Past Computational Mathematics and Applications Seminar

Nonlinear eigenvalue problem (NEP) is a class of eigenvalue problems where the matrix depends on the eigenvalue. We will first introduce some NEPs in real applications and some algorithms for general NEPs. Then we introduce our recent advances in NEPs, including second order Arnoldi algorithms for large scale quadratic eigenvalue problem (QEP), analysis and algorithms for symmetric eigenvalue problem with nonlinear rank-one updating, a new linearization for rational eigenvalue problem (REP).

This talk will introduce L1-regularized optimization problems that arise in image processing, and numerical methods for their solution. In particular, we will focus on methods of the split-Bregman type, which very efficiently solve large scale problems without regularization or time stepping. Applications include image

denoising, segmentation, non-local filters, and compressed sensing.

There is a widespread use of mathematical tools in finance and its

importance has grown over the last two decades. In this talk we

concentrate on optimization problems in finance, in particular on

numerical aspects. In this talk, we put emphasis on the mathematical problems and aspects, whereas all the applications are connected to the pricing of derivatives and are the

outcome of a cooperation with an international finance institution.

As one example, we take an in-depth look at the problem of hedging

barrier options. We review approaches from the literature and illustrate

advantages and shortcomings. Then we rephrase the problem as an

optimization problem and point out that it leads to a semi-infinite

programming problem. We give numerical results and put them in relation

to known results from other approaches. As an extension, we consider the

robustness of this approach, since it is known that the optimality is

lost, if the market data change too much. To avoid this effect, one can

formulate a robust version of the hedging problem, again by the use of

semi-infinite programming. The numerical results presented illustrate

the robustness of this approach and its advantages.

As a further aspect, we address the calibration of models being used in

finance through optimization. This may lead to PDE-constrained

optimization problems and their solution through SQP-type or

interior-point methods. An important issue in this context are

preconditioning techniques, like preconditioning of KKT systems, a very

active research area. Another aspect is the preconditioning aspect

through the use of implicit volatilities. We also take a look at the

numerical effects of non-smooth data for certain models in derivative

pricing. Finally, we discuss how to speed up the optimization for

calibration problems by using reduced order models.

Saddle-point problems occur frequently in liquid crystal modelling. For example, they arise whenever Lagrange multipliers are used for the pointwise-unit-vector constraints in director modelling, or in both general director and order tensor models when an electric field is present that stems from a constant voltage. Furthermore, in a director model with associated constraints and Lagrange multipliers, together with a coupled electric-field interaction, a particular ''double'' saddle-point structure arises. This talk will focus on a simple example of this type and discuss appropriate numerical solution schemes.

This is joint work with Eugene C. Gartland, Jr., Department of Mathematical Sciences, Kent State University.

We show that data assimilation using four-dimensional variation

(4DVar) can be interpreted as a form of Tikhonov regularisation, a

familiar method for solving ill-posed inverse problems. It is known from

image restoration problems that $L_1$-norm penalty regularisation recovers

sharp edges in the image better than the $L_2$-norm penalty

regularisation. We apply this idea to 4DVar for problems where shocks are

present and give some examples where the $L_1$-norm penalty approach

performs much better than the standard $L_2$-norm regularisation in 4DVar.

In dynamical systems given by an ODE, one is interested in the basin

of attraction of invariant sets, such as equilibria or periodic

orbits. The basin of attraction consists of solutions which converge

towards the invariant set. To determine the basin of attraction, one

can use a solution of a certain linear PDE which can be approximated

by meshless collocation.

The basin of attraction of an equilibrium can be determined through

sublevel sets of a Lyapunov function, i.e. a scalar-valued function

which is decreasing along solutions of the dynamical system. One

method to construct such a Lyapunov function is to solve a certain

linear PDE approximately using Meshless Collocation. Error estimates

ensure that the approximation is a Lyapunov function.

The basin of attraction of a periodic orbit can be analysed by Borg’s

criterion measuring the time evolution of the distance between

adjacent trajectories with respect to a certain Riemannian metric.

The sufficiency and necessity of this criterion will be discussed,

and methods how to compute a suitable Riemannian metric using

Meshless Collocation will be presented in this talk.

In the last few years, there has been renewed interest in stochastic

finite element methods (SFEMs), which facilitate the approximation

of statistics of solutions to PDEs with random data. SFEMs based on

sampling, such as stochastic collocation schemes, lead to decoupled

problems requiring only deterministic solvers. SFEMs based on

Galerkin approximation satisfy an optimality condition but require

the solution of a single linear system of equations that couples

deterministic and stochastic degrees of freedom. This is regarded as

a serious bottleneck in computations and the difficulty is even more

pronounced when we attempt to solve systems of PDEs with

random data via stochastic mixed FEMs.

In this talk, we give an overview of solution strategies for the

saddle-point systems that arise when the mixed form of the Darcy

flow problem, with correlated random coefficients, is discretised

via stochastic Galerkin and stochastic collocation techniques. For

the stochastic Galerkin approach, the systems are orders of

magnitude larger than those arising for deterministic problems. We

report on fast solvers and preconditioners based on multigrid, which

have proved successful for deterministic problems. In particular, we

examine their robustness with respect to the random diffusion

coefficients, which can be either a linear or non-linear function of

a finite set of random parameters with a prescribed probability

distribution.

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 Euler-Maruyama 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 diffusivities

will also be discussed.

A brief introduction to the field of complex networks is carried out by means of some examples. Then, we focus on the topics of defining and applying centrality measures to characterise the nodes of complex networks. We combine this approach with methods for detecting communities as well as to identify good expansion properties on graphs. All these concepts are formally defined in the presentation. We introduce the subgraph centrality from a combinatorial point of view and then connect it with the theory of graph spectra. Continuing with this line we introduce some modifications to this measure by considering some known matrix functions, e.g., psi matrix functions, as well as new ones introduced here. Finally, we illustrate some examples of applications in particular the identification of essential proteins in proteomic maps.

Future discovery and control in biology and medicine will come from

the mathematical modeling of large-scale molecular biological data,

such as DNA microarray data, just as Kepler discovered the laws of

planetary motion by using mathematics to describe trends in

astronomical data. In this talk, I will demonstrate that

mathematical modeling of DNA microarray data can be used to correctly

predict previously unknown mechanisms that govern the activities of

DNA and RNA.

First, I will describe the computational prediction of a mechanism of

regulation, by using the pseudoinverse projection and a higher-order

singular value decomposition to uncover a genome-wide pattern of

correlation between DNA replication initiation and RNA expression

during the cell cycle. Then, I will describe the recent

experimental verification of this computational prediction, by

analyzing global expression in synchronized cultures of yeast under

conditions that prevent DNA replication initiation without delaying

cell cycle progression. Finally, I will describe the use of the

singular value decomposition to uncover "asymmetric Hermite functions,"

a generalization of the eigenfunctions of the quantum harmonic

oscillator, in genome-wide mRNA lengths distribution data.

These patterns might be explained by a previously undiscovered asymmetry

in RNA gel electrophoresis band broadening and hint at two competing

evolutionary forces that determine the lengths of gene transcripts.

Regularization techniques based on the Golub-Kahan iterative bidiagonalization belong among popular approaches for solving large discrete ill-posed problems. First, the original problem is projected onto a lower dimensional subspace using the bidiagonalization algorithm, which by itself represents a form of regularization by projection. The projected problem, however, inherits a part of the ill-posedness of the original problem, and therefore some form of inner regularization must be applied. Stopping criteria for the whole process are then based on the regularization of the projected (small) problem.

We consider an ill-posed problem with a noisy right-hand side (observation vector), where the noise level is unknown. We show how the information from the Golub-Kahan iterative bidiagonalization can be used for estimating the noise level. Such information can be useful for constructing efficient stopping criteria in solving ill-posed problems.

This is joint work by Iveta Hn\v{e}tynkov\'{a}, Martin Ple\v{s}inger, and Zden\v{e}k Strako\v{s} (Faculty of Mathematics and Physics, Charles University, and Institute of Computer Science, Academy of Sciences, Prague)

We consider rational approximations to the Faddeeva or plasma dispersion function, defined

as

$w(z) = e^{-z^{2}} \mbox{erfc} (-iz)$.

With many important applications in physics, good software for

computing the function reliably everywhere in the complex plane is required. In this talk

we shall derive rational approximations to $w(z)$ via quadrature, M\"{o}bius transformations, and best approximation. The various approximations are compared with regard to speed of convergence, numerical stability, and ease of generation of the coefficients of the formula.

In addition, we give preference to methods for which a single expression yields uniformly

high accuracy in the entire complex plane, as well as being able to reproduce exactly the

asymptotic behaviour

$w(z) \sim i/(\sqrt{\pi} z), z \rightarrow \infty$

(in an appropriate sector).

This is Joint work with: Stephan Gessner, St\'efan van der Walt

Invariant subspaces are a well-established tool in the theory of linear eigenvalue problems. They are also computationally more stable objects than single eigenvectors if one is interested in a group of closely clustered eigenvalues. A generalization of invariant subspaces to matrix polynomials can be given by using invariant pairs.

We investigate some basic properties of invariant pairs and give perturbation results, which show that invariant pairs have similarly favorable properties for matrix polynomials than do invariant subspaces have for linear eigenvalue problems. In the second part of the talk we discuss computational aspects, namely how to extract invariant pairs from linearizations of matrix polynomials and how to do efficient iterative refinement on them. Numerical examples are shown using the NLEVP collection of nonlinear eigenvalue test problems.

This talk is joint work with Daniel Kressner from ETH Zuerich.

Molecular Dynamics Simulations are a tool to study the behaviour

of atomic-scale systems. The simulations themselves solve the

equations of motion for hundreds to millions of particles over

thousands to billions of time steps. Due to the size of the

problems studied, such simulations are usually carried out on

large clusters or special-purpose hardware.

At a first glance, there is nothing much of interest for a

Numerical Analyst: the equations of motion are simple, the

integrators are of low order and the computational aspects seem

to focus on hardware or ever larger and faster computer

clusters.

The field, however, having been ploughed mainly by domain

scientists (e.g. Chemists, Biologists, Material Scientists) and

a few Computer Scientists, is a goldmine for interesting

computational problems which have been solved either badly or

not at all. These problems, although domain specific, require

sufficient mathematical and computational skill to make finding

a good solution potentially interesting for Numerical Analysts.

The proper solution of such problems can result in speed-ups

beyond what can be achieved by pushing the envelope on Moore's

Law.

In this talk I will present three examples where problems

interesting to Numerical Analysts arise. For the first two

problems, Constraint Resolution Algorithms and Interpolated

Potential Functions, I will present some of my own results. For

the third problem, using interpolations to efficiently compute

long-range potentials, I will only present some observations and

ideas, as this will be the main focus of my research in Oxford

and therefore no results are available yet.

CFD is an indispensible part of the design process for all major gas turbine components. The growth in the use of CFD from single-block structured mesh steady state solvers to highly resolved unstructured mesh unsteady solvers will be described, with examples of the design improvements that have been achieved. The European Commission has set stringent targets for the reduction of noise, emissions and fuel consumption to be achieved by 2020. The application of CFD to produce innovative designs to meet these targets will be described. The future direction of CFD towards whole engine simulations will also be discussed.

The stability of our Solar System has been debated since Newton devised

the laws of gravitation to explain planetary motion. Newton himself

doubted the long-term stability of the Solar System, and the question

has remained unanswered despite centuries of intense study by

generations of illustrious names such as Laplace, Langrange, Gauss, and

Poincare. Finally, in the 1990s, with the advent of computers fast

enough to accurately integrate the equations of motion of the planets

for billions of years, the question has finally been settled: for the

next 5 billion years, and barring interlopers, the shapes of the

planetary orbits will remain roughly as they are now. This is called

"practical stability": none of the known planets will collide with each

other, fall into the Sun, or be ejected from the Solar System, for the

next 5 billion years.

Although the Solar System is now known to be practically stable, it may

still be "chaotic". This means that we may---or may not---be able

precisely to predict the positions of the planets within their orbits,

for the next 5 billion years. The precise positions of the planets

effects the tilt of each planet's axis, and so can have a measurable

effect on the Earth's climate. Although the inner Solar System is

almost certainly chaotic, for the past 15 years, there has been

some debate about whether the outer Solar System exhibits chaos or not.

In particular, when performing numerical integrations of the orbits of

the outer planets, some astronomers observe chaos, and some do not. This

is particularly disturbing since it is known that inaccurate integration

can inject chaos into a numerical solution whose exact solution is known

to be stable.

In this talk I will demonstrate how I closed that 15-year debate on

chaos in the outer solar system by performing the most carefully justified

high precision integrations of the orbits of the outer planets that has

yet been done. The answer surprised even the astronomical community,

and was published in _Nature Physics_.

I will also show lots of pretty pictures demonstrating the fractal nature

of the boundary between chaos and regularity in the outer Solar System.

Self adjusted meshes have important benefits approximating PDEs with solutions that exhibit nontrivial characteristics. When appropriately chosen, they lead to efficient, accurate and robust algorithms. Error control is also important, since appropriate analysis can provide guarantees on how accurate the approximate solution is through a posteriori estimates. Error control may lead to appropriate adaptive algorithms by identifying areas of large errors and adjusting the mesh accordingly. Error control and associated adaptive algorithms for important equations in Mathematical Physics is an open problem.

In this talk we consider the main structure of an algorithm which permits mesh redistribution with time and the nontrivial characteristics associated with it. We present improved algorithms and we discuss successful approaches towards error control for model problems (linear and nonlinear) of parabolic or hyperbolic type.

During the last two years, sparsity has become a key concept in various areas

of applied mathematics, computer science, and electrical engineering. Sparsity

methodologies explore the fundamental fact that many types of data/signals can

be represented by only a few non-vanishing coefficients when choosing a suitable

basis or, more generally, a frame. If signals possess such a sparse representation,

they can in general be recovered from few measurements using $\ell_1$ minimization

techniques.

One application of this novel methodology is the geometric separation of data,

which is composed of two (or more) geometrically distinct constituents -- for

instance, pointlike and curvelike structures in astronomical imaging of galaxies.

Although it seems impossible to extract those components -- as there are two

unknowns for every datum -- suggestive empirical results using sparsity

considerations have already been obtained.

In this talk we will first give an introduction into the concept of sparse

representations and sparse recovery. Then we will develop a very general

theoretical approach to the problem of geometric separation based on these

methodologies by introducing novel ideas such as geometric clustering of

coefficients. Finally, we will apply our results to the situation of separation

of pointlike and curvelike structures in astronomical imaging of galaxies,

where a deliberately overcomplete representation made of wavelets (suited

to pointlike structures) and curvelets/shearlets (suited to curvelike

structures) will be chosen. The decomposition principle is to minimize the

$\ell_1$ norm of the frame coefficients. Our theoretical results, which

are based on microlocal analysis considerations, show that at all sufficiently

fine scales, nearly-perfect separation is indeed achieved.

This is joint work with David Donoho (Stanford University).

Current community models in the geosciences employ a variety of numerical methods from finite-difference, finite-volume, finite- or spectral elements, to pseudospectral methods. All have specialized strengths but also serious weaknesses. The first three methods are generally considered low-order and can involve high algorithmic complexity (as in triangular elements or unstructured meshes). Global spectral methods do not practically allow for local mesh refinement and often involve cumbersome algebra. Radial basis functions have the advantage of being spectrally accurate for irregular node layouts in multi-dimensions with extreme algorithmic simplicity, and naturally permit local node refinement on arbitrary domains. We will show test examples ranging from vortex roll-ups, modeling idealized cyclogenesis, to the unsteady nonlinear flows posed by the shallow water equations to 3-D mantle convection in the earth’s interior. The results will be evaluated based on numerical accuracy, stability and computational performance.

This is a special talk outside the normal Computational Mathematics and Application seminar series. Please note it takes place on a Wednesday.

An iterative substructuring method with balancing Neumann-Neumann preconditioner is known as an efficient parallel algorithm for the elasticity equations. This method was extended to the Stokes equations by Pavarino and Widlund [2002]. In their extension, Q2/P0-discontinuous elements are used for velocity/pressure and a Schur complement system within "benign space", where incompressibility satisfied, is solved by CG method.

For the construction of the coarse space for the balancing preconditioner, some supplementary solvability conditions are considered. In our algorithm for 3-D computation, P1/P1 elements for velocity/pressure with pressure stabilization are used to save computational cost in the stiffness matrix. We introduce a simple coarse space similar to the one of elasticity equations. Owing to the stability term, solvabilities of local Dirichlet problem for a Schur complement system, of Neumann problem for the preconditioner, and of the coarse space problem are ensured. In our implementation, local Dirichlet and Neumann problems are solved by a 4x4-block modified Cholesky factorization procedure with an envelope method, which leads to fast computation with small memory requirement. Numerical result on parallel efficiency with a shared memory computer will be shown. Direct use of the Stokes solver in an application of Earth's mantle convection problem will be also shown.

Work with N.K. Nichols (Reading), C. Boess & A. Bunse-Gerstner (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 from applications in optimal control and state estimation. Variational data assimilation systems for weather, ocean and climate prediction 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 method 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 give a state estimation technique that retains more of the dynamical information of the full system. Numerical experiments using a shallow-water model illustrate the superior performance of model reduction to standard truncation techniques.

For the task of solving PDEs, finite difference (FD) methods are particularly easy to implement. Finite element (FE) methods are more flexible geometrically, but tend to be difficult to make very accurate. Pseudospectral (PS) methods can be seen as a limit of FD methods if one keeps on increasing their order of accuracy. They are extremely effective in many situations, but this strength comes at the price of very severe geometric restrictions. A more standard introduction to PS methods (rather than via FD methods of increasing orders of accuracy) is in terms of expansions in orthogonal functions (such as Fourier, Chebyshev, etc.).

Radial basis functions (RBFs) were first proposed around 1970 as a tool for interpolating scattered data. Since then, both our knowledge about them and their range of applications have grown tremendously. In the context of solving PDEs, we can see the RBF approach as a major generalization of PS methods, abandoning the orthogonality of the basis functions and in return obtaining much improved simplicity and flexibility. Spectral accuracy becomes now easily available also when using completely unstructured meshes, permitting local node refinements in critical areas. A very counterintuitive parameter range (making all the RBFs very flat) turns out to be of special interest. Computational cost and numerical stability were initially seen as serious difficulties, but major progress have recently been made also in these areas.

Quasicontinuum methods are a prototypical class of atomistic-to-continuum coupling methods. For example, we may wish to model a lattice defect (a vacancy or a dislocation) by an atomistic model, but the elastic far field by a continuum model. If the continuum model is consistent with the atomistic model (e.g., the Cauchy--Born model) then the main question is how the interface treatment affects the method.

In this talk I will introduce three of the main ideas how to treat the interface. I will explain their strengths and weaknesses by formulating the simplest possible non-trivial model problem and then simply analyzing the two classical concerns of numerical analysis: consistency and stability.

Inverse problems are often ill-posed, with solutions that depend sensitively on data. Regularization of some form is often used to counteract this. I will describe an approach to regularization, based on a Bayesian formulation of the problem, which leads to a notion of well-posedness for inverse problems, at the level of probability measures.

The stability which results from this well-posedness may be used as the basis for understanding approximation of inverse problems in finite dimensional spaces. I will describe a theory which carries out this program.

The ideas will be illustrated with the classical inverse problem for the heat equation, and then applied to so more complicated inverse problems arising in data assimilation, such as determining the initial condition for the Navier-Stokes equation from observations.

In recent years, the interdisciplinary field of imaging science has been experiencing an explosive growth in research activities including more models being developed, more publications generated, and above all wider applications attempted.
In this talk I shall first give an overview of the various imaging work carried out in our Liverpool group, some with collaborations with UCLA (T F Chan), CUHK (R H Chan) and Bergen (X C Tai) and several colleagues from other departments in Liverpool. Then I shall focus on two pieces of recent work, denoising and segmentation respectively:
(i) Image denoising has been a research topic deeply investigated within the last two decades. Even algorithmically the well-known ROF model (1992) can be solved efficiently. However less work has been done on models using high order regularization. I shall describe our first and successful attempt to develop a working multilevel algorithm for a 4th order nonlinear denoising model, and our work on solving the combined denoising and deblurring problem, different from the reformulation approach by M N Ng and W T Yin (2008) et al.
(ii) the image active contour model by Chan-Vese (2001) can be solved efficiently both by a geometric multigrid method and by an optimization based multilevel method. Surprisingly the new multilevel methods can find a solution closer to the global minimize than the existing unilevel methods. Also discussed are some recent work (jointly with N Badshah) on selective segmentation that has useful medical applications.

Geometric integration is the numerical integration of a differential equation, while preserving one or more of its geometric/physical properties exactly, i.e. to within round-off error.

Many of these geometric properties are of crucial importance in physical applications: preservation of energy, momentum, angular momentum, phase-space volume, symmetries, time-reversal symmetry, symplectic structure and dissipation are examples. The field has tantalizing connections to dynamical systems, as well as to Lie groups.

In this talk we first present a survey of geometric numerical integration methods for differential equations, and then exemplify this by discussing symplectic vs energy-preserving integrators for ODEs as well as for PDEs.

We discuss numerical methods for the solution of the palindromic eigenvalue problem Ax=λ A^Tx, where A is a complex matrix. Such eigenvalue problems occur, for example, in the vibration analysis of rail tracks.

The structure of palindromic eigenvalue problems leads to a symmetry in the spectrum: all eigenvalues occur in reciprocal pairs. The need for preservation of this symmetry in finite precision arithmetic requires the use of structure-preserving numerical methods. In this talk, we explain how such methods can be derived.

The aim of this talk is to render the power of (short-step) interior-point methods for linear programming (and by extension, convex programming) intuitively understandable to those who have a basic training in numerical methods for dynamical systems solving. The connection between the two areas is made by interpreting line-search methods in a forward Euler framework, and by analysing the algorithmic complexity in terms of the stiffness of the vector field of search directions. Our analysis cannot replicate the best complexity bounds, but due to its weak assumptions it also applies to inexactly computed search directions and has explanatory power for a wide class of algorithms.

Co-Author: Coralia Cartis, Edinburgh University School of Mathematics.

This talk is concerned with the probabilistic behaviour of a condition

number C(A) for the problem of deciding whether a system of n

homogeneous linear inequalities in m unknowns has a non-zero solution.

In the case where the input system is feasible, the exact probability

distribution of the condition number for random inputs is determined,

and a sharp bound for the general case. In particular, for the

expected value of the logarithm log C(A), an upper bound of order

O(log m) (m the number of variables) is presented which does not

depend on the number of inequalities.

The probability distribution of the condition number C(A) is closely

related to the probability of covering the m-sphere with n spherical

caps of a given radius. As a corollary, we obtain bounds on the

probability of covering the sphere with random caps.

The climate is largely determined by the ocean flow, which in itself is driven by wind and by gradients in temperature and salinity. Nowadays numerical models exist that are able to describe the occurring phenomena not only qualitatively but also quantitatively. At the Institute for Marine and Atmospheric research Utrecht (IMAU) a so-called thermohaline circulation model is developed in which methods of dynamical systems theory are used to study the stability of ocean flows. Here bifurcation diagrams are constructed by varying the strength of the forcing, for instance the amount of fresh water coming in from the north due to melting. For every value of the strength we have to solve a nonlinear system, which is handled by a Newton-type method. This produces many linear systems to be solved. 

In the talk the following will be addressed: the form of the system of equations, a special purpose method which uses Trilinos and MRILU. The latter is a multilevel ILU preconditioner developed at Groningen University. Results of the approach obtained on the Dutch national supercomputer will be shown.

For large--scale saddle point problems, the application of exact iterative schemes and preconditioners may be computationally expensive. In practical situations, only approximations to the inverses of the diagonal block or the related cross-product matrices are considered, giving rise to inexact versions of various solvers. Therefore, the approximation effects must be carefully studied. In this talk we study numerical behavior of several iterative Krylov subspace solvers applied to the solution of large-scale saddle point problems. Two main representatives of the segregated solution approach are analyzed: the Schur complement reduction method, based on an (iterative) elimination of primary variables and the null-space projection method which relies on a basis for the null-space for the constraints. We concentrate on the question what is the best accuracy we can get from inexact schemes solving either Schur complement system or the null-space projected system when implemented in finite precision arithmetic. The fact that the inner solution tolerance strongly influences the accuracy of computed iterates is known and was studied in several contexts.

In particular, for several mathematically equivalent implementations we study the influence of inexact solving the inner systems and estimate their maximum attainable accuracy. When considering the outer iteration process our rounding error analysis leads to results similar to ones which can be obtained assuming exact arithmetic. The situation is different when we look at the residuals in the original saddle point system. We can show that some implementations lead ultimately to residuals on the the roundoff unit level independently of the fact that the inner systems were solved inexactly on a much higher level than their level of limiting accuracy. Indeed, our results confirm that the generic and actually the cheapest implementations deliver the approximate solutions which satisfy either the second or the first block equation to the working accuracy. In addition, the schemes with a corrected direct substitution are also very attractive. We give a theoretical explanation for the behavior which was probably observed or it is already tacitly known. The implementations that we pointed out as optimal are actually those which are widely used and suggested in applications.

Multicore chips are nearly ubiquitous in modern machines, and to fully exploit this continuation of Moore's Law, numerical algorithms need to be able to exploit parallelism. We describe recent approaches to both dense and sparse parallel Cholesky factorization on shared memory multicore systems and present results from our new codes for problems arising from large real-world applications. In particular we describe our experiences using directed acyclic graph based scheduling in the dense case and retrofitting parallelism to a

sparse serial solver.

As research in topology optimisation has reached a level of maturity, two main classes of methods have emerged and their applications to real engineering design in industry are increasing. It has therefore become important to identify the limitations and challenges in order to ensure that topology optimisation is appropriately employed during the design process whilst research may continue to offer a more reliable and fast design tool to engineers.

The seminar will begin by introducing the topology optimisation problem and the two popular finite element based approaches. A range of numerical methods used in the typical implementations will be outlined. This will form the basis for the discussion on the short-comings and challenges as an easy-to-use design tool for engineers, particularly in the context of reliably providing the consistent optimum solutions to given problems with minimum a priori information. Another industrial requirement is a fast solution time to easy-to-set-up problems. The seminar will present the recent efforts in addressing some of these issues and the remaining challenges for the future.

Partial differential equations with a nonlinear pointwise constraint defined through a manifold occur in a variety of applications: The magnetization of a ferromagnet can be described by a unit length vector field and the orientation of the rod-like molecules that constitute a liquid crystal is often modeled by a vector field that attains its values in the real projective plane thus respecting the head-to-tail symmetry of the molecules. Other applications arise in geometric

modeling, quantum mechanics, and general relativity. Simple examples reveal that it is impossible to satisfy pointwise constraints exactly by lowest order finite elements. For two model problems we discuss the practical realization of the constraint, the efficient solution of the resulting nonlinear systems of equations, and weak accumulation of approximations at exact solutions.

We focus on domain decomposition methods for systems of PDEs (versus scalar PDEs). The Smith factorization (a "pure" algebra tool) is used systematically to derive new domain decompositions methods for symmetric and unsymmetric systems of PDEs: the compressible Euler equations, the Stokes and Oseen (linearized Navier-Stokes) problem. We will focus on the Stokes system. In two dimensions the key idea is the transformation of the Stokes problem into a scalar bi-harmonic problem. We show, how a proposed domain decomposition method for the bi-harmonic problem leads to a domain decomposition method for the Stokes equations which inherits the convergence behavior of the scalar problem. Thus, it is sufficient to study the convergence of the scalar algorithm. The same procedure can also be applied to the three-dimensional Stokes problem.

The butterfly-shaped Lorenz attractor is a fractal set made up of infinitely many periodic orbits. Ever since Lorenz (1963) introduced a system of three simple ordinary differential equations, much of the discussion of his system and its strange attractor has adopted a dynamical point of view. In contrast, we allow time to be a complex variable and look upon such solutions of the Lorenz system as analytic functions. Formal analysis gives the form and coefficients of the complex singularities of the Lorenz system. Very precise (> 500 digits) numerical computations show that the periodic orbits of the Lorenz system have singularities which obey that form exactly or very nearly so. Both formal analysis and numerical computation suggest that the mathematical analysis of the Lorenz system is a problem in analytic function theory. (Joint work with S. Sahutoglu).

The advent of multicore processors and other technologies like Graphical Processing Units (GPU) will considerably influence future research in High Performance Computing.

To take advantage of these architectures in dense linear algebra operations, new algorithms are

proposed that use finer granularity and minimize synchronization points.

After presenting some of these algorithms, we address the issue of pivoting and investigate randomization techniques to avoid pivoting in some cases.

In the particular case of GPUs, we show how linear algebra operations can be enhanced using

hybrid CPU-GPU calculations and mixed precision algorithms.

This is not intended to be a systematic History, but a selection of highlights, with some digressions, including:

The early days of the Computing Lab;

How the coming of the Computer changed some of the ways we do Computation;

A problem from the Study Groups;

Influence of the computing environment (hardware and software);

Convergence analysis for the heat equation, then and now.