Past

We consider the solution of a linear system of equations using the GMRES iterative method. In some applications, performing inexact matrix-vector products in this method may be interesting, provided that a reasonable convergence of GMRES is achieved. A GMRES algorithm where the matrix vector product is performed inexactly is termed ”inexact GMRES algorithm”. An application of this idea occurs in computational electromagnetics, where the fast multipole method provides approximations of the matrix-vector product within a user-defined precision, and where these inaccurate matrix-vector products are all the more cheaper (in terms of CPU time) as the user-defined precision is low. The key point is then to design a control strategy of the accuracy of the matrix-vector product so that the GMRES converges better in the sense that 1) the inexact method achieves a satisfactory limiting accuracy, 2) within a reasonable number of steps.

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In [1], a relaxation strategy is proposed for general systems and validated on a large set of numerical experiments. This work is based on heuristic considerations and proposes a strategy that enables a convergence of the GMRES iterates $x_{k}$ within a relative normwise backward error $\frac{\|b−Ax_{k}\|}{\|A\| \|x_{k}\| + \|b\|}$ less than a prescribed quantity $\eta$ > 0, on a significant number of numerical experiments. Similar strategies have been applied to the solution of device simulation problems using domain decomposition [2] and to the preconditioning of a radiation diffusion problem in [5].

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A step toward a theoretical explanation of the observed behaviour of the inexact GMRES is proposed in [3, 4]. In this talk, we show that in spite of this considerable theoretical study, the experimental work of [1] is not fully understood yet. We give an overview of the questions that still remains open both in exact arithmetic and in floating-point arithmetic, and we provide some insights into the solution of some of them. Possible applications of this work for the preconditioned GMRES method, when the matrix-vector product is accurate but the preconditioning operation is approximate, are also investigated, based on [3].

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References

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[1] A. Bouras and V. Frayss´e. Inexact matrix-vector products in Krylov methods for solving linear systems: a relaxation strategy. SIAM Journal on Matrix Analysis and Applications, 2004. To appear.

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[2] A. Bouras, V. Frayss´e, and L. Giraud. A relaxation strategy for inner-outer linear solvers in domain decomposition methods. Technical Report TR/PA/00/17, CERFACS, Toulouse, France, 2000.

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[3] V. Simoncini and D. B. Szyld. Theory of inexact Krylov subspace methods and applications to scientific computing. SIAM Journal Scientific Computing, 25:454–477, 2003.

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[4] J. van den Eshof and G. L. G. Sleijpen. Inexact Krylov subspace methods for linear systems. SIAM Journal on Matrix Analysis and Applications, February 2004. To appear.

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[5] J. S. Warsa, M. Benzi, T. A. Warein, and J. E. Morel. Preconditioning a mixed discontinuous finite element method for radiation diffusion. Numerical Linear Algebra with Applications, 2004. To appear.

The talk will discuss the variationnal problem on finite

dimensional normed spaces and Finsler manifolds.

We first review different notions of ellipticity (convexity) for

parametric integrands (densities) on normed spaces and compare them with

different minimality properties of affine subspaces. Special attention will

be given to Busemann and Holmes-Thompson k-area. If time permits, we will

then present the first variation formula on Finsler manifolds and exhibit a

class of minimal submanifolds.

A version of Lyons theory of rough path calculus which applies to a

subclass of rough paths for which more geometric interpretations are

valid will be presented. Application will be made to the Brownian and

to the (fractional) support theorem.

The convergence to stationarity of many finite ergodic Markov

chains presents a sharp cut-off: there is a time T such that before

time T the chain is far from its equilibrium and, after time T,

equilibrium is essentially reached. We will discuss precise

definitions of the cut-off phenomenon, examples, and some partial

results and conjectures.

In recent years, there has been considerable interest, especially in the context of

fluid-dynamics, in nonconforming finite element methods that are based on discontinuous

piecewise polynomial approximation spaces; such approaches are referred to as discontinuous

Galerkin (DG) methods. The main advantages of these methods lie in their conservation properties, their ability to treat a wide range of problems within the same unified framework, and their great flexibility in the mesh-design. Indeed, DG methods can easily handle non-matching grids and non-uniform, even anisotropic, polynomial approximation degrees. Moreover, orthogonal bases can easily be constructed which lead to diagonal mass matrices; this is particularly advantageous in unsteady problems. Finally, in combination with block-type preconditioners, DG methods can easily be parallelized.

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In this talk, we introduce DG discretizations of mixed field and potential-based formulations of

eddy current problems in the time-harmonic regime. For the electric field formulation, the

divergence-free constraint within non-conductive regions is imposed by means of a Lagrange

multiplier. This allows for the correct capturing of edge and corner singularities in polyhedral domains; in contrast, additional Sobolev regularity must be assumed in the DG formulation, and their conforming counterparts, when regularization techniques are employed. In particular, we present a mixed method involving discontinuous $P^\ell-P^\ell$ elements, which includes a normal jump stabilization term, and a non-stabilized variant employing discontinuous $P^\ell-P^{\ell+1}$ elements.The first formulation delivers optimal convergence rates for the vector-valued unknowns in a suitable energy norm, while the second (non-stabilized) formulation is designed to yield optimal convergence rates in both the $L^2$--norm, as well as in a suitable energy norm. For this latter method, we also develop the {\em a posteriori} error estimation of the mixed DG approximation of the Maxwell operator. Indeed, by employing suitable Helmholtz decompositions of the error, together with the conservation properties of the underlying method, computable upper bounds on the error, measured in terms of the energy norm, are derived.

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Numerical examples illustrating the performance of the proposed methods will be presented; here,

both conforming and non-conforming (irregular) meshes will be employed. Our theoretical and

numerical results indicate that the proposed DG methods provide promising alternatives to standard conforming schemes based on edge finite elements.

Here I would like to present how one can obtain SDYM and

heavenly equations in general Fedosov deformation quantisation scheme. I am

considering some aspects of integrability (conservation laws,Lax pair,dressing

operator and Riemann-Hilbert problem).Then, using an embedding of sl(N,C) in the

Moyal bracket algebra, I am going to show an example of a series of chiral

sl(N,C) fields tending to heavenly spacetime when N tends to infinity (Ward's

question). ( All this is a natural continuation of the works by Strachan and

Takasaki).