Approximating the exponential from a Lie algebra to a Lie group

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.