Past Computational Mathematics and Applications Seminar

9 October 2014
02:00
Professor Ke Chen
Abstract

Mathematical imaging is not only a multidisciplinary research area but also a major cross-discipline subject within mathematical sciences as  image analysis techniques involve differential geometry, optimization, nonlinear partial differential equations (PDEs), mathematical analysis, computational algorithms and numerical analysis. Segmentation refers to the essential problem in imaging and vision  of automatically detecting objects in an image.

 

In this talk I first review some various models and techniques in the variational framework that are used for segmentation of images, with the purpose of discussing the state of arts rather than giving a comprehensive survey. Then I introduce the practically demanding task of detecting local features in a large image and our recent segmentation methods using energy minimization and  PDEs. To ensure uniqueness and fast solution, we reformulate one non-convex functional as a convex one and further consider how to adapt the additive operator splitting method for subsequent solution. Finally I show our preliminary work to attempt segmentation of blurred images in the framework of joint deblurring and segmentation.

  

This talk covers joint work with Jianping Zhang, Lavdie Rada, Bryan Williams, Jack Spencer (Liverpool, UK), N. Badshah and H. Ali (Pakistan). Other collaborators in imaging in general include T. F. Chan, R. H. Chan, B. Yu,  L. Sun, F. L. Yang (China), C. Brito (Mexico), N. Chumchob (Thailand),  M. Hintermuller (Germany), Y. Q. Dong (Denmark), X. C. Tai (Norway) etc.

[Related publications from   http://www.liv.ac.uk/~cmchenke ]

  • Computational Mathematics and Applications Seminar
Dr Rachael Tappenden
Abstract
The accurate and efficient solution of linear systems Ax = b is very important in many engineering and technological applications, and systems of this form also arise as subproblems within other algorithms. In particular, this is true for interior point methods (IPM), where the Newton system must be solved to find the search direction at each iteration. Solving this system is a computational bottleneck of an IPM, and in this talk I will explain how preconditioning and deflation techniques can be used, to lessen this computational burden. This is joint work with Jacek Gondzio.
  • Computational Mathematics and Applications Seminar
12 June 2014
14:00
Professor Joachim Weickert
Abstract
Many successful methods in image processing and computer vision involve parabolic and elliptic partial differential equations (PDEs). Thus, there is a growing demand for simple and highly efficient numerical algorithms that work for a broad class of problems. Moreover, these methods should also be well-suited for low-cost parallel hardware such as GPUs. In this talk we show that two of the simplest methods for the numerical analysis of PDEs can lead to remarkably efficient algorithms when they are only slightly modified: To this end, we consider cyclic variants of the explicit finite difference scheme for approximating parabolic problems, and of the Jacobi overrelaxation method for solving systems of linear equations. Although cyclic algorithms have been around in the numerical analysis community for a long time, they have never been very popular for a number of reasons. We argue that most of these reasons have become obsolete and that cyclic methods ideally satisfy the needs of modern image processing applications. Interestingly this transfer of knowledge is not a one-way road from numerical analysis to image analysis: By considering a factorisation of general smoothing filters, we introduce novel, signal processing based ways of deriving cycle parameters. They lead to hitherto unexplored methods with alternative parameter cycles. These methods offer better smoothing properties than classical numerical concepts such as Super Time Stepping and the cyclic Richardson algorithm. We present a number of prototypical applications that demonstrate the wide applicability of our cyclic algorithms. They include isotropic and anisotropic nonlinear diffusion processes, higher dimensional variational problems, and higher order PDEs.
  • Computational Mathematics and Applications Seminar
29 May 2014
14:00
Christoph Ortner
Abstract
For many questions of scientific interest, all-atom molecular simulations are still out of reach, in particular in materials engineering where large numbers of atoms, and often expensive force fields, are required. A long standing challenge has been to construct concurrent atomistic/continuum coupling schemes that use atomistic models in regions of space where high accuracy is required, with computationally efficient continuum models (e.g., FEM) of the elastic far-fields. Many different mechanisms for achieving such atomistic/continuum couplings have been developed. To assess their relative merits, in particular accuracy/cost ratio, I will present a numerical analysis framework. I will use this framework to analyse in more detail a particularly popular scheme (the BQCE scheme), identifying key approximation parameters which can then be balanced (in a surprising way) to obtain an optimised formulation. Finally, I will demonstrate that this analysis shows how to remove a severe bottlenecks in the BQCE scheme, leading to a new scheme with optimal convergence rate.
  • Computational Mathematics and Applications Seminar
22 May 2014
14:00
Dr Colin Cotter
Abstract
We describe discretisations of the shallow water equations on the sphere using the framework of finite element exterior calculus. The formulation can be viewed as an extension of the classical staggered C-grid energy-enstrophy conserving and energy-conserving/enstrophy-dissipating schemes which were defined on latitude-longitude grids. This work is motivated by the need to use pseudo-uniform grids on the sphere (such as an icosahedral grid or a cube grid) in order to achieve good scaling on massively parallel computers, and forms part of the multi-institutional UK “Gung Ho” project which aims to design a next generation dynamical core for the Met Office Unified Model climate and weather prediction system. The rotating shallow water equations are a single layer model that is used to benchmark the horizontal component of numerical schemes for weather prediction models. We show, within the finite element exterior calculus framework, that it is possible to build numerical schemes with horizontal velocity and layer depth that have a con- served diagnostic potential vorticity field, by making use of the geometric properties of the scheme. The schemes also conserve energy and enstrophy, which arise naturally as conserved quantities out of a Poisson bracket formulation. We show that it is possible to modify the discretisation, motivated by physical considerations, so that enstrophy is dissipated, either by using the Anticipated Potential Vorticity Method, or by inducing stabilised advection schemes for potential vorticity such as SUPG or higher-order Taylor-Galerkin schemes. We illustrate our results with convergence tests and numerical experiments obtained from a FEniCS implementation on the sphere.
  • Computational Mathematics and Applications Seminar
15 May 2014
14:00
Andrea Moiola
Abstract
<p><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">Computer simulation of the propagation and interaction of linear waves</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">is a core task in computational science and engineering.</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">The finite element method represents one of the most common</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">discretisation techniques for Helmholtz and Maxwell's equations, which</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">model time-harmonic acoustic and electromagnetic wave scattering.</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">At medium and high frequencies, resolution requirements and the</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">so-called pollution effect entail an excessive computational effort</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">and prevent standard finite element schemes from an effective use.</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">The wave-based Trefftz methods offer a possible way to deal with this</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">problem: trial and test functions are special solutions of the</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">underlying PDE inside each element, thus the information about the</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">frequency is directly incorporated in the discrete spaces.</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">This talk is concerned with a family of those methods: the so-called</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">Trefftz-discontinuous Galerkin (TDG) methods, which include the</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">well-known ultraweak variational formulation (UWVF).</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">We derive a general formulation of the TDG method for Helmholtz</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">impedance boundary value problems and we discuss its well-posedness</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">and quasi-optimality.</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">A complete theory for the (a priori) h- and p-convergence for plane</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">and circular/spherical wave finite element spaces has been developed,</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">relying on new best approximation estimates for the considered</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">discrete spaces.</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">In the two-dimensional case, on meshes with very general element</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">shapes geometrically graded towards domain corners, we prove</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">exponential convergence of the discrete solution in terms of number of</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">unknowns.</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">This is a joint work with Ralf Hiptmair, Christoph Schwab (ETH Zurich,</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /><span style="text-transform: none; background-color: #fdfdfd; text-indent: 0px; display: inline !important; font: 13px monospace; white-space: normal; float: none; letter-spacing: normal; color: #333333; word-spacing: 0px;">Switzerland) and Ilaria Perugia (Vienna, Austria).</span><br style="text-transform: none; text-indent: 0px; font: 13px monospace; white-space: normal; letter-spacing: normal; color: #333333; word-spacing: 0px;" /></p>
  • Computational Mathematics and Applications Seminar
1 May 2014
14:00
Dr Matthew Juniper
Abstract
Thermoacoustic oscillations occur in combustion chambers when heat release oscillations lock into pressure oscillations. They were first observed in lamps in the 18th century, in rockets in the 1930s, and are now one of the most serious problems facing gas turbine manufacturers. This theoretical and numerical study concerns an infinite-rate chemistry diffusion flame in a tube, which is a simple model for a flame in a combustion chamber. The problem is linearized around the non-oscillating state in order to derive the direct and adjoint equations governing the evolution of infinitesimal oscillations. The direct equations are used to predict the frequency, growth rate, and mode shape of the most unstable thermoacoustic oscillations. The adjoint equations are then used to calculate how the frequency and growth rate change in response to (i) changes to the base state such as the flame shape or the composition of the fuel (ii) generic passive feedback mechanisms that could be added to the device. This information can be used to stabilize the system, which is verified by subsequent experiments. This analysis reveals that, as expected from a simple model, the phase delay between velocity and heat-release fluctuations is the key parameter in determining the sensitivities. It also reveals that this thermo-acoustic system is exceedingly sensitive to changes in the base state. This analysis can be extended to more accurate models and is a promising new tool for the analysis and control of thermo-acoustic oscillations.
  • Computational Mathematics and Applications Seminar
24 April 2014
14:00
Professor Eric Van den Eijnden
Abstract
Dynamics in nature often proceed in the form of reactive events, aka activated processes. The system under study spends very long periods of time in various metastable states; only very rarely does it transition from one such state to another. Understanding the dynamics of such events requires us to study the ensemble of transition paths between the different metastable states. Transition path theory (TPT) is a general mathematical framework developed for this purpose. It is also the foundation for developing modern numerical algorithms such as the string method for finding the transition pathways or milestoning to calculate the reaction rate, and it can also be used in the context of Markov State Models (MSMs). In this talk, I will review the basic ingredients of the transition path theory and discuss connections with transition state theory (TST) as well as approaches to metastability based on potential theory and large deviation theory. I will also discuss how the string method arises in order to find approximate solutions in the framework of the transition path theory, the connections between milestoning and TPT, and the way the theory help building MSMs. The concepts and methods will be illustrated using examples from molecular dynamics, material science and atmosphere/ocean sciences.
  • Computational Mathematics and Applications Seminar

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