Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2506.08308

An efficient Fourier spectral algorithm for the Bogoliubov-de Gennes excitation eigenvalue problem

Yu Li·Zhixuan Li·Manting Xie·Yong Zhang
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we propose an efficient Fourier spectral algorithm for an eigenvalue problem, that is, the Bogoliubov-de Gennes (BdG) equation arsing from spin-1 Bose-Einstein condensates (BEC) to describe the elementary/collective excitations around the mean-field ground state. The BdG equation is essentially a constrained eigenvalue/eigenfunction system. Firstly, we investigate its analytical properties, including exact eigenpairs, generalized nullspace, and bi-orthogonality of eigenspaces. Secondly, by combining the standard Fourier spectral method for spatial discretization and a stable Gram-Schmidt bi-orthogonal algorithm, we develop a subspace iterative solver for such a large-scale dense eigenvalue problem, and it proves to be numerically stable, efficient, and accurate. Our solver is matrix-free and the operator-function evaluation is accelerated by discrete Fast Fourier Transform (FFT) with almost optimal efficiency. Therefore, it is memory-friendly and efficient for large-scale problems. Furthermore, we give a rigorous and detailed numerical analysis on the stability and spectral convergence. Finally, we present extensive numerical results to illustrate the spectral accuracy and efficiency, and investigate the excitation spectrum and Bogoliubov amplitudes around the ground state in 1-3 spatial dimensions.

Explore related subjects

math.NAcs.NA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Yu Li, Zhixuan Li, Manting Xie, Yong Zhang. 2025-06-10. An efficient Fourier spectral algorithm for the Bogoliubov-de Gennes excitation eigenvalue problem. https://arxiv.org/abs/2506.08308

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

KEEP EXPLORING

Related papers

arXiv2026

How many continuous measurements are needed to learn a vector?

One can recover vectors from $\mathbb{R}^m$ with arbitrary precision, using only $\lceil \log_2(m)\rceil +1$ continuous measurements that are chosen adaptively. This surprising result is explained and discussed, and we present applications to infinite-dimensional approximation problems.

math.NA↗
arXiv2026

IterativeCUR: Large Rank-Adaptive Approximation From a Small Recycled Sketch

The computation of accurate low-rank matrix approximations is central to improving the scalability of various techniques in machine learning, uncertainty quantification, and control. Traditionally, low-rank approximations are constructed using SVD-based approaches such as truncated SVD or Randomized SVD. Although these SVD approaches---especially Randomized SVD---have proven to be very computationally efficient, other low-rank approximation methods can offer even greater performance. One such approach is the CUR decomposition, which forms a low-rank approximation using direct row and column subsets of a matrix. Because CUR uses direct matrix subsets, it is also often better able to preserve native matrix structures like sparsity or non-negativity than SVD-based approaches and can facilitate data interpretation in many contexts. This paper introduces IterativeCUR, which draws on previous work in randomized numerical linear algebra to build a new algorithm that is highly competitive compared to prior work. IterativeCUR is adaptive in the sense that it takes as an input parameter the desired tolerance $ε$ and outputs (with arbitrarily high probability) an approximation of error bounded by $ε$, rather than requiring an a priori guess of the numerical rank. IterativeCUR typically runs significantly faster than both existing CUR algorithms and techniques such as Randomized SVD. Its asymptotic complexity is $\mathcal{O}(mn + (m+n)r^2)$ for an $m\times n$ matrix of output rank $r$. IterativeCUR relies on a single small sketch from the matrix that is successively downdated as the algorithm proceeds. We demonstrate through extensive experiments that IterativeCUR achieves up to $4\times$ speed-up over state-of-the-art pivoting-on-sketch approaches with no loss of accuracy, and up to $40\times$ speed-up over rank-adaptive randomized SVD approaches.

math.NA↗
arXiv2026

Multigrid with Linear Storage Complexity

As the discretization error for the solution of a partial differential equation (PDE) decreases, the precision required to store the corresponding coefficients naturally increases. Storing the solution's finite element coefficients explicitly requires $\mathcal O(n \log n)$ bits of storage, where $n$ is the number of degrees of freedom (DoFs). This paper presents a full multigrid method to compute the solution in a compressed format that reduces the storage complexity of the solution and intermediate vectors to $\mathcal O(n)$ bits. This reduction allows a matrix-free implementation to solve elliptic PDEs with an overall linear space complexity. For problems limited by the memory capacity of current supercomputers, we expect a memory footprint reduction of about an order of magnitude compared to state-of-the-art mixed-precision methods. We demonstrate the applicability of our algorithm by solving two model problems. Depending on the PDE and polynomial degree, but irrespective of the problem size, the solution vector on the finest grid requires between 4 and 12 bits per DoF, and the residual and correction require 3 to 6 bits each. Additional data is stored on the coarse grids with modestly increasing bit widths toward coarser grids.

math.NA↗