Numerical Analysis and Scientific Computing Seminars 2010/11
Semester One (Autumn 2010)

08 Oct
2010
Building a better mesh: using spatial adaptivity in coupled problems
Andrew Hazel (University of Manchester)
3.00  Frank Adams Room 1, Alan Turing BuildingAbstract (click to view)Physical processes are generally nonuniform within a given spatial domain, often occurring on different length scales in different regions. Hence, a uniform numerical discretisation based on the smallest lengthscale in the problem can be extremely inefficient. One solution, known as spatial adaptivity, is to solve the problem using a coarse discretisation and then to refine selected regions based on a measure of the error. The process continues iteratively until the error measure is below a prescribed tolerance everywhere or the computer runs out of memory!
In problems with multiple physical processes (coupled problems), it is unlikely that all processes will operate over the same length scales in all regions of space. In these cases, using separate discretisations for each process can be the most efficient approach and obviates the need for construction of a combined error measure.
I shall explore these ideas by considering example transport problems in fluid mechanics and will also discuss recent work on unstructured adaptivity in regions with nontrivial curvilinear boundaries. 
15 Oct
2010
Modified equations and backward error analysis for stochastic differential equations
Konstantinos Zygalakis (University of Oxford)
3.00  Frank Adams Room 1, Alan Turing BuildingAbstract (click to view)In this talk we describe a general framework for deriving modified equations for stochastic differential equations with respect to weak convergence. We will start by quickly recapping of how to derive modified equations in the case of ODEs and describe how these ideas can be generalized in the case of SDEs. Results will be presented for first order methods such as the EulerMaruyama and the Milstein method. In the case of linear SDEs, using the Gaussianity of the underlying solutions, we will derive a SDE that the numerical method solves exactly in the weak sense. Applications of modified equations in the numerical study of Langevin equations and in the calculation of effective diffusivity will also be discussed.

29 Oct
2010
Polynomial approximations of regular and singular vector fields
with applications to electromagnetic problems
Alex Bespalov (University of Manchester)
3.00  Frank Adams Room 1, Alan Turing BuildingAbstract (click to view)In this talk we address the problem of conforming approximation of regular (in terms of Sobolev spaces) and singular vector fields by highorder polynomials. We will review earlier results on this topic as well as very recent developments, including optimal error estimates for H(div) and H(curl)conforming pinterpolation in two dimensions and precise error bounds for polynomial approximations of singularities on polyhedral surfaces. We will demonstrate how these results can be used in the analysis of numerical methods for timeharmonic problems of electromagnetics. In particular, as a model problem, we consider the electric field integral equation (EFIE), which is a boundary integral equation modelling the scattering of electromagnetic waves at a perfectly conducting body. The EFIE is discretised by divconforming RaviartThomas elements of order p, and we focus on the error analysis of highorder (p and hp) boundary element methods for this model problem on polyhedral surfaces. Throughout the talk we will emphasise the main theoretical ingredients of our analysis: the regularised Poincarétype integral operators, regular decompositions (splittings) of vector fields, and projectionbased pinterpolation operators satisfying the commuting diagram property (de Rham diagram).

05 Nov
2010
Challenges in seismic imaging
Paul Childs (Schlumberger Cambridge Research)
3.00  Frank Adams Room 1, Alan Turing BuildingAbstract (click to view)I will give a general overview of some of the challenges involved in seismic imaging for hydrocarbon exploration. The talk will be illustrated with a number of examples and I will describe some large scale problems arising in hydrocarbon exploration. Aspects of nonlinear optimization, linear algebra and the numerical solution of the wave equation will be discussed.

12 Nov
2010
The Northern British Differential Equations Seminar
(Replaces Numerical Analysis and Informal Applied seminars)
Entropy Stable Approximations of NavierStokes equations with no Artificial Numerical Viscosity
Eitan Tadmor (University of Maryland)
3.00  Frank Adams Room 1, Alan Turing BuildingAbstract (click to view)Entropy stability plays an important role in the dynamics of nonlinear systems of conservation laws and related convectiondiffusion equations. What about the corresponding numerical framework? We present a general theory of entropy stability for difference approximations of such nonlinear equations. Our approach is based on comparing numerical viscosities relative to certain entropy conservative schemes. It yields precise characterizations of entropy stability which is enforced in rarefactions while keeping sharp resolution of shocks.
We demonstrate this approach with a host of first and secondorder accurate schemes ranging from scalar examples to Euler and NavierStokes equations. In particular, we construct a new family of entropy stable schemes which retain the precise entropy decay of the NavierStokes equations. They contain no artificial numerical viscosity. Numerical experiments provide a remarkable evidence for the different roles of viscosity and heat conduction in forming sharp monotone profiles in the immediate neighborhoods of shocks and contacts. 
26 Nov
2010
Local spectral function of a selfadjoint analytic operator function
Heinz Langer (TU Wien)
3.00  Frank Adams Room 1, Alan Turing BuildingAbstract (click to view)We consider spectral properties of quadratic pencils L(λ) = λ^{2}I + λB + C (factorization, spectrum of definite type, variational principles) and some of their generalizations to selfadjoint analytic operator functions.

10 Dec
2010
On a surfacepotentialbased domain decomposition approach
Sergei Utyuzhnikov (MACE, University of Manchester)
3.00  Frank Adams Room 1, Alan Turing BuildingAbstract (click to view)A nonoverlapping domain decomposition approach is developed by the use of the CalderónRyaben´kii (CR) surface potentials [1, 2]. A key property of the CR potentials is that they are projections. A brief introduction to the CR potentials is given in application to linear and nonlinear boundary value problems (BVPs).
In particular, the approach allows us to transfer the boundary conditions from a real boundary to an intermediate one. This can be realised either exactly (for linear BVPs) or approximately (for nonlinear BVPs). The boundary conditions at the intermediate boundary are nonlocal and can be formulated via a pseudodifferential equation. The transfer of the boundary conditions is beneficial for practical applications in the reallife design. For example, it is often required to run serial computations while some input parameters, such as boundary conditions or the support of righthand side, are varied to find out their optimal values.
Three applications of the approach are discussed: absorbing boundary conditions for external problems, intermediate boundary conditions for nearwall turbulence modelling [3, 4], and the problem of active sound control and noise shielding [58].
References
1. Ryaben´kii, V.S., Method of Difference Potentials and Its Applications, SpringerVerlag, 2002.
2. Utyuzhnikov, S.V., Generalized CalderonRyaben´kii Potentials, IMA J. Appl. Math., 2009, 74 (1).
3. Utyuzhnikov, S.V., Domain decomposition for nearwall turbulent flows, Int. J. Computers & Fluids, 2009, 38 (9).
4. Utyuzhnikov, S.V., Robintype wall functions and their numerical implementation, J. Applied Numerical Mathematics, 2008, 58.
5. Lim, H., Utyuzhnikov, S. V., Lam, Y. W., Turan, A., Multidomain active sound control and noise shielding, JASA, 2010 (in press).
6. Utyuzhnikov, S.V., Nonlinear Problem of Active Sound Control, J. of Computational and Applied Mathematics, 2010, 234 (1).
7. Utyuzhnikov, S V., Nonstationary problem of active sound control in bounded domains, J. of Computational and Applied Mathematics, 2010, 234 (6).
8. Lim, H., Utyuzhnikov, S. V., Lam, Y. W., Turan, A., Avis, M. R., Ryaben´kii, V. S., Tsynkov, S. V., An experimental validation of the active noise control methodology based on difference potentials, AIAA J., 2009, 47 (4).
Semester Two (Spring 2011)

27 Jan
2011
A QRbased algorithm for computing the matrix polar decomposition and the SVD
Yuji Nakatsukasa (University of California, Davis, USA)
11.00 am  Frank Adams Room 1, Alan Turing Building
(Please note unusual time!)Abstract (click to view)Computing the matrix polar decomposition is required in many applications. The most widely used method is the scaled Newton iteration. The iteration involves explicit matrix inverses, which are expensive in communication costs. On the emerging multicore and heterogeneous computing systems, communication costs have exceeded arithmetic costs by orders of magnitude, and the gap is growing exponentially over time. This motivates some recent studies that seek linear algebra algorithms (e.g., for eigenproblems and the singular value decomposition (SVD)) that minimize communication. In this talk I will first present our new inversefree polar decomposition algorithm. The QR decompositionbased nature of our algorithm makes it a communicationfriendly method and offers an attractive alternative to the scaled Newton iteration. I will also introduce a QRbased algorithm for computing the SVD, which is based on a similar idea. The new SVD algorithm minimizes both communication and arithmetic costs, up to a small constant factor.

28 Jan
2011
An Elliptic Inverse Problem From Groundwater Flow
Andrew Stuart (University of Warwick)
3.00  Frank Adams Room 1, Alan Turing BuildingAbstract (click to view)There is a significant body of research work concerning the quantification of uncertainty in the solution of elliptic PDEs with random coefficients. Typically this work makes the assumption that the diffusion coefficient is a random field with a simple representation via a KarhunenLoeve or polynomial chaos representation. However, in many applications of interest this (prior) probabilistic information must be conditioned on observational data, leading to a posterior probability measure which has a much more complicated structure. For example in the study of groundwater flow it is natural to condition the conductivity (diffusion coefficient) on noisy observations of the solution to the elliptic PDE for the pressure.
In this talk I will show how the development of Bayesian statistics on function space provides a natural framework for the study of such problems. For illustrative pruposes I will concentrate on the problem of groundwater flow. I will develop prior probability meausures using KL expansions in wavelet or Fourier bases, and show how to condition these measures on data. I will develop a theory of wellposedness for the inverse problem and show how this leads to stability of the posterior measure with respect to changes in data, and finite truncation of the KL expansion.
Joint work with Masoumeh Dashti and Stephen Harris. 
11 Feb
2011
Backward Perturbation Analysis of Linear Least Squares Problems
David TitleyPeloquin (University of Oxford)
3.00  Frank Adams Room 1, Alan Turing BuildingAbstract (click to view)We consider the iterative solution of large sparse linear least squares (LS) problems. Specifically, we focus on the design and implementation of reliable stopping criteria for the widelyused algorithm LSQR of Paige and Saunders. First we perform a backward perturbation analysis of the LS problem. We show why certain projections of the residual vector are good measures of convergence, and we propose stopping criteria that use these quantities. These projections are too expensive to compute to be used directly in practice. We show how to estimate them efficiently at every iteration of the algorithm LSQR. Our proposed stopping criteria can therefore be used in practice.
This talk is based on joint work with XiaoWen Chang, Chris Paige, Pavel Jiranek, and Serge Gratton. 
25 Feb
2011
Optimally Blended SpectralFinite Element Scheme for Wave Propagation,
and NonStandard Reduced Integration
Mark Ainsworth (University of Strathclyde)
3.00  Frank Adams Room 1, Alan Turing BuildingAbstract (click to view)In an influential article, Marfurt suggested that the "best" scheme for computational wave propagation would involve an averaging of the consistent and lumped finite element approximations. Many authors have considered how this might be accomplished for first order approximation, but the case of higher orders remained unsolved.
We describe work on the dispersive and dissipative properties of a novel scheme for computational wave propagation obtained by averaging the consistent (finite element) mass matrix and lumped (spectral element) mass matrix. The objective is to obtain a hybrid scheme whose dispersive accuracy is superior to both of the schemes. We present the optimal value of the averaging constant for all orders of finite elements and prove that for this value, the scheme is two orders more accurate compared with finite and spectral element schemes and, in addition, the absolute accuracy is of this scheme is better than that of finite and spectral element methods.
Joint work with Hafiz Wajid, COMSATS Institute of Technology, Pakistan. 
17 Mar
2011
A Galerkin Method for Parametric Uncertainty Propagation in Hyperbolic Systems
Olivier Le Maître (LIMSICNRS, Orsay, France)
2.00  Frank Adams Room 1, Alan Turing Building
(Please note unusual day and time!)Abstract (click to view)We present a Galerkin method for the propagation of parametric uncertainties in systems of conservation laws. The method is based on a probabilistic treatment of the uncertainties, yielding a stochastic system of equations assumed hyperbolic almost surely. For the resolution of this system, we use a Galerkin technique with a stochastic discretization involving the expansion of the solution on a basis of orthonormal (uncorrelated) stochastic functionals. The Galerkin projection of the stochastic problem results in a large system of deterministic equations for the expansion coefficients of the solution, with a structure similar to conservation laws. We first study the properties of the Galerkin system and show, in particular, conditions ensuring its hyperbolic character.
Motivated by these theoretical results, we next propose a Finite Volume method with a Roetype approximation of the fluxes (involving a fast approximation of the upwinding matrix) for the resolution of the Galerkin system. One essential feature of the stochastic hyperbolic systems is their ability to develop discontinuous solutions along the uncertain parameter dimensions, requiring appropriate stochastic discretizations (piecewise polynomial). However, the discontinuities are localized in space, time, and in the stochastic domain. Therefore, we introduce an adaptive strategy, based on a multiresolution framework, to efficiently refine the stochastic discretization only where needed.
Application of the method on uncertain Burgers and Euler equations will be shown.
Joint work with J. Tryoen, M. Ndinjga and A. Ern. 
25 Mar
2011
Joint Manchester, Oxford and Edinburgh SIAM Student Chapters Event
2.00  5.00  Frank Adams Room 1, Alan Turing Building 
06 May
2011
Flow in a Channel with a Sudden Expansion
Andrew Cliffe (University of Nottingham)
3.00  Frank Adams Room 1, Alan Turing BuildingAbstract (click to view)Numerical calculations of laminar flow in a twodimensional channel with a sudden expansion exhibit a symmetry breaking bifurcation at Reynolds number 40.45 when the expansion ratio is 3:1. In the experiments reported by Fearn, Mullin and Cliffe (JFM, Vol. 211, pp. 595608, 1990) there is a large perturbation to this bifurcation and the agreement with the numerical calculations is surprisingly poor. The talk will describe an attempt to explain this discrepancy using techniques from uncertainty quantification and Bayesian inverse methods.

13 May
2011
HighOrder Approximations of Viscoelastic Flow Problems
Tim Phillips (Cardiff University)
3.00  Frank Adams Room 1, Alan Turing BuildingAbstract (click to view)This talk will consider the numerical approximation of the governing equations for viscoelastic fluids using highorder spectral techniques. Typically, mathematical models for complex fluids involve a nonlinear relationship between the stress and strain. The level of description of the model to be used in a particular application is determined by the available computational resources. Both macroscopic and microscopic models for viscoelastic fluids will be described. The former class of models include differential constitutive relationships derived from continuum mechanics considerations. The latter class of models include kinetic theory models for polymer solutions and reptation theory models for polymer melts. The benchmark problems considered are the flow past a cylinder and the extrudate swell problem. Mesh convergence studies are presented. Comparisons with available experimental results are made for materials that have been carefully characterized.

03 Jun
2011
Asynchronous Iterations for Performance Analysis
Jeremy Bradley (Imperial College London)
2.00 pm  Frank Adams Room 1, Alan Turing Building
(Please note earlier time!)Abstract (click to view)Performance analysis has it's foundations in modelling telephone exchanges with stochastic processes (Erlang) and aircraft takeoff and arrival patterns with queueing theory (Kendall). Although the applications have changed, the mathematical tools have stayed the same. However, the problems have increased in size to the point where we use clusters to analyse Markov and semiMarkov processes with 10^7+ states. To permit us to analyse these large problems, we use asynchronous numerical techniques to solve the underlying linear problems. Asynchronous iterative techniques allow large linear fixed point calculations to be divided among many processors without the overhead of maintaining high speed communication networks between the processors and without having processors stall waiting for neighbouring processors to complete their part of the computation. Instead, each processor is allowed to progress on it's subcomputation and, subject to some fairly weak conditions, occasionally transmit solution updates to neighbouring processors. The end result is (perhaps surprisingly) a convergent calculation in reasonable time with good scalability properties over potentially many different clusters of processors. In this talk, I will explore both the need for fast analysis capability in the performance analysis domain as well as an example of the use of asynchronous techniques to provide this capability.
Further information
For further information please contact the seminar organiser Alex Bespalov.