DTIC ADA433731: Applications of Lie Group Integrators and Exponential Schemes

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Applications of Lie group integrators
and exponential schemes
Workshop on Lie group methods and control
theory,
Edinburgh, 28.6-1.7, 2004,
Brynjulf Owren
Dept of Math Sci, NTNU
ApplicationsofLiegroupintegratorsandexponentialschemes –p.1/60
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Applications of Lie group integrators and exponential schemes
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Standard Form 298 (Rev. 8-98)
Prescribed by ANSI Std Z39-18
Outline
PART I (Introductory)
(cid:15) Linear IVPs, Eigenvalue problems, linear PDEs
(cid:15) Manifolds (“stay on manifold” principle)
(cid:15) Classical problems (“curved path” principle)
PART II (Recent results on exp ints)
(cid:15) A unified approach to exponential integrators
(cid:15) Order theory
(cid:15) Bounds for dimensions of involved function spaces
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I.1 Linear IVPs
One may for instance write
R Rn(cid:2)n
u_ = A(t) u; A : !
In literature, usually u 2 Rn.
LGI: Magnus series or related (Cayley etc)
When/Why use this scheme.
1. Highly oscillatory ODEs, large imaginary eigenvalues.
Iserles
2. PDEs, A(t) unbounded, classical example: Linear
Schrödinger equation (LSE)). Blanes & Moan,
Hochbruck & Lubich.
Recently also Landau-Lifschitz equation Sun, Qin, Ma
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I.1 Magnus works on LSE!
du
i = H(t) u; H(t) unbounded, selfadjoint
dt
Z
d exp is not invertible for 2k(cid:25)i 2 (cid:27)(u); k 2 nf0g.
u
Truncated series is still unbounded at 1.
H & L find error bounds of the form
p p(cid:0)1
ku (cid:0) u(t )k = C h t max kD u(t)k
m m m
0(cid:20)t(cid:20)t
m
D is a “differentiation operator” related to the LSE.
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Eigenvalue problems
Stability of travelling wave solutions to PDEs. Boils down
to eigenvalue problem
_
Y = A(t; (cid:21)) Y
where (cid:21) is a parameter.
Needs to be solved for several (cid:21).
Magnus integrators used with success by Malham, Oliver
and others.
Early work by Moan on such problems.
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I.2 Problems on (nonlinear) manifolds
A large part of the applications I know involves the
orthogonal group which acts transitively on either of
(cid:15) The orthogonal group itself (or its tangent bundle).
(cid:15) Stiefel manifold. (n (cid:2) p matrices with orthonormal
columns)
(cid:15) The n (cid:0) 1-sphere. (Stiefel with p = 1)
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I.2 Orthogonal group problems
Most used examples are on n = 3 (3D rotations): Free
rigid body, spinning top,. . .
Most LGIs work. RKMK, Crouch-Grossmann,. . .
Scheme.
combined with all possible “coordinates” exp, Cayley,
CCSK etc.
My evaluation
(cid:15) Most Lie group integrators do little else for you than
maintaining orthogonality.
(cid:15) Poor long-time behaviour.
(cid:15) Hard to get reversible / symplectic schemes.
(cid:15) There are exceptions (Lewis and Simo, Zanna et
al.) but these LGIs seem expensive.
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I.2 Stiefel manifolds
Some applications which involve computation on Stiefel
manifolds
(cid:15) Computation of Lyapunov exponents
(cid:15) Multivariate data analysis (optimisation, gradient
flows)
(cid:15) Neural networks, Independent Component Analysis
Maintain orthonormality. Inexpensive stepping,
Demands.
cost O(np2) per step.
Most LGIs work. RKMK,
Schemes.
Crouch-Grossmann,. . . combined with all possible
“coordinates” exp, Cayley, CCSK etc. Most of them
can be implemented in O(np2) ops per step, but
special care must be taken.
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I.2 Stiefel manifolds
My evaluation
(cid:15) Lie group integrators meet requirements specified
in literature
(cid:15) Long-time behaviour has not been an issue.
(cid:15) Overall judgement: Lie group integrators are
competitive, if not superior to classical integrators.
Sources
(cid:15) Dieci, Van Vleck [schemes, but also general
viewpoints, Lyapunov exponents]
(cid:15) Trendafilov. [Multivariate data analysis]
(cid:15) Celledoni, Fiori.[Neural nets, ICA]
(cid:15) LGIs for Stiefel, Krogstad, Celledoni + O
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