arXiv subjects

Xuanzhao Gao

Publications and source records attributed to Xuanzhao Gao.

11 recordsLinked to original sources

arXiv2026

An Adaptive Fast Algorithm for Periodic Coulomb Lattice Sums in Arbitrary Unit Cells

We present a fast algorithm for evaluating conditionally convergent Coulomb lattice sums, governed by the Laplace equation with periodic boundary conditions on arbitrary unit cells (oblique in 2D, triclinic in 3D) and arbitrary particle distributions. The algorithm extends the dual-space multilevel kernel-splitting (DMK) framework to this context. The root of the adaptive tree is now a rectangular grid of cubes consisting of an inner block covering the unit cell and a surrounding halo of image cubes, rather than a single cube, and the smooth top-level periodic kernel -- the only term that requires the consideration of conditional convergence issues -- is evaluated by the ``five-step procedure" used in fast Ewald summation: spreading, fast Fourier transform (FFT), diagonal scaling, inverse FFT, and interpolation. The resulting complexity is $O(N)$ for fixed cell shape. Benchmarked against the periodic fast multipole method on highly nonuniform source distributions, our 2D algorithm is roughly an order of magnitude faster across particle counts and target precisions; in three dimensions, it is often as fast as the free-space DMK on the same sources, even for triclinic cells with edge-length ratios up to roughly $17$.

math.NA↗

arXiv2026

Branch-and-Bound Tensor Networks for Exact Ground-State Characterization

Characterizing the ground-state properties of disordered systems, such as spin glasses and combinatorial optimization problems, is fundamental to science and engineering. However, computing exact ground states and counting their degeneracies are generally NP-hard and #P-hard problems, respectively, posing a formidable challenge for exact algorithms. Recently, Tensor Networks methods, which utilize high-dimensional linear algebra and achieve massive hardware parallelization, have emerged as a rapidly developing paradigm for efficiently solving these tasks. Despite their success, these methods are fundamentally constrained by the exponential growth of space complexity, which severely limits their scalability. To address this bottleneck, we introduce the Branch-and-Bound Tensor Network (BBTN) method, which seamlessly integrates the adaptive search framework of branch-and-bound with the efficient contraction of tropical tensor networks, significantly extending the reach of exact algorithms. We show that BBTN significantly surpasses existing state-of-the-art solvers, setting new benchmarks for exact computation. It pushes the boundaries of tractability to previously unreachable scales, enabling exact ground-state counting for $\pm J$ spin glasses up to $64 \times 64$ and solving Maximum Independent Set problems on King's subgraphs up to $100 \times 100$. For hard instances, BBTN dramatically reduces the computational cost of standard Tropical Tensor Networks, compressing years of runtime into minutes. Furthermore, it outperforms leading integer-programming solvers by over 30$\times$, establishing a versatile and scalable framework for solving hard problems in statistical physics and combinatorial optimization.

cond-mat.stat-mech↗

arXiv2026

An $O(\log N)$ Monte Carlo method for periodic Coulomb systems

Efficient Monte Carlo (MC) sampling of many-body systems with long-range electrostatics is often limited by the cost of per-move energy-difference evaluation under periodic boundary conditions. We present DMK-MC, an accelerated MC method that adapts the dual-space multilevel kernel-splitting (DMK) framework to single-particle Metropolis updates. DMK-MC computes the energy change and, upon acceptance, updates the stored incoming plane-wave fields with $O(1)$ work per tree level, yielding an overall $O(\log N)$ expected work per trial move for fixed accuracy. The method decomposes the Coulomb kernel into three components: a global, periodized smooth part; a multilevel sequence of smooth difference kernels whose interactions are restricted to same-level colleague boxes; and a singular residual kernel whose short-range interactions are evaluated directly. Benchmarks on uniform, highly nonuniform, and implicit-solvent electrolyte and colloidal configurations show that DMK-MC consistently outperforms a recent FMM-based $O(\log N)$ Monte Carlo method, delivering several-fold speedups at comparable tolerances.

physics.comp-ph↗

arXiv2026

An O(N) quasi-Ewald splitting method for nanoconfined electrostatics

Simulating the dynamics of charged particles in quasi-two-dimensional (quasi-2D) nanoconfined systems presents a significant computational challenge due to the long-range nature of electrostatic interactions and the geometric anisotropy. To address this, we introduce a novel quasi-Ewald splitting strategy tailored for particle-based simulations in such geometry. Our splitting strategy seamlessly integrates a collection of advanced numerical techniques, including optimal quadrature rules [L. N. Trefethen, SIAM Rev. 64(1)(2022), pp.132-150], fast pairwise kernel summation methods [S. Jiang and L. Greengard, Commun. Comput. Phys. 31(1)(2022), pp.1-26], and the random batch method with importance sampling in k-space [S. Jin, L. Li, Z. Xu et al., SIAM J. Sci. Comput. 43(4)(2021), pp.B937-B960]. The resulting algorithm achieves an O(N) overall computational complexity, where N denotes the total number of confined particles. Simulations of several prototype systems validate the accuracy and efficiency of our method. Furthermore, we present numerical observations specifically related to nanoconfined charged many-body systems, highlighting phenomena such as dielectric boundary effects, anisotropic diffusion, and the structure of the electrical double layer (EDL) under conditions of charge asymmetry.

cond-mat.soft↗

arXiv2025

Accurate Error Estimates and Optimal Parameter Selection in Ewald Summation for Dielectrically Confined Coulomb Systems

Dielectrically confined Coulomb systems are widely employed in molecular dynamics (MD) simulations. Despite extensive efforts in developing efficient and accurate algorithms for these systems, rigorous and accurate error estimates, which are crucial for optimal parameter selection for simulations, is still lacking. In this work, we present a rigorous error analysis in Ewald summation for electrostatic interactions in systems with two dielectric planar interfaces, where the polarization contribution is modeled by an infinitely reflected image charge series. Accurate error estimate is provided for the truncation error of image charge series, as well as decay rates of energy and force correction terms, as functions of system parameters such as vacuum layer thickness, dielectric contrasts, and image truncation levels. Extensive numerical tests conducted across several prototypical parameter settings validate our theoretical predictions. Additionally, our analysis elucidates the non-monotonic error convergence behavior observed in previous numerical studies. Finally, we provide an optimal parameter selection strategy derived from our theoretical insights, offering practical guidance for efficient and accurate MD simulations of dielectric-confined systems.

math.NA↗

arXiv2025

Fast Algorithm for Quasi-2D Coulomb Systems

Quasi-2D Coulomb systems are of fundamental importance and have attracted much attention in many areas nowadays. Their reduced symmetry gives rise to interesting collective behaviors, but also brings great challenges for particle-based simulations. Here, we propose a novel algorithm framework to address the $O(N^2)$ simulation complexity associated with the long-range nature of Coulomb interactions. First, we introduce an efficient Sum-of-Exponentials (SOE) approximation for the long-range kernel associated with Ewald splitting, achieving uniform convergence in terms of inter-particle distance, which reduces the complexity to $O(N^{7/5})$. We then introduce a random batch sampling method in the periodic dimensions, the stochastic approximation is proven to be both unbiased and with reduced variance via a tailored importance sampling strategy, further reducing the computational cost to $O(N)$. The performance of our algorithm is demonstrated via various numerical examples. Notably, it achieves a speedup of $2\sim 3$ orders of magnitude comparing with Ewald2D method, enabling molecular dynamics (MD) simulations with up to $10^6$ particles on a single core. The present approach is therefore well-suited for large-scale particle-based simulations of Coulomb systems under confinement, making it possible to investigate the role of Coulomb interaction in many practical situations.

math.NA↗

arXiv2024

Programming guide for solving constraint satisfaction problems with tensor networks

Constraint satisfaction problems (CSPs) are a class of problems that are ubiquitous in science and engineering. It features a collection of constraints specified over subsets of variables. A CSP can be solved either directly or by reducing it to other problems. This paper introduces the Julia ecosystem for solving and analyzing CSPs, focusing on the programming practices. We introduce some of the important CSPs and show how these problems are reduced to each other. We also show how to transform CSPs into tensor networks, how to optimize the tensor network contraction orders, and how to extract the solution space properties by contracting the tensor networks with generic element types. Examples are given, which include computing the entropy constant, analyzing the overlap gap property, and the reduction between CSPs.

physics.comp-ph↗

arXiv2024

A fast spectral sum-of-Gaussians method for electrostatic summation in quasi-2D systems

The quasi-2D electrostatic systems, characterized by periodicity in two dimensions with a free third dimension, have garnered significant interest in many fields. We apply the sum-of-Gaussians (SOG) approximation to the Laplace kernel, dividing the interactions into near-field, mid-range, and long-range components. The near-field component, singular but compactly supported in a local domain, is directly calculated. The mid-range component is managed using a procedure similar to nonuniform fast Fourier transforms in three dimensions. The long-range component, which includes Gaussians of large variance, is treated with polynomial interpolation/anterpolation in the free dimension and Fourier spectral solver in the other two dimensions on proxy points. Unlike the fast Ewald summation, which requires extensive zero padding in the case of high aspect ratios, the separability of Gaussians allows us to handle such case without any zero padding in the free direction. Furthermore, while NUFFTs typically rely on certain upsampling in each dimension, and the truncated kernel method introduces an additional factor of upsampling due to kernel oscillation, our scheme eliminates the need for upsampling in any direction due to the smoothness of Gaussians, significantly reducing computational cost for large-scale problems. Finally, whereas all periodic fast multipole methods require dividing the periodic tiling into a smooth far part and a near part containing its nearest neighboring cells, our scheme operates directly on the fundamental cell, resulting in better performance with simpler implementation. We provide a rigorous error analysis showing that upsampling is not required in NUFFT-like steps, achieving $O(N\log N)$ complexity with a small prefactor. The performance of the scheme is demonstrated via extensive numerical experiments.

math.NA↗

arXiv2024

Broken Symmetries in Quasi-2D Charged Systems via Negative Dielectric Confinement

We report spontaneous symmetry breaking (SSB) phenomena in symmetrically charged binary particle systems under planar nanoconfinement with negative dielectric constants.The SSB is triggered $solely$ via the dielectric confinement effect, without any external fields. The mechanism of SSB is found to be caused by the strong polarization field enhanced by nanoconfinement, giving rise to charge/field oscillations in the transverse directions. Interestingly, dielectric contrast can even determine the degree of SSB in transverse and longitudinal dimensions, forming charge-separated interfacial liquids and clusters on square lattices. Furthermore, we analytically show that the formed lattice constant is determined by the dielectric mismatch and the length scale of confinement, which is validated via molecular dynamics simulations. The novel broken symmetry mechanism may provide new insights in the study of quasi-2D systems and the design of future nanodevices.

cond-mat.soft↗

arXiv2024

Probabilistic Inference in the Era of Tensor Networks and Differential Programming

Probabilistic inference is a fundamental task in modern machine learning. Recent advances in tensor network (TN) contraction algorithms have enabled the development of better exact inference methods. However, many common inference tasks in probabilistic graphical models (PGMs) still lack corresponding TN-based adaptations. In this work, we advance the connection between PGMs and TNs by formulating and implementing tensor-based solutions for the following inference tasks: (i) computing the partition function, (ii) computing the marginal probability of sets of variables in the model, (iii) determining the most likely assignment to a set of variables, and (iv) the same as (iii) but after having marginalized a different set of variables. We also present a generalized method for generating samples from a learned probability distribution. Our work is motivated by recent technical advances in the fields of quantum circuit simulation, quantum many-body physics, and statistical physics. Through an experimental evaluation, we demonstrate that the integration of these quantum technologies with a series of algorithms introduced in this study significantly improves the effectiveness of existing methods for solving probabilistic inference tasks.

cs.LG↗

arXiv2024

Random Batch Ewald Method for Dielectrically Confined Coulomb Systems

Quasi two-dimensional Coulomb systems have drawn widespread interest. The reduced symmetry of these systems leads to complex collective behaviors, yet simultaneously poses significant challenges for particle-based simulations. In this paper, a novel method is presented for efficiently simulate a collection of charges confined in doubly-periodic slabs, with the extension to scenarios involving dielectric jumps at slab boundaries. Unlike existing methods, the method is insensitive to the aspect ratio of simulation box, and it achieves optimal O(N) complexity and strong scalability, thanks to the random batch Ewald (RBE) approach. Moreover, the additional cost for polarization contributions, represented as image reflection series, is reduced to a negligible cost via combining the RBE with an efficient structure factor coefficient re-calibration technique in k-space. Explicit formulas for optimal parameter choices of the algorithm are provided through error estimates, together with a rigorous proof. Finally, we demonstrate the accuracy, efficiency and scalability of our method, called RBE2D, via numerical tests across a variety of prototype systems. An excellent agreement between the RBE2D and the PPPM method is observed, with a significant reduction in the computational cost and strong scalability, demonstrating that it is a promising method for a broad range of charged systems under quasi-2D confinement.

math.NA↗