization, $\tau$ is a parameter of the strong rank revealing QR factorization, and $n$ is the number of columns of the matrix. For example, if the size of the panel is $b = 64$, and $\tau = 2$, then $(1+2b)^{(n/b)-1} = (1.079)^{n-64} \ll 2^{n-1}$, where $2^{n-1}$ is the upper bound of the growth factor of GEPP. Our extensive numerical experiments show that the new factorization scheme is as numerically stable as GEPP in practice, but it is more resistant to some pathological cases where GEPP fails. We note that the block LU_PRRP factorization does only $O(n^2 b)$ additional floating point operations compared to GEPP.
  • Computational Mathematics and Applications Seminar
21 November 2013
14:00
Dr Rémi Gribonval
Abstract

A popular approach within the signal processing and machine learning communities consists in modelling signals as sparse linear combinations of atoms selected from a learned dictionary. While this paradigm has led to numerous empirical successes in various fields ranging from image to audio processing, there have only been a few theoretical arguments supporting these evidences. In particular, sparse coding, or sparse dictionary learning, relies on a non-convex procedure whose local minima have not been fully analyzed yet. Considering a probabilistic model of sparse signals, we show that, with high probability, sparse coding admits a local minimum around the reference dictionary generating the signals. Our study takes into account the case of over-complete dictionaries and noisy signals, thus extending previous work limited to noiseless settings and/or under-complete dictionaries. The analysis we conduct is non-asymptotic and makes it possible to understand how the key quantities of the problem, such as the coherence or the level of noise, can scale with respect to the dimension of the signals, the number of atoms, the sparsity and the number of observations.

This is joint work with Rodolphe Jenatton & Francis Bach.

  • Computational Mathematics and Applications Seminar
14 November 2013
14:00
Professor Philippe Toint
Abstract

The context of data assimilation in oceanography will be described as well as the computational challenges associated with it. A class of numerical linear algebra methods is described whose purpose is to exploit the problem structure in order to reduce the computational burden and provide provable convergence results for what remains a (very large) nonlinear problem. This class belongs to the Krylov-space family of methods and the special structure used is the imbalance between the dimensions of the state space and the observation space. It is also shown how inexact matrix-vector products can be exploited. Finally, preconditioning issues and resulting adaptations of the trust-region methodology for nonlinear minimization will also be outlined.

By Serge Gratton, Selime Gurol, Philippe Toint, Jean Tshimanga and Anthony Weaver.

  • Computational Mathematics and Applications Seminar

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