Abstract not found

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 and their combinatorial properties. We also suggest how the reuse of flow calculations can be included in order to reduce the computational cost. In the case that at most two flow calculations are admitted in each stage, the order conditions simplify substantially. We derive families of fourth order schemes which effectively use only 5 flow calculations per step. 1

Most Lie group integrators can be expanded in series indexed by the set of ordered rooted trees. To each tree one can associate two distinct higher order derivation operators, which we call frozen and unfrozen operators. Composition of frozen operators induces a concatenation product on the trees, whereas composition of unfrozen operators induces a somewhat more complicated product known as the Grossman– Larson product. Both of these algebra structures can be supplemented by the same coalgebra structure and an antipode, the result being two distinct cocommutative graded Hopf algebras. We discuss the use of these structures and characterize subsets of the Hopf algebras corresponding to vector fields and mappings on manifolds. This is further relevant for deriving order conditions for a general class of Lie group integrators and for deriving the modified vector field in backward error analysis for these integrators.

In this paper we analyse optimality of the stable fixed point of the double bracket equations. We introduce di#erent types of optimality and prove local and global optimality results with respect to the Schatten p-norms. 1.

In this paper we consider the optimal structure of double bracket flows and generalisations of these flows to more complex gradient flows. We discuss different notions of optimality and the relationship of the flows to the structure of convex polytopes.

shown that the central paths of strictly feasible instances of linear programs generate curves on the Grassmannian that satisfy a universal ordinary differential equation. Instead of viewing the Grassmannian Gr(m, n) as the set of all n×n projection matrices of rank m, we view it as the set R n×m ∗ of all full column rank n×m matrices, quotiented by the right action of the general linear group GL(m). We propose a class of flows in R n×m ∗ that project to the flow on the Grassmannian. This approach requires much less storage space when n ≫ m (i.e., there are many more constraints than variables in the dual formulation). One of the flows in R n×m ∗ , that leaves invariant the set of orthonormal matrices, turns out to be a particular version of a matrix differential equation known as Oja’s flow. We also point out that the flow in the set of projection matrices admits a double bracket expression.