منابع مشابه
Lie Group Methods *
Definition 1.1. [Ol] An m-dimensional manifold M is a topological space covered by a collection of open subsets Wα ⊂M (coordinate charts) and maps Xα : Wα → Vα ⊂ R one-to-one and onto, where Vα is an open, connected subset of R. (Wα,Xα) define coordinates on M. M is a smooth manifold if the maps Xαβ = Xβ ◦X−1 α , are smooth where they are defined, i.e. on Xα(Wα ∩Wβ) to Xβ(Wα ∩Wβ). Example 1.2. ...
متن کاملCommutator-free Lie group methods
RKMK methods and Crouch-Grossman methods are two classes of Lie group methods. The former is using flows and commutators of a Lie algebra of vector fields as a part of the method definition. The latter uses only compositions of flows of such vector fields, but the number of flows which needs to be computed is much higher than in the RKMK methods. We present a new type of methods which avoids th...
متن کاملOptimization of Lie group methods for differential equations
In this paper we present a technique for reducing to a minimum the number of commutators required in the practical implementation of Lie group methods for integrating numerically matrix differential equations. This technique is subsequently applied to the linear and nonlinear case for constructing new geometric integrators, optimal with respect to the number of commutators. © 2002 Elsevier Scie...
متن کاملOrder conditions for commutator-free Lie group methods
We derive order conditions for commutator-free Lie group integrators. These schemes can for certain problems be good alternatives to the Runge-Kutta-Munthe-Kaas schemes, especially when applied to stiff problems or to homogeneous manifolds with large isotropy groups. The order conditions correspond to a certain subsets of the set of ordered rooted trees. We discuss ways to select these subsets ...
متن کاملLie Group Methods for Optimization with Orthogonality Constraints
Optimization of a cost function J(W) under an orthogonality constraint WW = I is a common requirement for ICA methods. In this paper, we will review the use of Lie group methods to perform this constrained optimization. Instead of searching in the space of n× n matrices W, we will introduce the concept of the Lie group SO(n) of orthogonal matrices, and the corresponding Lie algebra so(n). Using...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Acta Numerica
سال: 2000
ISSN: 0962-4929,1474-0508
DOI: 10.1017/s0962492900002154