最新科研成果

Efficient scaling and moving techniques for spectral methods in unbounded domains

Xia M, Shao S, Chou T. Efficient scaling and moving techniques for spectral methods in unbounded domains. SIAM Journal on Scientific Computing [Internet]. 2021;43(5):A3244–A3268. 访问链接Abstract
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 in order 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 fast spectral convergence. Our scaling algorithm employs an indicator in the frequency domain that both is used to determine when scaling is needed and informs the tuning of a scaling factor to redistribute collocation points in order 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. A 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. In addition to the Laguerre and Hermite basis functions with exponential decay at infinity, we also successfully apply the frequency-dependent scaling technique into rational basis functions with algebraic decay at infinity.

Nonsmooth critical point theory and applications to the spectral graph theory

Chang K-C, Shao S, Zhang D, Zhang W. Nonsmooth critical point theory and applications to the spectral graph theory. SCIENCE CHINA Mathematics [Internet]. 2021;64(1):1-32. 访问链接Abstract
Existing critical point theories including metric and topological critical point theories are difficult to be applied directly to some concrete problems in particular polyhedral settings, because the notions of critical sets could be either very vague or too large. To overcome these difficulties, we develop critical point theory for nonsmooth but Lipschitzian functions defined on convex polyhedrons. This yields natural extensions of classical results in critical point theory, such as the Liusternik-Schnirelmann multiplicity theorem. More importantly, eigenvectors for some eigenvalue problems involving graph 1-Laplacian coincide with critical points of the corresponding functions on polytopes, which indicates that the critical point theory proposed in the present paper can be applied to study the nonlinear spectral graph theory.

Branching random walk solutions to the Wigner equation

Shao S, Xiong Y. Branching random walk solutions to the Wigner equation. SIAM Journal on Numerical Analysis [Internet]. 2020;58(5):2589-2608. 访问链接Abstract
The stochastic solutions to the Wigner equation, which explain the nonlocal oscillatory integral operator $\Theta_V$ with an anti-symmetric kernel as the generator of two branches of jump processes,  are analyzed. All existing branching random walk solutions are formulated based on the Hahn-Jordan decomposition $\Theta_V=\Theta^+_V-\Theta^-_V$, i.e., treating $\Theta_V$ as the difference of two positive operators $\Theta^\pm_V$, each of which characterizes the transition of states for one branch of particles. Despite the fact that the first moments of such models solve the Wigner equation, we prove that the bounds of corresponding variances grow exponentially in time with the rate depending on the upper bound of $\Theta^\pm_V$, instead of $\Theta_V$. In other words, the decay of high-frequency components is totally ignored, resulting in a severe numerical sign problem. To fully utilize such decay property, we have recourse to the stationary phase approximation for $\Theta_V$, which captures essential contributions from the stationary phase points as well as the near-cancelation of positive and negative weights. The resulting branching random walk solutions are then proved to asymptotically solve the Wigner equation, but gain a substantial reduction in variances, thereby ameliorating the sign problem. Numerical experiments in 4-D phase space validate our theoretical findings. 

A branching random walk method for many-body Wigner quantum dynamics

Shao S, Xiong Y. A branching random walk method for many-body Wigner quantum dynamics. Numerical Mathematics: Theory, Methods and Applications [Internet]. 2019;12(1):21-71. 访问链接Abstract
A branching random walk algorithm for many-body Wigner equations and its numerical applications for quantum dynamics in phase space are proposed and analyzed in this paper. Using an auxiliary function, the truncated Wigner equation and its adjoint form are cast into integral formulations, which can be then reformulated into renewal-type equations with probabilistic interpretations. We prove that the first moment of a branching random walk is the solution for the adjoint equation. With the help of the additional degree of freedom offered by the auxiliary function, we are able to produce a weighted-particle implementation of the branching random walk. In contrast to existing signed-particle implementations, this weighted-particle one shows a key capacity of variance reduction by increasing the constant auxiliary function and has no time discretization errors. Several canonical numerical experiments on the 2D Gaussian barrier scattering and a 4D Helium-like system validate our theoretical findings, and demonstrate the accuracy, the efficiency, and thus the computability of the proposed weighted-particle Wigner branching random walk algorithm.

A high order efficient numerical method for 4-D Wigner equation of quantum double-slit interferences

Chen Z, Shao S, Cai W. A high order efficient numerical method for 4-D Wigner equation of quantum double-slit interferences. Journal of Computational Physics [Internet]. 2019;396:54-71. 访问链接Abstract
We propose a high order numerical method for computing time dependent 4-D Wigner equation with unbounded potential and study a canonical quantum double-slit interference problem. To address the difficulties of 4-D phase space computations and higher derivatives from the Moyal expansion of nonlocal pseudo-differential operator for unbounded potentials, an operator splitting technique is adopted to decompose the 4-D Wigner equation into two sub-equations, which can be computed analytically or numerically with high efficiency. The first sub-equation contains only linear convection term in $(\bm x, t)$-space and can be solved with an advective method, while the second involves the pseudo-differential term and can be approximated by a plane wave expansion in $\bm k$-space. By exploiting properties of Fourier transformation,  the expansion coefficients for the second sub-equation have explicit forms and the resulting scheme is shown to be unconditionally stable for any higher derivatives of the Moyal expansion, ensuring the feasibility of the 4-D Wigner numerical simulations for quantum double-slit interferences. Numerical experiments demonstrate the spectral convergence in $(\bm x, \bm k)$-space and provide highly accurate information on the number, position, and intensity of the interference fringes for different types of slits, quantum particle masses, and initial states (pure and mixed).

Nodal domains of eigenvectors for 1-Laplacian on graphs

Chang K-C, Shao S, Zhang D. Nodal domains of eigenvectors for 1-Laplacian on graphs. Advances in Mathematics [Internet]. 2017;308:529-574. 访问链接Abstract
The eigenvectors for graph 1-Laplacian possess some sort of localization property: On one hand, the characteristic function on any nodal domain of an eigenvector is again an eigenvector with the same eigenvalue; on the other hand, one can pack up an eigenvector for a new graph by several fundamental eigencomponents and modules with the same eigenvalue via few special techniques. The Courant nodal domain theorem for graphs is extended to graph 1-Laplacian for strong nodal domains, but for weak nodal domains it is false. The notion of algebraic multiplicity is introduced in order to provide a more precise estimate of the number of independent eigenvectors. A positive answer is given to a question raised in Chang (2016) [3], to confirm that the critical values obtained by the minimax principle may not cover all eigenvalues of graph 1-Laplacian.

An advective-spectral-mixed method for time-dependent many-body Wigner simulations

Xiong Y, Chen Z, Shao S. An advective-spectral-mixed method for time-dependent many-body Wigner simulations. SIAM Journal on Scientific Computing [Internet]. 2016;38(4):B491–B520. 访问链接Abstract
As a phase space language for quantum mechanics, the Wigner function approach bears a close analogy to classical mechanics and has been drawing growing attention, especially in simulating quantum many-body systems. However, deterministic numerical solutions have been almost exclusively confined to one-dimensional one-body systems and few results are reported even for one-dimensional two-body problems. This paper serves as the first attempt to solve the time-dependent many-body Wigner equation through a grid-based advective-spectral-mixed method. The main feature of the method is to resolve the linear advection in $(\bm{x},t)$-space by an explicit three-step characteristic scheme coupled with the piecewise cubic spline interpolation, while the Chebyshev spectral element method in $\bm k$-space is adopted for accurate calculation of the nonlocal pseudo-differential term. Not only the time step of the resulting method is not restricted by the usual CFL condition and thus a large time step is allowed, but also the mass conservation can be maintained. In particular, for the system consisting of identical particles,  the advective-spectral-mixed method can also rigorously preserve physical symmetry relations. The performance is validated through several typical numerical experiments, like the Gaussian barrier scattering, electron-electron interaction and a Helium-like system, where the third-order accuracy against both grid spacing and time stepping is observed.  

Comparison of deterministic and stochastic methods for time-dependent Wigner simulations

Shao S, Sellier JM. Comparison of deterministic and stochastic methods for time-dependent Wigner simulations. Journal of Computational Physics [Internet]. 2015;300:167-185. 访问链接Abstract
Recently a Monte Carlo method based on signed particles for time-dependent simulations of the Wigner equation has been proposed. While it has been thoroughly validated against physical benchmarks, no technical study about its numerical accuracy has been performed. To this end, this paper presents the first step towards the construction of firm mathematical foundations for the signed particle Wigner Monte Carlo method. An initial investigation is performed by means of comparisons with a cell average spectral element method, which is a highly accurate deterministic method and utilized to provide reference solutions. Several different numerical tests involving the time-dependent evolution of a quantum wave-packet are performed and discussed in deep details. In particular, this allows us to depict a set of crucial criteria for the signed particle Wigner Monte Carlo method to achieve a satisfactory accuracy.  
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