Past Computational Mathematics and Applications Seminar

23 October 2014
14:00
Professor Erik Burman
Abstract

In numerical analysis the design and analysis of computational methods is often based on, and closely linked to, a well-posedness result for the underlying continuous problem. In particular the continuous dependence of the continuous model is inherited by the computational method when such an approach is used. In this talk our aim is to design a stabilised finite element method that can exploit continuous dependence of the underlying physical problem without making use of a standard well-posedness result such as Lax-Milgram's Lemma or The Babuska-Brezzi theorem. This is of particular interest for inverse problems or data assimilation problems which may not enter the framework of the above mentioned well-posedness results, but can nevertheless satisfy some weak continuous dependence properties. First we will discuss non-coercive elliptic and hyperbolic equations where the discrete problem can be ill-posed even for well posed continuous problems and then we will discuss the linear elliptic Cauchy problem as an example of an ill-posed problem where there are continuous dependence results available that are suitable for the framework that we propose.

  • Computational Mathematics and Applications Seminar
16 October 2014
14:00
Professor Peter Schmid
Abstract

Gradient-based optimisation techniques have become a common tool in the analysis of fluid systems. They have been applied to replace and extend large-scale matrix decompositions to compute optimal amplification and optimal frequency responses in unstable and stable flows. We will show how to efficiently extract linearised and adjoint information directly from nonlinear simulation codes and how to use this information for determining common flow characteristics. We also extend this framework to deal with the optimisation of less common norms. Examples from aero-acoustics and mixing will be presented.

  • Computational Mathematics and Applications Seminar
9 October 2014
14:00
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
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

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