Approximating the exponential from a Lie algebra to a Lie group
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- by Elena Celledoni and Arieh Iserles PDF
- Math. Comp. 69 (2000), 1457-1480 Request permission
Abstract:
Consider a differential equation $y^{’}=A(t,y)y,\ y(0)=y_0$ with $y_ 0\in \mathrm {G}$ and $A:\mathbb {R}^{+}\times \mathrm {G}\rightarrow \mathfrak {g}$, where $\mathfrak {g}$ is a Lie algebra of the matricial Lie group $\mathrm {G}$. Every $B\in \mathfrak {g}$ can be mapped to $\mathrm {G}$ by the matrix exponential map $\operatorname {exp}{(tB)}$ with $t\in \mathbb {R}$. Most numerical methods for solving ordinary differential equations (ODEs) on Lie groups are based on the idea of representing the approximation $y_n$ of the exact solution $y (t_n)$, $t_n \in \mathbb {R}^{+}$, by means of exact exponentials of suitable elements of the Lie algebra, applied to the initial value $y_0$. This ensures that $y_n\in \mathrm {G}$. When the exponential is difficult to compute exactly, as is the case when the dimension is large, an approximation of $\operatorname {exp}{(tB)}$ plays an important role in the numerical solution of ODEs on Lie groups. In some cases rational or polynomial approximants are unsuitable and we consider alternative techniques, whereby $\operatorname {exp}{(tB)}$ is approximated by a product of simpler exponentials. In this paper we present some ideas based on the use of the Strang splitting for the approximation of matrix exponentials. Several cases of $\mathfrak {g}$ and $\mathrm {G}$ are considered, in tandem with general theory. Order conditions are discussed, and a number of numerical experiments conclude the paper.References
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Additional Information
- Elena Celledoni
- Affiliation: DAMTP, Cambridge University, Silver Street, England CB3 9EW
- Address at time of publication: Department of Mathematical Sciences, NTNU 7491 Trondheim, Norway
- MR Author ID: 623033
- Email: elenac@math.ntnu.no
- Arieh Iserles
- Affiliation: DAMTP, Cambridge University, Silver Street, England CB3 9EW
- Email: A.Iserles@damtp.cam.ac.uk
- Received by editor(s): May 20, 2021
- Received by editor(s) in revised form: January 1, 1998, and October 27, 1998
- Published electronically: March 15, 2000
- © Copyright 2000 American Mathematical Society
- Journal: Math. Comp. 69 (2000), 1457-1480
- MSC (1991): Primary 65D15; Secondary 22E99, 65F30
- DOI: https://doi.org/10.1090/S0025-5718-00-01223-0
- MathSciNet review: 1709149