arXiv subjects

Sihong Shao

Publications and source records attributed to Sihong Shao.

At least 37 records · Page 2Linked to original sources

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↗

arXiv2024

Continuous iterative algorithms for anti-Cheeger cut

As a judicious correspondence to the classical maxcut, the anti-Cheeger cut has more balanced structure, but few numerical results on it have been reported so far. In this paper, we propose a continuous iterative algorithm (CIA) for the anti-Cheeger cut problem through fully using an equivalent continuous formulation. It does not need rounding at all and has advantages that all subproblems have explicit analytic solutions, the objective function values are monotonically updated and the iteration points converge to a local optimum in finite steps via an appropriate subgradient selection. It can also be easily combined with the maxcut iterations for breaking out of local optima and improving the solution quality thanks to the similarity between the anti-Cheeger cut problem and the maxcut problem. The performance of CIAs is fully demonstrated through numerical experiments on G-set from two aspects: one is on the solution quality where we find that the approximate solutions obtained by CIAs are of comparable quality to those by the multiple search operator heuristic method; the other is on the computational cost where we show that CIAs always run faster than the often-used continuous iterative algorithm based on the rank-two relaxation.

math.OC↗

arXiv2024

The Boolean polynomial polytope with multiple choice constraints

We consider a class of $0$-$1$ polynomial programming termed multiple choice polynomial programming (MCPP) where the constraint requires exact one component per subset of the partition to be $1$ after all the entries are partitioned. Compared to the unconstrained counterpart, there are few polyhedral studies of MCPP in general form. This paper serves as the first attempt to propose a polytope associated with a hypergraph to study MCPP, which is the convex hull of $0$-$1$ vectors satisfying multiple choice constraints and production constraints. With the help of the decomposability property, we obtain an explicit half-space representation of the MCPP polytope when the underlying hypergraph is $α$-acyclic by induction on the number of hyperedges, which is an analogy of the acyclicity results on the multilinear polytope by Del Pia and Khajavirad (SIAM J Optim 28 (2018) 1049) when the hypergraph is $γ$-acyclic. We also present a necessary and sufficient condition for the inequalities lifted from the facet-inducing ones for the multilinear polytope to be still facet-inducing for the MCPP polytope. This result covers the particular cases by Bärmann, Martin and Schneider (SIAM J Optim 33 (2023) 2909).

math.OC↗

arXiv2024

A simple inverse power method for balanced graph cut

The existing inverse power ($\mathbf{IP}$) method for solving the balanced graph cut lacks local convergence and its inner subproblem requires a nonsmooth convex solver. To address these issues, we develop a simple inverse power ($\mathbf{SIP}$) method using a novel equivalent continuous formulation of the balanced graph cut, and its inner subproblem allows an explicit analytic solution, which is the biggest advantage over $\mathbf{IP}$ and constitutes the main reason why we call it $\mathit{simple}$. By fully exploiting the closed-form of the inner subproblem solution, we design a boundary-detected subgradient selection with which $\mathbf{SIP}$ is proved to be locally converged. We show that $\mathbf{SIP}$ is also applicable to a new ternary valued $θ$-balanced cut which reduces to the balanced cut when $θ=1$. When $\mathbf{SIP}$ reaches its local optimum, we seamlessly transfer to solve the $θ$-balanced cut within exactly the same iteration algorithm framework and thus obtain $\mathbf{SIP}$-$\mathbf{perturb}$ -- an efficient local breakout improvement of $\mathbf{SIP}$, which transforms some ``partitioned" vertices back to the ``un-partitioned" ones through the adjustable $θ$. Numerical experiments on G-set for Cheeger cut and Sparsest cut demonstrate that $\mathbf{SIP}$ is significantly faster than $\mathbf{IP}$ while maintaining approximate solutions of comparable quality, and $\mathbf{SIP}$-$\mathbf{perturb}$ outperforms $\mathtt{Gurobi}$ in terms of both computational cost and solution quality.

math.OC↗

arXiv2024

Adaptive Hyperbolic-cross-space Mapped Jacobi Method on Unbounded Domains with Applications to Solving Multidimensional Spatiotemporal Integrodifferential Equations

In this paper, we develop a new adaptive hyperbolic-cross-space mapped Jacobi (AHMJ) method for solving multidimensional spatiotemporal integrodifferential equations in unbounded domains. By devising adaptive techniques for sparse mapped Jacobi spectral expansions defined in a hyperbolic cross space, our proposed AHMJ method can efficiently solve various spatiotemporal integrodifferential equations such as the anomalous diffusion model with reduced numbers of basis functions. Our analysis of the AHMJ method gives a uniform upper error bound for solving a class of spatiotemporal integrodifferential equations, leading to effective error control.

math.NA↗

arXiv2024

Overcoming the numerical sign problem in the Wigner dynamics via adaptive particle annihilation

The infamous numerical sign problem poses a fundamental obstacle to particle-based stochastic Wigner simulations in high dimensional phase space. Although the existing particle annihilation via uniform mesh significantly alleviates the sign problem when dimensionality D $\le$ 4, the mesh size grows dramatically when D $\ge$ 6 due to the curse of dimensionality and consequently makes the annihilation very inefficient. In this paper, we propose an adaptive particle annihilation algorithm, termed Sequential-clustering Particle Annihilation via Discrepancy Estimation (SPADE), to overcome the sign problem. SPADE follows a divide-and-conquer strategy: Adaptive clustering of particles via controlling their number-theoretic discrepancies and independent random matching in each group, and it may learn the minimal amount of particles that can accurately capture the non-classicality of the Wigner function. Combining SPADE with the variance reduction technique based on the stationary phase approximation, we attempt to simulate the proton-electron couplings in 6-D and 12-D phase space. A thorough performance benchmark of SPADE is provided with the reference solutions in 6-D phase space produced by a characteristic-spectral-mixed scheme under a $73^3 \times 80^3$ uniform grid, which fully explores the limit of grid-based deterministic Wigner solvers.

physics.comp-ph↗

arXiv2023

An ODE approach to multiple choice polynomial programming

We propose an ODE approach to solving multiple choice polynomial programming (MCPP) after assuming that the optimum point can be approximated by the expected value of so-called thermal equilibrium as usually did in simulated annealing. The explicit form of the feasible region and the affine property of the objective function are both fully exploited in transforming the MCPP problem into the ODE system. We also show theoretically that a local optimum of the former can be obtained from an equilibrium point of the latter. Numerical experiments on two typical combinatorial problems, MAX-$k$-CUT and the calculation of star discrepancy, demonstrate the validity of our ODE approach, and the resulting approximate solutions are of comparable quality to those obtained by the state-of-the-art heuristic algorithms but with much less cost. This paper also serves as the first attempt to use a continuous algorithm for approximating the star discrepancy.

math.OC↗

arXiv2023

A simple iterative algorithm for maxcut

We propose a simple iterative (SI) algorithm for the maxcut problem through fully using an equivalent continuous formulation. It does not need rounding at all and has advantages that all subproblems have explicit analytic solutions, the cut values are monotonically updated and the iteration points converge to a local optima in finite steps via an appropriate subgradient selection. Numerical experiments on G-set demonstrate the performance. In particular, the ratios between the best cut values achieved by SI and those by some advanced combinatorial algorithms in [Ann. Oper. Res. 248 (2017) 365] are at least $0.986$ and can be further improved to at least $0.997$ by a preliminary attempt to break out of local optima.

math.OC↗

arXiv2023

Nonlocalization of singular potentials in quantum dynamics

Nonlocal modeling has drawn more and more attention and becomes steadily more powerful in scientific computing. In this paper, we demonstrate the superiority of a first-principle nonlocal model -- Wigner function -- in treating singular potentials which are often used to model the interaction between point charges in quantum science. The nonlocal nature of the Wigner equation is fully exploited to convert the singular potential into the Wigner kernel with weak or even no singularity, and thus highly accurate numerical approximations are achievable, which are hardly designed when the singular potential is taken into account in the local Schrödinger equation. The Dirac delta function, the logarithmic, and the inverse power potentials are considered. Numerically converged Wigner functions under all these singular potentials are obtained with an operator splitting spectral method, and display many interesting quantum behaviors as well.

quant-ph↗

arXiv2023

Application of Causal Inference Techniques to the Maximum Weight Independent Set Problem

A powerful technique for solving combinatorial optimization problems is to reduce the search space without compromising the solution quality by exploring intrinsic mathematical properties of the problems. For the maximum weight independent set (MWIS) problem, using an upper bound lemma which says the weight of any independent set not contained in the MWIS is bounded from above by the weight of the intersection of its closed neighbor set and the MWIS, we give two extension theorems -- independent set extension theorem and vertex cover extension theorem. With them at our disposal, two types of causal inference techniques (CITs) are proposed on the assumption that a vertex is strongly reducible (included or not included in all MWISs) or reducible (contained or not contained in a MWIS). One is a strongly reducible state-preserving technique, which extends a strongly reducible vertex into a vertex set where all vertices have the same strong reducibility. The other, as a reducible state-preserving technique, extends a reducible vertex into a vertex set with the same reducibility as that vertex and creates some weighted packing constraints to narrow the search space. Numerical experiments show that our CITs can help reduction algorithms find much smaller remaining graphs, improve the ability of exact algorithms to find the optimal solutions and help heuristic algorithms produce approximate solutions of better quality. In particular, detailed tests on $12$ representative graphs generated from datasets in Network Data Repository demonstrate that, compared to the state-of-the-art algorithms, the size of remaining graphs is further reduced by more than 32.6%, and the number of solvable instances is increased from 1 to 5.

math.OC↗

arXiv2023

Non-collinear density functional theory

An approach to generalize any kind of collinear functionals in density functional theory to non-collinear functionals is proposed. This approach, for the very first time, satisfies the correct collinear limit for any kind of functionals, guaranteeing that the exact collinear functional after generalized is still exact for collinear spins. Besides, it has well-defined and numerically stable functional derivatives, a desired feature for non-collinear and spin-flip time-dependent density functional theory. Furthermore, it provides local torque, hinting at its applications in spin dynamics.

quant-ph↗

arXiv2022

A higher-order accurate operator splitting spectral method for the Wigner-Poisson system

An accurate description of 2-D quantum transport in a double-gate metal oxide semiconductor filed effect transistor (dgMOSFET) requires a high-resolution solver to a coupled system of the 4-D Wigner equation and 2-D Poisson equation. In this paper, we propose an operator splitting spectral method to evolve such Wigner-Poisson system in 4-D phase space with high accuracy. After an operator splitting of the Wigner equation, the resulting two sub-equations can be solved analytically with spectral approximation in phase space. Meanwhile, we adopt a Chebyshev spectral method to solve the Poisson equation. Spectral convergence in phase space and a fourth-order accuracy in time are both numerically verified. Finally, we apply the proposed solver into simulating dgMOSFET, develop the steady states from long-time simulations and obtain numerically converged current-voltage (I-V) curves.

physics.comp-ph↗

arXiv2022

Adaptive Hermite Spectral Methods in Unbounded Domains

Recently, new adaptive techniques were developed that greatly improved the efficiency of solving PDEs using spectral methods. These adaptive spectral techniques are especially suited for accurately solving problems in unbounded domains and require the monitoring and dynamic adjustment of three key tunable parameters: the scaling factor, the displacement of the basis functions, and the spectral expansion order. There have been few analyses of numerical methods for unbounded domain problems. Specifically, there is no analysis of adaptive spectral methods to provide insight into how to increase efficiency and accuracy through dynamical adjustment of parameters. In this paper, we perform the first numerical analysis of the adaptive spectral method using generalized Hermite functions in both one- and multi-dimensional problems. Our analysis reveals why adaptive spectral methods work well when a "frequency indicator" of the numerical solution is controlled. We then investigate how the implementation of the adaptive spectral methods affects numerical results, thereby providing guidelines for the proper tuning of parameters. Finally, we further improve performance by extending the adaptive methods to allow bidirectional basis function translation, and the prospect of carrying out similar numerical analysis to solving PDEs arising from realistic difficult-to-solve unbounded models with adaptive spectral methods is also briefly discussed.

math.NA↗

arXiv2022

A characteristic-spectral-mixed scheme for six-dimensional Wigner-Coulomb dynamics

Numerical resolution for 6-D Wigner dynamics under the Coulomb potential faces with the combined challenges of high dimensionality, nonlocality, oscillation and singularity. In particular, the extremely huge memory storage of 6-D grids hinders the usage of all existing deterministic numerical scheme, which is well-known as the curse of dimensionality. To surmount these difficulties, we propose a massively parallel solver, termed the CHAracteristic-Spectral-Mixed (CHASM) scheme, by fully exploiting two distinct features of the Wigner equation: Locality of spatial advection and nonlocality of quantum interaction. Our scheme utilizes the local cubic B-spline basis to interpolate the local spatial advection. The key is to use a perfectly matched boundary condition to give a closure of spline coefficients, so that distributed pieces can recover the global one as accurately as possible owing to the rapid decay of wavelet basis in the dual space, and communication costs are significantly reduced. To resolve the nonlocal pseudodifferential operator with weakly singular symbol, CHASM further adopts the truncated kernel method to attain a highly efficient approximation. Several typical experiments including the quantum harmonic oscillator and Hydrogen 1s state demonstrate the accuracy and efficiency of CHASM. The non-equilibrium electron-proton couplings are also clearly displayed and reveal the uncertainty principle and quantum tunneling in phase space. Finally, the scalability of CHASM up to 16000 cores is presented.

quant-ph↗

arXiv2022

Performance evaluations on the parallel CHAracteristic-Spectral-Mixed (CHASM) scheme

Performance evaluations on the deterministic algorithms for 6-D problems are rarely found in literatures except some recent advances in the Vlasov and Boltzmann community [Dimarco et al. (2018), Kormann et al. (2019)], due to the extremely high complexity. Thus a detailed comparison among various techniques shall be useful to the researchers in the related fields. We try to make a thorough evaluation on a parallel CHAracteristic-Spectral-Mixed (CHASM) scheme to support its usage. CHASM utilizes the cubic B-spline expansion in the spatial space and spectral expansion in the momentum space, which many potentially overcome the computational burden in solving classical and quantum kinetic equations in 6-D phase space. Our purpose is three-pronged. First, we would like show that by imposing some effective Hermite boundary conditions, the local cubic spline can approximate to the global one as accurately as possible. Second, we will illustrate the necessity of adopting the truncated kernel method in calculating the pseudodifferential operator with a singular symbol, since the widely used pseudo-spectral method [Ringhofer (1990)] might fail to properly tackle the singularity. Finally, we make a comparison among non-splitting Lawson schemes and Strang operator splitting. Our numerical results demonstrate the advantage of the one-stage Lawson predictor-corrector scheme over multi-stage ones as well as the splitting scheme in both accuracy and stability.

math.NA↗

arXiv2020

A frequency-dependent $p$-adaptive technique for spectral methods

When using spectral methods, a question arises as how to determine the expansion order, especially for time-dependent problems in which emerging oscillations may require adjusting the expansion order. In this paper, we propose a frequency-dependent $p$-adaptive technique that adaptively adjusts the expansion order based on a frequency indicator. Using this $p$-adaptive technique, combined with recently proposed scaling and moving techniques, we are able to devise an adaptive spectral method in unbounded domains that can capture and handle diffusion, advection, and oscillations. As an application, we use this adaptive spectral method to numerically solve the Schrödinger equation in the whole domain and successfully capture the solution's oscillatory behavior at infinity.

math.NA↗

arXiv2020

Efficient Scaling and Moving Techniques for Spectral Methods in Unbounded Domains

When using Laguerre and Hermite spectral methods to numerically solve PDEs in unbounded domains, the number of collocation points assigned inside the region of interest is often insufficient, particularly when the region is expanded or translated to safely capture the unknown solution. Simply increasing the number of collocation points cannot ensure a fast convergence to spectral accuracy. In this paper, we propose a scaling technique and a moving technique to adaptively cluster enough collocation points in a region of interest in order to achieve a fast spectral convergence. Our scaling algorithm employs an indicator in the frequency domain that is used to determine when scaling is needed and informs the tuning of a scaling factor to redistribute collocation points to adapt to the diffusive behavior of the solution. Our moving technique adopts an exterior-error indicator and moves the collocation points to capture the translation. Both frequency and exterior-error indicators are defined using only the numerical solutions. We apply our methods to a number of different models, including diffusive and moving Fermi-Dirac distributions and nonlinear Dirac solitary waves, and demonstrate recovery of spectral convergence for time-dependent simulations. Performance comparison in solving a linear parabolic problem shows that our frequency scaling algorithm outperforms the existing scaling approaches. We also show our frequency scaling technique is able to track the blowup of average cell sizes in a model for cell proliferation.

math.NA↗