Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems

The present article presents a summarizing view at differential-algebraic equations (DAEs) and analyzes how new application fields and corresponding mathematical models lead to innovations both in theory and in numerical analysis for this problem class. Recent numerical methods for nonsmooth dynamical systems subject to unilateral contact and friction illustrate the topicality of this development.Comment: Preprint of Book Chapte

A Strictly Single-Site DMRG Algorithm with Subspace Expansion

We introduce a strictly single-site DMRG algorithm based on the subspace expansion of the Alternating Minimal Energy (AMEn) method. The proposed new MPS basis enrichment method is sufficient to avoid local minima during the optimisation, similarly to the density matrix perturbation method, but computationally cheaper. Each application of $\hat H$ to $|\Psi\rangle$ in the central eigensolver is reduced in cost for a speed-up of $\approx (d + 1)/2$, with $d$ the physical site dimension. Further speed-ups result from cheaper auxiliary calculations and an often greatly improved convergence behaviour. Runtime to convergence improves by up to a factor of 2.5 on the Fermi-Hubbard model compared to the previous single-site method and by up to a factor of 3.9 compared to two-site DMRG. The method is compatible with real-space parallelisation and non-abelian symmetries.Comment: 9 pages, 6 figures; added comparison with two-site DMR