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Online Talks

Durham Cathedral

London Mathematical Society Durham Symposium
Computational Linear Algebra for Partial Differential Equations
Monday 14th July - Thursday 24th July 2008
List of abstracts
Mark Allan (BAE Systems ATC)Friday 18th July 16:30
Accelerating simulations with heterogeneous clusters
It is likely that in the future heterogeneous compute clusters will become the norm. Compute clusters are likely to consist of various hardware (GPUs, accelerator cards, cell processors, etc) each with their own specific requirements for efficiency. In order to harness the power of such systems, the software developer will be required to take into account these requirements to develop efficient codes. This talk will describe some recent work to investigate future architectures and limitations of current software development in utilising the power of such systems. The requirement of an abstract layer which numerical libraries should conform to will be discussed, which will simplify the task of software development for such systems.

Gabriel Barrenechea (University of Strathclyde)Thursday 17th July 16:30
A two-level enriched finite element method for a mixed problem
In this work we present an enriched Petrov-Galerkin finite element method to solve the Darcy equation in mixed form. First, the trial finite element space is enriched with functions belonging to an apriori unknown space, while the test function space is enriched with bubble-like functions, in such a way that we allow static condensation. This process is applied in order to give a precise characterization of the enrichment function as being the solution of local Darcy problems including the residual of the strong equation, and this characterization is included in the method leading to a stabilized-alike method. Finally, since the local problems may not be solved analytically, we present a tow-level strategy to implement the method showing that the basis functions may be computed using a cheap and accurate method.

Michele Benzi (Emory University)Wednesday 16th July 16:00
Modified augmented Lagrangian preconditioning for incompressible flow problems
Augmented Lagrangian-based preconditioners have been shown to be highly effective and robust in the solution of the discrete Stokes and Oseen problem and in solving the linear systems arising from the stability analysis of (Newton) linearized solution to incompressible flow problems. So far, however, the use of this methodology has been limited to discretizations using structured meshes in simple, two-dimensional geometries. The main challenge is the (approximate) solution of the velocity subproblem in the augmented Lagrangian formulation: while effective geometric multigrid methods have been developed for 2D structured meshes, the question remains open for the case of 3D unstructured meshes. One possible solution to this problem is to approximate the coefficient matrix of the velocity subproblem in the preconditioner with a more manageable one, in particular, one for which there exist efficient algebraic solvers. The problem is how to do this in such a way that the overall effectiveness of the preconditioner is preserved. In this talk I will present some variants of the augmented Lagrangian preconditioner that are applicable to rather general discretizations and geometries and retain excellent overall robustness and effectiveness.

Martin Berggren (Umeå Universitet)Tuesday 22nd July 17:00
Megapixel topology optimization on a graphics processing unit

George Biros (University of Pennsylvania)Friday 18th July 13:45
Fast algorithms for inverse problems with elliptic and parabolic PDE constraints: applications to cardiac electrophysiology.
(Joint work with Santi S. Adavani)
The main goal of this project is to design and implement fast algorithms to solve inverse problems that arise in cardiac electrophysiology. We investigate the computational challenges involved in solving inverse electrophysiology problems and propose numerical techniques to address them. We consider the following two formulations : 1) a source identification problem with an elliptic PDE constraint, and 2) an inverse medium problem with a parabolic PDE constraint. We use L2 Tikhonov regularization in both the problems for stability. We use a reduced space approach in which we eliminate the state and the adjoint variables and we iterate in the inversion parameter space using Conjugate Gradients (CG). We propose SVD based preconditioners to accelerate the convergence of CG to solve the source identification problem. The overall complexity of reconstructing the source is O(N logN), where N is the number of grid points. We propose multigrid based preconditioners to accelerate the convergence of CG to solve the inverse medium problem. The overall complexity of recovering the medium properties is O(NtN +N log2 N), where N is the number of grid points and Nt is the number of time steps. We present numerical experiments to show mesh-independent convergence of our algorithms-even in the case of no regularization. This feature makes these methods algorithmically robust to the value of the regularization parameter, and thus, useful for the cases in which we seek high-fidelity reconstructions.

Matthias Bollhoefer (TU Braunschweig)Tuesday 22nd July 16:30
Algebraic multilevel methods for large scale symmetric indefinite problems
We propose an algebraic multilevel method that is devoted to the solution of large scale symmetric indefinite problems. In particular partial differential equations such as the Helmholtz equations but also saddle point problems like the Stokes equation are of special interest since they finally lead to symmetric indefinite systems with additional structure. The multilevel approach itself uses combinatorial approaches to improve the block diagonal dominance of the system. This is followed by coarsening strategy. Furthermore we present theoretical eigenvalue bounds for our coarsening strategy. Finally we discuss a multilevel approach for saddle point problems that preserves the saddle point structure at any level of the approximate factorization.

Coralia Cartis (University of Edinburgh)Thursday 17th July 16:00
Adaptive cubic overestimation methods for optimization problems

Andrew Cliffe (University of Nottingham)Saturday 19th July 10:15
Bifurcation phenomena in the flow through a sudden expansion in a pipe
(Joint work with Ed Hall and Paul Houston)
Despite its relative simplicity, the phenomena that occur in the flow of an incompressible viscous fluid in a pipe with a sudden expansion at moderate Reynolds number are not yet well understood. The talk will describe recent developments in goal-oriented adaptive meshing techniques for fluid dynamical stability calculations and their application to the pipe problem. The relationship of the numerical results to the experimental work of Tom Mullin and his co-workers will be discussed.

Sue Dollar (Rutherford Appleton Laboratory)Wednesday 16th July 17:00
Iterative methods designed for saddle-point systems
(Joint work with Nick Gould)
The solution of the humble saddle-point system still causes us many problems. Whilst the solution of such systems is often computed using iterative methods, many of the methods that we use are generally designed for more general systems. However, in some applications such as constrained optimization, we would also like our iterative method to tell us something about the inertia of the system we are solving because this will give us information on where the optimal point lies. Whilst the projected PCG method is popular, its use is limited for problems that have awkward constraints. Inspired by the Lanczos and SYMMBK algorithms, we aim to introduce a Krylov subspace method that provides information about the inertia of our saddle-point problem and uses a 5-term recursion formula.

Howard Elman (University of Maryland)Saturday 19th July 09:30
Effects of boundary conditions on preconditioning strategies for the Navier-Stokes equations
(Joint work with Ray Tuminaro)
Block preconditioners for the linearized Navier-Stokes equations that take advantage of the saddle-point structure of the algebraic systems have shown great potential as the basis of efficient iterative solution algorithms. In the derivation of these techniques, boundary conditions have not played a prominent role, in some cases because the methods have been developed from a purely algebraic point of view, and in others because of a lack of clarity concerning the impact of boundary conditions. In this talk, we look closely at boundary conditions and show that they can be used to construct improved preconditioners by improving the quality of certain commutators used in the derivation. This leads in particular to a new, Robin, boundary condition for preconditioners defined for problems with velocities satisfying inflow boundary conditions. We demonstrate the improved performance of the new block preconditioners, with mesh independent convergence rates for methods that previously exhibited some deterioration in performance as meshes are refined.

Oliver Ernst (TU Bergakademie Freiberg)Tuesday 15th July 15:00
PDEs, matrix functions and Krylov subspace methods
Many problems arising in the numerical solution of partial differential equations can be naturally phrased as the evaluation of a function of a matrix acting on a vector. In recent years there has been a resurgence of research activity in applying Krylov subspace methods to construct efficient solution algorithms for these problems. This talk will present an overview of the most important applications as well as recent innovations regarding restarting, deflation and error estimation techniques.

Melina Freitag (University of Bath)Tuesday 15th July 16:30
Shift-invert Arnoldi method with preconditioned iterative solves
(Joint work with Alastair Spence)
We consider the computation of a few eigenvectors and corresponding eigenvalues of a large sparse nonsymmetric matrix using shift-and-invert Arnoldi method with and without implicit restarts. For the inner iterations we use preconditioned GMRES as the iterative solver. The costs of the inexact solves are measured by the number of inner iterations needed by the iterative solver at each outer step of the algorithm. We first extend the relaxation strategy developed by Simoncini to implicitly restarted Arnoldi method which yields an improvement in the overall costs of the method. Secondly, we apply a new preconditioning strategy to the inner solver. We show examples, where the combination of the new preconditioner with the relaxation strategy in implicitly restarted Arnoldi produces enhancement in the overall costs of around 50 per cent.

David Gelder (Consultant, Mathematics for Manufacturers)Friday 18th July 14:30
Ill-posed problems in product and process design
Many industrial problems demand the solution of ill-posed PDEs, and a selection which are not appropriate for parametric optimisation are explored. They involve elliptic PDEs with initial conditions, hyperbolic PDEs with boundary conditions, parabolic PDEs with end conditions, and PDEs of mixed type - mostly non-linear. All the examples quoted have been of crucial importance in the development of products and processes in my personal experience, which is largely in the glass industry.

More usually inverse problems involving PDEs are dealing with inappropriate or noisy data in trying to find out more about an entity which undoubtedly exists. An attempt is made here to categorise the various routes to identifying robust industrial design procedures, given that it may prove impossible to achieve the required design or anything near it.

Max Gunzburger (Florida State)Monday 21st July 11:30
Numerical methods for stochastic partial differential equations and their control
A survey of computational approaches for the numerical solution of stochastic partial differential equations (SPDEs) is presented. For the most part, we take an algorithmic approach, focusing on methods that are useful for nonlinear problems. We begin by discussing the sources and types of uncertainty that enter into SPDEs. We then discuss methods for the three types of randomness that occur in practice: random parameters, white noise, and colored noise. Discussions of stochastic Galerkin methods (including polynomial chaos methods), stochastic collocation (including sparse grid methods), and stochastic sampling methods for solving SPDEs are provided. Finally, we consider methods for some simple control problems involving SPDEs.

Max Gunzburger (Florida State)Tuesday 22nd July 10:00
Numerical methods for stochastic partial differential equations and their control

Max Gunzburger (Florida State)Wednesday 23rd July 09:30
Numerical methods for stochastic partial differential equations and their control

Eldad Haber (Emory University)Monday 21st July 10:00
PDE constrained optimization
This talk aims to aid in introducing, experimenting and benchmarking algorithms for PDE-constrained optimization problems by presenting a set of model problems.

We specifically examine a type of PDE-constrained optimization problem, the parameter estimation problem. We present three model parameter estimation problems, each containing a different type of partial differential equation as the constraint.

We also shortly describe discretization and solution techniques for each problem. We further supply a simple to modify matlab code with the talk.

Eldad Haber (Emory University)Tuesday 22nd July 09:00
PDE constrained optimization

Eldad Haber (Emory University)Wednesday 23rd July 12:00
PDE constrained optimization

Matthias Heinkenschloss (Rice University)Friday 18th July 09:30
PDE constrained optimization
Optimization problems governed by partial differential equations (PDEs) arise in a growing number of science and engineering applications in the context of optimal control, optimal design, or parameter identification. The robust and efficient solution of these problems presents many challenges arising from the interactions among application specific structure, infinite dimensional problem structure, discretization, numerical solution of the underlying PDE, and numerical optimization.

In this talk I will give a brief overview of the design, theory and application of derivative based optimization algorithms for the solution of PDE constrained optimization problems. In particular, I will discuss the interaction between the optimization problem, discretization of the PDEs, and optimization algorithms. I will highlight some recent mathematical developments and research needs.

Peter Jimack (University of Leeds)Friday 18th July 10:15
Development of a high order discontinuous Galerkin method for the simulation of elastohydrodynamic lubrication phenomena
The use of lubricants to control friction and wear has been recognised as being of critical importance in the automotive industry from its very outset. Both the rising price of fuel and the desire to reduce CO2 emissions mean that this issue is more important today than ever before. Within the internal combustion engine there are a wide range of tribological contacts however one of the most important classes of problem involves so-called elastohydrodynamic lubrication (EHL). In this regime the pressure across the contact is so great that the lubricating fluids behave in a highly non-Newtonian manner and the contacting elements themselves are forced to deform (as the name suggests, such deformation is assumed to be elastic). This talk will describe one particular approach to the simulation of such problems that we have developed as part of a long-term collaboration with Shell Global Solutions. Shell have a lubricant division which develops and markets a wide range of products and they have a need to be able to assess the performance of new lubricants with the aid of numerical simulation, in addition to more expensive physical testing. After providing some essential mathematical background to EHL problems, this talk will describe a new numerical approach based upon the use of an adaptive high order discontinuous Galerkin discretization and a novel multilevel nonlinear solver. A selection of numerical results will also be presented.

David Kay (University of Oxford)Wednesday 16th July 16:30
Efficient numerical solution of the ALE Navier-Stokes equations: Modelling blood flow within the heart
In this talk we will present the Fp preconditioner applied to the Navier-Stokes equations on moving domains using the ALE formulation. Numerical results will show how GMRES convergence rates depend on domain velocity, mesh and time step size, and viscosity. Finally, we will apply this method to model blood flow within the heart. This will introduce a coupling between the ALE Navier-Stokes equations for the blod with solid elasticity equations for the tissue.

Michal Kocvara (University of Birmingham)Thursday 17th July 17:30
Solving nonlinear semidefinite programming by PENNON
I will present PENNON 1.0 - the first release of the code for solving mathematical optimization problems with real and matrix variables, smooth nonlinear (nonconvex) objective and smooth nonlinear equality and inequality constraints. All functions may depend on both types of variables. Additionally, the matrix variables may be subject to spectral constraints, in particular, to positive semidefinite constraints. Matlab and extended AMPL interface allow the user to formulate the problems easily. I will present a few of numerous applications of the code.

Angela Kunoth (Universität Paderborn)Tuesday 22nd July 16:00
Space-time adaptive wavelet methods for control problems constrained by parabolic PDEs
(Joint work with Max D. Gunzburger)
Optimization problems constrained by partial differential equations (PDEs) are particularly challenging from a computational point of view: the first order necessary conditions for optimality lead to a coupled system of PDEs. For these, adaptive methods which aim at distributing the available degrees of freedom in an a-posteriori-fashion to capture singularities in the data or domain appear to be most promising. For control problems constrained by a parabolic PDE, one needs to solve a system of PDEs coupled globally in time.

For such problems, an adaptive method based on wavelets is proposed. It builds on a recent paper by Schwab and Stevenson where a single linear parabolic evolution problem is formulated in a weak space-time form and where an adaptive wavelet method is designed for which optimal convergence rates can be shown.

Amos Lawless (University of Reading)Monday 21st July 16:30
Variable transformations and preconditioning for large-scale optimization problems in data assimilation.
(Joint work with D. Katz, N.K. Nichols, R.N. Bannister (Reading) and M.J.P. Cullen (Met Office))
Data assimilation is the process whereby observational data are combined with numerical models to estimate the state of a physical system. It is used on a daily basis in numerical weather prediction to provide initial conditions for computational forecasts. A common method of solving the assimilation problem is through the minimization of a nonlinear least-squares function, usually with several million degrees of freedom. In order to minimize this function in real time preconditioning is necessary. A first level preconditioning is achieved by transforming the problem to a new set of variables which are assumed to be uncorrelated and solving in the transformed space. In this work we compare a new method of performing this transformation with that currently used in operational forecasting centres. We show how the new transformation is better able to remove correlations over a wide range of dynamical regimes. Comment will be made on the effect of these transformations on the conditioning of the assimilation problem.

Sven Leyffer (Argonne National Laboratory)Saturday 19th July 11:30
FASTr: A Framework for Nonlinear Optimization
Filter methods have been introduced as an alternative to penalty functions as a globalization strategy for nonlinear optimization methods. A filter borrows ideas from multi-objective optimization and accepts a trial point whenever the objective or the constraint violation is improved compared to previous iterates.

We present a non-monotone filter method that offers greater freedom at accepting points. The methods are implemented in FASTr, our filter active-set trust-region framework. We present recent numerical experience comparing various globalization strategies, as well as different step-computation techniques.

Volker Mehrmann (TU Berlin)Tuesday 15th July 11:30
Nonlinear eigenvalue problems in practice: analysis and numerical methods

Volker Mehrmann (TU Berlin)Wednesday 16th July 10:00
Structure preservation in eigenvalue computation: a challenge and a chance.

Volker Mehrmann (TU Berlin)Thursday 17th July 09:00
Model reduction in the simulation, control and optimization of real world processes.

Agnieszka Miedlar (TU Berlin)Wednesday 16th July 17:30
Adaptive solution of eigenvalue problems for PDEs

Nancy Nichols (University of Reading)Thursday 17th July 17:00
Optimal state estimation for very large dynamical systems using reduced order models
(Joint work with A.S. Lawless (University of Reading) and C. Boess and A. Bunse-Gerstner (University of Bremen))

The Gauss-Newton (GN) method is a well known iterative technique for solving nonlinear least squares problems subject to dynamical system constraints. Such problems arise commonly in optimal state estimation where the systems may be stochastic. Variational data assimilation techniques for state estimation in very large environmental systems currently use approximate GN methods. The GN method solves a sequence of linear least squares problems subject to linearized system constraints. For very large systems, low resolution linear approximations to the model dynamics are used to improve the efficiency of the algorithm. We propose a new approach for deriving low order system approximations based on model reduction techniques from control theory. We show how this technique can be combined with the GN method to retain the response of the dynamical system more accurately and improve the performance of the approximate GN method.

Catherine Powell (University of Manchester)Monday 21st July 16:00
Preconditioning saddle-point systems arising in a stochastic mixed finite element problem
The approximate solution of PDEs with random data can be achieved via many mature numerical techniques. Stochastic Finite Element methods (SFEMs), based on stochastic Galerkin formulations of the underlying PDEs, provide a framework for incorporating statistical information about spatial variability in random parameters into computer simulations in such a way that comprehensive probabilistic information about solution variables is obtained.

SFEMs and mixed SFEMs have desirable convergence properties but have suffered from bad press due to a lack of robust solvers and preconditioners for the large coupled stochastic linear systems of equations that arise.

In this talk, we report on fast, robust linear algebra techniques and preconditioning schemes based on geometric and algebraic multigrid methods, for solving the stochastic Galerkin formulation of the Darcy flow problem (the steady-state diffusion problem in mixed form) which leads to massive systems of equations of saddle-point type. We consider stochastic formulations which necessitate a large number of deterministic solves, and more challenging models where one large saddle-point system has to be solved, in which the spatial and stochastic degrees of freedom are coupled.

Arnold Reusken (RWTH Aachen)Tuesday 15th July 16:00
An Eulerian finite element method for elliptic equations on moving surfaces
(Joint work with Maxim Olshanskii (Moscow University))
We present a new finite element approach for the discretization of elliptic partial differential equations on surfaces. The main idea is to use finite element spaces that are induced by triangulations of an ``outer'' domain to discretize the partial differential equation on the surface. The method is particularly suitable for problems in which there is a coupling with a flow problem in an outer domain that contains the surface, for example, two-phase incompressible flow problems. We give an analysis that shows that the method has optimal order of convergence. Results of numerical experiments are included that confirm this optimality. We finally address issues related to the resulting discrete problems, such as conditioning of the mass and stiffness matrices.

Robert Scheichl (University of Bath)Monday 21st July 17:00
Energy minimising coarse spaces for multiscale elliptic PDEs
More powerful (parallel) computers and better imaging and tomography tools have made it possible in recent years to carry out detailed simulations of biological, physical and engineering processes in complex geometries with highly varying heterogeneous material parameters. Consequently, the development of efficient and robust parallel solvers for such problems is of paramount importance. Here we are concerned with the convergence of two-level domain decomposition preconditioners in the context of mixed finite element discretisations of heterogeneous second-order elliptic problems, in both the deterministic and (Monte-Carlo simulated) stochastic cases. In contrast to standard analyses, we do not assume that the coefficients can be resolved by the subdomain partition or the coarse mesh. We perform a new analysis of the preconditioned matrix, which shows rather explicitly, how its condition number depends on the variable coefficient in the PDE, as well as on the coarse grid/overlap size. The classical estimates guarantee good conditioning only when the coefficients vary mildly inside the subdomains. By contrast, our new results show that, with some modifications, domain decomposition preconditioners can still be robust even for large coefficient variation inside subdomains, where the classical methods fail to be robust. In particular our estimates prove very precisely the previously made empirical observation in the AMG literature that the use of energy minimising coarse spaces can lead to robust preconditioners. Numerical experiments on a variety of two-scale model problems confirm our theoretical results, and the performance of the new preconditioners is greatly improved over standard preconditioners in the random coefficient cases.

Wil Schilders (NXP Semiconductors)Friday 18th July 11:30
Linear algebra in the electronics industry
The electronics industry is responsible for much of today's way-of-living, and it will remain in this position as long as the on-going miniaturisation can be continued. The industry is also responsible for the continued increase in computational power, which is vital for the complex simulations that are carried out nowadays. Not so well known is that linear algebra lies at the basis of these achievements. The virtual design environments used in the electronics industry rely heavily on continued developments in linear algebra, and computational speed of computers goes hand in hand with increases in the speed of the solution of linear systems. Linear algebra also plays an important role in the field of model order reduction, for which the electronics industry is one of the main fields of interest. In this presentation, these developments will be discussed in more detail, and many examples of the use of computational linear algebra in the electronics industry will be given. The talk will also discuss several open problems which could be a source of inspiration for further research.

Joachim Schoeberl (RWTH Aachen)Tuesday 22nd July 17:30
Divergence-free hybrid discontinuous Galerkin finite elements for incompressible Navier-Stokes equations, augmented Lagrangians, and robust preconditioners

Simon Shaw (Brunel University)Monday 21st July 17:30
Difficulties in computing the fundamental distortion mode in Coriolis mass flow meters.
The difficulty of computing accurate Coriolis distortion modes in Coriolis mass flow meters is presented and illustrated by several numerical results. It is tentatively suggested that the difficulty is due mainly to computer rounding error rather than any fundamental weakness in the finite element method or the eigen-solvers. An empirically evaluated successful method, using a shifted inverse iteration, for computing high quality results is given. This material is prsented in the spirit of a discussion. In particular, is there an existing method that can remove the difficulty? Or is this a new challenge for eigen-solvers?

David Silvester (University of Manchester)Saturday 19th July 12:15
Black-box multigrid preconditioning for unsteady incompressible flows
(Ongoing joint research with David Kay)
Simulation of the motion of an incompressible fluid remains an important but very challenging problem. The resources required for accurate three-dimensional simulation of practical flows test even the most advanced computer hardware. The necessity for reliable and efficient solvers is widely recognised. This talk will focus on two components of such a solver: the error control used for self-adaptive time stepping; and the linear solver used at each time level.

Valeria Simoncini ( Università di Bologna )Tuesday 15th July 09:00
Spectral properties of saddle point linear systems and relations to iterative solvers
In recently years there has been a significantly growing interest in the algebraic spectral analysis of matrices in the form

A=[A, BT; B, -C]

and of their preconditioned versions P-1A, with the nonsingular matrix P specifically chosen. These structured matrices often stem from saddle point problems in a variety of applications, that share some crucial algebraic properties. Under different hypotheses on A, B and C, most results aim at analyzing the spectrum of P-1A when P has a block structure similar to that of A, and such that P-1A has favorable spectral properties. In this series of lectures, we review some of the results in the literature, with special focus on:
  1. The influence of the spectral properties of the blocks of A and P on the spectrum of P-1A and the role of the proximity of P to A;
  2. The practical advantages and shortcomings of replacing the "ideal" P with a computationally more feasible preconditioning matrix PI;
  3. The actual usefulness of a spectral analysis in the prediction of the qualitative convergence behavior of iterative solvers based on Krylov subspaces.
The aim of this presentation is to emphasize some of the important aspects that should be taken into account when designing and analyzing a structured preconditioner for Krylov subspace methods. To this end, we try to avoid most technicalities and we are forced to limit our coverage of the wide body of available literature (see, e.g., the recent survey by Benzi, Golub and Liesen, Acta Numerica 2005).

Valeria Simoncini ( Università di Bologna )Wednesday 16th July 11:30
Spectral properties of saddle point linear systems and relations to iterative solvers

Valeria Simoncini ( Università di Bologna )Thursday 17th July 10:00
Spectral properties of saddle point linear systems and relations to iterative solvers

Alastair Spence (University of Bath)Tuesday 15th July 10:00
Eigenvalue calculation and stability assignment for discretised PDEs
An important problem in the analysis of time-dependent PDEs is the determination of the stability of steady state solutions, since this is often a prerequisite for a full understanding of the dynamical behaviour. We shall consider problems where the stability is determined by eigenvalues of certain linearised equations and concentrate of techniques based on bifurcation theory and eigenvalues.

We give an introduction to some basic concepts in numerical bifurcation theory and give examples of how these techniques have had spectacular success in explaining complicated flow behaviour in the Taylor problem in fluid dynamics. Next, we look in detail at the efficient numerical implementation of these bifurcation techniques for large sparse discretised PDEs. Finally, we consider some new developments in the preconditioning of linear systems arising from large sparse eigenvalue problems.

Alastair Spence (University of Bath)Wednesday 16th July 09:00
Eigenvalue calculation and stability assignment for discretised PDEs

Alastair Spence (University of Bath)Thursday 17th July 11:30
Eigenvalue calculation and stability assignment for discretised PDEs

Zdenek Strakos (Czech Academy of Sciences)Tuesday 15th July 14:00
On the RCWA method of computing solutions of Maxwell’s equation for a rectangular grating
Diffraction of light on a periodic media represents an important problem with numerous physical and engineering applications. The Rigorous Coupled Wave Analysis (RCWA) method assumes a specific form of gratings which enables a straightforward separation of space variables. Using Fourier expansions, the solutions of the resulting systems of ordinary differential equations for the Fourier amplitudes can be written in a form of matrix functions. They determine individual blocks of the subsequent linear algebraic system for integrating constants.

Derivation of the RCWA method must rigorously address two subtle points, in particular truncation of the Fourier expansions and physical interpretation of the solution of the discretized problem. The latter must take into account that in passive materials the signal in the direction of its propagation can only be damped, but not amplified. We briefly present the main points in derivation of the RCWA method and then examine a possible potential of matrix computations in building efficient RCWA-based solvers for practical problems.

Elisabeth Ullmann (TU Bergakademie Freiberg)Monday 21st July 18:00
An overview of solution strategies for stochastic Galerkin discretizations

Panayot Vassilevski (Lawrence Livermore National Laboratory)Tuesday 15th July 14:30
Finite volume/DG schemes based on constrained minimization function recovery
Traditional finite volume discretizations of time dependent PDEs allow for straightforward computation of averaged (piecewise constant) values of the physical quantities involved (such as pressure, velocity, density and energy). However, in order to close the overall discretization scheme, certain derivatives (gradient or divergence)of some of these quantities are needed. On the example of the Euler equations of gas dynamics,we study an approach based on minimizing TV (total variation) functionals subject to equality and inequality constraints to construct smooth function recovery of the pressure and velocity from their average values. The constraints have physical meaning; namely, positivity of pressure (or internal energy)and the recovered functions to preserve (approximately) their averages computed by the finite volume scheme. Extensions to higher order DG (discontinuous Galerkin) schemes can be derived in the same manner. We illustrate the overall finite volume scheme with some preliminary numerical results.

This work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

Ragner Winther (University of Oslo)Monday 21st July 09:00
Preconditioning saddle point problems arising from discretizations of partial differential equations

It seems to be an accepted fact that in order to properly design numerical methods for systems of partial differential equations one has to rely on the properties of the underlying differential systems themselves. In this series of talks we will argue that the same is true for the construction of preconditioners, i.e., we will argue that the structure of the preconditioners for the discrete systems are in some sense dictated by the properties of the corresponding continuous system.

In our first talk we will mainly discuss the concept of preconditioners for the continuous systems, and how these operators should be constructed. We will consider various examples. In particular, we will discuss parameter dependent problems, and how we can construct preconditioners that give rise to condition numbers which are independent of these parameters. As a consequence, we obtain iterative methods which converge with a rate which is bounded uniformly with respect to the parameters.

The second talk will be devoted to preconditioners for discrete problems obtained by finite element discretizations of systems of partial differential equations. Examples we will discuss include the stationary and time dependent Stokes problem, the Darcy--Stokes problem, and the Reissner--Mindlin plate model. We will also discuss examples of inverse problems and nonlinear problems. Furthermore, a disscussion of preconditioning in $H(div)$ and $H(curl)$ will also be given.

In the third talk will give a review of finite element exterior calculus and its connections to the Hellinger--Reissner mixed formulations of linear elasticity. We will discuss preconditioners for the general Hodge--Laplace problem. Furthermore, we will present various mixed finite elements for linear elasticity based on weak and strong symmetry requirements, and discuss how to construct preconditioners for these systems.

Ragner Winther (University of Oslo)Tuesday 22nd July 11:30
Preconditioning saddle point problems arising from discretizations of partial differential equations

Ragner Winther (University of Oslo)Wednesday 23rd July 10:30
Preconditioning saddle point problems arising from discretizations of partial differential equations

Walter Zulehner (Universität Linz)Tuesday 15th July 17:00
Multigrid methods for mixed variational problems and their convergence analysis
(Joint work with R. Simon (Duisburg-Essen) and J. Schöberl(Aachen))
We consider two different approaches for mixed problems to take advantage of the multigrid idea.

One way is to use an outer iteration, typically a block-preconditioned Richardson method, applied to the (discretized) mixed problem. Multigrid techniques (as an inner iteration or approximation) can be used for setting up the blocks of the block-preconditioner.

The other way is to use multigrid methods directly applied to the (discretized) mixed problem as an outer iteration based on appropriate smoothers (as a sort of inner iteration). This approach is also known as one-shot multigrid strategy. One of the most important ingredients of such a multigrid method is an appropriate smoother.

We consider typical examples for these two approaches applied to an elliptic optimal control problem and discuss the convergence analysis of these methods.