Magic 2009: Talks

Talks

Andreas Asheim

Title: The numerical method of steepest descent with approximated paths
Abstract:: The numerical method of steepest descent is an excellent method for evaluating highly oscillatory integrals. A drawback of the method is however that a generally non-linear equation for the path of steepest descent must be solved, a task that in some practical applications can be hard. We know from asymptotic theory that an exact path is not necessary, and we shall see that in the numerical setting even less so. In this talk I will present some rather surprising theoretical results regarding the asymptotic order of a steepest descent method with locally approximated paths. This is a joint work with Daan Huybrechs (KU Leuven).

Eivind Fonn

Title: Shape analysis using geodesic paths on manifolds
Abstract:: We represent a set of "shapes" as infinitely dimensional manifolds and develop some mathematical tools for working on these manifolds. We then use these to create algorithms for computing geodesics (i.e. morphing one shape into another). By suggestion of associate professor Elena Celledoni (NTNU). This work was carried out as a specialization project in preparation for a master thesis in the fall of 2008 under her supervision.

Sheehan Olver

Title: GMRES and Oscillatory Differential Equations
Abstract:: In this talk, we investigate using GMRES on differential operators to compute solutions to oscillatory differential equations. We will look at oscillatory integrals with stationary points, deriving an approximation to the Bessel function which converges everywhere in the complex plane and second order ordinary differential equations. Finally, we attempt to generalize the results to compute solutions to the Schrodinger equation in the semiclassical regime.

Kurusch Ebrahmi-Fard

Title: On the substitution law for B-series
Abstract:: Recently, Chartier, Hairer and Vilmart introduced the substitution law for B-series in the context of Backward Error Analysis. In this talk we present a Hopf algebraic approach to it by introducing a particular connected graded bialgebra of rooted trees with the grading given in terms of edges. This Hopf algebra has a natural coaction on Connes-Kreimer's Hopf algebra of rooted trees (graded by the number of vertices) which enables us to recover results due to the aforementioned authors as well as Murua. (This is joint work with D. Calaque and D. Manchon, preprint: arxiv:0806.2238.)

Per Martin Viddal

Title: A perceptual completion model
Abstract:: In the paper ''A Cortical Based Model of Perceptual Completion in the Roto-Translation Space'', G. Citti and A. Sarti present a model to complete so-called subjective contours. The model is inspired by the architecture of the visual cortex, and the natural space in which the completion is performed is the image-oriented manifold. The model will be discussed and results of chosen images with ''missing information'' will be presented. The work was carried out as a pre-master with Prof. Brynjulf Owren as supervisor.

Brett Ryland

Title: Multivariate Chebyshev polynomials for spectral elements methods
Abstract:: Multivariate Chebyshev polynomials are a generalisation of the Chebyshev polynomials of the first kind to multiple dimensions. They are constructed using a Fourier basis defined on a symmetric extension of the fundamental domain of the affine Weyl group associated with a root system. In this talk, I will present extremely quick spectrally accurate methods of approximating a gradient and an integral over a triangle using multivariate Chebyshev polynomials. I will also report on progress towards implementing a spectral element method on a periodic domain using multivariate Chebyshev polynomials.

Ilan Degani

Title: Control of quantum systems
Abstract:: We briefly overview some examples of quantum control problems appearing in different fields, ranging from existing technology (NMR/MRI) to research (control of nano devices, laser control of molecules). In all these examples the same type of problem emerges: to control a bilinear flow on a unitary (or orthogonal) group with piecewise constant controls. We review the existing optimal control approach, and point out some of its problems. Namely, the standard extremal control equations can not generally hold for piecewise constant controls; the only structural requirement from the controls is to have small $L_2$ norm. We then discuss our approach which derives the correct extremal equations, and which can prefer controls with desired structure. Particularly, we are able to compute optimal controls belonging to a desired subspace of control functions. Thus our methods may hopefuly be useful for computing controls which are producible by realistic laboratory laser pulse shapers. Preliminary results about convergence analysis will be shown.
Joint work with Antonella Zanna

Xavier Raynaud

Title:The Nonlinear Variational Wave equation
Abstract:: The Nonlinear Variational Wave (NVW) equation, u_tt-c(u)(c(u)u_x)_x, is the Euler-Lagrange equation of the variational principle $int_{t_1}^{t_2}\int_-\infty^\infty(u_t^2-c^2(u)u_x^2). When c(u)=c is constant, we recover the linear wave equation. In this case there are no interaction between left and right travelling waves but, in the general case, interactions exist and may lead to the blow up of the (derivative of the) solution where parts of the total energy concentrates in singular points. By using a coordinate transformation, we can construct a semigroup of solution for the NVW equation. In this talk, I will focus on the geometric properties of the coordinate transformation and show how this approach can be used to derive numerical schemes for the equation. This is work under progress in collaboration with Helge Holden.

Ferenc Bartha

Title: Computer aided proofs in Analysis
Abstract:: I would like to introduce the basics of interval arithmetics, then turn to automatic differentiation and the taylor method. This is a one-step method that is very efficient when it comes to ode-s. Finally combine these techniques to get the so-called Lohner algorithm, that tries to reduce the wrapping effect that arises naturally when we are counting in interval arithmetics.

Jitse Niesen

Title: Computing the phi-function of a matrix
Abstract:: Phi-functions play an important role in exponential integrators. We will give a brief introduction to exponential integrators and then focus on the computation of the phi-function of a matrix. The central ingredients in the algorithm we present are a Krylov-subspace method for reducing big sparse matrices to a smaller size and a time-stepping method akin to the scaling-and-squaring method for matrix exponentials. This is joint work with Will Wright (Melbourne).

Simon J Malham

Title: The algebraic structure of stochastic differential approximations
Abstract:: We study solutions to nonlinear stochastic differential systems driven by a multi-dimensional Wiener process. A useful algorithm for strongly simulating such stochastic systems is the Castell--Gaines method, which is based on the exponential Lie series. When the diffusion vector fields commute, it has been proved that at low orders this method is more accurate than corresponding stochastic Taylor methods. However it has also been shown that when the diffusion vector fields do not commute, this is not true for strong order one methods. Here we prove that when there is no drift, and the diffusion vector fields do not commute, the exponential Lie series is usurped by the sinh-log series. In other words, the mean-square error associated with a numerical method based on the sinh-log series, is always smaller than the corresponding stochastic Taylor error, to all orders. Our proof utilizes the underlying Hopf algebra structure of these series, and a two-alphabet associative algebra of shuffle and concatenation operations.

Haakon Marthinsen

Title: Optimisation on Lie Groups Applied to Optical Interference Filters.
Abstract:: An optical interference filter is needed when we would like to filter out only specific wavelengths of light, for example in anti-reflection coatings on camera lenses. The filter is composed of a stack of thin layers made of different materials, and each of these layers can be represented as an element of the Lie group SL(2,C). We will first develop the physical model, then look at how the problem of designing a filter with the desired spectral characteristics is naturally formulated as an optimisation problem on SL(2,C).

Bawfeh Kingsley

Title: Monotone interpolation in semi-Lagrangian advection schemes
Abstract:: Semi-Lagrangian (SL) advection schemes have recently been proposed [E.Celledoni,2005,2006] as a building block for the effective computation of matrix exponentials in commutator-free exponential Lie group integrators, and have been tested to have a great potential for convection dominated problems. However the effectiveness of these methods largely depend on the accuracy of the underlying SL schemes, which in turn depends on interpolation technique used. We hereby investigate the role of monotone interpolation schemes for SL advection.

Elena Celledoni

Title:
Abstract: On B-series and their structure-preserving properties: B-series are a powerful tool in the analysis of Runge-Kutta numerical integrators and some of their generalizations ("B-series methods"). Recently there has been a lot of activity for characterizing B-series methods in terms of their ability of preserving invariants, [1]. The ultimate goal is to understand what is possible to achieve with B-series methods in terms of structure preservation, and also to design practical numerical methods with the desired properties [4], [3]. In particular we have been focusing on the study of the linear subspaces of energy- preserving and symplectic modified vector fields which admit a B-series, their finite dimensional truncations, and their dual spaces. The manifold of B-series conjugate to symplectic and conjugate to energy-preserving have been characterized [2]. Joint work with R. McLachlan, R. Quispel and B. Owren.

[1] P. Chartier, E. Faou, and A. Murua. An algebraic approach to invariant preserving integrators: the case of quadratic and Hamiltonian invariants. Numer. Math., 103(4):575-590, 2006.
[2] D. McLaren B. Owren R. Quispel E. Celledoni, R. McLachlan. B-series preserving energy and symplecticity. 2008.
[3] G. R. W. Quispel and D. I. McLaren. A new class of energy-preserving numerical integration methods. Journal of Phys. A: Math. Theor., 41:xxx-xxx, 2008.
[4] J. E. Scully. A search for improved numerical integration methods using rooted trees and splittings. 2002.

Roman Kozlow

Title: Invariance and first integrals of continuous and discrete Hamiltonian equations
Abstract:: We consider the relation between symmetries and first integrals for both continuous canonical Hamiltonian equations and discrete Hamiltonian equations. The canonical Hamiltonian equations can be obtained by variational principle from an action functional. We consider invariance properties of this functional (as it is done in Lagrangian formalism) and rewrite the well-known Noether's identity in terms of the Hamiltonian function and symmetry operators. This approach, based on symmetries of the Hamiltonian action, provides a simple and clear way to construct first integrals of Hamiltonian equations without integration. A discrete analog of this identity is developed. The results are illustrated by a number of examples.

Markus Brunk

Title:
Abstract:: A class of transient drift-diffusion type equations occurring in semiconductor modeling is motivated. The parabolic PDE-system is space discretized by mixed hybrid finite elements leading to a positive DAE (or ODE) system. For time discretization a second order splitting scheme based on a combination of explicit exponential integration and implicit one-step methods is introduced. The scheme allows to increase the time step size restriction for positivity preservation. Coupling the splitting scheme to an iterative algorithm allows for efficient positivity preserving simulation of semiconductor devices.

Peter Rippis

Title: 2 + 2 Splitting of the Einstein Equation
Abstract:: In this talk I shall give a brief introduction to General Relativity and its Lagrangian formulation. We then consider a 2+2 decomposition of the spacetime manifold, and obtain the gravitational Lagrangian in the spherically symmetric case. Using the variational principle we derive the field equations in vacuum and for a scalar field. Also, this represents a reduction to a 2 dimensional gravitational theory with links to string theory. This is a work under progress with Snorre H. Christiansen.