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Sihong Shao

Publications and source records attributed to Sihong Shao.

At least 19 recordsLinked to original sources
arXiv2026

A Parallel Evolutionary Algorithm Framework for Graph $k$-CUT Problems

Graph k-CUT problems include many important variants whose objectives combine cut value, volume, and cardinality terms in different ways. Most existing algorithms are designed for individual formulations, which limits their transferability across related models. In this paper, we organize a broad family of graph partitioning problems into two classes, MaxGCP and MinGCP, according to their optimization orientation and balance-related structure. Based on this classification, we propose a unified Parallel Evolutionary Algorithm Framework (PEAF). This framework combines structure-inheriting crossover operators, a hierarchical mutation mechanism based on the Multiple Mutation Heuristic (MMH) and the Auxiliary Cut Mutation Heuristic (ACMH), and a diversity-preserving selection strategy. Extensive experiments on G-set with k \in\{2, 3, 4, 5\} show that PEAF-ACMH consistently outperforms Gurobi on nine representative k-CUT problems. For MaxGCP, PEAF-ACMH improves several best-known solutions for Max-k-Cut with k \geq 3, and through numerical bounds derived from its relation to Max-k-Cut, verifies the high quality of the obtained solutions for Judicious-k-Partition and AntiCheeger-k-Cut. The results further indicate that Judicious-k-Partition usually yields more balanced partitions than AntiCheeger-k-Cut. For MinGCP, theoretical and computational comparisons show that Cheeger-k-Cut and Sparsest-k-Cut produce more balanced partitions than Normalized-k-Cut and Ratio-k-Cut, respectively. PEAF-ACMH also obtains highly similar partitions for Min-k-Cut and MinMax-k-Cut within short running times, providing numerical evidence for their structural affinity. These results demonstrate that PEAF is both an effective unified solver and a useful tool for revealing structural properties of graph k-CUT models.

math.OC↗
arXiv2026

Weak Adversarial Neural Pushforward Method for the Wigner Transport Equation

We extend the Weak Adversarial Neural Pushforward Method to the Wigner transport equation governing the phase-space dynamics of quantum systems. The central contribution is a structural observation: integrating the nonlocal pseudo-differential potential operator against plane-wave test functions produces a Dirac delta that exactly inverts the Fourier transform defining the Wigner potential kernel, reducing the operator to a pointwise finite difference of the potential at two shifted arguments. This holds in arbitrary dimension, requires no truncation of the Moyal series, and treats the potential as a black-box function oracle with no derivative information. To handle the negativity of the Wigner quasi-probability distribution, we introduce a signed pushforward architecture that decomposes the solution into two non-negative phase-space distributions mixed with a learnable weight. The resulting method inherits the mesh-free, Jacobian-free, and scalable properties of the original framework while extending it to the quantum setting.

quant-ph↗
arXiv2026

Dynamical low-rank methods for the Wigner equation I: separable difference potential

Recent advances in dynamical low-rank approximation (DLRA) have demonstrated its effectiveness in high-dimensional simulations. However, existing DLRA algorithms still face significant challenges when handling systems that involve complex collision terms, including the pseudo-differential operator (${\rm Ψ}$) in the Wigner equation, a representative operator characterized by nonlocality. It is deserving to carry out a series of works to develop the DLRA algorithms for solving the Wigner equation. As the first step in this series of works, we propose an efficient DLRA algorithm for the Wigner equation, using a separable decomposition of the difference potential. We combine this separable assumption with two often-used truncations of ${\rm Ψ}$, namely $\mathcal{K}$-truncation and $\mathcal{Y}$-truncation, to obtain a kind of separated representation of ${\rm Ψ}$. Complexity analysis and several challenging experiments, including harmonic oscillators, Gaussian barrier scattering, electron-electron scattering, and a Helium-like system, all of which satisfy the separable assumption, confirm that the proposed DLRA algorithm has significant advantages, achieving a reduction in computational effort by one to two orders of magnitude in both runtime and memory requirements compared to the full-grid approach. It is worth noting that, even in the absence of a predetermined low-rank structure for the solution, DLRA can still serve as a numerical scheme that balances efficiency and accuracy.

math.NA↗
arXiv2026

An Adaptive Log-Laguerre Spectral Method for the Radial Dirac Equation: Resolving Asymptotic Decay and Core Singularities in Atomic Calculations

The high-precision solution of the radial Dirac equation is fundamental to relativistic quantum chemistry, essential for reliable pseudopotential generation and all-electron electronic structure methods. Capturing both the non-polynomial singularities at the origin and the state-dependent asymptotic decay on semi-infinite domains presents a significant computational challenge. In this work, we propose the Adaptive Log-Laguerre Spectral Method (ALLSM), a novel coupled spectral-element solver that seamlessly integrates three advanced mathematical methodologies into a unified framework. Specifically, Generalized Log-Orthogonal Functions (GLOFs) are deployed in the near-core region to intrinsically approximate complex $r^s$ singular behaviors without requiring prior knowledge of the exact analytical exponent $s$. Concurrently, an adaptive Laguerre spectral method is employed to dynamically capture diverse exponential tails on $[0, \infty)$, avoiding artificial domain truncation. To structurally guarantee spectral purity across this bipartite basis, the framework rigorously incorporates the Inverse Dirac Operator Method (IDOM), effectively eliminating variational collapse and spurious states. Validated across diverse physical regimes, including Coulomb, finite-nucleus, and screened potentials, the proposed solver restores exponential convergence and consistently achieves relative accuracies of $10^{-10}$. This work provides a robust, pollution-free computational kernel for atomic structure calculations, establishing a highly reliable numerical standard for complex molecular simulations.

math.NA↗
arXiv2026

Asymptotic Recovery in Fourier Spectral Methods for the Schrödinger Equation with Point Singularities

This paper studies the Fourier spectral method (FSM) for the Schrödinger equation with singular potentials $V \in H^{s}$, where $s > \max\{d/2-2,-1\}$ and $d$ denotes the spatial dimension. This setting includes a broad class of singular potentials, such as the 3D Coulomb potential and the 1D Dirac-delta potential. First, we combine the Feshbach-Schur map with a refined perturbation argument to derive sharp convergence orders for FSM, yielding order $2s+2$ for eigenvalues and order $s+1$ for eigenfunctions in the $H^1$ norm. More importantly, the $H^1$ error with respect to the projected eigenfunction converges with a higher order $s+1+b$, where $b=\min\{s+2-d/2-\varepsilon,\; s+1,\; 2\}>0$ for arbitrarily small $\varepsilon>0$, revealing a super-convergence phenomenon. Second, in the presence of potentials with isolated point singularities, we develop an asymptotic-recovery (AR) technique to post-process the FSM solutions. The resulting method, dubbed AR-FSM, fully exploits the super-convergence property and achieves convergence orders $2s+2+2b$ for eigenvalues and $s+1+b$ for eigenfunctions in the $H^1$ norm, while the AR post-processing requires only a computational cost that is linear in the number of FSM degrees of freedom. The analysis introduces a rigorous definition of point singularities and develops a foundational framework for their study. It further establishes an asymptotic expansion of eigenfunctions consisting of a regular component in $H^{s+4}$ together with $d+1$ asymptotic functions associated with each singular point. Numerical experiments confirm the sharpness of these theoretical bounds.

math.NA↗
arXiv2026

Eulerian-spanning set and coboundary operator: An investigation of maxcut beyond planar graphs

Using the concepts of Eulerian-spanning set and coboundary operator, we generalize Hadlock's conversion of the maxcut problem on planar graphs to one on general graphs with non-negative weights. Using our conversion, we can explore algorithms for maxcut beyond the class of planar graphs. We obtain a Fixed-Parameter Tractable algorithm for $k$-contraction apex graphs. Specifically, our algorithm can be applied to graphs with crossing number $k$, giving an $O(2^k(n+k)^{3/2}\log (n+k))$-time algorithm that matches the best known results when restricted to non-negative weights.

cs.DS↗
arXiv2026

Solving Vlasov-Poisson system with an adaptive Hermite spectral method

We propose an adaptive Hermite spectral method for the Vlasov-Poisson system based on a recently developed frequency indicator that measures the contribution of the high-order expansion coefficients. Precisely, the symmetrically weighted Hermite basis with a scaling factor is utilized to approximate the distribution function to satisfy the increasing resolution requirement, which, for example, is induced by filamentation. To implement the scaling adjustment, a fast conservative projection operator is constructed in two steps. The first step is to formulate the projection as a constrained optimization problem to preserve key invariants, including mass, momentum, energy, and the $L^2$ norm of the distribution function. The second step is an ODE-based approximation developed to compute the updated expansion coefficients with linear complexity. Numerical experiments with 1D1V and 2D2V settings validate the feasibility and efficiency of this proposed adaptive Hermite method.

math.NA↗
arXiv2026

Stochastic particle method with birth-death dynamics

In order to numerically solve high-dimensional nonlinear PDEs and alleviate the curse of dimensionality, a stochastic particle method (SPM) has been proposed to capture the relevant feature of the solution through the adaptive evolution of particles [J. Comput. Phys. 527 (2025) 113818]. In this paper, we introduce an active birth-death dynamics of particles to improve the efficiency of SPM. The resulting method, dubbed SPM-birth-death, sample new particles according to the nonlinear term and execute the annihilation strategy when the number of particles exceeds a given threshold. A rigorous error estimation for SPM-birth-death is established, elucidating the first-order convergence in time and space, as well as half-order accuracy in the initial sample size with explicit variance estimates. We also extend the analysis framework to SPM and provide theoretical justification for the existing numerical convergence study. Our theoretical results reveal that the introduced active birth-death dynamics of particles results into less frequent resampling and SPM-birth-death is thus able to achieve higher efficiency than SPM. Validating benchmarks are provided. In particular, preliminary numerical experiments on the Allen-Cahn equation demonstrate that SPM-birth-death can achieve smaller errors at the same computational cost compared with the original SPM.

math.NA↗
arXiv2026

Geometrization of Graphs: Towards Bounding the Chromatic Number via High-Dimensional Embedding

We establish a geometric framework by transforming a graph $G$ into a $(d-1)$-dimensional CW complex $U^{d-1}(G)$. This construction is achieved by systematically attaching $i$-spheres ($2 \le i \le d-1$) to $G$ according to specific rules, ensuring that the $j$-th homotopy group of $U^{d-1}(G)$ are trivial for $j = 0, 1, \dots, d-2$. Building upon this construction, we provide a necessary and sufficient condition for $U^{d-1}(G)$ to be embeddable into $\mathbb{R}^d$, which yields an upper bound for the chromatic number $χ(G)$. To be more specific, we prove that if $G$ does not contain $K_{d+3}$ and $K_{i, d+4-i}$ ($i \in \{2, 3, \dots, \lfloor \frac{d+4}{2} \rfloor \}$) as a minor, then $U^{d-1}(G)$ embeds into $\mathbb{R}^d$ and $χ(G) \leq 3\cdot 2^{d-1}$. Finally, as a preliminary attempt, we extend the Discharging method to $\mathbb{R}^d$ and investigate the coloring problem for $(d-2)$-faces in $\mathbb{R}^d$.

math.CO↗
arXiv2026

Equivalent spectral theory for fundamental graph cut problems

We introduce and develop equivalent spectral graph theory for several fundamental graph cut problems including maxcut, mincut, Cheeger cut, anti-Cheeger cut, dual Cheeger problem and their useful variants. A specified strategy for achieving an equivalent eigenproblem is proposed for a general graph cut problem via the set-pair Lovász extension and the Dinkelbach scheme. For a class of 2-cut and 3-cut problems, we reveal the intrinsic difference-of-submodularity for the fractional formulations and show that their set-pair Lovász extensions yield equivalent difference-of-convex structures. Building on the Dinkelbach scheme, we finally establish a unified research roadmap for nonlinear spectral theory that provides a one-to-one correspondence between certain eigenpairs and the optimal graph cut problems. The finer structure of the eigenvectors, the Courant nodal domain theorem and the graphic feature of eigenvalues are studied systematically in the setting of these new nonlinear eigenproblems.

math.CO↗
arXiv2025

Adaptive sampling accelerates the hybrid deviational particle simulations

To avoid ineffective collisions between the equilibrium states, the hybrid method with deviational particles (HDP) has been proposed to integrate the Vlasov-Poisson-Landau system, while leaving a new issue in sampling deviational particles from the high-dimensional source term. In this paper, we present an adaptive sampling (AS) strategy that first adaptively reconstructs a piecewise constant approximation of the source term based on sequential clustering via discrepancy estimation, and then samples deviational particles directly from the resulting adaptive piecewise constant function without rejection. The mixture discrepancy, which can be easily calculated thanks to its explicit analytical expression, is employed as a measure of uniformity instead of the star discrepancy the calculation of which is NP-hard. The resulting method, dubbed the HDP-AS method, samples deviational particles through adaptive sampling instead of the acceptance-rejection method in the original HDP method. In the Landau damping, two stream instability, bump on tail and Rosenbluth's test problems, the HDP-AS method runs approximately ten times faster than the HDP method while keeping the same accuracy.

physics.comp-ph↗
arXiv2025

Density estimation via mixture discrepancy and moments

With the aim of generalizing histogram statistics to higher dimensional cases, density estimation via discrepancy based sequential partition (DSP) has been proposed to learn an adaptive piecewise constant approximation defined on a binary sequential partition of the underlying domain, where the star discrepancy is adopted to measure the uniformity of particle distribution. However, the calculation of the star discrepancy is NP-hard and it does not satisfy the reflection invariance and rotation invariance either. To this end, we use the mixture discrepancy and the comparison of moments as a replacement of the star discrepancy, leading to the density estimation via mixture discrepancy based sequential partition (DSP-mix) and density estimation via moment-based sequential partition (MSP), respectively. Both DSP-mix and MSP are computationally tractable and exhibit the reflection and rotation invariance. Numerical experiments in reconstructing Beta mixtures, Gaussian mixtures and heavy-tailed Cauchy mixtures up to 30 dimension are conducted, demonstrating that MSP can maintain the same accuracy compared with DSP, while gaining an increase in speed by a factor of two to twenty for large sample size, and DSP-mix can achieve satisfactory accuracy and boost the efficiency in low-dimensional tests ($d \le 6$), but might lose accuracy in high-dimensional problems due to a reduction in partition level.

stat.ML↗
arXiv2025

A High-Dimensional Extension of Wagner's Theorem and the Geometrization of Hypergraphs

This paper introduces a geometric representation of hypergraphs by representing hyperedges as simplices. Building on this framework, we employ homotopy groups to analyze the topological structure of hypergraphs embedded in high-dimensional Euclidean spaces. Under the assumptions of the triangulation and that all $i$-th homotopy groups are trivial for $i \leq d-2$, we provide a necessary and sufficient condition for a $d$-uniform hypergraph to be embeddable in $\mathbb{R}^d$, which can be regarded as a kind of high-dimensional extension of Wagner's Theorem for planar graphs. Specifically, we establish that a triangulated $d$-uniform topological hypergraph embeds into $\mathbb{R}^d$ if and only if it contains neither $K_{d+3}^d$ nor $K_{3,d+1}^d$ as a minor. Here, a triangulated $d$-uniform topological hypergraph constitutes a geometrized form of a $d$-uniform hypergraph, while $K_{d+3}^d$ and $K_{3,d+1}^d$ are the high-dimensional generalizations of the complete graph $K_5$ and the complete bipartite graph $K_{3,3}$ in $\mathbb{R}^d$, respectively.

math.CO↗
arXiv2025

An adaptive Hermite spectral method for the Boltzmann equation

We propose an adaptive Hermite spectral method for the three-dimensional velocity space of the Boltzmann equation guided by a newly developed frequency indicator. For the homogeneous problem, the indicator is defined by the contribution of high-order coefficients in the spectral expansion. For the non-homogeneous problem, a Fourier-Hermite scheme is employed, with the corresponding frequency indicator formulated based on distributions across the entire spatial domain. The adaptive Hermite method includes scaling and p-adaptive techniques to dynamically adjust the scaling factor and expansion order according to the indicator. Numerical experiments cover both homogeneous and non-homogeneous problems in up to three spatial dimensions. Results demonstrate that the scaling adaptive method substantially reduces L2 errors at negligible computational cost, and the p-adaptive method achieves time savings of up to 74%.

math.NA↗
arXiv2025

Conductance Estimation in Digraphs: Submodular Transformation, Lovász Extension and Dinkelbach Iteration

Conventional spectral digraph partitioning methods typically symmetrize the adjacency matrix, thereby transforming the directed graph partitioning problem into an undirected one, where bipartitioning is commonly linked to minimizing graph conductance. However, such symmetrization approaches disregard the directional dependencies of edges in digraphs, failing to capture the inherent imbalance crucial to directed network modeling. Building on the parallels between digraph conductance and conductance under submodular transformations, we develop a generalized framework to derive their continuous formulations. By leveraging properties of the Lovász extension, this framework addresses the fundamental asymmetry problem in digraph partitioning. We then formulate an equivalent fractional programming problem, relax it via a three-step Dinkelbach iteration procedure, and design the Directed Simple Iterative ($\mathbf{DSI}$) algorithm for estimating digraph conductance. The subproblem within $\mathbf{DSI}$ is analytically solvable, and the algorithm is guaranteed to converge provably to a binary local optimum. Extensive experiments on synthetic and real-world networks demonstrate that our $\mathbf{DSI}$ algorithm significantly outperforms several state-of-the-art methods in digraph conductance minimization.

math.OC↗
arXiv2025

Dual Cheeger Constants, Signless 1-Laplacians and Maxcut

The first nontrivial lower bound of the worst-case approximation ratio for the maxcut problem was achieved via the dual Cheeger problem, whose optimal value is referred to the dual Cheeger constant $h^+$, and later improved through its modification $\widehat{h}^+$. However, the dual Cheeger problem and its modification themselves are relatively unexplored, especially lack of effective approximate algorithms. To this end, we first derive equivalent spectral formulations of $h^+$ and $\widehat{h}^+$ within the framework of the nonlinear spectral theory of signless 1-Laplacian, present their interactions with the Laplacian matrix and 1-Laplacian, and then use them to develop an inverse power algorithm that leverages the local linearity of the objective functions involved. We prove that the inverse power algorithm monotonically converges to a ternary-valued eigenvector, and provide the approximate values of $h^+$ and $\widehat{h}^+$ on G-set for the first time. The recursive spectral cut algorithm for the maxcut problem can be enhanced by integrating into the inverse power algorithms, leading to significantly improved approximate values on G-set. Finally, we show that the lower bound of the worst-case approximation ratio for the maxcut problem within the recursive spectral cut framework can not be improved beyond $0.769$.

math.SP↗
arXiv2024

An efficient stochastic particle method for high-dimensional nonlinear PDEs

Numerical resolution of high-dimensional nonlinear PDEs remains a huge challenge due to the curse of dimensionality. Starting from the weak formulation of the Lawson-Euler scheme, this paper proposes a stochastic particle method (SPM) by tracking the deterministic motion, random jump, resampling and reweighting of particles. Real-valued weighted particles are adopted by SPM to approximate the high-dimensional solution, which automatically adjusts the point distribution to intimate the relevant feature of the solution. A piecewise constant reconstruction with virtual uniform grid is employed to evaluate the nonlinear terms, which fully exploits the intrinsic adaptive characteristic of SPM. Combining both, SPM can achieve the goal of adaptive sampling in time. Numerical experiments on the 6-D Allen-Cahn equation and the 7-D Hamiltonian-Jacobi-Bellman equation demonstrate the potential of SPM in solving high-dimensional nonlinear PDEs efficiently while maintaining an acceptable accuracy.

math.NA↗