arXiv subjects

Hongji Gao

Publications and source records attributed to Hongji Gao.

3 recordsLinked to original sources

arXiv2025

Accelerating Correlated Wave Function Calculations with Hierarchical Matrix Compression of the Two-Electron Integrals

Leveraging matrix sparsity has proven a fruitful strategy for accelerating quantum chemical calculations. Here we present the hierarchical SOS-MP2 algorithm, which uses hierarchical matrix ($\mathcal{H}^{2}$) compression of the electron repulsion integral (ERI) tensor to reduce both time and space complexity. This approach is based on the atomic orbital Laplace transform MP2 calculations, leveraging the data sparsity of the ERI tensor and the element-wise sparsity of the energy-weighted density matrices. The $\mathcal{H}^{2}$ representation approximates the ERI tensor in a block low-rank form, taking advantage of the inherent low-rank nature of the repulsion integrals between distant sets of atoms. The resulting algorithm enables the calculation of the Coulomb-like term of the MP2 energy with a theoretical time complexity of $\mathcal{O}(N^{2}\log N)$ and a space complexity of $\mathcal{O}(N^{2}\log N)$, where $N$ denotes the number of basis functions. Numerical tests show asymptotic time and space complexities better than $\mathcal{O}(N^{2})$ for both linear alkanes and three-dimensional water clusters.

physics.chem-ph↗

arXiv2025

P-order: Unified Convergence Analysis for Nonlinear Iterative Methods

Measuring how quickly iterative methods converge is essential in computational mathematics, but current approaches have significant limitations. Q-order analysis requires strict smoothness conditions, while R-order analysis lacks precision and creates ambiguity, especially when analyzing convergence rates close to linear. We introduce P-order, a new framework that overcomes these limitations by using a power function $\psi(k)$ combined with asymptotic notation ($\Theta, o, \omega$). Our approach offers two key advantages: it works independently of the chosen norm while providing the precision needed to classify diverse convergence behaviors, including previously hard-to-characterize rates like fractional-power and linearithmic convergence. P-order also systematically accommodates weaker continuity conditions by naturally connecting mathematical assumptions to appropriate Taylor approximation forms. To enhance practical analysis, we develop two important subclasses, QUP-order and UP-order, which work effectively under different smoothness conditions. We demonstrate P-order's practical value through three applications: (1) refining fixed-point iteration analysis with minimal smoothness requirements (mere differentiability suffices where classical analysis required stronger conditions), (2) identifying previously unreported convergence rates for Newton's method and gradient descent algorithms, and (3) providing a unified analysis of $K$-point methods under $C^{K-1,\nu}$ (i.e., with H\"older continuous $(K-1)$th derivatives), yielding a new characteristic rate $q_{K}(\nu)$. Our P-order framework provides researchers and practitioners with a sharper, more comprehensive toolbox for convergence analysis, particularly valuable when classical assumptions fail or when analyzing complex convergence behaviors in modern computational applications.

math.NA↗

arXiv2024

Efficient and Scalable Wave Function Compression Using Corner Hierarchical Matrices

The exponential scaling of complete active space (CAS) and full configuration interaction (CI) calculations limits the ability of quantum chemists to simulate the electronic structures of strongly correlated systems. Herein, we present corner hierarchically approximated CI (CHACI), an approach to wave function compression based on corner hierarchical matrices (CH-matrices) -- a new variant of hierarchical matrices based on a block-wise low-rank decomposition. By application to dodecacene, a strongly correlated molecule, we demonstrate that CH matrix compression provides superior compression compared to a truncated global singular value decomposition. The compression ratio is shown to improve with increasing active space size. By comparison of several alternative schemes, we demonstrate that superior compression is achieved by a) using a blocking approach that emphasizes the upper-left corner of the CI vector, b) sorting the CI vector prior to compression, and c) optimizing the rank of each block to maximize information density.

physics.chem-ph↗