WSC Werkgemeenschap Scientific Computing

Spring Meeting WSC 2009

Friday April 24, 2009, the Werkgemeenschap Scientific Computing is organizing a spring meeting at the Technical University Eindhoven. A mixture of eight young and senior researchers have been selected to give a presentation on their research.
The filmhuis Zwarte Doos has been chosen as a perfect location for this meeting at the TU-Eindhoven. Here you can find more information about this location.


10.00-10.30 h.	Registration, coffee/tea, welcome

10.30-11.15 h.	Stefan Vandewalle (KU Leuven)
	Multigrid methods for partial differential equations with stochastic coefficients [abstract]
11.15-11.40 h.	Tammo Jan Dijkema (UU)
	Approximation in high dimensional product domains [abstract]
11.40-12.05 h.	Kim Volders (UA)
	Stability of finite difference schemes on nonuniform grids for the BlackScholes equation [abstract][pdf]
12.05-12.30 h.	Hisham bin Zubair (TUD-EWI)
	Multigrid Preconditioners for the indefinite Helmholtz Problems on Locally Refined Grids [abstract][pdf]
12.30-13.30 h.	Lunch at the lounge of the Zwarte Doos

13.30-14.15 h.	Michiel Hochstenbach (TU/e)
	Eigenvalue problems and DDE stability[abstract]
14.15-14.40 h.	Ricardo Reis da Silva (UvA-KdVI)
	Computable bounds for the smallest singular value of a rank-k perturbation of a matrix [abstract]
14.40-15.05 h.	Maria Ugryumova (TU/e)
	Stability and Passivity of the Super Node Algorithm for EM modelling of ICs [abstract]
15.05-15.15 h.	Break

15.15-16.00 h.	Kees Oosterlee (CWI,TUD-EWI)
	Numerical Mathematics Aspects in Computational Finance [abstract]

Organizing comittee:
Prof.dr. J.G. Verwer (CWI,UvA-KdvI), Prof.dr. W.H.A. Schilders (TU/e,NXP), Enna van Dijk (CASA), Drs. Margreet Nool (CWI, secretary).

Participation (including lunch) is free of charge but registration is obligatory. Please registrate before April 15 2009.
Questions? Please ask: Margreet Nool or Enna van Dijk

Stefan Vandewalle
Multigrid methods for partial differential equations with stochastic coefficients

The stochastic finite element method is an important technique for solving certain classes of stochastic partial differential equations (PDEs). This method approximates the solution of the PDE by a generalized polynomial chaos expansion. By using a Galerkin projection in the stochastic dimension, the stochastic PDE is transformed into a coupled set of deterministic PDEs. A finite element discretization converts this deterministic PDE system into a high dimensional algebraic system. Specialized iterative solvers are required to solve the resulting problem.
In this talk, we shall present an overview of iterative solution approaches. We start from iterative methods based on a block splitting of the system matrices. Next, we extend these methods for use as preconditioner for a Krylov method, and for use as smoother in a multilevel context. Then, the various solvers will be compared based on their convergence properties, computational cost and implementation effort. Our findings are illustrated by means of two numerical problems. The first one is a steady-state diffusion problem with a discontinuous random field as diffusion coefficient. The second is a deterministic diffusion problem defined on a random domain.
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Tammo Jan Dijkema
Approximation in high dimensional product domains

In this talk I will recall the concept of sparse grids, which can be used to approximate functions on product domains with a rate that is almost independent of the space dimension. So-called optimized sparse grids make this approximation rate optimal, and truly dimension independent. However nice these results may seem, the smoothness that is required for these methods to work, is rather high. I will show that the solution of the Poisson equation generally is not smooth enough.
As a solution to this problem, an adaptive method can be used, yielding the same optimal rate for a broader class of functions. I will introduce an adaptive wavelet method that was used to solve Poisson's equation in up to 10 space dimensions.
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Kim Volders
Stability of finite difference schemes on nonuniform grids for the BlackScholes equation

We consider the well-known Black-Scholes equation from financial option pricing theory,

$\displaystyle \frac{\partial u}{\partial t}=\frac{1}{2}\sigma^2s^2\frac{\partial^2 u} {\partial s^2}+rs\frac{\partial u}{\partial s}-ru\quad(s>0,\, t>0)$

with given real constants $r, \sigma > 0$ . The Black-Scholes equation is a time-dependent advection-diffusion-reaction equation and is supplemented with initial and boundary conditions.

A popular approach for the numerical solution of time-dependent partial differential equations is the method-of-lines. It consists of two steps:

Spatial discretization: the partial derivatives $\partial u/\partial s, \partial^2 u/\partial s^2$ are discretized on a finite spatial grid, yielding a (large) system of ordinary differential equations

$\displaystyle U'(t)=AU(t)+b(t) \quad (t>0)$ (1)

with given fixed matrix and vectors .
Temporal discretization: the above system of ordinary differential equations is numerically integrated in time.

Our research focuses on the stability analysis of second-order finite difference methods for the spatial discretization of the Black-Scholes equation. We first present practical upper bounds for

$\displaystyle \vert\vert e^{tA}\vert\vert _2 \quad (t>0)$

where $\vert\vert\cdot\vert\vert _2$ denotes a scaled version of the standard spectral norm. We subsequently present sufficient conditions for contractivity in the maximum-norm,

$\displaystyle \vert\vert e^{tA}\vert\vert _\infty\leq 1\quad (t>0).$

A virtue of our stability analysis is that it applies to spatial grids that are not uniform. Such grids are often used in actual applications. Numerical experiments are provided which support our theoretical results. Finally, we briefly discuss the stability of temporal discretization schemes for (1) w.r.t. $\vert\vert\cdot\vert\vert _2$ and $\vert\vert\cdot\vert\vert _\infty$
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Hisham bin Zubair
Multigrid Preconditioners for the indefinite Helmholtz Problems on Locally Refined Grids

In this talk we present the construction and performance of a geometric multigrid method for grids having two different layers of refinement. The method is employed to approximately invert the Krylovpreconditioner, i.e., the complex shifted Helmholtz operator, for the indefinite Helmholtz equation. The usual FAC and MLAT based grid coarsening techniques only coarsen the fine layer of the grid. The method presented here, in contrast, coarsens the whole grid simultaneously. Combined with a simple smoother, piece-wise constant restriction and bilinear prolongation, this gives an efficient multigrid method that works very well for the model problems, which are 2-d Helmholtz equations with strongly varying coefficients.
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Michiel Hochstenbach
Eigenvalue problems and DDE stability

We review some recent results in the field of stability of delay differential equations (DDEs) and show how various types of eigenvalue problems (including new types) play important roles. As these problems are often of a very high dimension, we also sketch structure-preserving methods to numerically solve them.
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Ricardo Reis da Silva
Computable bounds for the smallest singular value of a rank-k perturbation of a matrix

In this talk we look into the computation of lower bounds for the smallest singular values of perturbations of matrices. Our starting point is the smallest eigenvalue of a seemingly simple Hermitian rank-one perturbation H of a Hermitian matrix A. We will see how both Weyl's bound and a recent bound of Ipsen and Nadler are the two first terms of a sequence of bounds. This sequence is non-decreasing and, in general, the q-th term is the smallest eigenvalue of a q × q matrix. Similar bounds can be obtained for the smallest eigenvalue of rank-k perturbations of Hermitian matrices and for the smallest singular value of perturbations of arbitrary n×m matrices B.
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Maria Ugryumova
Stability and Passivity of the Super Node Algorithm for EM modelling of ICs

The super node algorithm is a model order reduction technique based on physical principles. Some of the properties of the reduced models generated by this algorithm, such as stability and passivity have not yet been studied thoroughly. The loss of passivity constitutes a serious problem because the reduced networks may show artificial behavior which renders the simulations useless. We investigate the stability and passivity properties of the algorithm. We explain why passivity is not guaranteed and we present a way to modify the algorithm in order to provide always passive reduced models.
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Kees Oosterlee
Numerical Mathematics Aspects in Computational Finance

In this presentation we will discuss some topics in Finance that require Mathematics, and, in particular, efficient numerical techniques. We will discuss option pricing, for example, for option contracts based on more than one underlying stock, and contracts with advanced stochastic models for the underlying stock price dynamics. We will focus on a mathematical framework in which we perform this research. One of the aims is to price the options as fast as possible.
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