Stochastic Lie group integrators

SIAM Journal on Scientific Computing

Volumn 30, Issue 2, 2007, Pages 597-617

  Malham, Simon J A ^a   Wiese, Anke ^a  

^a HERIOT WATT UNIVERSITY   (United Kingdom)

Author keywords

Stochastic differential equations on manifolds; Stochastic Lie group integrators

Indexed keywords

AUTONOMOUS UNDERWATER VEHICLES; DIFFERENTIAL EQUATIONS; NONLINEAR EQUATIONS; STOCHASTIC SYSTEMS;

CASTELL; LIE ALGEBRA; LIE-GROUPS; MUNTHE-KAAS METHOD; NON-COMMUTATIVE; STOCHASTIC DIFFERENTIAL EQUATION ON MANIFOLD; STOCHASTIC DIFFERENTIAL EQUATIONS; STOCHASTIC FLOWS; STOCHASTIC LIE GROUP INTEGRATOR; STOCHASTICS;

LIE GROUPS;

EID: 55349143595     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/060666743     Document Type: Article

References (49)

1. F. BAUDOIN, An Introduction to the Geometry of Stochastic Flows, Imperial College Press, London, 2004.
2. F. BAUDOIN AND L. COUTIN, Self-similarity and fractional Brownian motions on Lie groups, preprint, arXiv:math.PR/0603199 v1, 2006.
3. G. BEN AROUS, Flots et series de Taylor stochastiques, Probab. Theory Related Fields, 81 (1989), pp. 29-77.
4. F. BULLO AND R. M. MURRAY, Proportional Derivative (PD) Control on the Euclidean Group, CDS Technical Report 95-010, California Institute of Technology Pasedena, CA, 1995.
5. K. BURRAGE AND P. M. BURRAGE, High strong order methods for non-commutative stochastic ordinary differential equation systems and the Magnus formula, Phys. D, 133 (1999), pp. 34-48.
6. F. CASAS AND A. ISERLES, Explicit Magnus expansions for nonlinear equations, J. Phys. A: Math. Gen., 39 (2006), pp. 5445-5461.
7. F. CASTELL, Asymptotic expansion of stochastic flows, Probab. Theory Related Fields, 96 (1993), pp. 225-239.
8. F. CASTELL AND J. GAINES, An efficient approximation method for stochastic differential equations by means of the exponential Lie series. Math. Comp. Simulation, 38 (1995), pp. 13-19.
9. K. T. CHEN, Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula, Ann. Math., 65 (1957), pp. 163-178.
10. V. CHICKARMANE, A. RAY, H. M. SAURO, AND A. NADIM, A model for p53 dynamics triggered by DMA damage, SIAM J. Appl. Dyn. Sys., 6 (2007), pp. 61-78.
11. P. E. CROUCH AND R. GROSSMAN, Numerical integration of ordinary differential equations on manifolds, J. Nonlinear Sci., 3 (1993), pp. 1-33.
12. O. EGELAND, M. DALSMO, AND O. J. SøRDALEN, Feedback control of a nonholonomic underwater vehicle with a constant desired configuration, Int. J. Robotics Res., 15 (1996), pp. 24-35.
13. K. D. ELWORTHY, Stochastic Differential Equations on Manifolds, London Math. Soc. Lecture Note Ser., 70, Cambridge University Press, London, 1982.
14. M. EMERY, Stochastic Calculus on Manifolds, Universitext, Springer-Verlag, New York, Berlin, 1989.
15. M. EMERY, On two transfer principles in stochastic differential geometry, Séminaire de Probabilité 24 (1990), pp. 407-441.
16. A. ESTRADE AND M. PONTIER., Backward stochastic differential equations in a Lie group, Séminaire de Probabilité 35 (2001), pp. 241-259.
17. D. FLLIPOVIĆ AND J. TEICHMANN, On the geometry of the term structure of interest rates, Proc. R. Soc. London A, 460 (2004), pp. 129-167.
18. P. FRIZ, Continuity of the Itô-map for Hölder rough paths with applications to the support theorem in Hölder norm, in Probability and Partial Differential Equations in Modern Applied Mathematics, IMA Vol. Math. Appl. 140, Springer, New York, 2005, pp. 117-135.
19. P. FRIZ AND N. VICTOIR, Euler estimates for rough differential equations, J. Differential Equations, to appear.
20. J. G. GAINES AND T. J. LYONS, Variable step size control in the numerical solution of stochastic differential equations, SIAM J. Appl. Math., 57 (1997), pp. 1455-1484.
21. I. I. GIHMAN AND A. V. SKOROHOD, The Theory of Stochastic Processes III, Springer, New York, 1979.
22. E. HAIRER, C. LUBICH, AND G. WANNER, Geometric Numerical Integration, Springer Ser. Comput. Math., Springer, New York, 2002.
23. S. HELGASON, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, 1978.
24. P. HOLMES, J. JENKINS, AND N. E. LEONARD, Dynamics of the Kirchoff equations I: Coincident centers of gravity and buoyancy, Phys. D, 118 (1998), pp. 311-342.
25. A. ISERLES, H. Z. MUNTHE-KAAS, S. P. NøRSETT, AND A. ZANNA, Lie-group methods, Acta Numer., 9 (2000), pp. 215-365.
26. A. ISERLES AND A. ZANNA, Efficient computation of the matrix exponential by generalized polar decompositions, SIAM J. Numer. Anal., 42 (2005), pp. 2218-2256.
27. P. E. KLOEDEN AND E. PLATEN, Numerical Solution of Stochastic Differential Equations, Springer, New York, 1999.
28. M. KOHLMANN AND S. TANG, Multidimensional backward stochastic Riccati equations and applications, SIAM J. Control Optim., 41 (2003), pp. 1696-1721.
29. H. KUNITA, On the representation of solutions of stochastic differential equations, in Séminaire de Probabilités XIV, Lecture Notes in Math. 784, Springer-Verlag, New York, 1980, pp. 282-304.
30. H. KUNITA, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, London, 1990.
31. G. LORD, S. J. A. MALHAM, AND A. WIESE, Efficient strong integrators for linear stochastic systems, SIAM J. Numer. Anal., submitted.
32. T. LYONS, Differential equations driven by rough signals, Rev. Math. Iberoamericana, 14 (1998), pp. 215-310.
33. T. LYONS AND Z. QIAN, System Control and Rough Paths, Oxford University Press, London, 2002.
34. W. MAGNUS, On the exponential solution of differential equations for a linear operator, Comm. Pure Appl. Math., 7 (1954), pp. 649-673.
35. S. J. A. MALHAM AND A. WLESE, Universal Optimal Stochastic Expansions, in preparation, 2007.
36. P. MALLIAVIN, Stochastic Analysis, Grundlehren Math. Wiss. 313, Springer, New York, 1997.
37. J. E. MARSDEN AND T. S. RATIU, Introduction to Mechanics and Symmetry, 2nd ed., Springer, New York, 1999.
38. G. N. MILSTEIN AND M. V. TRETYAKOV, Stochastic Numerics for Mathematical Physics, Springer, New York, 2004.
39. T. MISAWA, A Lie algebraic approach to numerical integration of stochastic differential equations, SIAM J. Sci. Comput., 23 (2001), pp. 866-890.
40. H. MUNTHE-KAAS, High order Runge-Kutta methods on manifolds, Appl. Numer. Math., 29 (1999), pp. 115-127.
41. N. J. NEWTON, Asymptotically efficient Runge-Kutta methods for a class of Itô and Stratonovich equationsiu, SIAM J. Appl. Math., 51 (1991), pp. 542-567.
42. P. J. OLVER, Equivalence, Invariants, and Symmetry, Cambridge University Press, London, 1995.
43. J. SCHIFF AND S. SHNIDER, A natural approach to the numerical integration of Riccati differential equations, SIAM J. Numer. Anal., 36 (1999), pp. 1392-1413.
44. A. SRIVASTAVA, M. I. MILLER, AND U. GRENANDER, Jump-diffusion processes on matrix Lie groups for Bayesian inference, in Proceedings of the IEEE Signal Processing Workshop on Higher-order Statistics, Caesarea, Israel, 1999, pp. 126-129.
45. R. S. S'IRICHARTZ, The Campbell-Baker-Hausdorff-Dynkin formula and solutions of differential equations, J. Funct. Anal., 72 (1987), pp. 320-345.
46. H. J. SUSSMANN, Product expansions of exponential Lie series and the discretization of stochastic differential equations, in Stochastic Differential Systems, Stochastic Control Theory, and Applications, W. Fleming and J. Lions, eds., Springer IMA Series 10, Springer, New York 1988, pp. 563-582.
47. V. S. VARADARAJAN, Lie Groups, Lie Algebras, and Their Representations, Springer, New-York, 1984.
48. F. W. WARNER, Foundations of Differentiable Manifolds and Lie Groups, Grad. Texts in Math., Springer-Verlag, New York, 1983.
49. Y. YAMATO, Stochastic differential equations and nilpotent Lie algebras, Z. Wahrsch. Verw. Gebiete, 47 (1979), pp 213-229.