Magic 2008: Talks

Talks

Andreas Asheim

Title: High frequency wave scattering problems, part II
Abstract:: The discretization of an integral equation will normaly result in a dense matrix. In the context of scattering phenomena this is not consistent with the physics of the problem, namely the localization principle in high frequecy scattering. Huybrechs & Vandewalle have shown how to obtain a sparse discretization for such problems in 2D. A sparse representation implies that a partial solution to the problem could be computed. We will discuss the computation of such partial solutions, and how they can be implemented in for 2D and 3D-problems.

David Cohen

Title: Multi-symplectic integrators for the Camassa-Holm equation
Abstract:: We will present some initial attempts to develop geometric integrators for the Camassa-Holm equation. We will focus the talk on two new multi-symplectic formulations of this nonlinear partial differential equation. This is a joint work with Xavier Raynaud (NTNU) and Brynjulf Owren (NTNU).

Tore G. Halvorsen

Title: Lattice Gauge Theory and the Maxwell-Klein-Gordon equations
Abstract:: In this talk I will present a discretization of the Maxwell-Klein-Gordon equations, motivated by Lattice Gauge Theory, which preserves the local gauge invariance. Due to this symmetry, the electric charge of the system is conserved, which is not the case when using a standard leap-frog discretization. Lastly, some numerical results will be presented.

Daan Huybrechs

Title: High frequency wave scattering problems, part I
Abstract:: We discuss the scattering of acoustic waves by an obstacle, as modelled by the Helmholtz equation. The formulation leads to an integral equation that is defined on the boundary of the obstacle. At high frequencies, traditional numerical methods for these types of problems suffer severely from dispersion errors. High sampling requirements in addition make even the simplest problems intractable. In this talk, we remedy the situation with a combination of highly oscillatory quadrature and phase extraction. We explore the two-dimensional case and we show that higher frequencies actually make the problem easier.

Marianna Khanamiryan

Title: Quadrature methods for highly oscillatory vector-valued integrals
Abstract:: The work presents methods of efficient numerical approximation for linear and nonlinear systems of highly-oscillatory ordinary differential equations. We show how an appropriate choice of quadrature rule improves the accuracy of approximation as the frequency of oscillation grows. We present asymptotic and Filon-type methods to solve highly-oscillatory linear systems of ODEs, and WRF method, representing a special combination of the Filon-type methods and waveform relaxation methods, for nonlinear systems. Numerical examples support this paper.

Bawfeh Kometa

Title: Eigenvalues for the Dirac Equation
Abstract:: The need for the computation of a large number of eigenvalues of self-adjoint elliptic operator has become of paramount importance in recent years. Here I present a masters thesis work that attempts a problem of this nature for a stationary single-particle Dirac operator with a radial/Coulomb potential using FEM approximations. The resulting spectrum shows spurious states in the discrete band. How to resolve the spuriosity is still a task of great concern.
Keywords: Finite Element method (FEM), Dirac operator, Spurious states.

Roman Kozlov

Title: High-order conservative discretizations for some cases of the rigid body motion
Abstract:: Modified vector fields can be used to construct high-order structure--preserving numerical integrators for ordinary differential equations. We consider high-order integrators based on the implicit midpoint rule, which conserve quadratic first integrals. It is shown that these integrators are particularly suitable for the rigid body motion with an additional quadratic first integral. In this case high-order integrators preserve all four first integrals of motion. The approach is illustrated on the Lagrange top (a rotationally symmetric rigid body with a fixed point on the symmetry axis). The equations of motion are considered in the space fixed frame because in this frame Lagrange top admits a neat description. The Lagrange top motion includes the spherical pendulum and the planar pendulum, which swings in a vertical plane, as particular cases.

Alexander Lundervoll

Title: Magnus and Fer Expansions in Dendriform Algebras
Abstract:: I will present some recent work by K. Ebrahimi-Fard and D. Manchon on an approach to Magnus and Fer expansions using the language of dendriform algebras (arXiv:0707.0607v3)

Simon Malham

Title: Computing the Evans function using Grassmannians
Abstract:: We present a numerical method for computing the pure-point spectrum associated with the linear stability of coherent structures. Our method is based on the Evans function shooting and matching approach. The Grassmann representatives for the stable and unstable manifolds of the spectral problem suffice to construct the Evans function. Our idea is to fix a coordinate patch for the Grassmann representatives of each manifold and numerically compute in that representation. We are thus required to solve a nonlinear Riccati differential equation for each manifold. In practice the method is stable, robust, analytic in the spectral parameter and of complexity bounded by the order of the spectral problem. For large systems it represents a competitive method to that proposed by Humpherys and Zumbrun~\cite{HZ}. We demonstrate this by comparing the two methods in three applications: Boussinesq solitary waves, autocatalytic travelling waves and Ekman boundary layer.

joint work with Veerle Ledoux and Vera Thummler

Hans Munthe-Kaas

Title: Multidimensional pseudospectral methods on Lattice grid
Abstract:: In this talk we will discuss the use of Lattice integration rules applied in multidimensional pseudospectral methods. Given a periodic function on $R^n/Z^n$. If one truncate the spectral approximation to terms bounded in 1--norm in Fourier space, $|| {\bf k}||_1 \leq N$, the number of terms reduces to approximately $(2N+1)^2/s!$ as opposed to $(2N+1)^2$ if all terms with $|| {\bf k}||_{\infty} \leq N$ are included. A saving of $s!$ is thus potentially within reach provided we can compute expansion coeffisients with a quadrature rule with the same order of complexity, and find a fast algorithm associated with this quadrature rule for computing all the coefficients. In this talk we describe how lattice rules can be used for evaluating the expansion coeffisients and the modification necessary in order to apply the FFT-algorithm for this computation. We the apply the new technique to the 3 dimensional Poisson equation to demonstrate the benefits and problems of this approach. Joint work work with Tor Sørevik, UiB.

Christof Neuhauser

Title: Time-splitting for linear Schrödinger equations
Abstract:: In this talk we study linear Schrödinger equations, formulated as evolution equations involving two linear operators. We present convergence criteria for exponential operator splitting methods and confirm the theoretical results by numerical examples.

Jitse Niesen

Title: Exponential integrators using Krylov iteration
Abstract:: The biggest issue when implementing an exponential integrator is how to evaluate the matrix exponential. If the matrix is not small, as is the case when solving partial differential equations, then an iterative method needs to be used. Methods based on Krylov subspaces are a natural candidate. I will describe the efforts of Will Wright and myself to implement such a procedure and comment on our results.

Sheehan Olver

Title: Convergence acceleration of Fourier series
Abstract:: I will present some new methods for accelerating the Fourier series and modified Fourier series. First I will demonstrate how an asymptotic expansion for the error can be derived, which will lend itself to extrapolation schemes.

Alexander Ostermann

Title: High-order exponential splitting methods
Abstract:: In this talk, we are concerned with splitting methods for the time integration of abstract evolution equations. We introduce an analytic framework that allows to prove optimal convergence orders for various schemes. As the main result, we will show that the non-stiff order conditions are sufficient to prove the optimal convergence orders for exponential splitting methods in the stiff case. This is joint work with Eskil Hansen.

Xavier Raynaud

Title:Numerical methods based on multipeakons for the Camassa-Holm equation
Abstract:: The Camassa-Holm equation is a partial differential equation rich in geometric structures. In particular, it is a geodesic for the H^1 metric and it is bi-hamiltonian. Multipeakons are special solutions to the Camassa-Holm equation, which can also be obtained as solutions of an Hamiltonian ODE. In this talk, we will look at numerical schemes which rely on this special class of solutions to compute the solutions to the Camassa--Holm equation for any initial data in H^1.

Brett Ryland

Title: Multivariate Chebyshev approximation
Abstract:: I will discuss recent work on the use of multivariate Chebyshev polynomials on a deltoid for rapid approximation of the pseudospectral derivative and the integral of a function on a triangular surface patch.

Niklas Säfstrom

Title: Modelling of an offshore pipe installation, based on an elastic rod model
Abstract:: We will describe a system of an interconnected offshore pipe and a surface vessel. The pipe is suspended freely from the vessel, considered as a rigid body, to the seabed. The model of the pipe is based on the finite strain rod model by Simo et. al. We will also discuss some issues about existence and uniqueness of the model and numerical simulations.

Tatiana Shingel

Title: Splitting Methods for SU(N) Loop Approximation
Abstract:: The study of classes of periodic functions with values in a Lie group G (loops) is of theoretical importance as a simple example of infinite-dimensional Lie groups, but also occurs in a more practical context (e.g. optical FIR filter design, orthogonal wavelet constructions). We are interested in finding the correct asymptotic rate of approximating such functions by so-called polynomial loops in dependence of their smoothness. This can be viewed as a problem of nonlinearly constrained trigonometric approximation. Using the exponential map and ideas from splitting methods, we present the first to our knowledge non-trivial result concerning the Jackson-type estimate of an SU(N)-loop belonging to a Hölder-Zygmund class Lip_{alpha}, alpha>1/2.

Mechthild Thalhammer

Title: Time-splitting spectral methods for nonlinear Schrödinger equations
Abstract:: In this talk, I will consider numerical discretisations of nonlinear Schrödinger equations based on exponential operator splitting and spectral methods. In particular, I will discuss the convergence and energy conservation of time-splitting methods for the Gross-Pitaevskii equation describing a Bose-Einstein condensate. The theoretical results will be illustrated by numerical examples.

Antonella Zanna

Title: Image registration by matrix Lie groups actions
Abstract:: In this talk we consider the problem of registration of two images. We consider global transformations of the domain, viewed as an homogeneous manifold (subgroups of GL(n) acting on Omega, the image domain, subset of R^n). The registration problem is then rephrased as a minimization problem on a Lie group. Several intrinsic methods (like steepest descent and Newton's method on a manifold) will be considered.