arXiv subjects

Michel-Niklas Senn

Publications and source records attributed to Michel-Niklas Senn.

3 recordsLinked to original sources

arXiv2026

Structure-preserving Krylov Subspace Approximations for the Matrix Exponential of Hamiltonian Matrices: A Comparative Study

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 phi-function. Such computations are central to exponential integrators for Hamiltonian systems. Standard Krylov methods generally destroy the Hamiltonian structure under projection, motivating 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.

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arXiv2022

The Hamiltonian Extended Krylov Subspace Method

An algorithm for constructing a $J$-orthogonal basis of the extended Krylov subspace $\mathcal{K}_{r,s}=\operatorname{range}\{u,Hu, H^2u,$ $ \ldots, $ $H^{2r-1}u, H^{-1}u, H^{-2}u, \ldots, H^{-2s}u\},$ where $H \in \mathbb{R}^{2n \times 2n}$ is a large (and sparse) Hamiltonian matrix is derived (for $r = s+1$ or $r=s$). Surprisingly, this allows for short recurrences involving at most five previously generated basis vectors. Projecting $H$ onto the subspace $\mathcal{K}_{r,s}$ yields a small Hamiltonian matrix. The resulting HEKS algorithm may be used in order to approximate $f(H)u$ where $f$ is a function which maps the Hamiltonian matrix $H$ to, e.g., a (skew-)Hamiltonian or symplectic matrix. Numerical experiments illustrate that approximating $f(H)u$ with the HEKS algorithm is competitive for some functions compared to the use of other (structure-preserving) Krylov subspace methods.

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arXiv2020

On skew-Hamiltonian Matrices and their Krylov-Lagrangian Subspaces

It is a well-known fact that the Krylov space $\mathcal{K}_j(H,x)$ generated by a skew-Hamiltonian matrix $H \in \mathbb{R}^{2n \times 2n}$ and some $x \in \mathbb{R}^{2n}$ is isotropic for any $j \in \mathbb{N}$. For any given isotropic subspace $\mathcal{L} \subset \mathbb{R}^{2n}$ of dimension $n$ - which is called a Lagrangian subspace - the question whether $\mathcal{L}$ can be generated as the Krylov space of some skew-Hamiltonian matrix is considered. The affine variety $\mathbb{HK}$ of all skew-Hamiltonian matrices $H \in \mathbb{R}^{2n \times 2n}$ that generate $\mathcal{L}$ as a Krylov space is analyzed. Existence and uniqueness results are proven, the dimension of $\mathbb{HK}$ is found and skew-Hamiltonian matrices with minimal $2$-norm and Frobenius norm in $\mathbb{HK}$ are identified. In addition, a simple algorithm is presented to find a basis of $\mathbb{HK}$.

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