| ABSTRACTS | |
| Galerkin Lie Group Variational Integrators (James Hall, UCSD) | |
| Abstract: In this talk I will present a new method for constructing integrators for Lagrangian systems on Lie Groups. This method can be used to construct numerical methods that are symplectic, momentum preserving, and arbitrarily high order. Furthermore, it is possible to use the method to construct integrators that converge geometrically. We will present several numerical examples, and discuss possible applications and future work. | |
| Geometric properties of Kahan's method (Brynjulf Owren, NTNU) | |
| Abstract: There has been a fair amount of work in the last two decades characterizing one step integrationmethods for ODEs having certain geometric properties when applied to general Hamiltonian systems: symplectic integrators; energy/integral-preserving integrators; conjugate-to-symplectic integrators. A different point of view arises if we restrict the class of problems from the case of general Hamiltonian functions to the case of polynomial Hamiltonian functions and polynomial Hamiltonian vector fields. Then, it becomes often easier to preserve these geometric properties. A remarkable case is the one of the Kahan's method. We show that Kahan's discretization of quadratic vector fields is equivalent to a Runge-Kutta method. When the vector field is Hamiltonian on either a symplectic vector space or a Poisson vector space with constant Poisson structure, the map determined by this discretization has a conserved modied Hamiltonian and an invariant measure, a combination previously unknown amongst Runge-Kutta methods applied to nonlinear vector fields. This produces large classes of integrable rational mappings in two and three dimensions, explaining some of the integrable cases that were previously known. | |
| Mumford-Shan functional in a variational problem of segmentation (Irina Markina) | |
| Abstract: pdf | |
| Sub-Riemannian geometry, normal geodesic, cut locus, Stiefel manifolds, Grassmann manifold (Christian Autenried and Irina Markina) | |
| Abstract:In the paper we consider the Stiefel manifold $V_{n,k}$ as a principal $U(k)$-bundle over the Grassmann manifold and study the cut locus from the unit element. We gave the complete description of this cut locus on $V_{n,1}$ and presented the sufficient condition on the general case. | |
| On first integrals of lattice equations with symmetry (Roman Kozlov) | |
| Abstract: There will be presented a method which allows to find first integrals of lattice equations with symmetry. If sufficient amount of independent first integrals is found, the equation can be solved. | |
| Generating functions and volume preserving mappings (Huiyan Xue) | |
| Abstract: We study generating forms and functions for volume preserving mappings in Rn. We derive some parametric classes of volume preserving numerical schemes for divergence free vector fields. In passing, by extension of the Poincar´e generating function and a change of variables, we obtained symplectic equivalent of the theta-method for differential equations, which includes the implicit midpoint rule and symplectic Euler A and B methods as special cases. | |
| Topic of discussion: B-series and equivariance (Olivier Verdier and Hans Munthe-Kaas) | |
| Abstract: Let Aff(n) denote the affine linear group acting on Rn. It is well known that any numerical method expressible in a B-series is equivariant with respect to this action. This means that if you apply the numerical method and transform the result, you get exactly the same result as if you first pull back the vector field and then apply the numerical method. Some years ago Iserles, Quispel and Tse showed that no equivariant B-series method can be volume preserving. Other questions concerning equivariance have, however, been unanswered until now. Q1) Can *any* affine equivariant numerical method be developed in a B-series? Q2) What is the most general expansion of an equivariant method? Q3) Does there exist equivariant volume preserving numerical methods? We have very recently had a breakthrough in attacking these questions. Some answers will be given, and new open questions arise! Amusingly, an important tool for this investigation is a branch of representation theory called classical invariant theory. |
| Integral preserving Lie group methods (Elena Celledoni) | |
| Abstract:The discrete gradient approach is generalized to yield integral preserving methods for di erential equations in Lie groups. | |
| Some new insight in volume preserving forms for divergence free differential equations. (Antonella Zanna) | |
| Abstract: This is a follow-up of Huiyan's talk. I will discuss some new cases of generating 1-forms that we were recently able to understand and to associate to numerical methods (of the type: symplectic Euler and discrete-Lagrangian). Based on joint work with Huiyan and Olivier. | |
| Symplectic Lie Group Methods, part 2. (Geir Bogfjellmo) | |
| Abstract: Building on the framework presented in part 1, two classes of Symplectic Lie Group methods are developed, studied and tested numerically. Both classes contain Symplectic Lie Group methods of arbitrarily high order. | |
| Symplectic Lie Group Methods, part 1 (Håkon Marthinsen) | |
| Abstract: We want to study the numerical solution of Hamiltonian ODEs evolving on cotangent bundles. The base manifold of these are often Lie groups, so we can utilise Lie group integrators. With these methods, symplecticity is not automatically preserved. Variational methods are known to be symplectic, so our task is to combine these two. In part 1 of this talk, we will go through the background and the general framework of these methods. | |
| Shape Analysis by Optimization on Manifolds (Markus Eslitzbichler) | |
| Abstract: Certain problems in shape analysis can be elegantly formulated and solved in a differential geometry setting. We will take a look at a specific gradient descent method to determine the similarity of curves in R3. | |
| Averaged Lagrangian Methods (Eirik Hoel Høiseth) | |
| Abstract: The average Lagrangian method is generalized to construct higher order variational integrators. We present numerical examples, and discuss applications and future work. | |