The DNA Seminar

The Differential Equations and Numerical Analysis Seminar

This seminar is a continuation of two seminar series: One on numerical analysis and one on differential equations (DIFTA).

Fall 2012

Workshop in Algebraic Combinatorics and Numerical Analysis: 6-12th December 2012

In this workshop we will discuss algebraic structures related to numerical integrators for ODEs. The main topics are B-series expansions, pre-Lie algebras, post-Lie algebras, operads, geometric integration. See the webpage of the workshop for more information.

Next seminar:

Date	Speaker and title
5. Dec. 13.15–14.00 Room 734	Hans Lundmark, Linköping University Ghostpeakons The Camassa-Holm shallow water wave equation is famous for many reasons; one reason is that it admits travelling wave (weak) solutions with a peak-shaped crest, and also multisoliton solutions obtained as linear combinations of several such "peakons", each with its own time-dependent position and amplitude. The time-dependence of the positions and amplitudes is governed by a coupled system of nonlinear ODEs, which constitute an integrable finite-dimensional Hamiltonian system that can be solved exactly in terms of elementary functions (with the help of inverse spectral methods). However, this only covers the case when all the amplitudes are nonzero; of course this is the most interesting case from the point of view of the PDE, but if a system is integrable we should be able to solve it regardless of what the initial conditions happen to be, shouldn't we? In this talk, I will demonstrate a simple (but nontrivial) limiting procedure which from the known nonzero-amplitude solutions produces explicit solutions formulas also for the positions of invisible "ghostpeakons" with amplitude zero. (Incidentally, the trajectories of these ghostpeakons are the characteristic curves for the multipeakon solutions of the PDE.) With minor modifications, this procedure applies not only to the Camassa-Holm equation, but also to its relatives, the Degasperis-Procesi and Novikov equations. However, perhaps the most interesting application (which is still work in progress) concerns an integrable two-component peakon equation found by Geng and Xue. Via a lenghty inverse spectral procedure, the solution is known in the case where the peakons in one component interlace with the peakons in the other component. Now the idea is to study other configurations of peakons by letting selected amplitudes tend to zero, and here we expect the lessons learned from ghostpeakons to be very useful. This is joint work with my student Budor Shuaib.

Date

Speaker and title

5. Dec.

13.15–14.00
Room 734

Hans Lundmark, Linköping University
Ghostpeakons

The Camassa-Holm shallow water wave equation is famous for many reasons; one reason is that it admits travelling wave (weak) solutions with a peak-shaped crest, and also multisoliton solutions obtained as linear combinations of several such "peakons", each with its own time-dependent position and amplitude. The time-dependence of the positions and amplitudes is governed by a coupled system of nonlinear ODEs, which constitute an integrable finite-dimensional Hamiltonian system that can be solved exactly in terms of elementary functions (with the help of inverse spectral methods). However, this only covers the case when all the amplitudes are nonzero; of course this is the most interesting case from the point of view of the PDE, but if a system is integrable we should be able to solve it regardless of what the initial conditions happen to be, shouldn't we? In this talk, I will demonstrate a simple (but nontrivial) limiting procedure which from the known nonzero-amplitude solutions produces explicit solutions formulas also for the positions of invisible "ghostpeakons" with amplitude zero. (Incidentally, the trajectories of these ghostpeakons are the characteristic curves for the multipeakon solutions of the PDE.) With minor modifications, this procedure applies not only to the Camassa-Holm equation, but also to its relatives, the Degasperis-Procesi and Novikov equations. However, perhaps the most interesting application (which is still work in progress) concerns an integrable two-component peakon equation found by Geng and Xue. Via a lenghty inverse spectral procedure, the solution is known in the case where the peakons in one component interlace with the peakons in the other component. Now the idea is to study other configurations of peakons by letting selected amplitudes tend to zero, and here we expect the lessons learned from ghostpeakons to be very useful. This is joint work with my student Budor Shuaib.

Upcoming seminars:

Date	Speaker and Title

Earlier seminars of the fall semester

Date	Speaker and title
29. Nov 12:15-13:00 Room 656	Boris Kruglikov, UiT Hamiltonian integrability in geometry and relativity Resume: In describing dynamics of a Hamiltonian system it is very important to understand if it is chaotic or integrable. At the first part of the talk I will discuss how integrability constraints the geometry and topology of the phase space and how it influences the main dynamical characteristics. I will mention some examples from differential geometry. Then in the second half I consider a particular interesting family from general relativity: Zipoy-Voorhees metrics (including Minkowski and Schwarzschild spacetimes as partial cases). I will explain how usage of symbolic and numerical methods allows to show non-integrability of this model (contrary to some expectations that it could behave as the Kerr black hole).
28. Nov 13:15-14:00 room 734	Ishtiaq Ali, University of Warsaw Convergence analysis of spectral methods for integro-differential equations with vanishing proportional delays In this talk we describe the application of the spectral method to delay integro-differential equations with proportional delays. It is shown theoretically that the resulting numerical solutions exhibit the classical convergence order. A number of numerical examples confirm the exponential rate of convergence. Extensions to integro-delay differential equations with weakly singular kernels are also discussed.
31. Oct 13.15-14.00 room 734	Ferenc Bartha (UiB) Computer-aided proofs in analysis Computers are widely used for solving problems or more precisely for providing approximate solutions. This correction is necessary, since we use simplifications and the result may be influenced by numerical errors as well. On the other hand, one of the beauties of theoretical mathematics is that a proven theorem is true beyond doubt. The main question is if we can (and if yes, then how) use machine based computations in a proof. We will present one of the possibilities, namely Interval Arithmetic (IA). Our goal is to prove theorems about certain dynamical systems, therefore efficient derivation is a must. Another basic concept called Automatic Differentiation (AD) shall show us, that this is indeed possible for sufficiently regular equations. Combining these ideas we will present methods to handle discrete equations, ODEs, PDEs and DDEs (delay differential equations). In particular we will present a proof for the d = 2 case of the Levin and May conjecture '76, that states that whenever the non trivial fixed point of the difference-delay equation \[x_{k+1} = x_k e^{ \alpha - x_{k-d} }\] is locally stable, then it is globally stable as well.
24. Oct 13.15–14.00 room 734	Erik Wahlén, Lund University Existence and stability of solitary water waves with surface tension In this talk I will discuss the existence and stability of two-dimensional solitary water waves with surface tension. Here, the number of dimensions refers to the fluid domain. After giving an overview of the existence theory for various types of solitary waves, I will discuss how their stability can be proved by a variational method. This method requires a penalisation argument, due to the fact that the problem is quasilinear. When viewed in three-dimensions, these two-dimensional waves become so called line-solitary waves, that is, solitary waves which are spatially homogeneous in the transverse direction. I will discuss what is known about the transverse instability of line-solitary waves, that is, instability with respect to three-dimensional perturbations.
19. Oct 14.15–15.00 room 734	Yannick Sire Fractional Ginzburg Landau equation and Harmonic maps I will describe a joint work with Vincent Millot (Universite Paris 7). In this work, we prove a small energy result for a fractional version of the Ginzburg-Landau equation. As a consequence, it gives convergence of the minimizers to boundary harmonic and fractional harmonic maps.
26. Sept. 13.15–14.00 room 734	Joachim Escher, Gottfried Wilhelm Leibniz Universität Hannover, Germany Existence and stability of weak solutions for a degenerate parabolic system of thin film type Of concern is the evolution of two thin fluid films in a porous medium. Starting from the classical evolution equations modelling the Muskat problem with a free surface, the limit of small layer thickness is determined. It consists in a system of two coupled and degenerate parabolic equations for the two films height, respectively. In the absence of surface tension forces local well-posedness in the classical sense of the problem is established and the exponential stability of steady-states is shown. Finally, based on energy estimates and suitable regularizations, global weak solutions are constructed. These weak solutions are globally stable in the \(L_2\)-sense.
3. sept. 14:15–15:00 room 734	Arieh Iserles, DAMTP, Cambridge, UK Rapid expansion in orthogonal polynomials In this talk I will discuss O(n log n) algorithms that compute the first n coefficients of an expansion of a smooth function in ultraspherical polynomials. These algorithms are based on the theory of special functions and a number of rather surprising tricks. I will also describe a general approach to rapid calculation of general expansions of orthogonal polynomials on the real line and on the unit disc and of Laurent polynomials on the unit disc.
22. aug. 13:15–14:00 room 734	Alessandro Saccon, Institute for Systems and Robotics, Instituto Superior Técnico, Lisbon, Portugal Numerical optimal control on Lie groups with applications to motion planning of multiple autonomous underwater vehicles The Lie group Projection Operator Approach is a recently developed iterative algorithm for solving continuous-time optimal control problems on Lie groups. In the talk, we will discuss the main ideas lying behind this numerical method. In particular, we will detail the role played by the second covariant derivative of a map between Lie groups in obtaining a linear-quadratic approximation to the optimal control problem. This approximation is key to compute the descent direction at each iteration of the algorithm. The implementation of this numerical method on a computer requires the integration of the system dynamics and of a differential Riccati equation (DRE) associated with the linear-quadratic approximation of the optimal control problem. The integration of these differential equations is accomplished through the use of vector space and Lie group ODE solvers. Numerical examples will be provided to demonstrate the effectiveness of the numerical method and its quadratic converge rate to local minima satisfying the second order sufficient conditions (SSC) for optimality. The application of the algorithm to solving minimum energy constrained motion planning problems for multiple autonomous underwater vehicles will conclude the talk.
15. aug. 13:15–14:00 room 734	Richard Norton, La Trobe University, Melbourne Australia. Numerical computation of band gaps in photonic crystal fibres. Photonic crystal fibres are capable of special light guiding properties that ordinary optical fibres do not possess, and efforts have been made to numerically model these properties. The plane wave expansion method is one of the numerical methods that has been used. Unfortunately, the function that describes the material in the fibre n(x) is discontinuous, and convergence of the plane wave expansion method is adversely affected by this. For this reason, the plane wave expansion method may not be every applied mathematician's first choice method but we will show that it is comparable in implementation and convergence to the standard finite element method. In particular, an optimal preconditioner for the system matrix A can easily be obtained and matrix vector products with A can be computed in O(N logN) operations (where N is the size of A) using the Fast Fourier Transform. Although we are always interested in the efficiency of the method, the main contribution of this thesis is the development of convergence analysis for the plane wave expansion method applied to 4 different 2nd-order elliptic eigenvalue problems in R and R2 with discontinuous coefficients. To obtain the convergence analysis three issues must be confronted: regularity of the eigenfunctions; approximation error with respect to plane waves; and stability of the plane wave expansion method. We successfully tackle the regularity and approximation error issues but proving stability relies on showing that the plane wave expansion method is equivalent to a spectral Galerkin method, and not all of our problems allow this. However, stability is observed in all of our numerical computations. It has been proposed that replacing the discontinuous coefficients in the problem with smooth coefficients will improve the plane wave expansion method, despite the additional error. Our convergence analysis for the method shows that the overall rate of convergence is no faster than before. To define A we need the Fourier coefficients of n(x), and sometimes these must be approximated, thus adding an additional error. We analyse the errors for a method where n(x) is sampled on a uniform grid and the Fourier coefficients are computed with the Fast Fourier Transform. We then devise a strategy for setting the grid-spacing that will recover the convergence rate of the plane wave expansion method with exact Fourier coefficients.