arXiv subjects

Zhennan Zhou

Publications and source records attributed to Zhennan Zhou.

At least 55 records · Page 3Linked to original sources

arXiv2022

Error Analysis of Time-Discrete Random Batch Method for Interacting Particle Systems and Associated Mean-Field Limits

The random batch method provides an efficient algorithm for computing statistical properties of a canonical ensemble of interacting particles. In this work, we study the error estimates of the fully discrete random batch method, especially in terms of approximating the invariant distribution. Using a triangle inequality framework, we show that the long-time error of the method is $O(\sqrtτ + e^{-λt})$, where $τ$ is the time step and $λ$ is the convergence rate which does not depend on the time step $τ$ or the number of particles $N$. Our results also apply to the McKean-Vlasov process, which is the mean-field limit of the interacting particle system as the number of particles $N\rightarrow\infty$.

math.PR↗

arXiv2022

A structure preserving numerical scheme for Fokker-Planck equations of structured neural networks with learning rules

In this work, we are concerned with a Fokker-Planck equation related to the nonlinear noisy leaky integrate-and-fire model for biological neural networks which are structured by the synaptic weights and equipped with the Hebbian learning rule. The equation contains a small parameter $\varepsilon$ separating the time scales of learning and reacting behavior of the neural system, and an asymptotic limit model can be derived by letting $\varepsilon\to 0$, where the microscopic quasi-static states and the macroscopic evolution equation are coupled through the total firing rate. To handle the endowed flux-shift structure and the multi-scale dynamics in a unified framework, we propose a numerical scheme for this equation that is mass conservative, unconditionally positivity preserving, and asymptotic preserving. We provide extensive numerical tests to verify the schemes' properties and carry out a set of numerical experiments to investigate the model's learning ability, and explore the solution's behavior when the neural network is excitatory.

math.NA↗

arXiv2022

Dilating blow-up time: A generalized solution of the NNLIF neuron model and its global well-posedness

The nonlinear noisy leaky integrate-and-fire (NNLIF) model is a popular mean-field description of a large number of interacting neurons, which has attracted mathematicians to study from various aspects. A core property of this model is the finite time blow-up of the firing rate, which scientifically corresponds to the synchronization of a neuron network, and mathematically prevents the existence of a global classical solution. In this work, we propose a new generalized solution based on reformulating the PDE model with a specific change of variable in time. A firing rate dependent timescale is introduced, in which the transformed equation can be shown to be globally well-posed for any connectivity parameter even in the event of the blow-up. The generalized solution is then defined via the backward change of timescale, and it may have a jump when the firing rate blows up. We establish properties of the generalized solution including the characterization of blow-ups and the global well-posedness in the original timescale. The generalized solution provides a new perspective to understand the dynamics when the firing rate blows up as well as the continuation of the solution after a blow-up.

math.AP↗

arXiv2022

Efficient Frozen Gaussian Sampling Algorithms for Nonadiabatic Quantum Dynamics at Metal Surfaces

In this article, we propose a Frozen Gaussian Sampling (FGS) algorithm for simulating nonadiabatic quantum dynamics at metal surfaces with a continuous spectrum. This method consists of a Monte-Carlo algorithm for sampling the initial wave packets on the phase space and a surface-hopping type stochastic time propagation scheme for the wave packets. We prove that to reach a certain accuracy threshold, the sample size required is independent of both the semiclassical parameter $\varepsilon$ and the number of metal orbitals $N$, which makes it one of the most promising methods to study the nonadiabatic dynamics. The algorithm and its convergence properties are also validated numerically. Furthermore, we carry out numerical experiments including exploring the nuclei dynamics, electron transfer and finite-temperature effects, and demonstrate that our method captures the physics which can not be captured by classical surface hopping trajectories.

physics.comp-ph↗

arXiv2022

Ergodicity and long-time behavior of the Random Batch Method for interacting particle systems

We study the geometric ergodicity and the long time behavior of the Random Batch Method for interacting particle systems, which exhibits superior numerical performance in recent large-scale scientific computing experiments. We show that for both the interacting particle system (IPS) and the random batch interacting particle system (RB-IPS), the distribution laws converge to their respective invariant distributions exponentially, and the convergence rate does not depend on the number of particles $N$, the time step $τ$ for batch divisions or the batch size $p$. Moreover, the Wasserstein distance between the invariant distributions of the IPS and the RB-IPS is bounded by $O(\sqrtτ)$, showing that the RB-IPS can be used to sample the invariant distribution of the IPS accurately with greatly reduced computational cost.

math.PR↗

arXiv2022

A simplified voltage-conductance kinetic model for interacting neurons and its asymptotic limit

The voltage-conductance kinetic model for the collective behavior of neurons has been studied by scientists and mathematicians for two decades, but the rigorous analysis of its solution structure has been only partially obtained in spite of plenty of numerical evidence in various scenarios. In this work, we consider a simplified voltage-conductance model in which the velocity field in the voltage variable is in a separable form. The long time behavior of the simplified model is fully investigated leading to the following dichotomy: either the density function converges to the global equilibrium, or the firing rate diverges as time goes to infinity. Besides, the fast conductance asymptotic limit is justified and analyzed, where the solution to the limit model either blows up in finite time, or globally exists leading to time periodic solutions. An important implication of these results is that the non-separable velocity field, or physically the leaky mechanism, is a key element for the emergence of periodic solutions in the original model based on the available numerical evidence.

math.AP↗

arXiv2022

Bounds and long term convergence for the voltage-conductance kinetic system arising in neuroscience

The voltage-conductance equation determines the probability distribution of a stochastic process describing a fluctuation-driven neuronal network arising in the visual cortex. Its structure and degeneracy share many common features with the kinetic Fokker-Planck equation, which has attracted much attention recently. We prove an L $\infty$ bound on the steady state solution and the long term convergence of the evolution problem towards this stationary state. Despite the hypoellipticity property, the difficulty is to treat the boundary conditions that change type along the boundary. This leads us to use specific weights in the Alikakos iterations and adapt the relative entropy method.

math.AP↗

arXiv2022

Exponential convergence to equilibrium for a two-speed model with variant drift fields via the resolvent estimate

We study a two-speed model with variant drift fields, which generalizes the Goldstein-Taylor model but the two advection fields are not necessarily in proportion. Due to the lack of a local equilibrium structure of the steady state, prevailing hypocoercivity methods could not be directly applied to such a system. To prove the exponential convergence to equilibrium for this model, we use a recent generalization of the Gearhart-Prüss theorem [Sci. China Math. 64 (2021), no. 3, 507-518] to gain semigroup bounds, where the relative entropy estimate plays a central role in gaining desired resolvent estimates via a compactness argument.

math.AP↗

arXiv2021

A Mean Field Game Analysis of Consensus Protocol Design

A decentralized blockchain is a distributed ledger that is often used as a platform for exchanging goods and services. This ledger is maintained by a network of nodes that obeys a set of rules, called a consensus protocol, which helps to resolve inconsistencies among local copies of a blockchain. In this paper, we build a mathematical framework for the consensus protocol designer, specifying (a) the measurement of a resource which nodes strategically invest in and compete for to win the right to build new blocks in the blockchain; and (b) a payoff function for such efforts. Thus, the equilibrium of an associated stochastic differential game can be implemented by selecting nodes in proportion to this specified resource and penalizing dishonest nodes by its loss. This associated, induced game can be further analyzed using mean field games. The problem can be broken down into two coupled PDEs, where an individual node's optimal control path is solved using a Hamilton-Jacobi-Bellman equation, and where the evolution of states distribution is characterized by a Fokker-Planck equation. We develop numerical methods to compute the mean field equilibrium for both steady states at the infinite time horizon and evolutionary dynamics. As an example, we show how the mean field equilibrium can be applied to the Bitcoin blockchain mechanism design. We demonstrate that a blockchain can be viewed as a mechanism that operates in a decentralized setup and propagates properties of the mean field equilibrium over time, such as the underlying security of the blockchain.

math.OC↗

arXiv2021

Frozen Gaussian Sampling: A Mesh-free Monte Carlo Method For Approximating Semiclassical Schrödinger Equations

In this paper, we develop a Monte Carlo algorithm named the Frozen Gaussian Sampling (FGS) to solve the semiclassical Schrödinger equation based on the frozen Gaussian approximation. Due to the highly oscillatory structure of the wave function, traditional mesh-based algorithms suffer from "the curse of dimensionality", which gives rise to more severe computational burden when the semiclassical parameter $\ep$ is small. The Frozen Gaussian sampling outperforms the existing algorithms in that it is mesh-free in computing the physical observables and is suitable for high dimensional problems. In this work, we provide detailed procedures to implement the FGS for both Gaussian and WKB initial data cases, where the sampling strategies on the phase space balance the need of variance reduction and sampling convenience. Moreover, we rigorously prove that, to reach a certain accuracy, the number of samples needed for the FGS is independent of the scaling parameter $\ep$. Furthermore, the complexity of the FGS algorithm is of a sublinear scaling with respect to the microscopic degrees of freedom and, in particular, is insensitive to the dimension number. The performance of the FGS is validated through several typical numerical experiments, including simulating scattering by the barrier potential, formation of the caustics and computing the high-dimensional physical observables without mesh.

math.NA↗

arXiv2021

Investigating the integrate and fire model as the limit of a random discharge model: a stochastic analysis perspective

In the mean field integrate-and-fire model, the dynamics of a typical neuron within a large network is modeled as a diffusion-jump stochastic process whose jump takes place once the voltage reaches a threshold. In this work, the main goal is to establish the convergence relationship between the regularized process and the original one where in the regularized process, the jump mechanism is replaced by a Poisson dynamic, and jump intensity within the classically forbidden domain goes to infinity as the regularization parameter vanishes. On the macroscopic level, the Fokker-Planck equation for the process with random discharges (i.e. Poisson jumps) are defined on the whole space, while the equation for the limit process is on the half space. However, with the iteration scheme, the difficulty due to the domain differences has been greatly mitigated and the convergence for the stochastic process and the firing rates can be established. Moreover, we find a polynomial-order convergence for the distribution by a re-normalization argument in probability theory. Finally, by numerical experiments, we quantitatively explore the rate and the asymptotic behavior of the convergence for both linear and nonlinear models.

math.PR↗

arXiv2021

Rigorous justification of the Fokker-Planck equations of neural networks based on an iteration perspective

In this work, the primary goal is to establish rigorous connection between the Fokker-Planck equation of neural networks with its microscopic model: the diffusion-jump stochastic process that captures the mean field behavior of collections of neurons in the integrate-and-fire model. The proof is based on a novel iteration scheme: with an auxiliary random variable counting the firing events, both the density function of the stochastic process and the solution of the PDE problem admit series representations, and thus the difficulty in verifying the link between the density function and the PDE solution in each sub problem has been greatly mitigated. The iteration approach provides a generic frame in integrating the probability approach with PDE techniques, with which we prove that the density function of the diffusion-jump stochastic process is indeed the classical solution of the Fokker-Planck equation with a unique flux-shift structure.

math.AP↗

arXiv2021

Fast Algorithms and Error Analysis of Caputo Derivatives with Small Factional Orders

In this paper, we investigate fast algorithms in the small fraction order regime to approximate the Caputo derivative $^C_0D_t^αu(t)$ when $α$ is small. We focus on two fast algorithms, i.e. FIR and FIDR, both relying on the sum-of-exponential approximation to reduce the cost of evaluating the history part. FIR is the numerical scheme originally proposed in [16], and FIDR is an alternative scheme proposed in [26], and we show that the latter is superior when $α$ is small. With quantitative estimates, we prove that given a certain error threshold, the computational cost of evaluating the history part of the Caputo derivative can be decreased as $α$ gets small. Hence, only minimal cost for the fast evaluation is required in the small $α$ regime, which matches prevailing protocols in engineering practice. We also present improved stability and error analysis of FIDR for solving linear fractional diffusion equations, which achieves clear dependence of the error bound on the fraction order $α$. Finally, we carry out systematic numerical studies for the performances of both FIR and FIDR schemes, where we explore the trade-off between accuracy and efficiency when $α$ is small.

math.NA↗

arXiv2021

A novel spectral method for the semi-classical Schrödinger equation based on the Gaussian wave-packet transform

In this article, we develop and analyse a new spectral method to solve the semi-classical Schrödinger equation based on the Gaussian wave-packet transform (GWPT) and Hagedorn's semi-classical wave-packets (HWP). The GWPT equivalently recasts the highly oscillatory wave equation as a much less oscillatory one (the $w$ equation) coupled with a set of ordinary differential equations governing the dynamics of the so-called GWPT parameters. The Hamiltonian of the $ w $ equation consists of a quadratic part and a small non-quadratic perturbation, which is of order $ \mathcal{O}(\sqrt{\varepsilon }) $, where $ \varepsilon\ll 1 $ is the rescaled Planck's constant. By expanding the solution of the $ w $ equation as a superposition of Hagedorn's wave-packets, we construct a spectral method while the $ \mathcal{O}(\sqrt{\varepsilon}) $ perturbation part is treated by the Galerkin approximation. This numerical implementation of the GWPT avoids imposing artificial boundary conditions and facilitates rigorous numerical analysis. For arbitrary dimensional cases, we establish how the error of solving the semi-classical Schrödinger equation with the GWPT is determined by the errors of solving the $ w $ equation and the GWPT parameters. We prove that this scheme has the spectral convergence with respect to the number of Hagedorn's wave-packets in one dimension. Extensive numerical tests are provided to demonstrate the properties of the proposed method.

math.NA↗

arXiv2021

A tumor growth model with autophagy: the reaction-(cross-)diffusion system and its free boundary limit

In this paper, we propose a tumor growth model to incorporate and investigate the spatial effects of autophagy. The cells are classified into two phases: normal cells and autophagic cells, whose dynamics are also coupled with the nutrients. First, we construct a reaction-(cross-)diffusion system describing the evolution of cell densities, where the drift is determined by the negative gradient of the joint pressure, and the reaction terms manifest the unique mechanism of autophagy. Next, in the incompressible limit, such a cell density model naturally connects to a free boundary system, describing the geometric motion of the tumor region. Analyzing the free boundary model in a special case, we show that the ratio of the two phases of cells exponentially converges to a ``well-mixed" limit. Within this ``well-mixed" limit, we obtain an analytical solution of the free boundary system which indicates the exponential growth of the tumor size in the presence of autophagy in contrast to the linear growth without it. Numerical simulations are also provided to illustrate the analytical properties and to explore more scenarios.

math.AP↗

arXiv2021

Efficient Sampling of Thermal Averages of Interacting Quantum Particle Systems with Random Batches

An efficient sampling method, the pmmLang+RBM, is proposed to compute the quantum thermal average in the interacting quantum particle system. Benefiting from the random batch method (RBM), the pmmLang+RBM reduces the complexity due to the interaction forces per timestep from $O(NP^2)$ to $O(NP)$, where $N$ is the number of beads and $P$ is the number of particles. Although the RBM introduces a random perturbation of the interaction forces at each timestep, the long time effects of the random perturbations along the sampling process only result in a small bias in the empirical measure of the pmmLang+RBM from the target distribution, which also implies a small error in the thermal average calculation. We numerically study the convergence of the pmmLang+RBM, and quantitatively investigate the dependence of the error in computing the thermal average on the parameters including the batch size, the timestep, etc. We also propose an extension of the pmmLang+RBM, which is based on the splitting Monte Carlo method and is applicable when the interacting potential contains a singular part.

quant-ph↗

arXiv2021

Investigating the integrate and fire model as the limit of a random discharge model: a stochastic analysis perspective

math.PR↗

arXiv2020

Multi-level Monte Carlo path integral molecular dynamics for thermal average calculation in the nonadiabatic regime

With the path integral approach, the thermal average in a multi-electronic-state quantum systems can be approximated by the ring polymer representation on an extended configuration space, where the additional degrees of freedom are associated with the surface index of each bead. The primary goal of this work is to propose a more efficient sampling algorithm for the calculation of such thermal averages. We reformulate the extended ring polymer approximation according to the configurations of the surface indexes, and by introducing a proper reference measure, the reformulation is recast as a ratio of two expectations of function expansions. By quantitatively estimating the sub-estimators, and minimizing the total variance of the sampled average, we propose a multi-level Monte Carlo path integral molecular dynamics method (MLMC-PIMD) to achieve an optimal balance of computational cost and accuracy.

math.NA↗