Past Computational Mathematics and Applications Seminar

Based on an MPI library written over 10 years ago, OP2 is a new open-source library which is aimed at application developers using unstructured grids. Using a single API, it targets a variety of HPC architectures, including both manycore GPUs and multicore CPUs with vector units. The talk will cover the API design, key aspects of the parallel implementation on the different platforms, and preliminary performance results on a small but representative CFD test code.

Project homepage: http://people.maths.ox.ac.uk/gilesm/op2/

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 widely-used 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 Xiao-Wen Chang, Chris Paige, Pavel Jiranek, and Serge Gratton.

In various fields of application, one must solve very large scale boundary value problems using parallel solvers and supercomputers. The domain decomposition approach partitions the large computational domain into smaller computational subdomains. In order to speed up the convergence, we have several ``optimized'' algorithm that use Robin transmission conditions across the artificial interfaces (FETI-2LM). It is known that this approach alone is not sufficient: as the number of subdomains increases, the number of iterations required for convergence also increases and hence the parallel speedup is lost. A known solution for classical Schwarz methods as well as FETI algorithms is to incorporate a ``coarse grid correction'', which is able to transmit low-frequency information more quickly across the whole domain. Such algorithms are known to ``scale weakly'' to large supercomputers. A coarse grid correction is also necessary for FETI-2LM methods. In this talk, we will introduce and analyze coarse grid correction algorithms for FETI-2LM domain decomposition methods.

Can one of the most important Italian Renaissance frescoes reduced in hundreds of thousand fragments by a bombing during the Second World War be re-composed after more than 60 years from its damage? Can we reconstruct the missing parts and can we say something about their original color?

In this talk we would like to exemplify, hopefully effectively by taking advantage of the seduction of art, how mathematics today can be applied in real-life problems which were considered unsolvable only few years ago.

Implementations of the revised simplex method for solving large scale sparse linear programming (LP) problems are highly efficient for single-core architectures. This talk will discuss the limitations of the underlying techniques in the context of modern multi-core architectures, in particular with respect to memory access. Novel techniques for implementing the dual revised simplex method will be introduced, and their use in developing a dual revised simplex solver for multi-core architectures will be described.

We propose and analyze a primal-dual active set method for local and non-local vector-valued Allen-Cahn variational inequalities.

We show existence and uniqueness of a solution for the non-local vector-valued Allen-Cahn variational inequality in a formulation involving Lagrange multipliers for local and non-local constraints. Furthermore, convergence of the algorithm is shown by interpreting the approach as a semi-smooth Newton method and numerical simulations are presented.

Optimization problems with time-periodic parabolic PDE constraints can arise in important chemical engineering applications, e.g., in periodic adsorption processes. I will present a novel direct numerical method for this problem class. The main numerical challenges are the high nonlinearity and high dimensionality of the discretized problem. The method is based on Direct Multiple Shooting and inexact Sequential Quadratic Programming with globalization of convergence based on natural level functions. I will highlight the use of a generalized Richardson iteration with a novel two-grid Newton-Picard preconditioner for the solution of the quadratic subproblems. At the end of the talk I will explain the principle of Simulated Moving Bed processes and conclude with numerical results for optimization of such a process.

Efficient linear and infinitely smooth approximation of functions from equidistant samples is a fascinating problem, at least since Runge showed in 1901 that it is not delivered by the interpolating polynomial.

In 1988, I suggested to substitute linear rational for polynomial interpolation by replacing the denominator 1 with a polynomial depending on the nodes, though not on the interpolated function. Unfortunately the so-obtained interpolant converges merely as the square of the mesh size. In 2007, Floater and Hormann have given for every integer a denominator that yields convergence of that prescribed order.

In the present talk I shall present the corresponding interpolant as well as some of its applications to differentiation, integration and the solution of boundary value problems. This is joint work with Georges Klein and Michael Floater.

The Bi-Conjugate Gradient method (BiCG) is a well-known iterative solver (Krylov method) for linear systems of equations, proposed about 35 years ago, and the basis for some of the most successful iterative methods today, like BiCGSTAB. Nevertheless, the convergence behavior is poorly understood. The method satisfies a Petrov-Galerkin property, and hence its residual is constrained to a space of decreasing dimension (decreasing one per iteration). However, that does not explain why, for many problems, the method converges in, say, a hundred or a few hundred iterations for problems involving a hundred thousand or a million unknowns. For many problems, BiCG converges not much slower than an optimal method, like GMRES, even though the method does not satisfy any optimality properties. In fact, Anne Greenbaum showed that every three-term recurrence, for the first (n/2)+1 iterations (for a system of dimension n), is BiCG for some initial 'left' starting vector. So, why does the method work so well in most cases? We will introduce Krylov methods, discuss the convergence of optimal methods, describe the BiCG method, and provide an analysis of its convergence behavior.

Algebraic multigrid methods are nowadays popular to solve linear systems arising from the discretization of elliptic PDEs. They try to combine the efficiency of well tuned specific schemes like classical (geometric-based) multigrid methods, with the ease of use of general purpose preconditioning techniques. This requires to define automatic coarsening procedures, which set up an hierarchy of coarse representations of the problem at hand using only information from the system matrix.

In this talk, we focus on aggregation-based algebraic multigrid methods. With these, the coarse unknowns are simply formed by grouping variables in disjoint subset called aggregates.

In the first part of the talk, we consider symmetric M-matrices with nonnegative row-sum. We show how aggregates can then be formed in such a way that the resulting method satisfies a prescribed bound on its convergence rate. That is, instead of the classical paradigm that applies an algorithm and then performs its analysis, the analysis is integrated in the set up phase so as to enforce minimal quality requirements. As a result, we obtain one of the first algebraic multigrid method with full convergence proof. The efficiency of the method is further illustrated by numerical results performed on finite difference or linear finite element discretizations of second order elliptic PDEs; the set of problems includes problems with jumps, anisotropy, reentering corners and/or unstructured meshes, sometimes with local refinement.

In the second part of the talk, we discuss nonsymmetric problems. We show how the previous approach can be extended to M-matrices with row- and column-sum both nonnegative, which includes some stable discretizations of convection-diffusion equations with divergence free convective flow. Some (preliminary) numerical results are also presented.

This is joint work with Artem Napov.

Starting from a Navier-Stokes-Cahn-Hilliard equation for a two-phase flow problem we discuss efficient numerical approaches based on adaptive finite element methods. Various extensions of the model are discussed: a) we consider the model on implicitly described geometries, which is used to simulate the sliding of droplets over nano-patterned surfaces, b) we consider the effect of soluble surfactants and show its influence on tip splitting of droplets under shear flow, and c) we consider bijels as a new class of soft matter materials, in which colloidal particles are jammed on the fluid-fluid interface and effect the motion of the interface due to an elastic force.

The work is based on joint work with Sebastian Aland (TU Dresden), John Lowengrub (UC Irvine) and Knut Erik Teigen (U Trondheim).

We extend for the first time the linear discretization theory of Schaback, developed for meshfree methods, to nonlinear operator equations, relying heavily on methods of Böhmer, Vol I. There is no restriction to elliptic problems or to symmetric numerical methods like Galerkin techniques.

Trial spaces can be arbitrary, but have to approximate the solution well, and testing can be weak or strong. We present Galerkin techniques as an example. On the downside, stability is not easy to prove for special applications, and numerical methods have to be formulated as optimization problems. Results of this discretization theory cover error bounds and convergence rates. These results remain valid for the general case of fully nonlinear elliptic differential equations of second order. Some numerical examples are added for illustration.

We present an O(N logN) algorithm for the calculation of the first N coefficients in an expansion of an analytic function in Legendre polynomials. In essence, the algorithm consists of an integration of a suitably weighted function along an ellipse, a task which can be accomplished with Fast Fourier Transform, followed by some post-processing.

We discuss a class of high-order spectral-Galerkin surface integral algorithms with specific focus on simulating the scattering of electromagnetic waves by a collection of three dimensional deterministic and stochastic particles.

In the eighteenth century Gaspard Monge considered the problem of finding the best way of moving a pile of material from one site to another. This optimal transport problem has many applications such as mesh generation, moving mesh methods, image registration, image morphing, optical design, cartograms, probability theory, etc. The solution to an optimal transport problem can be found by solving the Monge-Amp\`{e}re equation, a highly nonlinear second order elliptic partial differential equation. Leonid Kantorovich, however, showed that it is possible to analyse optimal transport problems in a framework that naturally leads to a linear programming formulation. In recent years several efficient methods have been proposed for solving the Monge-Amp\`{e}re equation. For the linear programming problem, standard methods do not exploit the special properties of the solution and require a number of operations that is quadratic or even cubic in the number of points in the discretisation. In this talk I will discuss techniques that can be used to obtain more efficient methods.

Joint work with Chris Budd (University of Bath).

Meshless (or meshfree) methods are a relatively new numerical approach for the solution of ordinary- and partial differential equations. They offer the geometrical flexibility of finite elements but without requiring connectivity from the discretization support (ie a mesh). Meshless methods based on the collocation of radial basis functions (RBF methods) are particularly easy to code, and have a number of theoretical advantages as well as practical drawbacks. In this talk, an adaptive RBF scheme is presented for a novel application, namely the solution of (a rather broad class of) delayed- and neutral differential equations.

We consider saddle point problems arising as (linearized) optimality conditions in elliptic optimal control problems. The efficient solution of such systems is a core ingredient in second-order optimization algorithms. In the spirit of Bramble and Pasciak, the preconditioned systems are symmetric and positive definite with respect to a suitable scalar product. We extend previous work by Schoeberl and Zulehner and consider problems with control and state constraints. It stands out as a particular feature of this approach that an appropriate symmetric indefinite preconditioner can be constructed from standard preconditioners for those matrices which represent the inner products, such as multigrid cycles.

Numerical examples in 2D and 3D are given which illustrate the performance of the method, and limitations and open questions are addressed.

Interior-point methods for linear and convex quadratic programming

require the solution of a sequence of symmetric indefinite linear

systems which are used to derive search directions. Safeguards are

typically required in order to handle free variables or rank-deficient

Jacobians. We propose a consistent framework and accompanying

theoretical justification for regularizing these linear systems. Our

approach is akin to the proximal method of multipliers and can be

interpreted as a simultaneous proximal-point regularization of the

primal and dual problems. The regularization is termed "exact" to

emphasize that, although the problems are regularized, the algorithm

recovers a solution of the original problem. Numerical results will be

presented. If time permits we will illustrate current research on a

matrix-free implementation.

This is joint work with Michael Friedlander, University of British Columbia, Canada

Shifted Laplace preconditioners have attracted considerable attention as

a technique to speed up convergence of iterative solution methods for the

Helmholtz equation. In the talk we present a comprehensive spectral

analysis of the discrete Helmholtz operator preconditioned with a shifted

Laplacian. Our analysis is valid under general conditions. The propagating

medium can be heterogeneous, and the analysis also holds for different types

of damping, including a radiation condition for the boundary of the computational

domain. By combining the results of the spectral analysis of the

preconditioned Helmholtz operator with an upper bound on the GMRES-residual

norm we are able to derive an optimal value for the shift, and to

explain the mesh-depency of the convergence of GMRES preconditioned

with a shifted Laplacian. We will illustrate our results with a seismic test

problem.

Joint work with: Yogi Erlangga (University of British Columbia) and Kees Vuik (TU Delft)

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.