Speaker
Description
It is well known that the matrix exponential $\exp(H)$ is symplectic whenever
$H \in \mathbb{R}^{2n \times 2n}$ is a Hamiltonian matrix. A matrix $H$ is called
Hamiltonian if it satisfies $HJ = (HJ)^{T},$
while a matrix $S$ is called symplectic (or $J$-orthogonal) if $S^{T} J S = J.$
Here, $J \in \mathbb{R}^{2n \times 2n}$ denotes $J =
\left [\begin{smallmatrix} 0 & I_n \\ -I_n & 0\end{smallmatrix}\right].$
In this talk, we study structure-preserving Krylov subspace methods for approximating the matrix–vector products $f(H)b,$ where $H$ is a large Hamiltonian matrix and $f$ denotes either the matrix exponential or the related $\varphi$-function ($\varphi(z) = \frac{e^{z} - 1}{z}$). Such computations are central to exponential integrators for Hamiltonian systems.
Assume that a suitable projection $\Pi = V W^{T} \in \mathbb{R}^{2 n \times 2 n}$ is given, where $V, W \in \mathbb{R}^{2n \times 2m}$, $m \leq n$, have full column rank and $W^{T} V = I_{2m}$.
Then $f(H)b$ for $b \in \mathbb{R}^{2n}$ can be approximated by $f(H)b \approx V f(\widetilde{H}) W^{T} b$ with $\widetilde{H} = W^{T} H V \in \mathbb{R}^{2m \times 2m}.$ When $m\ll n$, evaluating $f(\widetilde{H})b$ is far less computationally expensive than evaluating $f(H)b$. Typically, $V=W$ and the columns of $V$ form an orthonormal basis of the standard Krylov subspace $\mathcal{K}_{2m}(H,b)=\operatorname{span}\{b,Hb,\ldots,H^{2m-1}b\}$ of order $2m$, but then $\widetilde{H}$ is in general not a Hamiltonian matrix. This motivates the use of Krylov bases with $J$-orthogonal columns that yield Hamiltonian projected matrices and symplectic reduced exponentials. We compare several such structure-preserving Krylov methods on representative Hamiltonian test problems, focusing on accuracy, efficiency, and structure preservation, and briefly discuss adaptive strategies for selecting the Krylov subspace dimension.