Monday June 28th
| Time | Speaker | Title |
| 9:00--9:30 | --- | Registration at ICMS |
| 9:30--10:30 | Brunjulf Owren | Applications of Lie group integrators and exponential schemes |
| 10:30--11:00 | coffee | |
| 11:00--12:00 | Hans Munthe--Kaas | Symmetries and the efficiency of Lie group integrators |
| 12:00--14:00 | lunch | |
| 14:00--15:00 | Uwe Helmke | Control Theoretic Aspects of Matrix Factorizations |
| 15:00--16:00 | Clyde Martin | tba |
| 16:00--16:30 | tea | |
| 16:30--17:30 | Nicoletta Del Buono | Numerical techniques for approximating the solution of matrix ordinary differential equations on the general linear group |
Tuesday June 29th
| Time | Speaker | Title |
| 9:30--10:30 | Arieh Iserles | Highly oscillatory quadrature and its applications |
| 10:30--11:00 | coffee | |
| 11:00--12:00 | Antonella Zanna | The discrete Moser--Veselov algorithm for the rigid body |
| 12:00--14:00 | lunch | |
| 14:00--15:00 | P.S. Krishnaprasad | Geometry of Steering Laws in Cooperative Control |
| 15:00--16:00 | Claudia Wulff | Numerical Continuation of Symmetric Periodic Orbits |
| 16:00--16:30 | tea | |
| 16:30--17:30 | Knut Hueper | Newton-like Methods on differentiable manifolds |
Wednesday June 30th
| Time | Speaker | Title |
| 9:30--10:30 | Tony Bloch | Nonholonomic Flows on Lie Groups |
| 10:30--11:00 | coffee | |
| 11:00--12:00 | Moody Chu | Group Theory, Linear Transformations, and Flows: Dynamical Systems on Manifolds |
| 12:00--14:00 | lunch | |
| 14:00--15:00 | Elena Celledoni | Lie group techniques for Neural Learning |
| 15:00--16:00 | Anders Hansen | C-* algebras and geometric integration |
| 16:00--16:30 | tea | |
| 16:30--17:30 | Fernando Casas | New numerical integrators based on solvability and splitting |
| 18:30-- | conference dinner | Howies |
Thursday July 1st
| Time | Speaker | Title |
| 9:30--10:30 | Liz Mansfield | Noether's theorem for smooth, finite difference and finite element models |
| 10:30--11:00 | coffee | |
| 11:00--12:00 | Sergio Blanes | Composition Magnus Integrators |
| 12:00--14:00 | lunch | |
| 14:00--15:00 | Nicoleta Bila | Application of symmetry analysis to a PDE arising in the car windscreen design |
| 15:00--16:00 | Marcel Oliver | Numerical evaluation of the Evans function by Magnus integration |
| 16:00--16:30 | tea and close |
A new approach to parameter identification problems from the point of view of symmetry analysis theory is given. A mathematical model that arises in the design of car windshield represented by a linear second order mixed type partial differential equation is considered. Following a particular case of the direct method (due to Clarkson and Kruskal), we introduce a method to study the group invariance between the parameter and the data. The equivalence transformations associated with this inverse problem are also found. As a consequence, the symmetry reductions relate the inverse and the direct problem and lead us to a reduced order model.
We consider the Magnus expansion for the numerical solution of linear and non-linear ODEs with an explicitly time-dependent vector field. A Magnus integrator for a linear system can be considered as a map associated to an autonomous linear system (a time averaged matrix involving commutators of matrices evaluated at different times). This map is just the exponential (of a matrix) which can be efficiently approximated following different procedures. The same technique can also be used for the non-linear system. The corresponding map, for one time-step, is associated to the solution of an autonomous ODE. Instead of commutators, the new vector field involves Lie brackets of vector fields, and this turns into complicated maps to evaluate. However, in general, those complicated maps can be approximated by considering compositions of simpler maps. Then, we end up with new integrators which can take the benefits of the Magnus expansion with respect to the number of time-dependent function evaluations (which appear on the vector field) as well as the preservation of the geometric properties. The new composition Magnus integrators can also have several free parameters to be chosen according to different criterion of stability, accuracy, etc. (This is a joint work with Per Christian Moan.)
In this talk I will discuss the dynamics of mechanical systems with nonholonomic constraints on Lie groups. In particular I shall consider when such flows preserve a measure and when such a measure is preserved when one has in addition internal degrees of freedom or a shape space. I will contrast such flows to the flows of Hamiltonian systems. I will discuss some physical examples of interest and also some related control problems.
In this talk we review briefly some Lie group integrators applied to linear systems of ODEs, with special emphasis on methods based on the Magnus expansion. This class of schemes requires the evaluation of matrix exponentials, and this leads to a reduction in the computational efficiency when the dimension of the matrix is very large. For quadratic Lie groups it is possible to approximate the matrix exponential with a rational function and still preserve the Lie-group structure, but this is not longer true, for instance, for the special linear group. In the second part of the talk we propose a new class of integration algorithms specially designed to avoid this problem. They are based on a factorization of the solution as product of upper and lower triangular matrices obtained explicitly in terms of quadratures. We analyse the main features of the procedure and discuss the practical aspects of its implementation as a numerical method.
We consider orthonormal learning of linear neural networks with n inputs and p outputs. The context is that of a new technique in statistical signal processing named Independent Component Analysis (ICA). Geometric integration of the neural learning equations via Lie group techniques is considered, resulting in new and efficient algorithms.
It is known that there is a close relationship between matrix groups and linear transformations. The purpose of this exposition is to explore that relationship and to bridge it to the area of applied linear algebra. Some known connections between discrete algorithms and differential systems will be used to motivate a larger framework that allows embracing more general matrix groups. Different types of group actions will also be considered. The inherited topological structure of the Lie groups makes it possible to design various flows to approximate or to effect desired canonical forms of linear transformations. While the group action on a fixed matrix usually preserves a certain internal properties or structures along the orbit, the action alone often is not sufficient to drive the orbit home to the desired canonical form. Various means to further control these actions will be introduced. These controlled group actions on linear transformations often can be characterized by a certain dynamical systems on a certain matrix manifolds. Wide range of applications starting from eigenvalue computation to structured low rank approximation, and to some inverse problems are demonstrated. A number of open problems will be identified.
This talk concerns the numerical solution of matrix differential systems evolving on the general linear group GL(n,R) which come up frequently in the context of control and system theory, as well as in multivariate data analysis and inverse eigenvalue problems. The presence in the differential equation of the inverse of the solution and the topological structure of general linear group make the numerical solution of this ODE somewhat worrisome. In this talk we will suggest some techniques to approximate the solution of this kind of equation. Particularly, we show how the numerical solution is strictly connected with a general Riccati algebraic equation; we also suggest the use of a continuous singular value decomposition to detect when the numerical solution is approaching a singular matrix.
We discuss the problem of finding factorizations of arbitrary elements in a Lie group that minimize the total sum of a cost function, associated with certain of the factors. This includes time optimal control on Lie groups, where the time spend by the drift term defines the essential cost. Other applications include quantum computing and, potentially, numerical linear algebra. The talk will include a tutorial on controllability of systems on Lie groups. (This is joint work with Gunther Dirr and Martin Kleinsteuber.)
Since the early work by Gabay in '82 Newton's method on differentiable manifolds was treated in different ways by Smith '93/94, Udriste '94, Mahony '96, Owren/Welfert '00, Mahony/Manton '02. We present new ideas how to design Newton-like methods on differentiable manifolds, where the Riemannian structure is less important. We apply these methods to examples like Rayleigh quotient minimization on S^n or essential matrix estimation in vision. Local convergence results are discussed.
The implementation of Lie group methods for highly oscillatory equations requires the quadrature of multivariate rapidly oscillating integrals. This talk will address itself to this issue, developing a far-ranging asymptotic and numerical theory of quadrature in a highly oscillatory setting and applying it to integral, ordinary and partial differential equations.
The idea of studying the Frenet-Serret equations as a control system goes back to the seventies and perhaps earlier. From this vantage point, they constitute a fundamental example of a control system on a Lie group. In this talk we consider these and other moving frames adapted to trajectories of a collective of unit speed particles. We discuss the geometry of interactions between the adapted frames via coupling laws between the frame invariants (curvatures). We show that there exist interesting choices of such coupling laws that cause coherent bundling of the curves. We also discuss extensions of such laws to include non-collisional interactions of particles with surfaces. (This is joint work with Eric Justh, and Fumin Zhang.)
A key physical property of a physical model with a Lagrangian, that a geometric integrator might emulate, are the conservation laws that arise from symmetries of the Lagrangian. These include conservation of energy, which arises when the Lagrangian is invariant with respect to translation in time, linear momenta (translation with respect to independent variables), angular momenta (rotations with respect to independent variables), and so on. One problem to solve is how a smooth group action carries over to a discretised space. Another is the actual calculation of the conserved quantities.
I shall talk generally about the issues involved. My immediate conclusion will be that the key to solving the problem is to keep the underlying algebraic constructions for discrete models in strict alignment with those of the smooth. In this way, whether a system is inherently discrete or a discretisation of some kind, variational systems, their symmetries and their conservation laws can be studied in a clear, coherent and rigorous way.
The first half of this talk is a general introduction to work over the recent years on optimizing the performance of Lie group integrators; by minimizing the number of commutators, efficient computation of matrix exponentials and employing efficiently computable local coordinates on Lie groups.
In the second half of the talk we will present recent work on the applications of non-commutative Fourier analysis in the computation of matrix exponentials. This has applications to Lie group integrators for physical problems with domain symmetries, or approximate domain symmetries.
We use Magnus methods to compute the Evans function for spectral problems as arise when determining the linear stability of travelling wave solutions to reaction-diffusion and related partial differential equations. In a typical application scenario, we need to repeatedly sample the solution to a system of linear non-autonomous ordinary differential equations for different values of one or more parameters as we detect and locate the zeros of the Evans function in the right half of the complex plane.
In this situation, a substantial portion of the computational effort---the numerical evaluation of the iterated integrals which appear in the Magnus series---can be performed independent of the parameters and hence needs to be done only once. More importantly, for any given tolerance Magnus integrators possess lower bounds on the step size which are uniform across large regions of parameter space and which can be estimated \emph{a priori}. We demonstrate, analytically as well as through numerical experiment, that these features render Magnus integrators extremely robust and, depending on the regime of interest, efficient in comparison with standard ODE solvers.
In the first part of this talk, we address the most important applications of Lie group integrators up to date, which we choose to divide into three main cases.
1. Linear problems. Schemes based on Magnus series and variants based on the same principle. These schemes are important in solving and understanding highly oscillatory problems. In PDEs the schemes have been applied with success to the linear Schrodinger equation.
2. Nonlinear problems on manifolds. In principle, all sorts of nonlinear Lie group integrators are applicable here, as for instance the Runge-Kutta-Munthe-Kaas schemes. In some applications there are manifolds with a natural group action which ensures that the numerical approximation remains on some submanifold of Rn. The predominant cases in applications are when the acting group is the orthogonal group, and the manifold is either the orthogonal group, a Stiefel manifold or a sphere.
3. Problems which benfit from the principle of curved path building blocks. These problems are in many cases formulated on Euclidean space, there is no underlying nonlinear manifold. But the use of Lie group integrators allow basic movements that are better suited to the problem at hand. One may use this to deal with stiffness or high oscillations, or simply to obtain high accuracy with a large stepsize. This is similar to the effects seen in highly oscillatory linear problems solved with Magnus series schemes. In the PDE setting, it is popular to use Lie group integrators where the action is by the affine group, and this leads to schemes which are very similar to the exponential integrators first developed by Certaine, Lawson and Norsett.
In the second part of the talk I will focus on some recent developments in exponential integrators. The classical order theory seems straightforward on the outset, but there are particular issues that need to be addressed for exponential integrators. We introduce a general format of schemes which seem to include the majority of the known exponential integrators, as those derived from the Lie group scheme formalism, and the ones recently proposed by Cox and Matthews, and Krogstad. One of the hidden features of exponential integrators is the underlying function spaces from where to choose the coefficient functions of the schemes. Some theorems will be presented which give sharp lower bounds for the dimensions of these spaces. The results will significantly simplify the construction of exponential integrators, to illustrate this we present a few new schemes of high order.
The bifurcation theory and numerics of periodic orbits of general dynamical systems is well-developed, and in recent years there has been a rapid progress in the development of a bifurcation theory for dynamical systems with structure, like symmetry or symplecticity. But there are hardly any results on the numerical computation of those bifurcations yet. In this talk we will show spatio-temporal symmetries of periodic orbits can be exploited numerically, we will describe methods for the computation of symmetry-breaking bifurcations for free group actions and will show how bifurcations increasing the spatiotemporal symmetry (period-halving bifurcations and Hopf bifurcations) can be detected numerically. We will moreover present a method for the numerical continuation of non-degenerate Hamiltonian relative periodic orbits with regular drift-momentum pair. We will apply our methods to coupled cells and present a new family of rotating choreographies of the 3-body system. Our numerical algorithm is based on a multiple shooting algorithm for the numerical computation of periodic orbits via an adaptive Poincare-section and a pathfollowing algorithm using a tangential continuation method with implicit reparametrization. (Joint work with A. Schebesch (Freie Universitat Berlin))
In this talk we consider the discrete Moser--Veselov algorithm for the reduced equations of the rigid body. Moser and Veselov derived this second order algorithm and proved its integrability by using certain matrix polynomial factorizations first introduced by Peter Lax for the KdV equations and other isospectral flows. By backward error analysis, we show that, in the 3x3 case, the discrete Moser--Veselov algorithm is a time-reparametrization of the continuous rigid body equations. By determining some parameters, depending on the Hamiltonian and the other constants of the problem, we construct new higher order, explicit, symplectic and integrable algoritithms for the 3x3 rigid body.