Past

We are interested in a microscopic stochastic description of a

population of discrete individuals characterized by one adaptive

trait. The population is modeled as a stochastic point process whose

generator captures the probabilistic dynamics over continuous time of

birth, mutation and death, as influenced by each individual's trait

values, and interactions between individuals. An offspring usually

inherits the trait values of her progenitor, except when a mutation

causes the offspring to take an instantaneous mutation step at birth

to new trait values. Once this point process is in place, the quest

for tractable approximations can follow different mathematical paths,

which differ in the normalization they assume (taking limit on

population size , rescaling time) and in the nature of the

corresponding approximation models: integro or integro-differential

equations, superprocesses. In particular cases, we consider the long

time behaviour for the stochastic or deterministic models.

The role of electron emission (either thermionic, secondary or

photoelectric) in charging an object immersed in a plasma is

investigated, both theoretically and numerically.

In fact, recent work [1] has shown how electron emission can

fundamentally affect the shielding potential around the object. In

particular, depending on the physical parameters of the system (that

were chosen such to correspond to common experimental conditions), the

shielding potential can develop an attractive potential well.

The conditions for the formation of the well will be reviewed, based

on a theoretical model of electron emission from the

grain. Furthermore, simulations will be presented regarding specific

laboratory, space and astrophysical applications.

[1] G.L. Delzanno, G. Lapenta, M. Rosenberg, Phys. Rev.

Lett., 92, 035002 (2004).

Time delays are associated with rock fracture and earthquakes. The delay associated with the initiation of a single fracture can be attributed to stress corrosion and a critical stress intensity factor [1]. Usually, however, the fracture of a brittle material, such as rock, results from the coalescence and growth of micro cracks. Another example of time delays in rock is the systematic delay before the occurrence of earthquake aftershocks. There is also a systematic time delay associated with rate-and-state friction. One important question is whether these time delays are related. Another important question is whether the time delays are thermally activated. In many cases systematic scaling laws apply to the time delays. An example is Omori92s law for the temporal decay of after shock activity. Experiments on the fracture of fiber board panels, subjected instantaneously to a load show a systematic power-law decrease in the delay time to failure as a function of the difference between the applied stress and a yield stress [2,3]. These experiments also show a power-law increase in energy associated with acoustic emissions prior to rupture. The frequency-strength statistics of the acoustic emissions also satisfy the power-law Gutenberg-Richter scaling. Damage mechanics and dynamic fibre-bundle models provide an empirical basis for understanding the systematic time delays in rock fracture and seismicity [4-7]. We show that these approachesgive identical results when applied to fracture, and explain the scaling obtained in the fibre board experiments. These approaches also give Omori92s type law. The question of precursory activation prior to rock bursts and earthquakes is also discussed. [1] Freund, L. B. 1990. Dynamic Fracture Mechanics, Cambridge University Press, Cambridge.20
[2] Guarino, A., Garcimartin, A., and Ciliberto, S. 1998. An experimental test of the critical behaviour of fracture precursors. Eur. Phys. J.; B6:13-24.20
[3] Guarino, A., Ciliberto, S., and Garcimartin, A. 1999. Failure time and micro crack nucleation. Europhys. Lett.; 47: 456.20
[4] Kachanov, L. M. 1986. Introduction to Continuum Damage Mechanics, Martinus Nijhoff, Dordrecht, Netherlands.20
[5] Krajcinovic, D. 1996. Damage Mechanics, Elsevier, Amsterdam.20
[6] Turcotte, D. L., Newman, W. I., and Shcherbakov, R. 2002. Micro- and macroscopic models of rock fracture, Geophys. J. Int.; 152: 718-728.
[7] Shcherbakov, R. and Turcotte, D. L. 2003. Damage and self-similarity in fracture. Theor. and Appl. Fracture Mech.; 39: 245-258.

In this talk we discuss the analytic approximation to the loss

distribution of large conditionally independent heterogeneous portfolios. The

loss distribution is approximated by the expectation of some normal

distributions, which provides good overall approximation as well as tail

approximation. The computation is simple and fast as only numerical

integration is needed. The analytic approximation provides an excellent

alternative to some well-known approximation methods. We illustrate these

points with examples, including a bond portfolio with correlated default risk

and interest rate risk. We give an analytic expression for the expected

shortfall and show that VaR and CVaR can be easily computed by solving a

linear programming problem where VaR is the optimal solution and CVaR is the

optimal value.

An essential first step in many problems of numerical analysis and

computer graphics is to cover a region with a reasonably regular mesh.

We describe a short MATLAB code that begins with a "distance function"

to describe the region: $d(x)$ is the distance to the boundary

(with d

The talk will start with an introduction to recent development in Mathematica, with emphasis on numerical computing. This will be followed by a discussion of graph drawing algorithms for the display of relational information, in particular force directed algorithms. The talk will show that by employing multilevel approach and octree data structure, it is possible to achieve fast display of very large relational information, without compromising the quality.

Complete stochastic volatility models provide prices and

hedges. There are a number of complete models which jointly model an

underlying and one or more vanilla options written on it (for example

see Lyons, Schonbucher, Babbar and Davis). However, any consistent

model describing the volatility of options requires a complex

dependence of the volatility of the option on its strike. To date we

do not have a clear approach to selecting a model for the volatility

of these options

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.