arXiv subjects

Zhiwen Zhang

Publications and source records attributed to Zhiwen Zhang.

At least 73 records · Page 4Linked to original sources

arXiv2021

speechocean762: An Open-Source Non-native English Speech Corpus For Pronunciation Assessment

This paper introduces a new open-source speech corpus named "speechocean762" designed for pronunciation assessment use, consisting of 5000 English utterances from 250 non-native speakers, where half of the speakers are children. Five experts annotated each of the utterances at sentence-level, word-level and phoneme-level. A baseline system is released in open source to illustrate the phoneme-level pronunciation assessment workflow on this corpus. This corpus is allowed to be used freely for commercial and non-commercial purposes. It is available for free download from OpenSLR, and the corresponding baseline system is published in the Kaldi speech recognition toolkit.

cs.CL↗

arXiv2021

Convergence analysis of a Lagrangian numerical scheme in computing effective diffusivity of 3D time-dependent flows

In this paper, we study the convergence analysis for a robust stochastic structure-preserving Lagrangian numerical scheme in computing effective diffusivity of time-dependent chaotic flows, which are modeled by stochastic differential equations (SDEs). Our numerical scheme is based on a splitting method to solve the corresponding SDEs in which the deterministic subproblem is discretized using structure-preserving schemes while the random subproblem is discretized using the Euler-Maruyama scheme. We obtain a sharp and uniform-in-time convergence analysis for the proposed numerical scheme that allows us to accurately compute long-time solutions of the SDEs. As such, we can compute the effective diffusivity for time-dependent flows. Finally, we present numerical results to demonstrate the accuracy and efficiency of the proposed method in computing effective diffusivity for the time-dependent Arnold-Beltrami-Childress (ABC) flow and Kolmogorov flow in three-dimensional space.

math.NA↗

arXiv2021

A convergent interacting particle method and computation of KPP front speeds in chaotic flows

In this paper, we study the propagation speeds of reaction-diffusion-advection (RDA) fronts in time-periodic cellular and chaotic flows with Kolmogorov-Petrovsky-Piskunov (KPP) nonlinearity. We first apply the variational principle to reduce the computation of KPP front speeds to a principal eigenvalue problem of a linear advection-diffusion operator with space-time periodic coefficients on a periodic domain. To this end, we develop efficient Lagrangian particle methods to compute the principal eigenvalue through the Feynman-Kac formula. By estimating the convergence rate of Feynman-Kac semigroups and the operator splitting methods for approximating the linear advection-diffusion solution operators, we obtain convergence analysis for the proposed numerical methods. Finally, we present numerical results to demonstrate the accuracy and efficiency of the proposed method in computing KPP front speeds in time-periodic cellular and chaotic flows, especially the time-dependent Arnold-Beltrami-Childress (ABC) flow and time-dependent Kolmogorov flow in three-dimensional space.

math.NA↗

arXiv2021

A data-driven and model-based accelerated Hamiltonian Monte Carlo method for Bayesian elliptic inverse problems

In this paper, we consider a Bayesian inverse problem modeled by elliptic partial differential equations (PDEs). Specifically, we propose a data-driven and model-based approach to accelerate the Hamiltonian Monte Carlo (HMC) method in solving large-scale Bayesian inverse problems. The key idea is to exploit (model-based) and construct (data-based) the intrinsic approximate low-dimensional structure of the underlying problem which consists of two components - a training component that computes a set of data-driven basis to achieve significant dimension reduction in the solution space, and a fast solving component that computes the solution and its derivatives for a newly sampled elliptic PDE with the constructed data-driven basis. Hence we achieve an effective data and model-based approach for the Bayesian inverse problem and overcome the typical computational bottleneck of HMC - repeated evaluation of the Hamiltonian involving the solution (and its derivatives) modeled by a complex system, a multiscale elliptic PDE in our case. We present numerical examples to demonstrate the accuracy and efficiency of the proposed method.

math.NA↗

arXiv2020

Sharp error estimates on a stochastic structure-preserving scheme in computing effective diffusivity of 3D chaotic flows

In this paper, we study the problem of computing the effective diffusivity for particles moving in chaotic flows. Instead of solving a convection-diffusion type cell problem in the Eulerian formulation (arising from homogenization theory for parabolic equations), we compute the motion of particles in the Lagrangian formulation, which is modeled by stochastic differential equations (SDEs). A robust numerical integrator based on a splitting method was proposed to solve the SDEs and a rigorous error analysis for the numerical integrator was provided using the backward error analysis (BEA) technique [35]. However, the upper bound in the error estimate is not sharp. To improve our result, we propose a new and uniform in time error analysis for the numerical integrator that allows us to get rid of the exponential growth factor in our previous error estimate. Our new error analysis is based on a probabilistic approach, which interprets the solution process generated by our numerical integrator as a Markov process. By exploring the ergodicity of the solution process, we prove the convergence analysis of our method in computing effective diffusivity over infinite time. We present numerical results to verify the accuracy and efficiency of the proposed method in computing effective diffusivity for several chaotic flows, especially the Arnold-Beltrami-Childress (ABC) flow and Kolmogorov flow in three-dimensional space.

math.NA↗

arXiv2020

A QMC-deep learning method for diffusivity estimation in random domains

Exciton diffusion plays a vital role in the function of many organic semiconducting opto-electronic devices, where an accurate description requires precise control of heterojunctions. This poses a challenging problem because the parameterization of heterojunctions in high-dimensional random space is far beyond the capability of classical simulation tools. Here, we develop a novel method based on quasi-Monte Carlo sampling to generate the training data set and deep neural network to extract a function for exciton diffusion length on surface roughness with high accuracy and unprecedented efficiency, yielding an abundance of information over the entire parameter space. Our method provides a new strategy to analyze the impact of interfacial ordering on exciton diffusion and is expected to assist experimental design with tailored opto-electronic functionalities.

physics.comp-ph↗

arXiv2020

Error estimates for a POD method for solving viscous G-equations in incompressible cellular flows

The G-equation is a well-known model for studying front propagation in turbulent combustion. In this paper, we develop an efficient model reduction method for computing \textcolor{black}{regular solutions} of viscous G-equations in incompressible steady and time-periodic cellular flows. Our method is based on the Galerkin proper orthogonal decomposition (POD) method. To facilitate the algorithm design and convergence analysis, we decompose the solution of the viscous G-equation into a mean-free part and a mean part, where their evolution equations can be derived accordingly. We construct the POD basis from the solution snapshots of the mean-free part. With the POD basis, we can efficiently solve the evolution equation for the mean-free part of the solution to the viscous G-equation. After we get the mean-free part of the solution, the mean of the solution can be recovered. We also provide rigorous convergence analysis for our method. Numerical results for \textcolor{black}{viscous G-equations and curvature G-equations} are presented to demonstrate the accuracy and efficiency of the proposed method. In addition, we study the turbulent flame speeds of the viscous G-equations in incompressible cellular flows.

math.NA↗

arXiv2020

Convergence of stochastic structure-preserving schemes for computing effective diffusivity in random flows

In this paper, we propose stochastic structure-preserving schemes to compute the effective diffusivity for particles moving in random flows. We first introduce the motion of particles using the Lagrangian formulation, which is modeled by stochastic differential equations (SDEs). We also discuss the definition of the corrector problem and effective diffusivity. Then we propose stochastic structure-preserving schemes to solve the SDEs and provide a sharp convergence analysis for the numerical schemes in computing effective diffusivity. The convergence analysis follows a probabilistic approach, which interprets the solution process generated by our numerical schemes as a Markov process. By using the central limit theorem for the solution process, we obtain the convergence analysis of our method in computing long time solutions. Most importantly our convergence analysis reveals the connection of discrete-type and continuous-type corrector problems, which is fundamental and interesting. We present numerical results to demonstrate the accuracy and efficiency of the proposed method and investigate the convection-enhanced diffusion phenomenon in two- and three-dimensional incompressible random flows.

math.NA↗

arXiv2019

Efficient multiscale methods for the semiclassical Schrödinger equation with time-dependent potentials

The semiclassical Schrödinger equation with time-dependent potentials is an important model to study electron dynamics under external controls in the mean-field picture. In this paper, we propose two multiscale finite element methods to solve this problem. In the offline stage, for the first approach, the localized multiscale basis functions are constructed using sparse compression of the Hamiltonian operator at the initial time; for the latter, basis functions are further enriched using a greedy algorithm for the sparse compression of the Hamiltonian operator at later times. In the online stage, the Schrödinger equation is approximated by these localized multiscale basis in space and is solved by the Crank-Nicolson method in time. These multiscale basis have compact supports in space, leading to the sparsity of stiffness matrix, and thus the computational complexity of these two methods in the online stage is comparable to that of the standard finite element method. However, the spatial mesh size in multiscale finite element methods is $ H=\mathcal{O}(ε) $, while $H=\mathcal{O}(ε^{3/2})$ in the standard finite element method, where $ε$ is the semiclassical parameter. By a number of numerical examples in 1D and 2D, for approximately the same number of basis, we show that the approximation error of the multiscale finite element method is at least two orders of magnitude smaller than that of the standard finite element method, and the enrichment further reduces the error by another one order of magnitude.

cs.CE↗

arXiv2019

A class of robust numerical methods for solving dynamical systems with multiple time scales

In this paper, we develop a class of robust numerical methods for solving dynamical systems with multiple time scales. We first represent the solution of a multiscale dynamical system as a transformation of a slowly varying solution. Then, under the scale separation assumption, we provide a systematic way to construct the transformation map and derive the dynamic equation for the slowly varying solution. We also provide the convergence analysis of the proposed method. Finally, we present several numerical examples, including ODE system with three and four separated time scales to demonstrate the accuracy and efficiency of the proposed method. Numerical results verify that our method is robust in solving ODE systems with multiple time scale, where the time step does not depend on the multiscale parameters.

math.NA↗

arXiv2019

Solving high-dimensional nonlinear filtering problems using a tensor train decomposition method

In this paper, we propose an efficient numerical method to solve high-dimensional nonlinear filtering (NLF) problems. Specifically, we use the tensor train decomposition method to solve the forward Kolmogorov equation (FKE) arising from the NLF problem. Our method consists of offline and online stages. In the offline stage, we use the finite difference method to discretize the partial differential operators involved in the FKE and extract low-dimensional structures in the solution space using the tensor train decomposition method. In addition, we approximate the evolution of the FKE operator using the tensor train decomposition method. In the online stage using the pre-computed low-rank approximation tensors, we can quickly solve the FKE given new observation data. Therefore, we can solve the NLF roblem in a real-time manner. Under some mild assumptions, we provide convergence analysis for the proposed method. Finally, we present numerical results to show the efficiency and accuracy of the proposed method in solving high-dimensional NLF problems.

math.NA↗

arXiv2019

A data-driven approach for multiscale elliptic PDEs with random coefficients based on intrinsic dimension reduction

We propose a data-driven approach to solve multiscale elliptic PDEs with random coefficients based on the intrinsic low dimension structure of the underlying elliptic differential operators. Our method consists of offline and online stages. At the offline stage, a low dimension space and its basis are extracted from the data to achieve significant dimension reduction in the solution space. At the online stage, the extracted basis will be used to solve a new multiscale elliptic PDE efficiently. The existence of low dimension structure is established by showing the high separability of the underlying Green's functions. Different online construction methods are proposed depending on the problem setup. We provide error analysis based on the sampling error and the truncation threshold in building the data-driven basis. Finally, we present numerical examples to demonstrate the accuracy and efficiency of the proposed method.

math.NA↗

arXiv2019

A multiscale reduced basis method for Schrödinger equation with multiscale and random potentials

The semiclassical Schrödinger equation with multiscale and random potentials often appears when studying electron dynamics in heterogeneous quantum systems. As time evolves, the wavefunction develops high-frequency oscillations in both the physical space and the random space, which poses severe challenges for numerical methods. In this paper, we propose a multiscale reduced basis method, where we construct multiscale reduced basis functions using an optimization method and the proper orthogonal decomposition method in the physical space and employ the quasi-Monte Carlo method in the random space. Our method is verified to be efficient: the spatial gridsize is only proportional to the semiclassical parameter and the number of samples in the random space is inversely proportional to the same parameter. Several theoretical aspects of the proposed method, including how to determine the number of samples in the construction of multiscale reduced basis and convergence analysis, are studied with numerical justification. In addition, we investigate the Anderson localization phenomena for Schrödinger equation with correlated random potentials in both 1D and 2D.

math.NA↗

arXiv2019

An adaptive dynamically low-dimensional approximation method for multiscale stochastic diffusion equations

In this paper, we propose a dynamically low-dimensional approximation method to solve a class of time-dependent multiscale stochastic diffusion equations. A dynamically bi-orthogonal (DyBO) method was developed to explore low-dimensional structures of stochastic partial differential equations (SPDEs) and solve them efficiently. However, when the SPDEs have multiscale features in physical space, the original DyBO method becomes expensive. To address this issue, we construct multiscale basis functions within each coarse grid block for dimension reduction in the physical space. To further improve the accuracy, we also perform online procedure to construct online adaptive basis functions. In the stochastic space, we use the generalized polynomial chaos (gPC) basis functions to represent the stochastic part of the solutions. Numerical results are presented to demonstrate the efficiency of the proposed method in solving time-dependent PDEs with multiscale and random features.

math.NA↗

arXiv2019

A mesh-free method for interface problems using the deep learning approach

In this paper, we propose a mesh-free method to solve interface problems using the deep learning approach. Two interface problems are considered. The first one is an elliptic PDE with a discontinuous and high-contrast coefficient. While the second one is a linear elasticity equation with discontinuous stress tensor. In both cases, we formulate the PDEs into variational problems, which can be solved via the deep learning approach. To deal with the inhomogeneous boundary conditions, we use a shallow neuron network to approximate the boundary conditions. Instead of using an adaptive mesh refinement method or specially designed basis functions or numerical schemes to compute the PDE solutions, the proposed method has the advantages that it is easy to implement and mesh-free. Finally, we present numerical results to demonstrate the accuracy and efficiency of the proposed method for interface problems.

physics.comp-ph↗

arXiv2019

A multiscale finite element method for the Schrödinger equation with multiscale potentials

In recent years, an increasing attention has been paid to quantum heterostructures with tailored functionalities, such as heterojunctions and quantum matematerials, in which quantum dynamics of electrons can be described by the Schrödinger equation with multiscale potentials. The model, however, cannot be solved by asymptoics-based approaches where an additive form of different scales in the potential term is required to construct the prescribed approximate solutions. In this paper, we propose a multiscale finite element method to solve this problem in the semiclassical regime. The localized multiscale basis are constructed using sparse compression of the Hamiltonian operator, and thus are "blind" to the specific form of the potential term. After an one-shot eigendecomposition, we solve the resulting system of ordinary differential equations explicitly for the time evolution. In our approach, the spatial mesh size is $ H=\mathcal{O}(ε) $ where $ε$ is the semiclassical parameter and the time stepsize $ k$ is independent of $ε$. Numerical examples in one dimension with a periodic potential, a multiplicative two-scale potential, and a layered potential, and in two dimension with an additive two-scale potential and a checkboard potential are tested to demonstrate the robustness and efficiency of the proposed method. Moreover, first-order and second-order rates of convergence are observed in $H^1$ and $L^2$ norms, respectively.

math.NA↗

arXiv2018

Critical slowing down and attractive manifold: a mechanism for dynamic robustness in yeast cell-cycle process

The biological processes that execute complex multiple functions, such as cell cycle, must ensure the order of sequential events and keep the dynamic robustness against various fluctuations. Here, we examine the dynamic mechanism and the fundamental structure to achieve these properties in the cell-cycle process of budding yeast Saccharomyces cerevisiae. We show that the budding yeast cell-cycle process behaves like an excitable system containing three well-coupled saddle-node bifurcations to execute DNA replication and mitosis events. The yeast cell-cycle regulatory network can be separated into G1/S phase module, early M module and late M phase module, where the positive feedbacks in each module and the interactions among the modules play important role. If the cell-cycle process operates near the critical points of the saddle-node bifurcations, there is a critical slowing down or ghost effect. This can provide the cell-cycle process with a sufficient duration for each event and an attractive manifold for the state checking of the completion of DNA replication and mitosis; moreover, the fluctuation in the early module/event is forbidden to transmit to the latter module/event. Our results suggest both a fundamental structure of cell-cycle regulatory network and a hint for the evolution of eukaryotic cell-cycle processes, from the dynamic checking mechanism to the molecule checkpoint pathway.

q-bio.MN↗

arXiv2018

A Minimal Mechanosensing Model Predicts Keratocyte Evolution on Flexible Substrates

A mathematical model is proposed for shape evolution and locomotion of fish epidermal keratocytes on elastic substrates. The model is based on mechanosensing concepts: cells apply contractile forces onto the elastic substrate, while cell shape evolution depends locally on the substrate stress generated by themselves or external mechanical stimuli acting on the substrate. We use the level set method to study the behavior of the model numerically, and predict a number of distinct phenomena observed in experiments, such as (i) symmetry breaking from the stationary centrosymmetric to the well-known steadily propagating crescent shape, (ii) asymmetric bipedal oscillations and traveling waves in the lamellipodium leading edge (iii) response to mechanical stress externally applied to the substrate (tensotaxis), (iv) changing direction of motion towards an interface with a rigid substrate (durotaxis) and (v) the configuration of substrate wrinkles induced by contractile forces applied by the keratocyte.

physics.bio-ph↗