arXiv subjects

Zhenning Cai

Publications and source records attributed to Zhenning Cai.

At least 37 records · Page 2Linked to original sources

arXiv2024

Simulation of Spin Chains with off-diagonal Coupling Using Inchworm Method

We study the dynamical simulation of open quantum spin chain with nearest neighboring coupling, where each spin in the chain is associated with a harmonic bath. This is an extension of our previous work [G. Wang and Z. Cai, J. Chem. Theory Comput., 19, 8523--8540, 2023] by generalizing the application of the inchworm method and the technique of modular path integrals from diagonally coupled cases to off-diagonally coupled cases. Additionally, to reduce computational and memory cost in long time simulation, we apply tensor-train representation to efficiently represent the reduced density matrix of the spin chains, and employ the transfer tensor method (TTM) to avoid exponential growth of computational cost with respect to time. Abundant numerical experiments are performed to validate our method.

quant-ph↗

arXiv2023

Tree-based Implementation of the Small Matrix Path Integral for System-Bath Dynamics

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. The method relies on a sequence of kernel matrices that defines the non-Markovian dynamics of the quantum system. In the original SMatPI method, these kernels are computed indirectly through the QuAPI method. Instead, we focus on the definition of the kernel matrices and reveal a recurrence relation in these matrices. Using such a relationship, a tree based algorithm (t-SMatPI) is developed, which is shown to be much faster than straightforward computation of the kernel matrices based on their definitions. This algorithm bypasses the step to compute the SMatPI matrices by other path integral methods and provides more understanding of the SMatPI matrices themselves. Meanwhile, it keeps the memory cost and computational cost low. Numerical experiments show that the t-SMatPI algorithm gives exactly the same result as i-QuAPI and SMatPI. In spite of this, our method may indicate some new properties of open quantum systems, and has the potential to be generalized to higher-order numerical schemes.

quant-ph↗

arXiv2023

A symmetric Gauss-Seidel method for the steady-state Boltzmann equation

We introduce numerical solvers for the steady-state Boltzmann equation based on the symmetric Gauss-Seidel (SGS) method. Due to the quadratic collision operator in the Boltzmann equation, the SGS method requires solving a nonlinear system on each grid cell, and we consider two methods, namely Newton's method and the fixed-point iteration, in our numerical tests. For small Knudsen numbers, our method has an efficiency between the classical source iteration and the modern generalized synthetic iterative scheme, and the complexity of its implementation is closer to the source iteration. A variety of numerical tests are carried out to demonstrate its performance, and it is concluded that the proposed method is suitable for applications with moderate to large Knudsen numbers.

physics.comp-ph↗

arXiv2023

Some extensions of the $ϕ$-divergence moment closures for the radiative transfer equation

The $ϕ$-divergence-based moment method was recently introduced Abdelmalik et al. (2023) for the discretization of the radiative transfer equation. At the continuous level, this method is very close to the entropy-based MN methods and possesses its main properties, i.e. entropy dissipation, rotational invariance and energy conservation. However, the $ϕ$-divergence based moment systems are easier to resolve numerically due to the improved conditioning of the discrete equations. Moreover, exact quadrature rules can be used to compute moments of the distribution function, which enables the preservation of energy conservation, entropy dissipation and rotational invariants, discretely. In this paper we consider different variants of the $ϕ$-divergence closures that are based on different approximations of the exponential function and the Planck function. We compare the approximation properties of the proposed closures in the numerical benchmarks.

math.NA↗

arXiv2023

Real-Time Simulation of Open Quantum Spin Chains with Inchworm Method

We study the real-time simulation of open quantum systems, where the system is modeled by a spin chain, with each spin associated with its own harmonic bath. Our method couples the inchworm method for the spin-boson model and the modular path integral methodology for spin systems. In particular, the introduction of the inchworm method can significantly suppress the numerical sign problem. Both methods are tweaked to make them work seamlessly with each other. We represent our approach in the language of diagrammatic methods, and analyze the asymptotic behavior of the computational cost. Extensive numerical experiments are done to validate our method.

quant-ph↗

arXiv2023

A positive and moment-preserving Fourier spectral method

This paper presents a novel Fourier spectral method that utilizes optimization techniques to ensure the positivity and conservation of moments in the space of trigonometric polynomials. We rigorously analyze the accuracy of the new method and prove that it maintains spectral accuracy. To solve the optimization problem, we propose an efficient Newton solver that has quadratic convergence rate. Numerical examples are provided to demonstrate the high accuracy of the proposed method. Our method is also integrated into the spectral solver of the Boltzmann equation, showing the benefit of our approach in applications.

math.NA↗

arXiv2023

Moment Methods for the 3D Radiative Transfer Equation Based on $φ$-Divergences

The method of moments is widely used for the reduction of kinetic equations into fluid models. It consists in extracting the moments of the kinetic equation with respect to a velocity variable, but the resulting system is a priori underdetermined and requires a closure relation. In this paper, we adapt the $φ$-divergence based closure, recently developed for rarefied gases i.e. with a velocity variable describing $\mathbb{R}^d$, to the radiative transfer equation where velocity describes the unit sphere $\mathbb{S}^2$. This closure is analyzed and a numerical method to compute it is provided. Eventually, it provides the main desirable properties to the resulting system of moments: Similarily to the entropy minimizing closure ($M_N$), it dissipates an entropy and captures exactly the equilibrium distribution. However, contrarily to $M_N$, it remains computationnally tractable even at high order and it relies on an exact quadrature formula which preserves exactly symmetry properties, i.e. it does not trigger ray effects. The purely anisotropic regimes (beams) are not captured exactly but they can be approached as close as desired and the closures remains again tractable in this limit.

physics.comp-ph↗

arXiv2023

Linear regularized 13-moment equations with Onsager boundary conditions for general gas molecules

We develop the steady-state regularized 13-moment equations in the linear regime for rarefied gas dynamics with general collision models. For small Knudsen numbers, the model is accurate up to the super-Burnett order, and the resulting system of moment equations is shown to have a symmetric structure. We also propose Onsager boundary conditions for the moment equations that guarantees the stability of the equations. The validity of our model is verified by benchmark examples for the one-dimensional channel flows.

physics.flu-dyn↗

arXiv2022

The Bold-Thin-Bold Diagrammatic Monte Carlo Method for Open Quantum Systems

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 Dyson series to accelerate its convergence. The underlying mechanism of the governing equations associated with the two methods lies in the recurrence relation of the path integrals, which is the most costly part in the numerical methods. The proposed algorithms give an extension to the work ["Fast algorithms of bath calculations in simulations of quantum system-bath dynamics", Computer Physics Communications, to appear], where the algorithms are designed based on reusing the previous calculations of bath influence functionals. Compared with the algorithms therein, our methods further include the reuse of system associated functionals and show better performance in terms of computational efficiency and memory cost. We demonstrate the two methods in the framework of spin-boson model, and numerical experiments are carried out to verify the validity of the methods.

quant-ph↗

arXiv2022

Fast Algorithms of Bath Calculations in Simulations of Quantum System-Bath Dynamics

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 design fast algorithms based on reusing the bath influence functionals computed in the previous time steps to reduce the number of calculations. It is proven that the proposed fast algorithms reduce the number of such calculations by a factor of $O(N)$, where $N$ is the total number of time steps. Numerical experiments are carried out to show the efficiency of the method and to verify the theoretical results.

quant-ph↗

arXiv2022

Asymptotic analysis of diabatic surface hopping algorithm in the adiabatic and non-adiabatic limits

Surface hopping algorithms, as an important class of quantum dynamics simulation algorithms for non-adiabatic dynamics, are typically performed in the adiabatic representation, which can break down in the presence of ill-defined adiabatic potential energy surfaces (PESs) and adiabatic coupling term. Another issue of surface hopping algorithms is the difficulty in capturing the correct scaling of the transition rate in the Marcus (weak-coupling/non-adiabatic) regime. Though the first issue can be circumvented by exploiting the diabatic representation, diabatic surface hopping algorithms usually lack justification on the theoretical level. We consider the diabatic surface hopping algorithm proposed in [Fang, Lu. Multiscale Model. Simul. 16:4, 1603-1622, 2018] and provide the asymptotic analysis of the transition rate in the Marcus regime that justifies the correct scaling for the spin-boson model. We propose two conditions that guarantee the correctness for general potentials. In the opposite (strong-coupling/adiabatic) regime, we derive the asymptotic behavior of the algorithm that interestingly matches a type of mean-field description. The techniques used here may shed light on the analysis for other diabatic-based algorithms.

math.NA↗

arXiv2021

Numerical analysis for inchworm Monte Carlo method: Sign problem and error growth

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 method to the classical Dyson series. To better understand the underlying mechanism of the inchworm Monte Carlo method, we distinguish two types of exponential error growth, which are known as the numerical sign problem and the error amplification. The former is due to the fast growth of variance in the stochastic method, which can be observed from the Dyson series, and the latter comes from the evolution of the numerical solution. Our analysis demonstrates that the technique of partial resummation can be considered as a tool to balance these two types of error, and the inchwormMonte Carlo method is a successful case where the numerical sign problem is effectively suppressed by such means. We first demonstrate our idea in the context of ordinary differential equations, and then provide complete analysis for the inchworm Monte Carlo method. Several numerical experiments are carried out to verify our theoretical results.

math.NA↗

arXiv2021

Numerical Solver for the Boltzmann Equation With Self-Adaptive Collision Operators

We use the Burnett spectral method to solve the Boltzmann equation whose collision term is modeled by separate treatments for the low-frequency part and high-frequency part of the solution. For the low-frequency part representing the sketch of the distribution function, the binary collision is applied, while for the high-frequency part representing the finer details, the BGK approximation is applied. The parameter controlling the ratio of the high-frequency part and the low-frequency part is selected adaptively on every grid cell at every time step. This self-adaptation is based on an error indicator describing the difference between the model collision term and the original binary collision term. The indicator is derived by controlling the quadratic terms in the modeling error with linear operators. Our numerical experiments show that such an error indicator is effective and computationally affordable.

math.NA↗

arXiv2021

Regularization of Complex Langevin Method

The complex Langevin method, a numerical method used to compute the ensemble average with a complex partition function, often suffers from runaway instability. We study the regularization of the complex Langevin method via augmenting the action with a stabilization term. Since the regularization introduces biases to the numerical result, two approaches, named 2R and 3R methods, are introduced to recover the unbiased result. The 2R method supplements the regularization with regression to estimate the unregularized ensemble average, and the 3R method reduces the computational cost by coupling the regularization with a reweighting strategy before regression. Both methods can be generalized to the SU(n) theory and are assessed from several perspectives. Several numerical experiments in the lattice field theory are carried out to show the effectiveness of our approaches.

physics.comp-ph↗

arXiv2021

Differential Equation Based Path Integral for System-Bath Dynamics

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 system has infinite equations, we introduce a reasonable closure to obtain a series of finite systems. New numerical schemes can be derived by discretizing these differential equations. It is numerically verified that in certain cases, by selecting appropriate systems and applying suitable numerical schemes, the memory cost required in the i-QuAPI method can be significantly reduced.

quant-ph↗

arXiv2021

An entropic method for discrete systems with Gibbs entropy

We consider general systems of ordinary differential equations with monotonic Gibbs entropy, and introduce an entropic scheme that simply imposes an entropy fix after every time step of any existing time integrator. It is proved that in the general case, our entropy fix has only infinitesimal influence on the numerical order of the original scheme, and in many circumstances, it can be shown that the scheme does not affect the numerical order. Numerical experiments on the linear Fokker-Planck equation and nonlinear Boltzmann equation are carried out to support our numerical analysis.

math.NA↗

arXiv2021

Moment method as a numerical solver: Challenge from shock structure problems

We survey a number of moment hierarchies and test their performances in computing one-dimensional shock structures. It is found that for high Mach numbers, the moment hierarchies are either computationally expensive or hard to converge, making these methods questionable for the simulation of highly non-equilibrium flows. By examining the convergence issue of Grad's moment methods, we propose a new moment hierarchy to bridge the hydrodynamic models and the kinetic equation, allowing nonlinear moment methods to be used as a numerical tool to discretize the velocity space for high-speed flows. For the case of one-dimensional velocity, the method is formulated for odd number of moments, and it can be extended seamlessly to the three-dimensional case. Numerical tests show that the method is capable of predicting shock structures with high Mach numbers accurately, and the results converge to the solution of the Boltzmann equation as the number of moments increases. Some applications beyond the shock structure problem are also considered, indicating that the proposed method is suitable for the computation of transitional flows.

physics.flu-dyn↗

arXiv2021

Inclusion-Exclusion Principle for Open Quantum Systems with Bosonic Bath

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 diagrammatics", Phys. Rev. B, 98:115152, 2018] from fermionic to bosonic systems. We prove that the proposed fast algorithms reduce the computational complexity from double factorial to exponential. Numerical experiments are carried out to verify the theoretical results and to compare the efficiency of the methods.

quant-ph↗