Past Computational Mathematics and Applications Seminar

13 November 2014
14:00
Professor Rob Scheichl
Abstract

The coefficients in mathematical models of physical processes are often impossible to determine fully or accurately, and are hence subject to uncertainty. It is of great importance to quantify the uncertainty in the model outputs based on the (uncertain) information that is available on the model inputs. This invariably leads to very high dimensional quadrature problems associated with the computation of statistics of quantities of interest, such as the time it takes a pollutant plume in an uncertain subsurface flow problem to reach the boundary of a safety region or the buckling load of an airplane wing. Higher order methods, such as stochastic Galerkin or polynomial chaos methods, suffer from the curse of dimensionality and when the physical models themselves are complex and computationally costly, they become prohibitively expensive in higher dimensions. Instead, some of the most promising approaches to quantify uncertainties in continuum models are based on Monte Carlo sampling and the “multigrid philosophy”. Multilevel Monte Carlo (MLMC) Methods have been introduced recently and successfully applied to many model problems, producing significant gains. In this talk I want to recall the classical MLMC method and then show how the gains can be improved further (significantly) by using quasi-Monte Carlo (QMC) sampling rules. More importantly the dimension independence and the improved gains can be justified rigorously for an important model problem in subsurface flow. To achieve uniform bounds, independent of the dimension, it is necessary to work in infinite dimensions and to study quadrature in sequence spaces. I will present the elements of this new theory for the case of lognormal random coefficients in a diffusion problem and support the theory with numerical experiments.

  • Computational Mathematics and Applications Seminar
Bill Lionheart
Abstract

For many tomographic imaging problems there are explicit inversion formulas, and depending on the completeness of the data these are unstable to differing degrees. Increasingly we are solving tomographic problems as though they were any other linear inverse problem using numerical linear algebra. I will illustrate the use of numerical singular value decomposition to explore the (in)stability for various problems. I will also show how standard techniques from numerical linear algebra, such as conjugate gradient least squares, can be employed with systematic regularization compared with the ad hoc use of slowly convergent iterative methods more traditionally used in computed tomography. I will mainly illustrate the talk with examples from three dimensional x-ray tomography but I will also touch on tensor tomography problems.
 

  • Computational Mathematics and Applications Seminar
30 October 2014
14:00
Professor Olavi Nevanlinna
Abstract

We outline a path from polynomial numerical hulls to multicentric calculus for evaluating f(A). Consider
$$Vp(A) = {z ∈ C : |p(z)| ≤ kp(A)k}$$
where p is a polynomial and A a bounded linear operator (or matrix). Intersecting these sets over polynomials of degree 1 gives the closure of the numerical range, while intersecting over all polynomials gives the spectrum of A, with possible holes filled in.
Outside any set Vp(A) one can write the resolvent down explicitly and this leads to multicentric holomorphic functional calculus.
The spectrum, pseudospectrum or the polynomial numerical hulls can move rapidly in low rank perturbations. However, this happens in a very controlled way and when measured correctly one gets an identity which shows e.g. the following: if you have a low-rank homotopy between self-adjoint and quasinilpotent, then the identity forces the nonnormality to increase in exact compensation with the spectrum shrinking.
In this talk we shall mention how the multicentric calculus leads to a nontrivial extension of von Neumann theorem
$$kf(A)k ≤ sup |z|≤1
kf(z)k$$
where A is a contraction in a Hilbert space, and conclude with some new results on (nonholomorphic) functional calculus for operators for which p(A) is normal at a nontrivial polynomial p. Notice that this is always true for matrices.

 

  • 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

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