# Zhenning Cai's research while affiliated with National University of Singapore and other places

## Publications (32)

Article
We present fast algorithms for the summation of Dyson series and the inchworm Monte Carlo method for quantum systems that are coupled with harmonic baths. The algorithms are based on evolving the integro-differential equations where the most expensive part comes from the computation of bath influence functionals. To accelerate the computation, we d...
Preprint
The small matrix path integral (SMatPI) method is an efficient numerical approach to simulate the evolution of a quantum system coupled to a harmonic bath. We focus on the computation of SMatPI matrices and perform some rigorous study of its computational cost. Finding that the computational cost for each path scales exponentially with the memory l...
Preprint
We present two diagrammatic Monte Carlo methods for quantum systems coupled with harmonic baths, whose dynamics are described by integro-differential equations. The first approach can be considered as a reformulation of Dyson series, and the second one, called "bold-thin-bold diagrammatic Monte Carlo", is based on resummation of the diagrams in the...
Preprint
We present fast algorithms for the summation of Dyson series and the inchworm Monte Carlo method for quantum systems that are coupled with harmonic baths. The algorithms are based on evolving the integro-differential equations where the most expensive part comes from the computation of bath influence functionals. To accelerate the computation, we d...
Article
In this paper, the ground state Wigner function of a many-body system is explored theoretically and numerically. First, an eigenvalue problem for Wigner function is derived based on the energy operator of the system. The validity of finding the ground state through solving this eigenvalue problem is obtained by building a correspondence between its...
Preprint
We propose the differential equation based path integral (DEBPI) method to simulate the real-time evolution of open quantum systems. In this method, a system of partial differential equations is derived based on the continuation of a classical numerical method called iterative quasi-adiabatic propagator path integral (i-QuAPI). While the resulting...
Article
Full-text available
We present two fast algorithms which apply inclusion–exclusion principle to sum over the bosonic diagrams in bare diagrammatic quantum Monte Carlo and inchworm Monte Carlo method, respectively. In the case of inchworm Monte Carlo, the proposed fast algorithm gives an extension to the work [2018 Inclusion–exclusion principle for many-body diagrammat...
Preprint
Full-text available
In this paper, the ground state Wigner function of a many-body system is explored theoretically and numerically. First, an eigenvalue problem for Wigner function is derived based on the energy operator of the system. The validity of finding the ground state through solving this eigenvalue problem is obtained by building a correspondence between its...
Preprint
We present two fast algorithms which apply inclusion-exclusion principle to sum over the bosonic diagrams in bare diagrammatic quantum Monte Carlo (dQMC) and inchworm Monte Carlo method, respectively. In the case of inchworm Monte Carlo, the proposed fast algorithm gives an extension to the work ["Inclusion-exclusion principle for many-body diagram...
Preprint
We solve the Boltzmann equation whose collision term is modeled by the hybridization of the binary collision and the BGK approximation. The parameter controlling the ratio of these two collision mechanisms is selected adaptively on every grid cell at every time step. This self-adaptation is based on a heuristic error indicator describing the differ...
Article
We study the structure of stationary channel flows predicted by the regularized 13-moment equations. Compared with the work of Taheri et al. [“Couette and Poiseuille microflows: Analytical solutions for regularized 13-moment equations,” Phys. Fluids 21, 017102 (2009)], we focus on gases whose molecules satisfy the general inverse power law. The ana...
Preprint
We study the structure of stationary channel flows predicted by the regularized 13-moment equations. Compared with the previous work [P. Taheri et al., Phys. Fluids, 21 (2009), 017102], we focus on gases whose molecules satisfy the general inverse power law. The analytical solutions are obtained for the semi-linear equations, and the structures of...
Article
Regularized 13-moment equations for inverse power law models - Volume 894 - Zhenning Cai, Yanli Wang
Preprint
We consider the numerical analysis of the inchworm Monte Carlo method, which is proposed recently to tackle the numerical sign problem for open quantum systems. We focus on the growth of the numerical error with respect to the simulation time, for which the inchworm Monte Carlo method shows a flatter curve than the direct application of Monte Carlo...
Article
We develop a spectral method for the spatially homogeneous Boltzmann equation using Burnett polynomials in the basis functions. Using the sparsity of the coefficients in the expansion of the collision term, we reduce the computational cost by one order of magnitude for general collision kernels and by two orders of magnitude for Maxwell molecules....
Article
We investigate in this work a recently proposed diagrammatic quantum Monte Carlo method—the inchworm Monte Carlo method—for open quantum systems. We establish its validity rigorously based on resummation of Dyson series. Moreover, we introduce an integro‐differential equation formulation for open quantum systems, which illuminates the mathematical...
Preprint
We introduce a numerical solver for the spatially inhomogeneous Boltzmann equation using the Burnett spectral method. The modelling and discretization of the collision operator are based on the previous work [Z. Cai, Y. Fan, and Y. Wang, Burnett spectral method for the spatially homogeneous Boltzmann equation, arXiv:1810.07804], which is the hybrid...
Preprint
We propose a systematic methodology to derive the regularized thirteen-moment equations in the rarefied gas dynamics for a general class of linearized collision models. Detailed expressions of the moment equations are written down for all inverse power law models as well as the hard-sphere model. By linear analysis, we show that the equations are s...
Preprint
We investigate in this work a recently proposed diagrammatic quantum Monte Carlo method --- the inchworm Monte Carlo method --- for open quantum systems. We establish its validity rigorously based on resummation of Dyson series. Moreover, we introduce an integro-differential equation formulation for open quantum systems, which illuminates the mathe...
Preprint
Full-text available
We develop a spectral method for the spatially homogeneous Boltzmann equation using Burnett polynomials in the basis functions. Using the sparsity of the coefficients in the expansion of the collision term, the computational cost is reduced by one order of magnitude for general collision kernels and by two orders of magnitude for Maxwell molecules....
Preprint
Full-text available
We propose a Hermite spectral method for the spatially inhomogeneous Boltzmann equation. For the inverse-power-law model, we generalize an approximate quadratic collision operator defined in the normalized and dimensionless setting to an operator for arbitrary distribution functions. An efficient algorithm with a fast transform is introduced to dis...
Article
Based on the Hermite expansion of the distribution function, we introduce a Galerkin spectral method for the spatially homogeneous Boltzmann equation with the realistic inverse- power-law models. A practical algorithm is proposed to evaluate the coefficients in the spectral method with high accuracy, and these coefficients are also used to construc...
Article
In this work, we derive a discrete analog of the Wigner transform over the space $(\mathbb{C}^p)^{\otimes N}$ for any prime $p$ and any positive integer $N$. We show that the Wigner transform over this space can be constructed as the inverse Fourier transform of the standard Pauli matrices for $p=2$ or more generally of the Heisenberg-Weyl group el...
Article
Full-text available
We propose an entropic Fourier method for the numerical discretization of the Boltzmann collision operator. The method, which is obtained by modifying a Fourier Galerkin method to match the form of the discrete velocity method, can be viewed both as a discrete velocity method and as a Fourier method. As a discrete velocity method, it preserves the...
Article
Full-text available
We study the unphysical recurrence phenomenon arising in the numerical simulation of the transport equations using Hermite-spectral method. From a mathematical point of view, the suppression of this numerical artifact with filters is theoretically analyzed for two types of transport equations. It is rigorously proven that all the non-constant modes...
Article
Full-text available
We propose a general framework of quantum kinetic Monte Carlo algorithm, based on a stochastic representation of a series expansion of the quantum evolution. Two approaches have been developed in the context of quantum many-body spin dynamics, using different decomposition of the Hamiltonian. The effectiveness of the methods is tested for many-body...
Article
Full-text available
We propose a spectral method that discretizes the Boltzmann collision operator and satisfies a discrete version of the H-theorem. The method is obtained by modifying the existing Fourier spectral method to match a classical form of the discrete velocity method. It preserves the positivity of the solution on the Fourier collocation points and as a r...
Article
Full-text available
We propose a surface hopping Gaussian beam method to numerically solve a class of high frequency linear transport systems in high spatial dimensions, based on asymptotic analysis. The stochastic surface hopping is combined with Gaussian beam method to deal with the multiple characteristic directions of the transport system in high dimensions. The M...

## Citations

... However, the i-QuAPI method still has to record the contribution of all the paths within the memory length, leading to prohibitively large memory cost when the time nonlocality is long. Some improvements have been developed in the past decades to reduce the memory cost, among which the blip-summed decomposition [17,18] reduces the memory cost by ignoring the paths with small contributions, and the differential equation based path integral (DEBPI) studies the continuous form of i-QuAPI and formulates a differential equation system [30]. Recently, the small matrix decomposition of the path integral (SMatPI) [19,20,21,22] successfully overcomes the exponential scaling of the memory cost while preserving the accuracy of i-QuAPI. ...
... For real-time simulations, Monte Carlo based methods suffer from numerical sign problem [12,3]. Different techniques are developed recently to relieve the numerical sign problem and accelerate the computation [26,7,2,32,4,5]. ...
... (1. 6) with µ ij = m (i) m (j) m (i) +m (j) , Γ the Gamma function, ω ij the viscosity index, and d ref,ij , T ref,ij are the reference diameter and temperature. Particularly, we could obtain the variable hard sphere model (VHS) when α ij = 1 and 0.5 ω ij 1 (especially, the Maxwell molecules and hard sphere model (HS) are corresponding to ω = 1 and ω = 0.5 respectively); and the VSS model when 1 < α ij 2 and 0.5 ω ij 1. ...
... Indeed, it has already taken over thirty years to develop efficient Wigner solvers, including both deterministic and stochastic algorithms. In contrast to the relatively newer branch of particle-based stochastic methods, [12][13][14] which usually exhibit slower convergence rate, grid-based deterministic solvers allow highly accurate numerical resolutions in the light of their concise principle and solid mathematical foundation, ranging from the finite difference scheme 15 and the spectral collocation method combined with the operator splitting 16,17 to the recent advanced techniques such as the spectral element method, 18-20 the spectral decomposition 21 and the Hermite spectral method, 22,23 as well as those for advection such as the discontinuous Galerkin method, 24 WENO scheme 25 and exponential integrators. 22 Unfortunately, there still remains a huge gap in terms of the applicability of even the state-of-the-art deterministic scheme to full 6-D problems, and the foremost problem is definitely the storage of 6-D grid mesh. ...
... For real-time simulations, Monte Carlo based methods suffer from numerical sign problem [12,3]. Different techniques are developed recently to relieve the numerical sign problem and accelerate the computation [26,7,2,32,4,5]. ...
... Many of the solutions which we derive refer to a flow with constant velocity and pressure. This is obviously quite a restrictive condition, but does correspond to some physically realistic scenarios of current interest (for example, Fourier flow, where one has a static gas between two stationary parallel plates and the flow structure as it exists is due only to the temperature difference between the plates) [29]. We will refer to these types of flow as heat flows to emphasise that the medium is not moving. ...
... For the general choice of ̟ and ϑ, the macroscopic physical quantities of interest can also be related directly to these two parameters and the low-order coefficients, i.e., f α with |α| ≤ 3. We refer to [21] for the detailed relations in this general case. Additionally, as pointed out in [20], the convergence of the series (9) could usually be expected for a smooth distribution function if θ < 2ϑ. Due to the importance of the Grad method, we will restrict ourselves to the case of ̟ = u and ϑ = θ for the approximation of every distribution function in this paper. ...
... First, the moment equations proposed by Grad [14] are regarded as an efficient reduction model [7] of the Boltzmann equation. As an extension of the celebrated Navier-Stokes equations, the moment equations have gained much more attention in recent years [31,6,18,9]. The half-space problem 2 Basic Equations and Main Results ...
... For real-time simulations, Monte Carlo based methods suffer from numerical sign problem [12,3]. Different techniques are developed recently to relieve the numerical sign problem and accelerate the computation [26,7,2,32,4,5]. ...
... Evolution for g ROM (t, v) is given by (14), with the projection matrix H kj and arrayŝ A k ,k ,k and B k ,k corresponding to the ZCMP basis. Following the ideas of [17,56,18], evolution of g ⊥ (t, v) is simply an exponential decay ...