Numerical Analysis and Scientific Computing Seminars 2013/14
Semester Two (Spring 2013)

31 January 2014
Numerical method for ergodic stochastic differential equations
Konstantinos Zygalakis (University of Southampton)
3.00 pm  Frank Adams Room 1, Alan Turing BuildingAbstract (click to view)In this talk we will discuss recent results regarding the accuracy of numerical integrators for ergodic stochastic differential equations (SDEs). We introduce new sufficient conditions for a numerical method to approximate with high order of accuracy the invariant measure of an ergodic system of SDEs, independently of the weak order of accuracy of the method. Using these conditions and the framework of modified differential equations we construct numerical integrators that capture the invariant measure of Brownian dynamics with an accuracy independent of the weak order of the underlying method. Using similar ideas we will then investigate the accuracy of Lie Trotter splitting method for Langevin dynamics.

14 February 2014
Numerical methods for stiff and multiscale stochastic problems
Assyr Abdulle (Echole Polytechnique Federale de Lausanne)
3.00 pm  Frank Adams Room 1, Alan Turing BuildingAbstract (click to view)In this talk we discuss various numerical methods recently introduced for the stable and efficient integration of stiff (meansquare stable) stochastic differential equations. We will show how these techniques can be integrated in a versatile ``swissknife'' integrator for diffusionadvectionreaction problems with noise or used for multilevel MonteCarlo methods. We will conclude the talk with a brief discussion on other classes of stochastic methods designed to accurately capture the invariant measure of ergodic SDEs. These methods are for example essential for the numerical integration of fastslow stochastic systems and stiff solvers usually fail to approximate efficiently the correct dynamics of such problems.

21 February 2014
Disk brake squeal. A challenge
for FEM Analysis and Numerical Linear Algebra.
Volker Mehrmann (Technische Universitat Berlin)
3.00 pm  Frank Adams Room 1, Alan Turing BuildingAbstract (click to view)We discuss the modelling and simulation of disk brake squeal. We derive a nonlinear parametric dynamical system and the associated eigenvalue problem that allows to indicate the reasons for squeal. To use this model in optimization and control leads to the desire to derive model reduction methods for such systems. It is shown that traditional methods fail and that also current eigenvalue methods for such nonlinear problems face many difficulties. To deal with these problems we discuss adaptive finite element methods for eigenvalue problems and show that they offer large potential in improving efficiency and accuracy of computed eigenvalues.

14 March 2014
Solving PDEs on surfaces with radial basis functions: from global to local methods
Grady Wright (Boise State University)
3.00 pm  Frank Adams Room 1, Alan Turing BuildingAbstract (click to view)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 (RBFFD). In this talk, we first describe our recent work on developing a new, highorder, global RBF method for numerically solving PDEs on relatively general surfaces, with a specific focus on reactiondiffusion 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 RBFFD approach with this same flexibility. This is the first application of the RBFFD method to general surfaces. We conclude with applications of these methods to some biologically relevant problems. This talk represents joint work with Ed Fuselier (High Point University), Aaron Fogelson, Mike Kirby, and Varun Shankar (all at the University of Utah).

28 March 2014
Fast Solvers and Error Estimation for Stochastic Galerkin Finite Element Methods.
Catherine Powell (University of Manchester)
3.00 pm  Frank Adams Room 1, Alan Turing BuildingAbstract (click to view)This talks focuses on efficient numerical methods for solving PDE models that typically arise in fluid flow applications (e.g., Darcy, Stokes and NavierStokes equations) with uncertain inputs modelled as random variables. Socalled stochastic Galerkin finite element methods (SGFEMs) yield approximate solutions that take the form of generalised polynomial chaos expansions with coefficient functions belonging to finite element spaces. Despite having highly favourable approximation properties, such methods cannot be used in their standard form for challenging applications (such as flow in porous media). To improve the efficiency of standard SGFEMs, and increase the size of problems that can reasonably be solved on modest computer architectures, we focus attention on two tasks. 1) Reducing the cost of solving the discrete linear systems, via the use of appropriate iterative methods and preconditioners. 2) Developing adaptive refinement schemes that can be driven by cheaptocompute a posteriori error estimators.

11 April 2014
Highly oscillatory problems in wave scattering theory: their asymptotic and numerical approximation
Samuel Groth (The University of Reading)
3.00 pm  Frank Adams Room 1, Alan Turing BuildingAbstract (click to view)Rapid oscillations arise in wave scattering problems when the wavelength of the incident field is small relative to the size of the scattering obstacles. In this “high frequency” regime the solution is highly oscillatory and hence notoriously expensive to approximate numerically. On the other hand, the presence of such rapid oscillations may be exploited using asymptotic techniques. In this talk I will begin by outlining some of the current stateoftheart numerical methods in high frequency scattering in order to highlight the benefits of a recently developed purely asymptotic technique. This new technique is based on a reformulation of the problem as a boundary integral equation in which a WKB ansatz is made for the unknown. The resulting oscillatory integrals are then evaluated using the method of steepest descent. I will discuss how this method is extended to multiple scattering configurations and how, ultimately, it may form the basis of numerical technique which inherits the best of both the asymptotic and numerical worlds.

29 April 2014
Diagonally Dominant Matrices: Surprising recent results on a classical type of matrices
Froilan Dopico (Universidad Carlos III)
10.00 am  Frank Adams Room 1, Alan Turing Building (Please note the unusual date and time.)Abstract (click to view)Diagonally dominant matrices are among the most important classes of matrices arising in applications and enjoy excellent theoretical and numerical properties, which have been extensively studied in the last 50 years and are presented in many standard texts on Numerical Analysis and Matrix Analysis. Therefore, it is difficult to believe that something really new can be said on these matrices. However, in the last four years a number of highly structured perturbation results have been proved for these matrices. Apart from being interesting by themselves, these results have been used to prove that certain algorithms, recently developed, allow us to perform computations with diagonally dominant matrices with much more accuracy than traditional matrix algorithms, even in the case the traditional condition number of a diagonally dominant matrix is extremely large. The new results rely in looking at diagonally dominant matrices from a completely different perspective that avoids to work directly with the matrix entries. This is joint work with Megan Dailey and Qiang Ye.

9 May 2014
Continuous analogues of matrix factorizations
Alex Townsend (University of Oxford)
3.00 pm  Frank Adams Room 1, Alan Turing BuildingAbstract (click to view)A fundamental idea in matrix linear algebra is the factorization of a matrix into simpler matrices, such as orthogonal, tridiagonal, and triangular. In this talk we extend this idea to a continuous setting, asking: "What are the continuous analogues of matrix factorizations?" The answer we develop involves functions of two variables, an iterative variant of Gaussian elimination, and sufficient conditions for convergence. This leads to a test for nonnegative definite kernels, a continuous definition of a triangular quasimatrix (a matrix whose columns are functions), and a fresh perspective on a classic subject.

23 May 2014
Numerical inversion of the geodesic Xray transform
Sean Holman (University of Manchester)
3.00 pm  Frank Adams Room 1, Alan Turing BuildingAbstract (click to view)The standard Xray transform integrates a function in $\mathbb{R}^n$ over a collection of straight lines. The problem of Xray tomography is then to recover the function from these integrals, and this inversion forms the mathematical basis for Xray CT. In some applications, such as the linearisation of travel time tomography in seismology, the straight lines are replaced by other families of curves which are often the geodesics of a Riemannian metric. In this case we have the geodesic Xray transform. I will review some numerical methods for the inversion of the geodesic Xray transform, and also discuss ideas for future research in this direction.
Semester One (Autumn 2013)

27 September 2013
A fast Jacobitype SVD for the graphics processors
Vedran Novaković (University of Zagreb)
3.00 pm  Frank Adams Room 1, Alan Turing BuildingAbstract (click to view)Singular value decomposition presents a challenge for the graphics processors and similar massively parallel architectures. We present a hierarchically blocked Jacobi SVD algorithm, for both a single and the multiple GPUs. Unlike common hybrid approaches, our algorithm needs a CPU for the controlling purposes only. The algorithm is noticeably faster than Magma's DGESVD on a Tesla C2070 GPU, while retaining the high relative accuracy. In this talk we shall discuss the key components of the algorithm: the blocking structure that reflects the levels of GPU's memory hierarchy, and some new parallel pivoting strategies. The algorithm, in principle, scales to an arbitrary number of GPU nodes, by adding the appropriate blocking levels. The scalability is demonstrated by more than twofold speedup on a Tesla S2050 system with four GPUs.

11 October 2013
Phase Transitions in Convex Geometry and Optimization
Dennis Amelunxen (The University of Manchester)
3.00 pm  Frank Adams Room 1, Alan Turing BuildingAbstract (click to view)Convex optimization provides a powerful approach to solving a wide range of problems under structural assumptions on the solutions. Examples include solving linear inverse problems or separating signals with mutually incoherent structures. A curious phenomenon arises when studying such problems; as the underlying parameters in the optimization program shift, the convex relaxation can change quickly from success to failure. We reduce the analysis of these phase transitions to a summary parameter, the statistical dimension, associated to the problem. We prove a new concentration of measure phenomenon for some integral geometric invariants, and deduce from this the existence of phase transitions for a wide range of problems; the phase transition being located at the statistical dimension. We furthermore calculate the statistical dimension in concrete problems of interest, and use it to relate previously existing  but seemingly unrelated  approaches to compressed sensing by Donoho and Rudelson & Vershynin. (joint work with Martin Lotz, Michael B. McCoy, Joel A. Tropp)

25 October 2013
Calculation of the Heat Transfer Coefficient for Quenching Probes
Saša Singer (University of Zagreb)
3.00 pm  Frank Adams Room 1, Alan Turing BuildingAbstract (click to view)The purpose of the Liscic/Petrofer probe is to determine the cooling intensity during liquid quenching in laboratory and workshop environments. The heat transfer coefficient calculation is based on temperatures measured in time near the surface of the probe, and involves two interesting numerical problems: monotone smoothing of measured temperatures, and solution of (onedimensional) inverse heat conduction problem to determine the surface temperature and the heat transfer coefficient.

8 November 2013
Sparse multifrontal QR factorization on the GPU
Tim Davis (University of Florida)
3.00 pm  Frank Adams Room 1, Alan Turing BuildingAbstract (click to view)Sparse matrix factorization involves a mix of regular and irregular computation, which is a particular challenge when trying to obtain highperformance on the highly parallel generalpurpose computing cores available on graphics processing units (GPUs). We present a sparse multifrontal QR factorization method that meets this challenge, and is up to ten times faster than a highly optimized method on a multicore CPU. Our method is unique compared with prior methods, since it factorizes many frontal matrices in parallel, and keeps all the data transmitted between frontal matrices on the GPU. A novel bucket scheduler algorithm extends the communicationavoiding QR factorization for dense matrices, by exploiting more parallelism and by exploiting the staircase form present in the frontal matrices of a sparse multifrontal method. This is joint work with Nuri Yeralan and Sanjay Ranka (also both at the Univ. of Florida).

22 November 2013
Extending GMRES to allow multiple preconditioners
Tyrone Rees (Rutherford Appleton Laboratory)
3.00 pm  Frank Adams Room 1, Alan Turing BuildingAbstract (click to view)The fast solution of linear systems by iterative methods generally requires effective preconditioners, and these preconditioners are usually highly tuned to the problem. However, in many cases there is more than one possible choice of preconditioner, and the different preconditioners often complement each other. In spite of this, existing methods only allow us to use one preconditioner at a time, or at best naively cycle through them. Furthermore, almost all computers available today have multiple cores, allowing us to apply more than one preconditioner to a vector simultaneously for minimal extra time cost. In this talk I will introduce MPGMRES, an extension of GMRES which allows the use of more than one preconditioner. This algorithm is available as the HSL subroutine HSL_MI29 and, as well as describing some of the theory behind the algorithm, I will give some numerical examples to illustrate its applicability and potential.

6 December 2013
Fast Iterative Solution of PDEConstrained Optimization Problems
John Pearson (University of Edinburgh)
3.00 pm  Frank Adams Room 1, Alan Turing BuildingAbstract (click to view)The solution of PDEconstrained optimization problems is a topic of much recent interest within the numerical analysis and applied mathematics communities. In this presentation we examine the numerical solution of matrix systems arising from finite element discretizations of these problems, by devising preconditioned iterative methods for the systems. Our methods for these problems utilize the fact that the matrix systems are large, sparse, and of saddle point structure. This enables us to create effective techniques by devising good approximations of the (1,1)block and Schur complement of the matrices involved. In this talk we investigate a range of PDEconstrained optimization problems, including Poisson control, problems motivated by fluid dynamics, timedependent formulations, and reactiondiffusion control problems resulting from chemical processes. For each problem we motivate the preconditioners that we employ, and carry out numerical experiments to demonstrate the performance of our methods.
Further information
For further information please contact the seminar organiser Vanni Noferini.