University of Bath

When steady solutions of complex physical problems are computed numerically it is often crucial to compute their stability in order to, for example, check that the computed solution is "physical", or carry out a sensitivity analysis, or help understand complex nonlinear phenomena near a bifurcation point. Usually a stability analysis requires the solution of an eigenvalue problem which may arise in its own right or as an appropriate linearisation. In the case of discretized PDEs the corresponding matrix eigenvalue problem will often be of generalised form: \\ $Ax=\lambda Mx$ (1) \\ with $A$ and $M$ large and sparse. In general $A$ is unsymmetric and $M$ is positive semi-definite. Only a small number of "dangerous" eigenvalues are usually required, often those (possibly complex) eigenvalues nearest the imaginary axis. In this context it is usually necessary to perform "shift-invert" iterations, which require repeated solution of systems of the form \\ $(A - \sigma M)y = Mx$, (2) \\ for some shift $\sigma$ (which may be near a spectral point) and for various right-hand sides $x$. In large applications systems (2) have to be solved iteratively, requiring "inner iterations". \\ \\ In this talk we will describe recent progress in the construction, analysis and implementation of fast algorithms for finding such eigenvalues, utilising algebraic domain decomposition techniques for the inner iterations. \\ \\ In the first part we will describe an analysis of inverse iteration techniques for (1) for a model problem in the presence of errors arising from inexact solves of (2). The delicate interplay between the convergence of the (outer) inverse iteration and the choice of tolerance for the inner solves can be used to determine an efficient iterative method provided a good preconditioner for $A$ is available. \\ \\ In the second part we describe an application to the computation of bifurcations in Navier-Stokes problems discretised by mixed finite elements applied to the velocity-pressure formulation. We describe the construction of appropriate preconditioners for the corresponding (3 x 3 block) version of (2). These use additive Schwarz methods and can be applied to any unstructured mesh in 2D or 3D and for any selected elements. An important part of the preconditioner is the adaptive coarsening strategy. At the heart of this are recent extensions of the Bath domain decomposition code DOUG, carried out by Eero Vainikko. \\ \\ An application to the computation of a Hopf bifurcation of planar flow around a cylinder will be given. \\ \\ This is joint work with Jörg Berns-Müller, Andrew Cliffe, Alastair Spence and Eero Vainikko and is supported by EPSRC Grant GR/M59075.

The simulation of sedimentary basins aims at reconstructing its historical

evolution in order to provide quantitative predictions about phenomena

leading to hydrocarbon accumulations. The kernel of this simulation is the

numerical solution of a complex system of time dependent, three

dimensional partial differential equations of mixed parabolic-hyperbolic

type in highly heterogeneous media. A discretisation and linearisation of

this system leads to large ill-conditioned non-symmetric linear systems

with three unknowns per mesh element.

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In the seminar I will look at different preconditioning approaches for

these systems and at their parallelisation. The most effective

preconditioner which we developed so far consists in three stages: (i) a

local decoupling of the equations which (in addition) aims at

concentrating the elliptic part of the system in the "pressure block'';

(ii) an efficient preconditioning of the pressure block using AMG; (iii)

the "recoupling'' of the equations. Numerical results on real case

studies, exhibit (i) a significant reduction of sequential CPU times, up

to a factor 5 with respect to the current ILU(0) preconditioner, (ii)

robustness with respect to physical and numerical parameters, and (iii) a

speedup of up to 4 on 8 processors.

In my talk I will introduce the concept of spectral discrete solitons

(SDSs): solutions of nonlinear Schroedinger type equations, which are localized on a regular grid in frequency space. In time domain such solitons correspond to periodic trains of pulses. SDSs play important role in cascaded four-wave-mixing processes (frequency comb generation) in optical fibres, where initial excitation by a two-frequency pump leads to the generation of multiple side-bands. When free space diffraction is taken into consideration, a non-trivial generalization of 1D SDSs will be discussed, in which every individual harmonic is an optical vortex with its own topological charge. Such excitations correspond to spatio-temporal helical beams.

TBA

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.