Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2308.15683

A spectrum adaptive kernel polynomial method

Tyler Chen
View original source ↗Read source document ↗Export metadata ↓

Abstract

The kernel polynomial method (KPM) is a powerful numerical method for approximating spectral densities. Typical implementations of the KPM require an a prior estimate for an interval containing the support of the target spectral density, and while such estimates can be obtained by classical techniques, this incurs addition computational costs. We propose an spectrum adaptive KPM based on the Lanczos algorithm without reorthogonalization which allows the selection of KPM parameters to be deferred to after the expensive computation is finished. Theoretical results from numerical analysis are given to justify the suitability of the Lanczos algorithm for our approach, even in finite precision arithmetic. While conceptually simple, the paradigm of decoupling computation from approximation has a number of practical and pedagogical benefits which we highlight with numerical examples.

Explore related subjects

physics.comp-phcs.NAmath.NA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Tyler Chen. 2023-08-30. A spectrum adaptive kernel polynomial method. https://doi.org/10.1063/5.0166678

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Optimal Bias Potentials via Ergodic Optimal Control and Generator Learning

We investigate the computation of optimal bias potentials for accelerating transitions between metastable states and for computation of equilibrium properties in molecular dynamics simulations. We formulate optimal biasing as an ergodic optimal control problem (OCP), which can be recast as a linear eigenvalue problem for the infinitesimal generator of the unbiased dynamics. We demonstrate that data-driven learning methods for the generator enable reliable solution of the OCP, computation of biasing potentials, extraction of equilibrium properties, and acceleration of state transitions. We also explore the relation of the control problem to coarse grained representations and learning of coarse grained dynamics.

physics.comp-ph↗
arXiv2026

Optimal limits on weak integrability breaking and protected thermal memory near qutrit exchange

Although integrability does not universally require a continuous one-site symmetry, we rigorously prove that every jointly analytic, regular Yang-Baxter deformation of the qutrit exchange interaction necessarily retains a nontrivial, analytically varying one-site charge. Breaking this local symmetry imposes a fundamental physical constraint on approximate conservation, governed by the optimal uniform bound $δ^3 \le C\varepsilon$ that explicitly relates the minimal one-site symmetry defect $δ$ to the local current-conservation residual $\varepsilon$. While breaking all one-site charges strictly forbids an exact integrable completion, an optimally compensated nearest-neighbor interaction saturates this cubic limit and anomalously extends the guaranteed infinite-temperature energy-current correlation window to order $|λ|^{-3}$ in the perturbation strength $λ$. Furthermore, we reveal a fundamental resonance obstruction for intrinsic conversion perturbations that strictly prevents any exact first-order repair of a broken one-site charge on any finite ring. Nevertheless, we demonstrate that the complete eight-dimensional charge memory matrix remains thermodynamically protected and approaches the identity for timescales $t=o(|λ|^{-3/2})$, a robust feature of the full infinite-temperature dynamics when the thermodynamic limit is taken before weak coupling.

physics.comp-ph↗
arXiv2026

Bi-Hamiltonian in Semiflexible Polymers built upon Overdamping Process

Quantifying the interaction between a system of interest and its ambient conditions, the memory effect links the states of two distinct Hamiltonians: one for the target system and one for the environment. In this paper, we propose the diffusion process derived from the Smoluchowski equation that can derive the evolution process described by the memory effect integration in a non Markovian regime. The Smoluchowski picture, within the framework of stochastic thermodynamics, justifies a diffusion process incorporated into the equations of motion, and the result of the derivation enables a coarse-grained molecular dynamics simulation with the modified equation of motion to reproduce attenuation from collisions between single walled carbon nanotubes (SWCNTs) under far from equilibrium conditions. The results of the numerical experiments on the collision confirm that heat diffusion compensates for the correlated momentum arising from the memory effect between the two Hamiltonians in both equilibrium and far from equilibrium states.

physics.comp-ph↗