A. Iserles, H. Z. Munthe-Kaas, S. P. Nrsett, and A. Zanna, Lie-group methods, Acta Numerica, 9(2000), 215-365. Home/Search Document Not in Database Summary Related Articles Check |
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On Magnus Integrators for Time-Dependent Schrödinger Equations - Hochbruck, Lubich (Correct)
....potential. As the discretization of an unbounded operator, H(t) can be of arbitrarily large norm. Magnus integrators are an ecient class of numerical methods for such problems [2, 10] Though the error behavior of these methods is well understood in the case of moderately bounded H(t) [5, 6], no results are so far available when kH(t)k becomes large. The present paper gives optimal order estimates for situations where the product of the time step h with kH(t)k can be of arbitrary size. Even more interesting than the error bounds themselves are the mechanisms which lead to these ....
....; A(t) Picard iteration yields the Magnus expansion t) A( d ; A( d 1 Z A( d ; A( d ; A( d (2. 6) 1 A( d ; A( d ; A( d : Numerical methods based on this expansion are reviewed by Iserles, Munthe Kaas, N rsett and Zanna [5]. They are of the form yn 1 = exp( n )y n (2.7) to give an approximation to y(t n 1 ) at t n 1 = t n h. Here n is a suitable approximation of h) given by (2.6) with A(tn ) instead of A( This approximation rst involves truncating the expansion, and second approximating the ....
A. Iserles, H.Z. Munthe-Kaas, S.P. N rsett, and A. Zanna, Lie group methods, Acta Numerica, 9 (2000), pp. 215-365.
Accurate Long-Time-Step Integrators for Almost-Adiabatic Quantum.. - Jahnke (2002) (Correct)
....Y ] XY Y X. Unfortunately, the terms of the Magnus series soon become very complicated, because the number of commutators and integrals grows rapidly, but truncating the series enables an approximation up to any order and hence can be used for the construction of very precise numerical schemes [11, 12, 10]. This requires calculating integrals over A, which is not a trivial task if A is highly oscillatory and only one evaluation of A per time step is desired, as is the case in our situation. However, we can overcome this diculty by applying again the technique presented in the previous section. ....
A. Iserles, H. Z. Munthe-Kaas, S. P. N rsett, and A. Zanna, Lie-group methods, Acta Numerica 2000, pp. 215-365.
Means and Averaging in the Group of Rotations - Moakher (2002) (2 citations) (Correct)
....R in SO(3) is the matrix in so(3) given by Log R = 0, if # = 0, if # (2. 4) where # satisfies tr R = 1 2 cos # and # # (this formula breaks down when #) An alternative expression for the logarithm of a matrix in SO(3) where the parameter # does not appear, is given in [14]. Solutions in SO(3) of the matrix equation Q = R with k a positive integer will be called kth roots of R. These kth roots are given by exp 1 2l# Log R , l = 0, k 1, where # is the angle of rotation of R. The kth root exp( Log R) is the one for which the ....
A. Iserles, H. Z. Munthe-Kaas, S. P. Nrsett, and A. Zanna, Lie-group methods, Acta Numerica, 9 (2000), pp. 215--365.
Geometric Integration Algorithms on Homogeneous Manifolds - Lewis, Olver (2001) (Correct)
....the group structure, the symplectic structure, and all point invariants of the generalized rigid body. Geometric integration schemes for general Lie groups and their associated bundles have been developed by Lewis and Simo [25, 26] and Munthe Kaas, Iserles, N rsett and their collaborators, [32, 15]. In particular, Munthe Kaas [32] extended the classical Runge Kutta algorithms to arbitrary Lie groups, creating a large, versatile family of geometric integrators. In general, if a Lie group G acts freely and transitively on a manifold M , then a di erential equation on M uniquely determines a ....
....conditions and inertia tensors, show that the energy oscillates about a slow drift away from the correct value. We consider six fourth order geometric methods. Four utilize a series expansion for the generator along a solution curve, while the other two use the RKMK4 algorithm of MuntheKaas, [32, 15], with the Cayley transform as the algorithmic exponential) Using the Cayley transform, the map sy4 b determined by the basic generator b for the rigid body system on S 2 13 t sy4 b sy4 o sy4 ib sy4 io RK4 b RK4 o Triaxial 10 5:67 10 3 3:37 10 2 6:06 10 3 1:90 ....
Iserles, A., Munthe{Kaas, H.Z., N rsett, S.P., and Zanna, A., Lie group methods, Acta Numerica (2000), 215-365. 26
On the Role of Mathematical Abstractions for Scientific .. - Åhlander, Haveraaen.. (Correct)
....There are many related initiatives in the scienti c computing community that address software abstractions for PDE solvers, for instance [1, 5, 6, 20] The emphasis on continuous abstractions is also noted in [16] regarding the modeling of computational geometries. For geometric integration [12] of ordinary di erential equations, Di man provides a coordinate free software package [7] Coordinate free numerical optimization is also emphasized in [8] To some extent, coordinate free di erentiation of PDEs is also addressed in Overture [4] concerning composite curvilinear grids. The ....
A. Iserles et al. Lie-group methods. Acta Numerica, 9:215-365, 2000.
Expansions That Grow on Trees - Arieh Iserles Linear (2002) Self-citation (Iserles) (Correct)
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A. Iserles, H. Z. Munthe-Kaas, S. P. Nrsett, and A. Zanna. Lie-group methods. Acta Numerica, 9:215--365, 2000.
On the Foundations of Computational Mathematics - Baxter, Iserles (2002) Self-citation (Iserles) (Correct)
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A. Iserles, H. Z. Munthe-Kaas, S. P. Nrsett, and A. Zanna. Lie-group methods. Acta Numerica, 9:215--365, 2000.
On The Numerical Analysis Of Rapid Oscillation - Iserles (2003) Self-citation (Iserles) (Correct)
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A. Iserles, H. Z. Munthe-Kaas, S. P. Nrsett, and A. Zanna, Lie-group methods, Acta Numerica 9 (2000), 215--365.
On the Method of Neumann Series for Highly Oscillatory Equations - Iserles (2004) Self-citation (Iserles) (Correct)
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A. Iserles, H. Munthe-Kaas, S. P. Nrsett, and A. Zanna, Lie-group methods, Acta Numerica, 9 (2000), pp. 215--365.
On The Optimality Of Double-Bracket Flows - Bloch, Iserles (2003) Self-citation (Iserles) (Correct)
....of nn real, skew symmetric matrices. It easily follows that Y (t) Q(t)Y 0 Q # (t) t 0, 1.3) Q # = B(t, QY 0 Q # ) t 0, Q(0) I . Since the latter is a Lie group equation, we deduce that Q evolves in SO(n) therefore, by (1. 3) Y is a similarity transformation of the initial value Y 0 [IMKNZ00]. In other words, the eigenvalues of Y (t) in (1.2) and, by implication, in the DBF (1.1) are invariants of the flow and do not vary as t increases. We note in passing that the equation (1.3) is the key to practical computation of the solution of (1.2) whilst respecting its invariants [CIZ97, ....
A. Iserles, H. Z. Munthe-Kaas, S. P. Nrsett, and A. Zanna, Lie-group methods, Acta Numerica 9 (2000), 215--365.
On the Spectra of Certain Matrices Generated by Involutory.. - Iserles, Zanna (2003) Self-citation (Iserles Zanna) (Correct)
....are important in motivating and setting the backdrop for our work, hence we commence by reviewing them briefly. Let g be a matrix Lie algebra. The approximation of exp A, where A g, is a central step in most numerical methods for the solution of di#erential equations evolving in Lie groups [5]. The purpose of such Lie group solvers is to propagate the solution within the Lie group G, say, whose Lie algebra is g. Therefore, it is of critical importance that the approximate exponential resides in G whenever the argument lives in g. Unfortunately, many standards technique to ....
A. Iserles, H. Munthe-Kaas, S. P. Nrsett, and A. Zanna, Lie-group methods, Acta Numerica, 9 (2000), pp. 215--365.
Numerical analysis in Lie groups - Iserles (2000) Self-citation (Iserles) (Correct)
....classical numerical method can respect isospectral structure for m 3 [4] while methods that evolve in SO(m) are widely available, not least in this paper. An extensive survey of Lie group methods is available and it goes into considerable detail in the many aspects of this fast evolving subject [14]. Our purpose in this paper is considerably more modest, to introduce a mathematical reader to this area by highlighting a limited number of themes and focussing our narrative on principles, rather than details. To this end, we will make two important simplifying assumptions. Firstly, we consider ....
....ordinary differential equations. Secondly, we consider mainly two expansion methods, namely Magnus and Cayley series. It goes without saying that other methods are available and that the scope of Lie group solvers is restricted neither to linear problems nor to ordinary differential equations [14]. Having said this, the relative simplicity of linear ODEs renders the theory more succinct and complete, makes for cleaner exposition and avoids issues which might be important in their own right but are marginal to our argument, e.g. solution of nonlinear algebraic equations in Lie groups, ....
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Iserles, A., Munthe-Kaas, H.Z., Nrsett, S.P. and Zanna, A. (2000). Lie-group methods, Acta Numerica 9, 215--365.
Total Decoupling Of General Quadratic Pencils, - Part Ii Structure (Correct)
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A. Iserles, H. Z. Munthe-Kaas, S. P. Nrsett, and A. Zanna, Lie-group methods, Acta Numerica, 9(2000), 215-365.
Reachable Matrices By The Qr Iteration With Shift - Delin Chu And (Correct)
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A. Iserles, H. Z. Munthe-Kass, S. P. Nrsett, and A. Zanna, Lie-group methods, Acta Numerica, 2000, 215-365.
On The Low-Rank Approximation Of Data On The Unit Sphere - Chu Del Buono (Correct)
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A. Iserles, H. Z. Munthe-Kass, S. P. Nrsett, and A. Zanna, Lie-group methods, Acta Numerica, 2000, 215-365.
On Multivariate Interpolation - Olver (2004) (Correct)
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Iserles, A., Munthe--Kaas, H.Z., Nrsett, S.P., and Zanna, A., Lie group methods, Acta Numerica (2000), 215--365.
On Magnus Integrators for Time-Dependent Schrödinger Equations - Hochbruck, Lubich (Correct)
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A. Iserles, H.Z. Munthe-Kaas, S.P. N rsett, and A. Zanna, Lie group methods, Acta Numerica, 9 (2000), pp. 215-365.
Geometric Foundations of Numerical Algorithms and Symmetry - Olver (1 citation) (Correct)
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Iserles, A., Munthe{Kaas, H.Z., N rsett, S.P., and Zanna, A., Lie group methods, Acta Numerica (2000), 215-365.
A low complexity Lie group method on the Stiefel manifold - Krogstad (2001) (2 citations) (Correct)
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A. Iserles, H.Z. Munthe-Kaas, S.P. Nrsett and A. Zanna. Lie-group methods. Acta Numerica 9, CUP, 215-365, 2000.
Geometric Integration and Its Applications - Budd, Piggott (2000) (4 citations) (Correct)
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A. Iserles, H.Z. Munthe-Kaas, S.P. N rsett and A. Zanna, Lie-group methods. Acta Numerica (2000), pp. 215-365.
Newton's Method on Riemannian Manifolds: - Covariant Alpha-Theory Jean-Pierre (Correct)
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Iserles A.,H.Z. Munthe-Kaas, S.P. N rset, and A.Zanna, Liegroup methods, Acta Numerica (2001) vol. 9, pp. 215-365.
Integrators for Quantum Dynamics: a numerical analyst's brief.. - Lubich (Correct)
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A. Iserles, H.Z. Munthe-Kaas, S.P. Nrsett, A. Zanna, Lie-group methods, Acta Numerica 2000, 215--365.
A low complexity Lie group method on the Stiefel manifold - Krogstad (2001) (2 citations) (Correct)
No context found.
A. Iserles, H.Z. Munthe-Kaas, S.P. Nrsett and A. Zanna. Lie-group methods. Acta Numerica 9, CUP, 215-365, 2000.
Moving Frames: A Brief Survey - Olver (Correct)
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Iserles, A., Munthe{Kaas, H.Z., N rsett, S.P., and Zanna, A., Lie group methods, Acta Numerica (2000), 215-365.
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