# Past Computational Mathematics and Applications Seminar

7 February 2013
14:00
Dr Winnifried Wollner
Abstract
Subtitle: And applications to problems involving pointwise constraints on the gradient of the state on non smooth polygonal domains \\ \\ In this talk we are concerned with an analysis of Moreau-Yosida regularization of pointwise state constrained optimal control problems. As recent analysis has already revealed the convergence of the primal variables is dominated by the reduction of the feasibility violation in the maximum norm. \\ \\ We will use a new method to derive convergence of the feasibility violation in the maximum norm giving improved the known convergence rates. \\ \\ Finally we will employ these techniques to analyze optimal control problems governed by elliptic PDEs with pointwise constraints on the gradient of the state on non smooth polygonal domains. For these problems, standard analysis, fails because the control to state mapping does not yield sufficient regularity for the states to be continuously differentiable on the closure of the domain. Nonetheless, these problems are well posed. In particular, the results of the first part will be used to derive convergence rates for the primal variables of the regularized problem.
• Computational Mathematics and Applications Seminar
31 January 2013
14:00
Professor Martin Gander
Abstract
Domain decomposition methods have been developed in various contexts, and with very different goals in mind. I will start my presentation with the historical inventions of the Schwarz method, the Schur methods and Waveform Relaxation. I will show for a simple model problem how all these domain decomposition methods function, give precise results for the model problem, and also explain the most general convergence results available currently for these methods. I will conclude with the parareal algorithm as a new variant for parallelization of evolution problems in the time direction.
• Computational Mathematics and Applications Seminar
24 January 2013
14:00
Dr David May
Abstract
Over million year time scales, the evolution and deformation of rocks on Earth can be described by the equations governing the motion of a very viscous, incompressible fluid. In this regime, the rocks within the crust and mantle lithosphere exhibit both brittle and ductile behaviour. Collectively, these rheologies result in an effective viscosity which is non-linear and may exhibit extremely large variations in space. In the context of geodynamics applications, we are interested in studying large deformation processes both prior and post to the onset of material failure. \\ \\ Here I introduce a hybrid finite element (FE) - Lagrangian marker discretisation which has been specifically designed to enable the numerical simulation of geodynamic processes. In this approach, a mixed FE formulation is used to discretise the incompressible Stokes equations, whilst the markers are used to discretise the material lithology. \\ \\ First I will show the a priori error estimates associated with this hybrid discretisation and demonstrate the convergence characteristics via several numerical examples. Then I will discuss several multi-level preconditioning strategies for the saddle point problem which are robust with respect to both large variations in viscosity and the underlying topological structure of the viscosity field. \\ Finally, I will describe an extension of the multi-level preconditioning strategy that enables high-resolution, three-dimensional simulations to be performed with a small memory footprint and which is performant on multi-core, parallel architectures.
• Computational Mathematics and Applications Seminar
17 January 2013
14:00
Professor Massimiliano Pontil
Abstract
We discuss the problem of estimating a structured matrix with a large number of elements. A key motivation for this problem occurs in multi-task learning. In this case, the columns of the matrix correspond to the parameters of different regression or classification tasks, and there is structure due to relations between the tasks. We present a general method to learn the tasks' parameters as well as their structure. Our approach is based on solving a convex optimization problem, involving a data term and a penalty term. We highlight different types of penalty terms which are of practical and theoretical importance. They implement structural relations between the tasks and achieve a sparse representations of parameters. We address computational issues as well as the predictive performance of the method. Finally we discuss how these ideas can be extended to learn non-linear task functions by means of reproducing kernels.
• Computational Mathematics and Applications Seminar
10 January 2013
14:00
Professor Stephen Wright
Abstract
Problems of packing shapes with maximal density, sometimes into a container of restricted size, are classical in discrete mathematics. We describe here the problem of packing a given set of ellipsoids of different sizes into a finite container, in a way that allows overlap but that minimizes the maximum overlap between adjacent ellipsoids. We describe a bilevel optimization algorithm for finding local solutions of this problem, both the general case and the simpler special case in which the ellipsoids are spheres. Tools from conic optimization, especially semidefinite programming, are key to the algorithm. Finally, we describe the motivating application - chromosome arrangement in cell nuclei - and compare the computational results obtained with this approach to experimental observations. \\ \\ This talk represents joint work with Caroline Uhler (IST Austria).
• Computational Mathematics and Applications Seminar
29 November 2012
14:00
Dr Donna Calhoun
Abstract
We describe our current efforts to develop finite volume schemes for solving PDEs on logically Cartesian locally adapted surfaces meshes. Our methods require an underlying smooth or piecewise smooth grid transformation from a Cartesian computational space to 3d surface meshes, but does not rely on analytic metric terms to obtain second order accuracy. Our hyperbolic solvers are based on Clawpack (R. J. LeVeque) and the parabolic solvers are based on a diamond-cell approach (Y. Coudi\`ere, T. Gallou\"et, R. Herbin et al). If time permits, I will also discuss Discrete Duality Finite Volume methods for solving elliptic PDEs on surfaces. \\ \\ To do local adaption and time subcycling in regions requiring high spatial resolution, we are developing ForestClaw, a hybrid adaptive mesh refinement (AMR) code in which non-overlapping fixed-size Cartesian grids are stored as leaves in a forest of quad- or oct-trees. The tree-based code p4est (C. Burstedde) manages the multi-block connectivity and is highly scalable in realistic applications. \\ \\ I will present results from reaction-diffusion systems on surface meshes, and test problems from the atmospheric sciences community.</p>
• Computational Mathematics and Applications Seminar
Dr Carola-Bibiane Schönlieb
Abstract

Domain decomposition methods were introduced as techniques for solving partial differential equations based on a decomposition of the spatial domain of the problem into several subdomains. The initial equation restricted to the subdomains defines a sequence of new local problems. The main goal is to solve the initial equation via the solution of the local problems. This procedure induces a dimension reduction which is the major responsible of the success of such a method. Indeed, one of the principal motivations is the formulation of solvers which can be easily parallelized.

In this presentation we shall develop a domain decomposition algorithm to the minimization of functionals with total variation constraints. In this case the interesting solutions may be discontinuous, e.g., along curves in 2D. These discontinuities may cross the interfaces of the domain decomposition patches. Hence, the crucial difficulty is the correct treatment of interfaces, with the preservation of crossing discontinuities and the correct matching where the solution is continuous instead. I will present our domain decomposition strategy, including convergence results for the algorithm and numerical examples for its application in image inpainting and magnetic resonance imaging.

• Computational Mathematics and Applications Seminar
15 November 2012
14:00
Professor Mark Ainsworth
Abstract
We study the dispersion and dissipation of the numerical scheme obtained by taking a weighted averaging of the consistent (finite element) mass matrix and lumped (spectral element) mass matrix for the small wave number limit. We find and prove that for the optimum blending the resulting scheme (a) provides $2p+4$ order accuracy for $p$th order method (two orders more accurate compared with finite and spectral element schemes); (b) has an absolute accuracy which is $\mathcal{O}(p^{-3})$ and $\mathcal{O}(p^{-2})$ times better than that of the pure finite and spectral element schemes, respectively; (c) tends to exhibit phase lag. Moreover, we show that the optimally blended scheme can be efficiently implemented merely by replacing the usual Gaussian quadrature rule used to assemble the mass and stiffness matrices by novel nonstandard quadrature rules which are also derived.
• Computational Mathematics and Applications Seminar
8 November 2012
14:00
Dr Irene Kyza
Abstract

ALE formulations are useful when approximating solutions of problems in deformable domains, such as fluid-structure interactions. For realistic simulations involving fluids in 3d, it is important that the ALE method is at least of second order of accuracy. Second order ALE methods in time, without any constraint on the time step, do not exist in the literature and the role of the so-called geometric conservation law (GCL) for stability and accuracy is not clear. We propose discontinuous Galerkin (dG) methods of any order in time for a time dependent advection-diffusion model problem in moving domains. We prove that our proposed schemes are unconditionally stable and that the conservative and non conservative formulations are equivalent. The same results remain true when appropriate quadrature is used for the approximation of the involved integrals in time. The analysis hinges on the validity of a discrete Reynolds' identity and generalises the GCL to higher order methods. We also prove that the computationally less intensive Runge-Kutta-Radau (RKR) methods of any order are stable, subject to a mild ALE constraint. A priori and a posteriori error analysis is provided. The final estimates are of optimal order of accuracy. Numerical experiments confirm and complement our theoretical results.

This is joint work with Andrea Bonito and Ricardo H. Nochetto.

• Computational Mathematics and Applications Seminar
1 November 2012
14:00
Dr Andreas Dedner
Abstract

The Discontinuous Galerkin (DG) method has been used to solve a wide range of partial differential equations. Especially for advection dominated problems it has proven very reliable and accurate. But even for elliptic problems it has advantages over continuous finite element methods, especially when parallelization and local adaptivity are considered.

In this talk we will first present a variation of the compact DG method for elliptic problems with varying coefficients. For this method we can prove stability on general grids providing a computable bound for all free parameters. We developed this method to solve the compressible Navier-Stokes equations and demonstrated its efficiency in the case of meteorological problems using our implementation within the DUNE software framework, comparing it to the operational code COSMO used by the German weather service.

After introducing the notation and analysis for DG methods in Euclidean spaces, we will present a-priori error estimates for the DG method on surfaces. The surface finite-element method with continuous ansatz functions was analysed a few years ago by Dzuik/Elliot; we extend their results to the interior penalty DG method where the non-smooth approximation of the surface introduces some additional challenges.

• Computational Mathematics and Applications Seminar