arXiv subjects

Zhennan Zhou

Publications and source records attributed to Zhennan Zhou.

At least 37 records · Page 2Linked to original sources

arXiv2024

Dimension-free Ergodicity of Path Integral Molecular Dynamics

The quantum thermal average plays a central role in describing the thermodynamic properties of a quantum system. Path integral molecular dynamics (PIMD) is a prevailing approach for computing quantum thermal averages by approximating the quantum partition function as a classical isomorphism on an augmented space, enabling efficient classical sampling, but the theoretical knowledge of the ergodicity of the sampling is lacking. Parallel to the standard PIMD with $N$ ring polymer beads, we also study the Matsubara mode PIMD, where the ring polymer is replaced by a continuous loop composed of $N$ Matsubara modes. Utilizing the generalized $Γ$ calculus, we prove that both the Matsubara mode PIMD and the standard PIMD have uniform-in-$N$ ergodicity, i.e., the convergence rate towards the invariant distribution does not depend on the number of modes or beads $N$.

math.NA↗

arXiv2024

Statistical Error of Numerical Integrators for Underdamped Langevin Dynamics with Deterministic And Stochastic Gradients

We propose a novel discrete Poisson equation approach to estimate the statistical error of a broad class of numerical integrators for the underdamped Langevin dynamics. The statistical error refers to the mean square error of the estimator to the exact ensemble average with a finite number of iterations. With the proposed error analysis framework, we show that when the potential function $U(x)$ is strongly convex in $\mathbb R^d$ and the numerical integrator has strong order $p$, the statistical error is $O(h^{2p}+\frac1{Nh})$, where $h$ is the time step and $N$ is the number of iterations. Besides, this approach can be adopted to analyze integrators with stochastic gradients, and quantitative estimates can be derived as well. Our approach only requires the geometric ergodicity of the continuous-time underdamped Langevin dynamics, and relaxes the constraint on the time step.

math.NA↗

arXiv2024

Efficient bound preserving and asymptotic preserving semi-implicit schemes for the fast reaction-diffusion system

We consider a special type of fast reaction-diffusion systems in which the coefficients of the reaction terms of the two substances are much larger than those of the diffusion terms while the diffusive motion to the substrate is negligible. Specifically speaking, the rate constants of the reaction terms are $O(1/ε)$ while the diffusion coefficients are $O(1)$ where the parameter $ε$ is small. When the rate constants of the reaction terms become highly large, i.e. $ε$ tends to 0, the singular limit behavior of such a fast reaction-diffusion system is inscribed by the Stefan problem with latent heat, which brings great challenges in numerical simulations. In this paper, we adopt a semi-implicit scheme, which is first-order accurate in time and can accurately approximate the interface propagation even when the reaction becomes extremely fast, that is to say, the parameter $ε$ is sufficiently small. The scheme satisfies the positivity, bound preserving properties and has $L^2$ stability and the linearized stability results of the system. For better performance on numerical simulations, we then construct a semi-implicit Runge-Kutta scheme which is second-order accurate in time. Numerous numerical tests are carried out to demonstrate the properties, such as the order of accuracy, positivity and bound preserving, the capturing of the sharp interface with various $ε$ and to simulate the dynamics of the substances and the substrate, and to explore the heat transfer process, such as solid melting or liquid solidification in two dimensions.

math.NA↗

arXiv2024

Classical solutions of a mean field system for pulse-coupled oscillators: long time asymptotics versus blowup

We introduce a novel reformulation of the mean-field system for pulse-coupled oscillators. It is based on writing a closed equation for the inverse distribution function associated to the probability density of oscillators with a given phase in a suitable time scale. This new framework allows to show a hidden contraction/expansion of certain distances leading to a full clarification of the long-time behavior, existence of steady states, rates of convergence, and finite time blow-up of classical solutions for a large class of monotone phase response functions. In the process, we get insights about the origin of obstructions to global-in-time existence and uniform in time estimates on the firing rate of the oscillators.

math.AP↗

arXiv2024

A unified Bayesian inversion approach for a class of tumor growth models with different pressure laws

In this paper, we use the Bayesian inversion approach to study the data assimilation problem for a family of tumor growth models described by porous-medium type equations. The models contain uncertain parameters and are indexed by a physical parameter $m$, which characterizes the constitutive relation between density and pressure. Based on these models, we employ the Bayesian inversion framework to infer parametric and nonparametric unknowns that affect tumor growth from noisy observations of tumor cell density. We establish the well-posedness and the stability theories for the Bayesian inversion problem and further prove the convergence of the posterior distribution in the so-called incompressible limit, $m \rightarrow \infty$. Since the posterior distribution across the index regime $m\in[2,\infty)$ can thus be treated in a unified manner, such theoretical results also guide the design of the numerical inference for the unknown. We propose a generic computational framework for such inverse problems, which consists of a typical sampling algorithm and an asymptotic preserving solver for the forward problem. With extensive numerical tests, we demonstrate that the proposed method achieves satisfactory accuracy in the Bayesian inference of the tumor growth models, which is uniform with respect to the constitutive relation.

math.NA↗

arXiv2024

Uncertainty Quantification of Phase Transition Problems with an Injection Boundary

We develop an enthalpy-based modeling and computational framework to quantify uncertainty in Stefan problems with an injection boundary. Inspired by airfoil icing studies, we consider a system featuring an injection boundary inducing domain changes and a free boundary separating phases, resulting in two types of moving boundaries. Our proposed enthalpy-based formulation seamlessly integrates thermal diffusion across the domain with energy fluxes at the boundaries, addressing a modified injection condition for boundary movement. Uncertainty then stems from random variations in the injection boundary. The primary focus of our Uncertainty Quantification (UQ) centers on investigating the effects of uncertainty on free boundary propagation. Through mapping to a reference domain, we derive an enthalpy-based numerical scheme tailored to the transformed coordinate system, facilitating a simple and efficient simulation. Numerical and UQ studies in one and two dimensions validate the proposed model and the extended enthalpy method. They offer intriguing insights into ice accretion and other multiphysics processes involving phase transitions.

math.NA↗

arXiv2023

Asymptotic analysis for Bloch electrons with Weyl nodes

In this paper, we study the semiclassical behavior of Bloch electrons in the presence of Weyl nodes, which are singular points in the band structure of certain materials. We carry out asymptotic analysis and present a rigorous derivation of the semiclassical asymptotic expansion of the current of Bloch electrons with the presence of Weyl nodes. The analysis shows that the current contains two parts, one independent of the Weyl nodes and the other a contribution from the singular points. This work provides a theoretical foundation towards a rigorous justification of recent scientific discoveries in Weyl semimetals. The main innovation of this paper is a new strategy to deal with the singular points with quantitative estimates, which may have broader applications in multiscale models with singularities.

math.AP↗

arXiv2023

Prompt Engineering Through the Lens of Optimal Control

Prompt Engineering (PE) has emerged as a critical technique for guiding Large Language Models (LLMs) in solving intricate tasks. Its importance is highlighted by its potential to significantly enhance the efficiency and effectiveness of human-machine interaction. As tasks grow increasingly complex, recent advanced PE methods have extended beyond the limitations of single-round interactions to embrace multi-round interactions, which allows for a deeper and more nuanced engagement with LLMs. In this paper, we propose an optimal control framework tailored for multi-round interactions with LLMs. This framework provides a unified mathematical structure that not only systematizes the existing PE methods but also sets the stage for rigorous analytical improvements. Furthermore, we extend this framework to include PE via ensemble methods and multi-agent collaboration, thereby enlarging the scope of applicability. By adopting an optimal control perspective, we offer fresh insights into existing PE methods and highlight theoretical challenges that warrant future research. Besides, our work lays a foundation for the development of more effective and interpretable PE methods.

cs.LG↗

arXiv2023

Tumor growth with a necrotic core as an obstacle problem in pressure

Motivated by the incompressible limit of a cell density model, we propose a free boundary tumor growth model where the pressure satisfies an obstacle problem on an evolving domain $Ω(t)$, and the coincidence set $Λ(t)$ captures the emerging necrotic core. We contribute to the analytical characterization of the solution structure in the following two aspects. By deriving a semi-analytical solution and studying its dynamical behavior, we obtain quantitative transitional properties of the solution separating phases in the development of necrotic cores and establish its long time limit with the traveling wave solutions. Also, we prove the existence of traveling wave solutions incorporating non-zero outer densities outside the tumor bulk, provided that the size of the outer density is below a threshold.

math.AP↗

arXiv2023

Stochastic Schrödinger equation approach to real-time dynamics of Anderson-Holstein impurities: an open quantum system perspective

We develop a stochastic Schrödinger equation (SSE) framework to simulate real-time dynamics of Anderson-Holstein (AH) impurities coupled to a continuous fermionic bath. The bath degrees of freedom are incorporated through fluctuating terms determined by exact system-bath correlations, which is derived in an ab initio manner. We show that such an SSE treatment provides a middle ground between numerically expansive microscopic simulations and macroscopic master equations. Computationally, the SSE model enables efficient numerical methods for propagating stochastic trajectories. We demonstrate that this approach not only naturally provides microscopically-detailed information unavailable from reduced models, but also captures effects beyond master equations, thus serves as a promising tool to study open quantum dynamics emerging in physics and chemistry.

quant-ph↗

arXiv2023

Frozen Gaussian Sampling for Scalar Wave Equations

In this article, we introduce the frozen Gaussian sampling (FGS) algorithm to solve the scalar wave equation in the high-frequency regime. The FGS algorithm is a Monte Carlo sampling strategy based on the frozen Gaussian approximation, which greatly reduces the computation workload in the wave propagation and reconstruction. In this work, we propose feasible and detailed procedures to implement the FGS algorithm to approximate scalar wave equations with Gaussian initial conditions and WKB initial conditions respectively. For both initial data cases, we rigorously analyze the error of applying this algorithm to wave equations of dimensionality $d \geq 3$. In Gaussian initial data cases, we prove that the sampling error due to the Monte Carlo method is independent of the typical wave number. We also derive a quantitative bound of the sampling error in WKB initial data cases. Finally, we validate the performance of the FGS and the theoretical estimates about the sampling error through various numerical examples, which include using the FGS to solve wave equations with both Gaussian and WKB initial data of dimensionality $d = 1, 2$, and $3$.

math.NA↗

arXiv2023

Transition behavior of the waiting time distribution in a jumping model with the internal state

It has been noticed that when the waiting time distribution exhibits a transition from an intermediate time power law decay to a long-time exponential decay in the continuous time random walk model, a transition from anomalous diffusion to normal diffusion can be observed at the population level. However, the mechanism behind the transition of waiting time distribution is rarely studied. In this paper, we provide one possible mechanism to explain the origin of such transition. A jump model terminated by a state-dependent Poisson clock is studied by a formal asymptotic analysis for the time evolutionary equation of its probability density function. The waiting time behavior under a more relaxed setting can be rigorously characterized by probability tools. Both approaches show the transition phenomenon of the waiting time $T$, which is further verified numerically by particle simulations. Our results indicate that a small drift and strong noise in the state equation and a stiff response in the Poisson rate are crucial to the transitional phenomena.

math.AP↗

arXiv2023

A synchronization-capturing multi-scale solver to the noisy integrate-and-fire neuron networks

The noisy leaky integrate-and-fire (NLIF) model describes the voltage configurations of neuron networks with an interacting many-particles system at a microscopic level. When simulating neuron networks of large sizes, computing a coarse-grained mean-field Fokker-Planck equation solving the voltage densities of the networks at a macroscopic level practically serves as a feasible alternative in its high efficiency and credible accuracy. However, the macroscopic model fails to yield valid results of the networks when simulating considerably synchronous networks with active firing events. In this paper, we propose a multi-scale solver for the NLIF networks, which inherits the low cost of the macroscopic solver and the high reliability of the microscopic solver. For each temporal step, the multi-scale solver uses the macroscopic solver when the firing rate of the simulated network is low, while it switches to the microscopic solver when the firing rate tends to blow up. Moreover, the macroscopic and microscopic solvers are integrated with a high-precision switching algorithm to ensure the accuracy of the multi-scale solver. The validity of the multi-scale solver is analyzed from two perspectives: firstly, we provide practically sufficient conditions that guarantee the mean-field approximation of the macroscopic model and present rigorous numerical analysis on simulation errors when coupling the two solvers; secondly, the numerical performance of the multi-scale solver is validated through simulating several large neuron networks, including networks with either instantaneous or periodic input currents which prompt active firing events over a period of time.

math.NA↗

arXiv2023

A voltage-conductance kinetic system from neuroscience: probabilistic reformulation and exponential ergodicity

The voltage-conductance kinetic equation for an ensemble of neurons has been studied by many scientists and mathematicians, while its rigorous analysis is still at a premature stage. In this work, we obtain for the first time the exponential convergence to the steady state of this kinetic model in the linear setting. Our proof is based on a probabilistic reformulation, which allows us to investigate microscopic trajectories and bypass the difficulties raised by the special velocity field and boundary conditions in the macroscopic equation. We construct an associated stochastic process, for which proving the minorization condition becomes tractable, and the exponential ergodicity is then proved using Harris' theorem.

math.PR↗

arXiv2023

A spectral method for a Fokker-Planck equation in neuroscience with applications in neural networks with learning rules

In this work, we consider the Fokker-Planck equation of the Nonlinear Noisy Leaky Integrate-and-Fire (NNLIF) model for neuron networks. Due to the firing events of neurons at the microscopic level, this Fokker-Planck equation contains dynamic boundary conditions involving specific internal points. To efficiently solve this problem and explore the properties of the unknown, we construct a flexible numerical scheme for the Fokker-Planck equation in the framework of spectral methods that can accurately handle the dynamic boundary condition. This numerical scheme is stable with suitable choices of test function spaces, and asymptotic preserving, and it is easily extendable to variant models with multiple time scales. We also present extensive numerical examples to verify the scheme properties, including order of convergence and time efficiency, and explore unique properties of the model, including blow-up phenomena for the NNLIF model and learning and discriminative properties for the NNLIF model with learning rules.

math.NA↗

arXiv2023

An infinite-times renewal equation

In neuroscience, the time elapsed since the last discharge has been used to predict the probability of the next discharge. Such predictions can be improved taking into account the last two discharge times, and possibly more. Such multi-time processes arise in many other areas and there is no universal limitation on the number of times to be used. This observation leads us to study the infinite-times renewal equation as a simple model to understand the meaning and properties of such partial differential equations depending on an infinite number of variables.We define two notions of solutions, prove existence and uniqueness of solutions, possibly measures. We also prove the long time convergence, with exponential rate, to the steady state in different, strong or weak, topologies depending on assumptions on the coefficients.

math.AP↗

arXiv2022

A convergent numerical algorithm for the stochastic growth-fragmentation problem

The stochastic growth-fragmentation model describes the temporal evolution of a structured cell population through a discrete-time and continuous-state Markov chain. The simulations of this stochastic process and its invariant measure are of interest. In this paper, we propose a numerical scheme for both the simulation of the process and the computation of the invariant measure, and show that under appropriate assumptions, the numerical chain converges to the continuous growth-fragmentation chain with an explicit error bound. With a triangle inequality argument, we are also able to quantitatively estimate the distance between the invariant measures of these two Markov chains.

math.NA↗

arXiv2022

Tumor boundary instability induced by nutrient consumption and supply

We investigate the tumor boundary instability induced by nutrient consumption and supply based on a Hele-Shaw model derived from taking the incompressible limit of a cell density model. We analyze the boundary stability/instability in two scenarios: 1) the front of the traveling wave; 2) the radially symmetric boundary. In each scenario, we investigate the boundary behaviors under two different nutrient supply regimes, in vitro, and in vivo. Our main conclusion is that for either scenario, the in vitro regime always stabilizes the tumor's boundary regardless of the nutrient consumption rate. However, boundary instability may occur when the tumor cells aggressively consume nutrients, and the nutrient supply is governed by the in vivo regime.

math.AP↗