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Algebraic Structures on Ordered Rooted Trees and Their Significance to Lie Group Integrators
, 2003
"... 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, w ..."
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Cited by 8 (1 self)
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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.
Order conditions for commutatorfree Lie group methods
 J. Phys. A: Math. Gen
, 2006
"... We derive order conditions for commutatorfree Lie group integrators. These schemes can for certain problems be good alternatives to the RungeKuttaMuntheKaas schemes, especially when applied to stiff problems or to homogeneous manifolds with large isotropy groups. The order conditions correspond ..."
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Cited by 7 (2 self)
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We derive order conditions for commutatorfree Lie group integrators. These schemes can for certain problems be good alternatives to the RungeKuttaMuntheKaas 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
The Magnus expansion and some of its applications
, 2008
"... Approximate resolution of linear systems of differential equations with varying coefficients is a recurrent problem shared by a number of scientific and engineering areas, ranging from Quantum Mechanics to Control Theory. When formulated in operator or matrix form, the Magnus expansion furnishes an ..."
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Cited by 4 (0 self)
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Approximate resolution of linear systems of differential equations with varying coefficients is a recurrent problem shared by a number of scientific and engineering areas, ranging from Quantum Mechanics to Control Theory. When formulated in operator or matrix form, the Magnus expansion furnishes an elegant setting to built up approximate exponential representations of the solution of the system. It provides a power series expansion for the corresponding exponent and is sometimes referred to as TimeDependent Exponential Perturbation Theory. Every Magnus approximant corresponds in Perturbation Theory to a partial resummation of infinite terms with the important additional property of preserving at any order certain symmetries of the exact solution. The goal of this review is threefold. First, to collect a number of developments scattered through half a century of scientific literature on Magnus expansion. They concern the methods for the generation of terms in the expansion, estimates of the radius of convergence of the series, generalizations and related nonperturbative
On The Optimality Of DoubleBracket Flows
 Int. J. Math. Math. Sci
, 2003
"... 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 pnorms. 1. ..."
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Cited by 2 (1 self)
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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 pnorms. 1.
Optimality of Double Bracket and Generalized Double Bracket Flows
, 2003
"... 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. ..."
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Cited by 1 (0 self)
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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.
RECURRENCE RELATIONS AND CONVERGENCE THEORY OF THE GENERALIZED POLAR DECOMPOSITION ON LIE GROUPS
"... Abstract. The subject matter of this paper is the analysis of some issues related to generalized polar decompositions on Lie groups. This decomposition, depending on an involutive automorphism σ, is equivalent to a factorization of z ∈ G, G being a Lie group, as z = xy with σ(x) =x −1 and σ(y) =y, a ..."
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Abstract. The subject matter of this paper is the analysis of some issues related to generalized polar decompositions on Lie groups. This decomposition, depending on an involutive automorphism σ, is equivalent to a factorization of z ∈ G, G being a Lie group, as z = xy with σ(x) =x −1 and σ(y) =y, and was recently discussed by MuntheKaas, Quispel and Zanna together with its many applications to numerical analysis. It turns out that, contrary to X(t) =log(x), an analysis of Y (t) =log(y)isa very complicated task. In this paper we derive the series expansion for Y (t) = log(y), obtaining an explicit recurrence relation that completely defines the function Y (t) in terms of projections on a Lie triple system pσ and a subalgebra kσ of the Lie algebra g, and obtain bounds on its region of analyticity. The results presented in this paper have direct application, among others, to linear algebra, integration of differential equations and approximation of the exponential. 1.
NUMERICAL REPRESENTATIONS OF A UNIVERSAL SUBSPACE FLOW FOR LINEAR PROGRAMS ∗
"... 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 ∗ ..."
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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.