arXiv subjects

Zhiwen Zhang

Publications and source records attributed to Zhiwen Zhang.

At least 37 records · Page 2Linked to original sources

arXiv2025

Functional tensor train neural network for solving high-dimensional PDEs

Discrete tensor train decomposition is widely employed to mitigate the curse of dimensionality in solving high-dimensional PDEs through traditional methods. However, the direct application of the tensor train method typically requires uniform grids of regular domains, which limits its application on non-uniform grids or irregular domains. To address the limitation, we develop a functional tensor train neural network (FTTNN) for solving high-dimensional PDEs, which can represent PDE solutions on non-uniform grids or irregular domains. An essential ingredient of our approach is to represent the PDE solutions by the functional tensor train format whose TT-core functions are approximated by neural networks. To give the functional tensor train representation, we propose and study functional tensor train rank and employ it into a physics-informed loss function for training. Because of tensor train representation, the resulting high-dimensional integral in the loss function can be computed via one-dimensional integrals by Gauss quadrature rules. Numerical examples including high-dimensional PDEs on regular or irregular domains are presented to demonstrate that the performance of the proposed FTTNN is better than that of Physics Informed Neural Networks (PINN).

math.NA↗

arXiv2025

Regularity estimate and sparse approximation of pathwise robust Duncan-Mortensen-Zakai equation

In this paper, we establish an \textit{a priori} estimate for arbitrary-order derivatives of the solution to the pathwise robust Duncan-Mortensen-Zakai (DMZ) equation within the framework of weighted Sobolev spaces. The weight function, which vanishes on the physical boundary, is crucial for the \textit{a priori} estimate, but introduces a loss of regularity near the boundary. Therefore, we employ the Sobolev inequalities and their weighted analogues to sharpen the regularity bound, providing improvements in both classical Sobolev spaces and H{ö}lder continuity estimates. The refined regularity estimate reinforces the plausibility of the quantized tensor train (QTT) method in [S. Li, Z. Wang, S. S.-T. Yau, and Z. Zhang, IEEE Trans. Automat. Control, 68 (2023), pp. 4405--4412] and provides convergence guarantees of the method. To further enhance the capacity of the method to solve the nonlinear filtering problem in a real-time manner, we reduce the complexity of the method under the assumption of a functional polyadic state drift $f$ and observation $h$. Finally, we perform numerical simulations to reaffirm our theory. For high-dimensional cubic sensor problems, our method demonstrates superior efficiency and accuracy in comparison to the particle filter (PF) and the extended Kalman filter (EKF). Beyond this, for multi-mode problems, while the PF exhibits a lack of precision due to its stochastic nature and the EKF is constrained by its Gaussian assumption, the enhanced method provides an accurate reconstruction of the multi-mode conditional density function.

math.NA↗

arXiv2025

Estimation of Cointegration Vectors in Time Series via Global Optimisation

Time Series Analysis has been given a great amount of study in which many useful tests were developed. The phenomenal work of Engle and Granger in 1987 and Johansen in 1988 has paved the way for the most commonly used cointegration tests so far. Even though cointegrating relationships focus on long-term behaviour and correlation of multiple nonstationary time series, oftentimes we encounter statistical data with limited sample sizes and other information. Thus other tests with empirical advantages may also be of considerable importance. In this paper, we provide an optimisation approach motivated by the Blind Source Separation, or also known as Independent Component Analysis, for cointegration between financial time series. Two methods for cointegration tests are introduced, namely decorrelation for the bivariate case and maximisation of nongaussianity for higher-dimensions. We highlight the empirical preponderances of independent components and also the computational simplicity, compared to common practices of cointegration such as the Johansen's Cointegration Test. The advantages of our methods, especially the better performances in limited sample size, enable a wider range of application and accessibility for researchers and practitioners to identify cointegrating relationships.

math.NA↗

arXiv2025

A quasi-Monte Carlo multiscale method for the wave propagation in random media

In this paper, we propose and analyze an accurate numerical approach to simulate the Helmholtz problem in a bounded region with a random refractive index, where the random refractive index is denoted using an infinite series parameterized by stochastic variables. To calculate the statistics of the solution numerically, we first truncate the parameterized model and adopt the quasi-Monte Carlo (qMC) method to generate stochastic variables. We develop a boundary-corrected multiscale method to discretize the truncated problem, which allows us to accurately resolve the Robin boundary condition with randomness. The proposed method exhibits superconvergence rates in the physical space (theoretical analysis suggests $\mathcal{O}(H^4)$ for $L^2$-error and $\mathcal{O}(H^2)$ for a defined $V$-error). Owing to the employment of the qMC method, it also exhibits almost the first-order convergence rate in the random space. We provide the wavenumber explicit convergence analysis and conduct numerical experiments to validate key features of the proposed method.

math.NA↗

arXiv2025

Numerical approximation of effective diffusivities in homogenization of nondivergence-form equations with large drift by a Lagrangian method

In this paper, we study numerical methods for the homogenization of linear second-order elliptic equations in nondivergence-form with periodic diffusion coefficients and large drift terms. Upon noting that the effective diffusion matrix can be characterized through the long-time variance of an associated diffusion process, we construct a Lagrangian numerical scheme based on a direct simulation of the underlying stochastic differential equation and utilizing the framework of modified equations, thereby avoiding the need to solve the Fokker--Planck--Kolmogorov equation. Through modified equation analysis, we derive higher-order weak convergence rates for our method. Finally, we conduct numerical experiments to demonstrate the accuracy of the proposed method. The results show that the method efficiently computes effective diffusivities, even in high dimensions.

math.NA↗

arXiv2025

Adaptive Nonoverlapping Preconditioners for the Helmholtz Equation

The Helmholtz equation poses significant computational challenges due to its oscillatory solutions, particularly for large wavenumbers. Inspired by the Schur complement system for elliptic problems, this paper presents a novel substructuring approach to mitigate the potential ill-posedness of local Dirichlet problems for the Helmholtz equation. We propose two types of preconditioners within the framework of nonoverlapping spectral additive Schwarz (NOSAS) methods. The first type of preconditioner focuses on the real part of the Helmholtz problem, while the second type addresses both the real and imaginary components, providing a comprehensive strategy to enhance scalability and reduce computational cost. Our approach is purely algebraic, which allows for adaptability to various discretizations and heterogeneous Helmholtz coefficients while maintaining theoretical convergence for thresholds close to zero. Numerical experiments confirm the effectiveness of the proposed preconditioners, demonstrating robust convergence rates and scalability, even for large wavenumbers.

math.NA↗

arXiv2025

Operator Inference for Elliptic Eigenvalue Problems

Eigenvalue problems for elliptic operators play an important role in science and engineering applications, where efficient and accurate numerical computation is essential. In this work, we propose a novel operator inference approach for elliptic eigenvalue problems based on neural network approximations that directly maps computational domains to their associated eigenvalues and eigenfunctions. Motivated by existing neural network architectures and the mathematical characteristics of eigenvalue problems, we represent computational domains as pixelated images and decompose the task into two subtasks: eigenvalue prediction and eigenfunction prediction. For the eigenvalue prediction, we design a convolutional neural network (CNN), while for the eigenfunction prediction, we employ a Fourier Neural Operator (FNO). Additionally, we introduce a critical preprocessing module that integrates domain scaling, detailed boundary pixelization, and main-axis alignment. This preprocessing step not only simplifies the learning task but also enhances the performance of the neural networks. Finally, we present numerical results to demonstrate the effectiveness of the proposed method.

math.NA↗

arXiv2025

A Bidirectional DeepParticle Method for Efficiently Solving Low-dimensional Transport Map Problems

This paper aims to efficiently compute transport maps between probability distributions arising from particle representation of bio-physical problems. We develop a bidirectional DeepParticle (BDP) method to learn and generate solutions under varying physical parameters. Solutions are approximated as empirical measures of particles that adaptively align with the high-gradient regions. The core idea of the BDP method is to learn both forward and reverse mappings (between the uniform and a non-trivial target distribution) by minimizing the discrete 2-Wasserstein distance (W2) and optimizing the transition map therein by a minibatch technique. We present numerical results demonstrating the effectiveness of the BDP method for learning and generating solutions to Keller-Segel chemotaxis systems in the presence of laminar flows and Kolmogorov flows with chaotic streamlines in three space dimensions. The BDP outperforms two recent representative single-step flow matching and diffusion models (rectified flow and shortcut diffusion models) in the generative AI literature. However when the target distribution is high-dimensional (4 and above), e.g. a mixture of two Gaussians, the single-step diffusion models scale better in dimensions than BDP in terms of W2-accuracy.

physics.comp-ph↗

arXiv2025

Efficient finite element methods for semiclassical nonlinear Schrödinger equations with random potentials

In this paper, we propose two time-splitting finite element methods to solve the semiclassical nonlinear Schrödinger equation (NLSE) with random potentials. We then introduce the multiscale finite element method (MsFEM) to reduce the degrees of freedom in the physical space. We construct multiscale basis functions by solving optimization problems and rigorously analyze two time-splitting MsFEMs for the semiclassical NLSE with random potentials. We provide the $L^2$ error estimate of the proposed methods and show that they achieve second-order accuracy in both spatial and temporal spaces and an almost first-order convergence rate in the random space. Additionally, we present a multiscale reduced basis method to reduce the computational cost of constructing basis functions for solving random NLSEs. Finally, we carry out several 1D and 2D numerical examples to validate the convergence of our methods and investigate wave propagation behaviors in the NLSE with random potentials.

math.NA↗

arXiv2025

A model reduction method for solving the eigenvalue problem of semiclassical random Schrödinger operators

In this paper, we compute the eigenvalue problem (EVP) for the semiclassical random Schrödinger operators, where the random potentials are parameterized by an infinite series of random variables. After truncating the series, we introduce the multiscale finite element method (MsFEM) to approximate the resulting parametric EVP. We then use the quasi-Monte Carlo (qMC) method to calculate empirical statistics within a finite-dimensional random space. Furthermore, using a set of low-dimensional proper orthogonal decomposition (POD) basis functions, the referred degrees of freedoms for constructing multiscale basis are independent of the spatial mesh. Given the bounded assumption on the random potentials, we then derive and prove an error estimate for the proposed method. Finally, we conduct numerical experiments to validate the error estimate. In addition, we investigate the localization of eigenfunctions for the Schrödinger operator with spatially random potentials. The results show that our method provides a practical and efficient solution for simulating complex quantum systems governed by semiclassical random Schrödinger operators.

math.NA↗

arXiv2025

A Novel Stochastic Interacting Particle-Field Algorithm for 3D Parabolic-Parabolic Keller-Segel Chemotaxis System

We introduce an efficient stochastic interacting particle-field (SIPF) algorithm with no history dependence for computing aggregation patterns and near singular solutions of parabolic-parabolic Keller-Segel (KS) chemotaxis system in three space dimensions (3D). The KS solutions are approximated as empirical measures of particles coupled with a smoother field (concentration of chemo-attractant) variable computed by the spectral method. Instead of using heat kernels causing history dependence and high memory cost, we leverage the implicit Euler discretization to derive a one-step recursion in time for stochastic particle positions and the field variable based on the explicit Green's function of an elliptic operator of the form Laplacian minus a positive constant. In numerical experiments, we observe that the resulting SIPF algorithm is convergent and self-adaptive to the high gradient part of solutions. Despite the lack of analytical knowledge (e.g. a self-similar ansatz) of the blowup, the SIPF algorithm provides a low-cost approach to study the emergence of finite time blowup in 3D by only dozens of Fourier modes and through varying the amount of initial mass and tracking the evolution of the field variable. Notably, the algorithm can handle at ease multi-modal initial data and the subsequent complex evolution involving the merging of particle clusters and formation of a finite time singularity.

math.NA↗

arXiv2025

A convergent interacting particle method for computing KPP front speeds in random flows

We aim to efficiently compute spreading speeds of reaction-diffusion-advection (RDA) fronts in divergence free random flows under the Kolmogorov-Petrovsky-Piskunov (KPP) nonlinearity. We study a stochastic interacting particle method (IPM) for the reduced principal eigenvalue (Lyapunov exponent) problem of an associated linear advection-diffusion operator with spatially random coefficients. The Fourier representation of the random advection field and the Feynman-Kac (FK) formula of the principal eigenvalue (Lyapunov exponent) form the foundation of our method implemented as a genetic evolution algorithm. The particles undergo advection-diffusion, and mutation/selection through a fitness function originated in the FK semigroup. We analyze convergence of the algorithm based on operator splitting, present numerical results on representative flows such as 2D cellular flow and 3D Arnold-Beltrami-Childress (ABC) flow under random perturbations. The 2D examples serve as a consistency check with semi-Lagrangian computation. The 3D results demonstrate that IPM, being mesh free and self-adaptive, is simple to implement and efficient for computing front spreading speeds in the advection-dominated regime for high-dimensional random flows on unbounded domains where no truncation is needed.

math.NA↗

arXiv2024

Random Batch Method with Momentum Correction

The Random Batch Method (RBM) is an effective technique to reduce the computational complexity when solving certain stochastic differential problems (SDEs) involving interacting particles. It can transform the computational complexity from O(N^2) to O(N), where N represents the number of particles. However, the traditional RBM can only be effectively applied to interacting particle systems with relatively smooth kernel functions to achieve satisfactory results. To address the issue of non-convergence of the RBM in particle interaction systems with significant singularities, we propose some enhanced methods to make the modified algorithm more applicable. The idea for improvement primarily revolves around a momentum-like correction, and we refer to the enhanced algorithm as the Random Batch Method with Momentum Correction ( RBM-M). We provide a theoretical proof to control the error of the algorithm, which ensures that under ideal conditions it has a smaller error than the original algorithm. Finally, numerical experiments have demonstrated the effectiveness of the RBM-M algorithm.

math.NA↗

arXiv2024

Finite element approximation of stationary Fokker--Planck--Kolmogorov equations with application to periodic numerical homogenization

We propose and rigorously analyze a finite element method for the approximation of stationary Fokker--Planck--Kolmogorov (FPK) equations subject to periodic boundary conditions in two settings: one with weakly differentiable coefficients, and one with merely essentially bounded measurable coefficients under a Cordes-type condition. These problems arise as governing equations for the invariant measure in the homogenization of nondivergence-form equations with large drifts. In particular, the Cordes setting guarantees the existence and uniqueness of a square-integrable invariant measure. We then suggest and rigorously analyze an approximation scheme for the effective diffusion matrix in both settings, based on the finite element scheme for stationary FPK problems developed in the first part. Finally, we demonstrate the performance of the methods through numerical experiments.

math.NA↗

arXiv2024

A Stochastic Interacting Particle-Field Algorithm for a Haptotaxis Advection-Diffusion System Modeling Cancer Cell Invasion

The investigation of tumor invasion and metastasis dynamics is crucial for advancements in cancer biology and treatment. Many mathematical models have been developed to study the invasion of host tissue by tumor cells. In this paper, we develop a novel stochastic interacting particle-field (SIPF) algorithm that accurately simulates the cancer cell invasion process within the haptotaxis advection-diffusion (HAD) system. Our approach approximates solutions using empirical measures of particle interactions, combined with a smoother field variable - the extracellular matrix concentration (ECM) - computed by the spectral method. We derive a one-step time recursion for both the positions of stochastic particles and the field variable using the implicit Euler discretization, which is based on the explicit Green's function of an elliptic operator characterized by the Laplacian minus a positive constant. Our numerical experiments demonstrate the superior performance of the proposed algorithm, especially in computing cancer cell growth with thin free boundaries in three-dimensional (3D) space. Numerical results show that the SIPF algorithm is mesh-free, self-adaptive, and low-cost. Moreover, it is more accurate and efficient than traditional numerical techniques such as the finite difference method (FDM) and spectral methods.

math.NA↗

arXiv2024

A note on Landau damping of two-species Vlasov-Poisson system

In this note we adopt an approach by Grenier, Nguyen and Rodnianski in \cite{GNR} for studying the nonlinear Landau damping of the two-species Vlasov-Poisson system in the phase space $\mathbb{T}^d_x \times \mathbb{R}^d_v$ with the dimension $d\geq 1$. The main goal is twofold: one is to extend the one-species case to the two-species case where the electron mass is finite and the ion mass is sufficiently large, and the other is to modify the $G$-functional such that it involves the norm in $L^{d+1}$ instead of $L^2$ as well as derivatives up to only the first order.

math.AP↗

arXiv2024

A novel model reduction method to solve inverse problems of parabolic type

In this paper, we propose novel proper orthogonal decomposition (POD)--based model reduction methods that effectively address the issue of inverse crime in solving parabolic inverse problems. Both the inverse initial value problems and inverse source problems are studied. By leveraging the inherent low-dimensional structures present in the data, our approach enables a reduction in the forward model complexity without compromising the accuracy of the inverse problem solution. Besides, we prove the convergence analysis of the proposed methods for solving parabolic inverse problems. Through extensive experimentation and comparative analysis, we demonstrate the effectiveness of our method in overcoming inverse crime and achieving improved inverse problem solutions. The proposed POD model reduction method offers a promising direction for improving the reliability and applicability of inverse problem-solving techniques in various domains.

math.NA↗

arXiv2024

A structure-preserving scheme for computing effective diffusivity and anomalous diffusion phenomena of random flows

This paper aims to investigate the diffusion behavior of particles moving in stochastic flows under a structure-preserving scheme. We compute the effective diffusivity for normal diffusive random flows and establish the power law between spatial and temporal variables for cases with anomalous diffusion phenomena. From a Lagrangian approach, we separate the corresponding stochastic differential equations (SDEs) into sub-problems and construct a one-step structure-preserving method to solve them. Then by modified equation systems, the convergence analysis in calculating the effective diffusivity is provided and compared between the structure-preserving scheme and the Euler-Maruyama scheme. Also, we provide the error estimate for the structure-preserving scheme in calculating the power law for a series of super-diffusive random flows. Finally, we calculate the effective diffusivity and anomalous diffusion phenomena for a series of 2D and 3D random fields.

math.NA↗