arXiv subjects

Jian-Guo Liu

Publications and source records attributed to Jian-Guo Liu.

At least 37 records · Page 2Linked to original sources

arXiv2025

Data Assimilation Models for Computing Probability Distributions of Complex Multiscale Systems

We introduce a data assimilation strategy aimed at accurately capturing key non-Gaussian structures in probability distributions using a small ensemble size. A major challenge in statistical forecasting of nonlinearly coupled multiscale systems is mitigating the large errors that arise when computing high-order statistical moments. To address this issue, a high-order stochastic-statistical modeling framework is proposed that integrates statistical data assimilation into finite ensemble predictions. The method effectively reduces the approximation errors in finite ensemble estimates of non-Gaussian distributions by employing a filtering update step that incorporates observation data in leading moments to refine the high-order statistical feedback. Explicit filter operators are derived from intrinsic nonlinear coupling structures, allowing straightforward numerical implementations. We demonstrate the performance of the proposed method through extensive numerical experiments on a prototype triad system. The triad system offers an instructive and computationally manageable platform mimicking essential aspects of nonlinear turbulent dynamics. The numerical results show that the statistical data assimilation algorithm consistently captures the mean and covariance, as well as various non-Gaussian probability distributions exhibited in different statistical regimes of the triad system. The modeling framework can serve as a useful tool for efficient sampling and reliable forecasting of complex probability distributions commonly encountered in a wide variety of applications involving multiscale coupling and nonlinear dynamics.

math.NA↗

arXiv2024

Oscillatory solutions at the continuum limit of Lorenz 96 systems

In this paper, we study the generation and propagation of oscillatory solutions observed in the widely used Lorenz 96 (L96) systems. First, period-two oscillations between adjacent grid points are found in the leading-order expansions of the discrete L96 system. The evolution of the envelope of period-two oscillations is described by a set of modulation equations with strictly hyperbolic structure. The modulation equations are found to be also subject to an additional reaction term dependent on the grid size, and the period-two oscillations will break down into fully chaotic dynamics when the oscillation amplitude grows large. Then, similar oscillation solutions are analyzed in the two-layer L96 model including multiscale coupling. Modulation equations for period-three oscillations are derived based on a weakly nonlinear analysis in the transition between oscillatory and nonoscillatory regions. Detailed numerical experiments are shown to confirm the analytical results.

nlin.CD↗

arXiv2024

Geometric ergodicity of SGLD via reflection coupling

We consider the geometric ergodicity of the Stochastic Gradient Langevin Dynamics (SGLD) algorithm under nonconvexity settings. Via the technique of reflection coupling, we prove the Wasserstein contraction of SGLD when the target distribution is log-concave only outside some compact set. The time discretization and the minibatch in SGLD introduce several difficulties when applying the reflection coupling, which are addressed by a series of careful estimates of conditional expectations. As a direct corollary, the SGLD with constant step size has an invariant distribution and we are able to obtain its geometric ergodicity in terms of $W_1$ distance. The generalization to non-gradient drifts is also included.

math.PR↗

arXiv2024

Some properties on the reversibility and the linear response theory of Langevin dynamics

Linear response theory is a fundamental framework studying the macroscopic response of a physical system to an external perturbation. This paper focuses on the rigorous mathematical justification of linear response theory for Langevin dynamics. We give some equivalent characterizations for reversible overdamped/underdamped Langevin dynamics, which is the unperturbed reference system. Then we clarify sufficient conditions for the smoothness and exponential convergence to the invariant measure for the overdamped case. We also clarify those sufficient conditions for the underdamped case, which corresponds to hypoellipticity and hypocoercivity. Based on these, the asymptotic dependence of the response function on the small perturbation is proved in both finite and infinite time horizons. As applications, Green-Kubo relations and linear response theory for a generalized Langevin dynamics are also proved in a rigorous fashion.

math.AP↗

arXiv2024

Fluctuations in Wasserstein dynamics on Graphs

In this paper, we propose a drift-diffusion process on the probability simplex to study stochastic fluctuations in probability spaces. We construct a counting process for linear detailed balanced chemical reactions with finite species such that its thermodynamic limit is a system of ordinary differential equations (ODE) on the probability simplex. This ODE can be formulated as a gradient flow with an Onsager response matrix that induces a Riemannian metric on the probability simplex. After incorporating the induced Riemannian structure, we propose a diffusion approximation of the rescaled counting process for molecular species in the chemical reactions, which leads to Langevin dynamics on the probability simplex with a degenerate Brownian motion constructed from the eigen-decomposition of Onsager's response matrix. The corresponding Fokker-Planck equation on the simplex can be regarded as the usual drift-diffusion equation with the extrinsic representation of the divergence and Laplace-Beltrami operator. The invariant measure is the Gibbs measure, which is the product of the original macroscopic free energy and a volume element. When the drift term vanishes, the Fokker-Planck equation reduces to the heat equation with the Laplace-Beltrami operator, which we refer to as canonical Wasserstein diffusions on graphs. In the case of a two-point probability simplex, the constructed diffusion process is converted to one dimensional Wright-Fisher diffusion process, which leads to a natural boundary condition ensuring that the process remains within the probability simplex.

math.PR↗

arXiv2024

Coupled Stochastic-Statistical Equations for Filtering Multiscale Turbulent Systems

We present a new strategy for filtering high-dimensional multiscale systems characterized by high-order non-Gaussian statistics using observations from leading-order moments. A closed stochastic-statistical modeling framework suitable for systematic theoretical analysis and efficient numerical simulations is designed. Optimal filtering solutions are derived based on the explicit coupling structures of stochastic and statistical equations subject to linear operators, which satisfy an infinite-dimensional Kalman-Bucy filter with conditional Gaussian dynamics. To facilitate practical implementation, we develop a finite-dimensional stochastic filter model that approximates the optimal filter solution. We prove that this approximating filter effectively captures key non-Gaussian features, demonstrating consistent statistics with the optimal filter first in its analysis step update, then at the long-time limit guaranteeing stable convergence to the optimal filter. Finally, we build a practical ensemble filter algorithm based on the approximating filtering model, which enables accurate recovery of the true model statistics. The proposed modeling and filtering strategies are applicable to a wide range challenging problems in science and engineering, particularly for statistical prediction and uncertainty quantification of multiscale turbulent states.

math-ph↗

arXiv2024

A Three-dimensional tumor growth model and its boundary instability

In this paper, we investigate the tumor instability by employing both analytical and numerical techniques to validate previous results and extend the analytical findings presented in a prior study by Feng et al 2023. Building upon the insights derived from the analytical reconstruction of key results in the aforementioned work in one dimension (1D) and two dimensions (2D), we extend our analysis to three dimensions (3D). Specifically, we focus on the determination of boundary instability using perturbation and asymptotic analysis along with spherical harmonics. Additionally, we have validated our analytical results in a two-dimensional framework by implementing the Alternating Directional Implicit (ADI) method, as detailed in Witelski and Bowen (2003). Our primary focus has been on ensuring that the numerical simulation of the propagation speed aligns accurately with the analytical findings. Furthermore, we have matched the simulated boundary stability with the analytical predictions derived from the evolution function, which will be defined in subsequent sections of our paper. These alignment is essential for accurately determining the stability or instability of tumor boundaries.

math.AP↗

arXiv2024

Binary particle collisions with mass exchange

We investigate a kinetic model for interacting particles whose masses are integer multiples of an elementary mass. These particles undergo binary collisions which preserve momentum and energy but during which some number of elementary masses can be exchanged between the particles. We derive a Boltzmann collision operator for such collisions and study its conservation properties. Under some adequate assumptions on the collision rates, we show that it satisfies a H-theorem and exhibit its equilibria. We formally derive the system of fluid equations that arises from the hydrodynamic limit of this Boltzmann equation. We compute the viscous corrections to the leading order hydrodynamic equations on a simplified collision operator of BGK type. We show that this diffusive system can be put in the formalism of nonequilibrium thermodynamics. In particular, it satisfies Onsager's reciprocity relation and entropy decay.

math-ph↗

arXiv2024

Some Grönwall inequalities for a class of discretizations of time fractional equations on nonuniform meshes

We consider the completely positive discretizations of fractional ordinary differential equations (FODEs) on nonuniform meshes. Making use of the resolvents for nonuniform meshes, we first establish comparison principles for the discretizations. Then we prove some discrete Grönwall inequalities using the comparison principles and careful analysis of the solutions to the time continuous FODEs. Our results do not have any restrictions on the step size ratio. The Grönwall inequalities for dissipative equations can be used to obtain the uniform-in-time error control and decay estimates of the numerical solutions. The Grönwall inequalities are then applied to subdiffusion problems and the time fractional Allen-Cahn equations for illustration.

math.NA↗

arXiv2023

A personalized Uncertainty Quantification framework for patient survival models: estimating individual uncertainty of patients with metastatic brain tumors in the absence of ground truth

TodevelopanovelUncertaintyQuantification (UQ) framework to estimate the uncertainty of patient survival models in the absence of ground truth, we developed and evaluated our approach based on a dataset of 1383 patients treated with stereotactic radiosurgery (SRS) for brain metastases between January 2015 and December 2020. Our motivating hypothesis is that a time-to-event prediction of a test patient on inference is more certain given a higher feature-space-similarity to patients in the training set. Therefore, the uncertainty for a particular patient-of-interest is represented by the concordance index between a patient similarity rank and a prediction similarity rank. Model uncertainty was defined as the increased percentage of the max uncertainty-constrained-AUC compared to the model AUC. We evaluated our method on multiple clinically-relevant endpoints, including time to intracranial progression (ICP), progression-free survival (PFS) after SRS, overall survival (OS), and time to ICP and/or death (ICPD), on a variety of both statistical and non-statistical models, including CoxPH, conditional survival forest (CSF), and neural multi-task linear regression (NMTLR). Our results show that all models had the lowest uncertainty on ICP (2.21%) and the highest uncertainty (17.28%) on ICPD. OS models demonstrated high variation in uncertainty performance, where NMTLR had the lowest uncertainty(1.96%)and CSF had the highest uncertainty (14.29%). In conclusion, our method can estimate the uncertainty of individual patient survival modeling results. As expected, our data empirically demonstrate that as model uncertainty measured via our technique increases, the similarity between a feature-space and its predicted outcome decreases.

cs.LG↗

arXiv2023

A selection principle for weak KAM solutions via Freidlin-Wentzell large deviation principle of invariant measures

This paper reinterprets Freidlin-Wentzell's variational construction of the rate function in the large deviation principle for invariant measures from the weak KAM perspective. Through a one-dimensional irreversible diffusion process on a torus, we explicitly characterize essential concepts in the weak KAM theory, such as the Peierls barrier and the projected Mather/Aubry/Mañé sets. The weak KAM representation of Freidlin-Wentzell's variational construction of the rate function is discussed based on the global adjustment for the boundary data on the Aubry set and the local trimming from the lifted Peierls barriers. This rate function gives the maximal Lipschitz continuous viscosity solution to the corresponding stationary Hamilton-Jacobi equation (HJE), satisfying Freidlin-Wentzell's variational formula for the boundary data. Choosing meaningful self-consistent boundary data at each local attractor are essential to select a unique weak KAM solution to stationary HJE. This selected viscosity solution also serves as the global energy landscape of the original stochastic process. This selection for stationary HJEs can be described by first taking the long time limit and then taking the zero noise limit, which also provides a special construction of vanishing viscosity approximation.

math.AP↗

arXiv2023

Master equations for finite state mean field games with nonlinear activations

We formulate a class of mean field games on a finite state space with variational principles resembling those in continuous-state mean field games. We construct a controlled continuity equation featuring a nonlinear activation function on graphs induced by finite-state reversible continuous time Markov chains. In these graphs, each edge is weighted by the transition probability and invariant measure of the original process. Using these controlled dynamics on the graph and the dynamic programming principle for the value function, we derive several key components: the mean field game systems, the functional Hamilton-Jacobi equations, and the master equations on a finite probability space for potential mean field games. The existence and uniqueness of solutions to the potential mean field game system are ensured through a convex optimization reformulation in terms of the density-flux pair. We also derive variational principles for the master equations of both non-potential games and mixed games on a continuous state space. Finally, we offer several concrete examples of discrete mean field game dynamics on a two-point space, complete with closed-formula solutions. These examples include discrete Wasserstein distances, mean field planning, and potential mean field games.

math.OC↗

arXiv2023

Large deviation principle and thermodynamic limit of chemical master equation via nonlinear semigroup

Chemical reactions can be modeled by a random time-changed Poisson process on countable states. The macroscopic behaviors, such as large fluctuations, can be studied via the WKB reformulation. The WKB reformulation for the backward equation is Varadhan's discrete nonlinear semigroup and is also a monotone scheme that approximates the limiting first-order Hamilton-Jacobi equations (HJE). The discrete Hamiltonian is an m-accretive operator, which generates a nonlinear semigroup on countable grids and justifies the well-posedness of the chemical master equation (CME) and the backward equation with 'no reaction' boundary conditions. The convergence from the monotone schemes to the viscosity solution of HJE is proved by constructing barriers to overcome the polynomial growth coefficients in the Hamiltonian. This implies the convergence of Varadhan's discrete nonlinear semigroup to the continuous Lax-Oleinik semigroup and leads to the large deviation principle for the chemical reaction process at any single time. Consequently, the macroscopic mean-field limit reaction rate equation is recovered with a concentration rate estimate. Furthermore, we establish the convergence from a reversible invariant measure to an upper semicontinuous viscosity solution of the stationary HJE.

math.AP↗

arXiv2023

High-order Moment Closure Models with Random Batch Method for Efficient Computation of Multiscale Turbulent Systems

We propose a high-order stochastic-statistical moment closure model for efficient ensemble prediction of leading-order statistical moments and probability density functions in multiscale complex turbulent systems. The statistical moment equations are closed by a precise calibration of the high-order feedbacks using ensemble solutions of the consistent stochastic equations, suitable for modeling complex phenomena including non-Gaussian statistics and extreme events. To address challenges associated with closely coupled spatio-temporal scales in turbulent states and expensive large ensemble simulation for high-dimensional systems, we introduce efficient computational strategies using the random batch method (RBM). This approach significantly reduces the required ensemble size while accurately capturing essential high-order structures. Only a small batch of small-scale fluctuation modes is used for each time update of the samples, and exact convergence to the full model statistics is ensured through frequent resampling of the batches during time evolution. Furthermore, we develop a reduced-order model to handle systems with really high dimension by linking the large number of small-scale fluctuation modes to ensemble samples of dominant leading modes. The effectiveness of the proposed models is validated by numerical experiments on the one-layer and two-layer Lorenz '96 systems, which exhibit representative chaotic features and various statistical regimes. The full and reduced-order RBM models demonstrate uniformly high skill in capturing the time evolution of crucial leading-order statistics, non-Gaussian probability distributions, while achieving significantly lower computational cost compared to direct Monte-Carlo approaches.

math.NA↗

arXiv2023

Random walk approximation for irreversible drift-diffusion process on manifold: ergodicity, unconditional stability and convergence

Irreversible drift-diffusion processes are very common in biochemical reactions. They have a non-equilibrium stationary state (invariant measure) which does not satisfy detailed balance. For the corresponding Fokker-Planck equation on a closed manifold, using Voronoi tessellation, we propose two upwind finite volume schemes with or without the information of the invariant measure. Both schemes possess stochastic $Q$-matrix structures and can be decomposed as a gradient flow part and a Hamiltonian flow part, enabling us to prove unconditional stability, ergodicity and error estimates. Based on the two upwind schemes, several numerical examples - including sampling accelerated by a mixture flow, image transformations and simulations for stochastic model of chaotic system - are conducted. These two structure-preserving schemes also give a natural random walk approximation for a generic irreversible drift-diffusion process on a manifold. This makes them suitable for adapting to manifold-related computations that arise from high-dimensional molecular dynamics simulations.

math.NA↗

arXiv2023

Existence of weak solutions to $p$-Navier-Stokes equations

We study the existence of weak solutions to the $p$-Navier-Stokes equations with a symmetric $p$-Laplacian on bounded domains. We construct a particular Schauder basis in $W_0^{1,p}(Ω)$ with divergence free constraint and prove existence of weak solutions using the Galerkin approximation via this basis. Meanwhile, in the proof, we establish a chain rule for the $L^p$ norm of the weak solutions, which fixes a gap in our previous work. The equality of energy dissipation is also established for the weak solutions considered.

math.AP↗

arXiv2023

Diffusion approximations of Oja's online principal component analysis

Oja's algorithm of principal component analysis (PCA) has been one of the methods utilized in practice to reduce dimension. In this paper, we focus on the convergence property of the discrete algorithm. To realize that, we view the algorithm as a stochastic process on the parameter space and semi-group. We approximate it by SDEs, and prove large time convergence of the SDEs to ensure its performance. This process is completed in three steps. First, the discrete algorithm can be viewed as a semigroup: $S^kφ=\mathbb{E}[φ(\mathbf W(k))]$. Second, we construct stochastic differential equations (SDEs) on the Stiefel manifold, i.e. the diffusion approximation, to approximate the semigroup. By proving the weak convergence, we verify that the algorithm is 'close to' the SDEs. Finally, we use the reversibility of the SDEs to prove long-time convergence.

math.NA↗

arXiv2022

Interaction phenomenon for variable coefficient Kadomtsev-Petviashvili equation by utilizing variable coefficient bilinear neural network method

In this paper, a variable coefficient Bilinear neural network method is proposed to deal with the analytical solutions of variable coefficient nonlinear partial differential equations. As an example, a Kadomstev-Petviashvili equation with variable coefficients is investigated by using the variable coefficient Bilinear neural network method. By establishing "3-2-2-1" and "3-3-2-1" models respectively, rich analytical solutions of the variable coefficient Kadomstev-Petviashvili equation are obtained. By choosing some special values of the parameters, the dynamics properties are demonstrated in some three-dimensional and density graphics.

nlin.SI↗