Peter Oehme | Searcharxiv

arXiv2026

Randomized Methods for Kernelized DMD

Dynamic Mode Decomposition (DMD) is a data-driven method related to Koopman operator theory that extracts information about dominant dynamics from data snapshots. In this paper we examine techniques to accelerate the application of DMD to large-scale data sets with an eye on randomized techniques. Randomized techniques exploit low-rank matrix approximations at a much smaller computational cost, therefore permitting the use of increased data set sizes. In particular, we propose the application of the RPCholesky algorithm in the setting of kernelized DMD (KDMD). This algorithm relies on adaptive randomized sampling to approximate positive semidefinite kernel matrices and provides better stability guarantees than previously implemented randomized methods for KDMD. Differences between existing competitive randomized techniques and our proposed implementation are discussed with a focus on numerical stability and tradeoff between exploration and exploitation of information obtained from data. The efficacy of this new combination of algorithms is demonstrated on well-established benchmark problems from DMD literature increasing in problem dimension.

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arXiv2026

A novel Krylov subspace method for approximating Fr\'echet derivatives of large-scale matrix functions

We present a novel Krylov subspace method for approximating $L_f(A, E) \vc{b}$, the matrix-vector product of the Fr\'echet derivative $L_f(A, E)$ of a large-scale matrix function $f(A)$ in direction $E$, a task that arises naturally in the sensitivity analysis of quantities involving matrix functions, such as centrality measures for networks. It also arises in the context of gradient-based methods for optimization problems that feature matrix functions, e.g., when fitting an evolution equation to an observed solution trajectory. In principle, the well-known identity \[ f\left( \begin{bmatrix} A & E \\ 0 & A \end{bmatrix} \right) \begin{bmatrix} 0 \\ \vc{b} \end{bmatrix} = \begin{bmatrix} L_f(A, E) \vc{b} \\ f(A) \vc{b} \end{bmatrix}, \] allows one to directly apply any standard Krylov subspace method, such as the Arnoldi algorithm, to address this task. However, this comes with the major disadvantage that the involved block triangular matrix has unfavorable spectral properties, which impede the convergence analysis and, to a certain extent, also the observed convergence. To avoid these difficulties, we propose a novel modification of the Arnoldi algorithm that aims at better preserving the block triangular structure. In turn, this allows one to bound the convergence of the modified method by the best polynomial approximation of the derivative $f^\prime$ on the numerical range of $A$. Several numerical experiments illustrate our findings.

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