Munthe-Kaas, H. and Owren, B. (1999). Computations in a free Lie algebra, Phil. Trans Royal Society A 357, 957--982.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....1 2 ) l Gamma1 B l = A k ; k = 1; 2; We rewrite (3.8) in the new basis, Q(h) X k2C s b k L(B k1 ; B k2 ; B ks ) 3. 9) where b k = Z S s Y i=1 ( i Gamma 1 2 ) k i Gamma1 d: The self adjoint representation has been introduced by Munthe Kaas and Owren [21], who have pointed out three distinct mechanisms that reduce the number of terms that need be incorporated into (3.9) We consider a free graded Lie algebra generated by the alphabet fC 1 ; C 2 ; C g, where each C k is equipped with the grade (C k ) The grades propagate in a natural ....

....the Jacobi identity [A; B; C] B; C; A] C; A; B] O, allow us to express many terms as linear combinations or other terms. This is well known in the classical Numerical analysis in Lie groups 13 case (C k ) j 1 (which corresponds to (3. 8) and has been extended to general grades in [21]. In particular, for grades (B k ) k, we let 1 ; 2 ; be the zeros of q(z) 1 Gamma 2z z 1 1 Gamma z ; whence dim K r = 1 r X kjr (k) X i=1 r=k i ; 3.10) where is the Mobius function. Since a basis of K r can be constructed similarly to the classical Hall ....

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Munthe-Kaas, H. and Owren, B. (1999). Computations in a free Lie algebra, Phil. Trans Royal Society A 357, 957--982.

....quadrature is required. All the integrals in a Magnus expansion can be approximated in a surprisingly small number of function evaluations (Iserles Nrsett 1999) and this advantage is equally valid in the more general framework. However, unless graded free Lie algebraic techniques, pioneered in (Munthe Kaas Owren 1999), are used, this advantage is offset by a large number of commutator calculations. To produce the requisite free Lie algebra for (1.4) and a method of order 2 we need to take in (1.4) generators with the grades f1; 2; g twice (once for A and once for B) while (2.11) requires a single ....

Munthe-Kaas, H. & Owren, B. (1999), `Computations in a free Lie algebra', Phil.

....can be obtained recursively (Iserles Nrsett 1999) Practical implementation of the Magnus expansion requires the discretization of the integrals in (4. 2) Surprisingly effective and affordable quadrature methods have been developed in (Blanes, Casas Ros 1999, Iserles Nrsett 1999, Munthe Kaas Owren 1999) and they rely on the special form of the integrals and the theory of free graded Lie algebras. Local error consists of the contribution of two phenomena: the truncation of the infinite Magnus expansion and the replacement of integrals by quadrature. In this section we assume, though, that the ....

Munthe-Kaas, H. & Owren, B. (1999), `Computations in a free Lie algebra', Phil.

....back to the turn of the century. The results are applicable to time varying di erential equations in a much broader sense, which includes many recent applications in chemistry and physics, deformations and quantum groups, as well as novel algorithms for numerical integration on manifolds see e.g. [7, 8, 25]. They basically just ask for interpreting the function u as e.g. a perturbation (or general time varying function) rather than thinking of it as a control that may be freely chosen. Clearly, the solution formulas and di erential geometric features are una ected by such re interpretation. However, ....

H. Munthe-Kaas and B. Owren, Computations in a Free Lie Algebra, report no. 148, Dept. Informatics, Univ. Bergen, 1998.

....t t t #ops ky1 ,y1;exactk ky2 ,y2;exactk stepsize global error Figure 9: The solution of the Mathieu problem #4.12#. We have not addressed ourselves to the important issue of how to approximate the matrix exponential. This problem is common to a number of Lie group solvers #Crouch Grossman 1993,Munthe Kaas 1998,Owren Marthinsen 1997,Zanna 1996#. It is known that the standard approach of replacing the exponential function by a rational approximant is unsuitable for general Lie groups. For example, it is possible to prove that the only analytic function mapping sl#n#into SL#n# and consistent with e z is the exponential ....

Munthe-Kaas, H. & Owren, B. #n.d.#, Computations in a free Lie algebra, to appear. Owren, B. & Marthinsen, A. #1997#, In tegration methods based on rigid frames, Technical Report Numerics No. 1#1997, Department of Mathematical Sciences, The Norwegian University of Science and Technology.

.... u 2 , both as elements in g, are su#ciently small, then # exp(u 1 ) #(exp(u 2 ) p) # exp(u 1 ) exp(u 2 ) p = # exp(B(u 1 , u 2 ) p , and hence #(u 1 , #(u 2 , p) #(B(u 1 , u 2 ) p) where B : g g # g is the well known Baker Campbell Hausdor# formula (see, e.g. [33, 23]) 3.1. Crouch Grossman methods. The Crouch Grossman methods are described in a number of texts; see, e.g. 2, 25, 19] Much of the discussion in this section is based on notation and results from [25] Letting E 1 , E n be smooth vector fields (a frame) on the manifold M, we may ....

H. Munthe-Kaas and B. Owren, Computations in a free Lie algebra, Philos. Trans. Roy. Soc. London Ser. A, 357 (1999), pp. 957--981. INTERPOLATION IN LIE GROUPS 285

....however, they are related to the the Fer [7] and Magnus [19] expansions: in fact, both expansions are based on the same product integral as we used above. Recently, the use of the Fer and Magnus expansions for numerically solving differential equations has been investigated by a number of people [15, 16, 17, 27, 36]. 27] discusses the use of graded Lie algebras and implementation issues for deriving Lie bracket identities in computational software (Matlab) 6 Experimental results We ran a number of experiments to compare the performance of the various simulation methods described above. All the ....

....to the the Fer [7] and Magnus [19] expansions: in fact, both expansions are based on the same product integral as we used above. Recently, the use of the Fer and Magnus expansions for numerically solving differential equations has been investigated by a number of people [15, 16, 17, 27, 36] [27] discusses the use of graded Lie algebras and implementation issues for deriving Lie bracket identities in computational software (Matlab) 6 Experimental results We ran a number of experiments to compare the performance of the various simulation methods described above. All the experiments ....

H. Munthe-Kaas and B. Owren, Computations in a free Lie algebra, Tech. Rep. 148, University of Bergen, 1998.

.... The Crouch Grossman method was introduced by Crouch and Grossman in [2] A thorough analysis of the method, in the setting usual for Runge Kutta methods, was performed by Owren and Marthinsen [14] and also other papers discuss and analyze this method as a numerical integrator on Lie groups [13, 10, 12]. In this paper we want to focus on the issue of time symmetry for the Crouch Grossman method. Time symmetric numerical integrators are advantageous in many applications, and we consider it important and necessary to investigate this property also for the family of Crouch Grossman methods. Our ....

H. MUNTHE-KAAS AND B. OWREN, Computations in a Free Lie Algebra, Phil. Trans Royal Soc. A, 357 (1999), pp. 957--981.

....a Stiefel manifold with respect to a product by a member of O(n; R) the projective space of all lines through the origin in R n or C , and the set of all elements of gl(n; R) similar to a given matrix. A crucial feature of Lie group solvers is that they can be extended to homogeneous spaces (Munthe Kaas 1999). Differential equations evolving on a homogeneous space M can be locally written in the form y 0 = y (F (t; y) t 0; y(t 0 ) y 0 2 M; 2.3) where F : R Theta M g is Lipschitz (g being the Lie algebra of G) while the map y : g TM is defined as y (X) d d (exp( X) y)j =0 : ....

.... and map the solution all the way to M in accordance with the diagram Theta N Theta N 1 YN 6 YN 1 y N 6 y N 1 Lie algebraic method X Theta 0 = dexp Gamma1 Theta F (t; e Theta Y 0 ; y 0 ) Y 0 = F (t; Y; y 0 ) Y y 0 = y (F (t; y) Technical details were given by Munthe Kaas (1999) but the reader might easily work out explicitly the case of isospectral equations, whereby (X; Z) XZX Gamma1 , X;Z 2 GL(n; R) Calvo, Iserles Zanna 1997, Zanna 1999) 3 Lie group solvers Inasmuch as we wish to focus in this paper on methods that adhere to the general scheme (1.2) it is ....

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Munthe-Kaas, H. & Owren, B. (1999), `Computations in a free Lie algebra', Phil.

....v; q) 2. In formulas like the Runge Kutta method, one nds that the u to be inserted in d exp 1 u (v) always satis es u = O(h) where h is the time step. Thus ad k u = O(h k ) and one may discard terms from the series of higher order than the integration method itself. Munthe Kaas and Owren [22] have found ways to make the number of commutators even smaller than what comes from the crude discussion above by applying techniques from the theory of free Lie algebras. Coming back to so(3) once again, it is possible to derive an exact expression for the map d exp 1 u , one obtains the ....

H. Munthe-Kaas and B. Owren. Computations in a free Lie algebra. In C. J. Budd and A. Iserles, editors, Geometric Integration: Numerical Solution of Dierential Equations on Manifolds, volume 357 of Philosophical Transactions of the Royal Society A, pages 957-982. London Mathematical Society, 1999.

....[15] In a recent investigation on Lie group numerical methods [12] it was discovered that, for linear problems, i.e. when fl j fl(t) Magnus type methods based on symmetric collocation points, a la Gauss Legendre, were selfadjoint. Similar results were obtained for the Munthe Kaas type methods [17], again when the RK methods employed in the numerical integration of (1:3) were time symmetric (like in the classical theory of RK methods [9] However, for genuine nonlinear problems, when fl j fl(y) such schemes (with due exceptions, for instance the implicit midpoint rule) are not generally ....

H. Munthe-Kaas and B. Owren. Computations in a free Lie algebra. Phil. Trans. R. Soc. Lond. A, 357:957--982, 1999.

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