# Manifolds and Geometric Integration Colloquium 2014

## February 24th – 27th in Meråker, Norway

The annual informal workshop, Manifolds and Geometric Integration Colloquia (MaGIC) is organized by the Numerical Analysis groups in Bergen and Trondheim.

The workshop is intended mainly for researchers and research students in Geometric Integration, primarily from Trondheim and Bergen. The scope of the colloquium is to provide an informal atmosphere to exchange recent developments in the field of GI as well as time to establish collaborations. A restricted number of participants from abroad are usually invited to attend the meeting.

The colloquium will be held this year in Meråker, Norway.

The conference will start on Tuesday, February 25th 2014 in the morning and will end on Thursday, 27th, in the afternoon.

We expect that most people arrive on Monday, February 24th in the afternoon.

We will stay at Kyrkebyfjellet Konferansesenter in Meråker Nord Trondelag.

There should be possibilities both for cross country and down hill skiing. Meråker is quite convinently placed with respect to the Værnes Airport (Trondheim Airport), just a buss ride away. Here you find the timetable: http://www.rutebok.no/nriiisstatictables/tables/ruter/t/23-651.htm and http://www.nettbuss.no/rutetilbud/lokalbuss/nord-trondelag for the general time table.

Alternatively you can also take the train: https://www.nsb.no/en/

The organizers | |
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Antonella Zanna Munthe-Kaas | Antonella.Zanna (at) math.uib.no |

Hans Z. Munthe-Kaas | hans.munthe-kaas (at) math.uib.no |

Brynjulf Owren | brynjulf.owren (at) math.ntnu.no |

Elena Celledoni | elenac (at) math.ntnu.no |

### Speakers

Name | Affiliation | Talk |
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Geir Bogfjellmo | NTNU Trondheim | TBA |

Håkon Marthinsen | NTNU Trondheim | |

Eirik Hoel Høiseth | NTNU Trondheim | A class of explicit splitting methods for a marine vessel model with control. |

Markus Eslitzbichler | NTNU Trondheim | Modelling animations on infite-dimensional manifolds. |

Roman Kozlov | NHH, Bergen | First integrals of ODEs admitting symmetries |

Hans Munthe-Kaas | UiB, Bergen | Equivariant B-series. Recent developments. |

Lu Li | NTNU | An overview on geometric methods for the Maxwell equations. |

Eivind Fonn | ETH | Alternative discretizations for the Boltzmann equation: hyperbolic cross Fourier and polar Laguerre polynomials |

Klas Modin | Chalmers | Spherical midpoint method |

Tim Cardilin | Chalmers | |

Kurusch Ebrahimi Fard | ICMAT | Post-Lie algebras and matrix factorization algorithms |

Antonella Zanna | UiB, Bergen | TBA |

Brynjulf Owren | NTNU Trondheim | Manin transformations and the integrability of the Kanah map |

Elena Celledoni | NTNU Trondheim | Port-Hamiltonian systems |

Olivier Vedier | Umeå | Integrators on homogeneous spaces: Isotropy choices and connections |

ABSTRACTS |
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Hans Munthe-Kaas Equivariant B-series. Recent developments. |

Abstract:The last year has brought many new insights into the algebraic and geometric structures of B-series (and family and friends, such as Lie-Butcher and equivariant series).It is exciting to see that the classical theory is now tied closely both to central concepts of differential geometry (connections on Lie algebroids) as well as to representation theory (Schur-Weyl duality and invariant tensor theorem). We will give an exposition of these new developments, and discuss (still) open problems. |

Eivind Fonn Alternative discretizations for the Boltzmann
equation: hyperbolic cross Fourier and polar Laguerre polynomials |

Abstract: We give a brief introduction to Boltzmann equation theory (motivation, conservation laws and equilibrium), and present two deterministic spectral velocity-space discretization methods for the spatially homogeneous Boltzmann equation: a sparse variety of the de facto standard Fourier discretization, and a novel idea for 2D based on polar representations and Laguerre polynomials, which can be seen to outperform the existing methods under certain conditions. |

Roman Kozlov First integrals of ODEs admitting symmetries |

Abstract:The talk is devoted to a method for finding conservation laws of differential and discrete equations which admit symmetries. It can be used for equations which do not possess a variational formulation. For simplicity the method will be presented for ODEs. There will mentioned a recent extension of this approach to ODEs admitting nonstandard symmetries. |

Klas Modin Spherical midpoint method |

Abstract: In this talk I will discuss a novel symplectic integrator for Hamiltonian systems on direct products of 2–spheres. Such systems are called \emph{spin systems} and occur frequently in physics; examples include the free rigid body, point vortex dynamics on the sphere, the classical Heisenberg spin chain, and the Landau-Lifshitz equation of micromagnetics. The new method is simple, works for all Hamiltonians, and is $O(3)$–equivariant. I will explain how the method is related to the classical midpoint method and to the recent concept of collective symplectic integrators. |

Kurusch Ebrahimi Fard Post-Lie algebras and matrix factorization algorithms |

Abstract: In this talk we would like to discuss a link between post-Lie algebras and abstract matrix factorization algorithms. It is motivated by the quest for understanding several apparently unrelated factorizations which arise in the context of quantum field theory, combinatorial Hopf algebras, integrable systems, and numerical methods for differential equations. Key is a particular recursion formula which is based on the classical Baker-Campbell-Hausdorff identity. We will describe the algebraic structures underlying this approach, i.e., pre- and post-Lie algebras. |

Markus Eslitzbichler Modelling Animations on Infinite-Dimensional Manifolds |

Abstract: In this talk, we will look at how concepts and techniques from the field of geometric shape analysis can be applied to problems in computer animation. |

Eirik hoel Hoiseth A class of explicit splitting methods for a marine vessel model with control. |

Abstract: We introduce a class of explicit splitting methods for a marine vessel model with control. Controlled systems commonly satisfy a passivity condition which we would like our chosen integrator to preserve. We therefore compare these methods to other often used integrators based on their passivity properties. |

Lu Li An overview on geometric methods for the Maxwell equations |

Abstract: In this talk, I will discuss the numerical behavior of different geometric numerical methods applied to Maxwell equations. Specifically, I will focus on the following numerical methods: Yee scheme, Stormer/Verlet method, midpoint rule. |

Elena Celledoni Port-Hamiltonian systems |

Abstract: In this talk I will discuss port-Hamiltonian systems and their structure preserving discretization. |

Olivier Verdier Integrators on homogeneous spaces: Isotropy choices and connections |

Abstract: Homogeneous spaces are spaces with built-in symmetry, such as planes, spheres, or hyperbolic spaces. We consider numerical integrators of ODEs on homogeneous spaces. One obtain such integrators by combining a Lie group integrator with an isotropy map. In order for the integrators to preserve the intrinsic symmetry of the homogeneous space, we show that the isotropy map must be equivariant. In order to show that, we introduce a novel, all encompassing description of Lie group integrators, comprising most of the existing Lie group methods. These equivariant isotropy maps can be identified with invariant connections. Such connections may or may not exist, and can be given either implicitly or explicitly. Examples include Lie groups, affine spaces, Stiefel, Grassmann, isospectral manifolds, and fixed rank matrices. Finally, one can consider higher order isotropy choices. In the affine case, they have in fact highest order one, and form an affine space of dimension two. These isotropy choices are related to exponential integrators. |

Brynjulf Owren Manin transformation and the integrability of the Kahan map |

Abstract: The Kahan method for the numerical integration of quadratic Hamiltonian systems in 2D is completely integrable. The map can be explicitly integrated by means of a construction called the Manin transformation. |

##### Programme

Tuesday 25th | Wednesday 26th | Thursday 27th | |
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9.00 | E. Fonn | K. E. Fard | A. Zanna |

K. Modin | O. Verdier | M. Eslizbichler | |

R. Kozlov | H. Munthe-Kaas | Lu Li | |

16.00 | B. Owren | G.Bogfjellmo | |

E. Celledoni | E. Høiseth |