arXiv subjects

Jiwei Zhang

Publications and source records attributed to Jiwei Zhang.

At least 37 records · Page 2Linked to original sources

arXiv2023

Second order convergence of a modified MAC scheme for Stokes interface problems

Stokes flow equations have been implemented successfully in practice for simulating problems with moving interfaces. Though computational methods produce accurate solutions and numerical convergence can be demonstrated using a resolution study, the rigorous convergence proofs are usually limited to particular reformulations and boundary conditions. In this paper, a rigorous error analysis of the marker and cell (MAC) scheme for Stokes interface problems with constant viscosity in the framework of the finite difference method is presented. Without reformulating the problem into elliptic PDEs, the main idea is to use a discrete Ladyzenskaja-Babuska-Brezzi (LBB) condition and construct auxiliary functions, which satisfy discretized Stokes equations and possess at least second order accuracy in the neighborhood of the moving interface. In particular, the method, for the first time, enables one to prove second order convergence of the velocity gradient in the discrete $\ell^2$-norm, in addition to the velocity and pressure fields. Numerical experiments verify the desired properties of the methods and the expected order of accuracy for both two-dimensional and three-dimensional examples.

math.NA↗

arXiv2023

High performance implementation of 3D FEM for nonlocal Poisson problem with different ball approximation strategies

Nonlocality brings many challenges to the implementation of finite element methods (FEM) for nonlocal problems, such as large number of queries and invoke operations on the meshes. Besides, the interactions are usually limited to Euclidean balls, so direct numerical integrals often introduce numerical errors. The issues of interactions between the ball and finite elements have to be carefully dealt with, such as using ball approximation strategies. In this paper, an efficient representation and construction methods for approximate balls are presented based on combinatorial map, and an efficient parallel algorithm is also designed for assembly of nonlocal linear systems. Specifically, a new ball approximation method based on Monte Carlo integrals, i.e., the fullcaps method, is also proposed to compute numerical integrals over the intersection region of an element with the ball.

math.NA↗

arXiv2022

Stability and convergence analysis of high-order numerical schemes with DtN-type absorbing boundary conditions for nonlocal wave equations

The stability and convergence analysis of high-order numerical approximations for the one- and two-dimensional nonlocal wave equations on unbounded spatial domains are considered. We first use the quadrature-based finite difference schemes to discretize the spatially nonlocal operator, and apply the explicit difference scheme to approximate the temporal derivative to achieve a fully discrete infinity system. After that, we construct the Dirichlet-to-Neumann (DtN)-type absorbing boundary conditions (ABCs) to reduce the infinite discrete system into a finite discrete system. To do so, we first adopt the idea in [Du, Zhang and Zheng, \emph{Commun. Comput. Phys.}, 24(4):1049--1072, 2018 and Du, Han, Zhang and Zheng, \emph{SIAM J. Sci. Comp.}, 40(3):A1430--A1445, 2018] to derive the Dirichlet-to-Dirichlet (DtD)-type mappings for one- and two-dimensional cases, respectively. We then use the discrete nonlocal Green's first identity to achieve the discrete DtN-type mappings from the DtD-type mappings. The resulting DtN-type mappings make it possible to perform the stability and convergence analysis of the reduced problem. Numerical experiments are provided to demonstrate the accuracy and effectiveness of the proposed approach.

math.NA↗

arXiv2022

A unconditionally energy dissipative, adaptive IMEX BDF2 scheme and its error estimates for Cahn-Hilliard equation on generalized SAV approach

An adaptive implicit-explicit (IMEX) BDF2 scheme is investigated on generalized SAV approach for the Cahn-Hilliard equation by combining with Fourier spectral method in space. It is proved that the modified energy dissipation law is unconditionally preserved at discrete levels. Under a mild ratio restriction, i.e., \Ass{1}: $0<r_k:=τ_k/τ_{k-1}< r_{\max}\approx 4.8645$, we establish a rigorous error estimate in $H^1$-norm and achieve optimal second-order accuracy in time. The proof involves the tools of discrete orthogonal convolution (DOC) kernels and inequality zoom. It is worth noting that the presented adaptive time-step scheme only requires solving one linear system with constant coefficients at each time step. In our analysis, the first-consistent BDF1 for the first step does not bring the order reduction in $H^1$-norm. The $H^1$ bound of the numerical solution under periodic boundary conditions can be derived without any restriction (such as zero mean of the initial data). Finally, numerical examples are provided to verify our theoretical analysis and the algorithm efficiency.

math.NA↗

arXiv2022

Learning to Prune Instances of Steiner Tree Problem in Graphs

We consider the Steiner tree problem on graphs where we are given a set of nodes and the goal is to find a tree sub-graph of minimum weight that contains all nodes in the given set, potentially including additional nodes. This is a classical NP-hard combinatorial optimisation problem. In recent years, a machine learning framework called learning-to-prune has been successfully used for solving a diverse range of combinatorial optimisation problems. In this paper, we use this learning framework on the Steiner tree problem and show that even on this problem, the learning-to-prune framework results in computing near-optimal solutions at a fraction of the time required by commercial ILP solvers. Our results underscore the potential of the learning-to-prune framework in solving various combinatorial optimisation problems.

cs.DS↗

arXiv2022

On perfectly matched layers of nonlocal wave equations in unbounded multi-scale media

A nonlocal perfectly matched layer (PML) is formulated for the nonlocal wave equation in the whole real axis and numerical discretization is designed for solving the reduced PML problem on a bounded domain. The nonlocal PML poses challenges not faced in PDEs. For example, there is no derivative in nonlocal models, which makes it impossible to replace derivates with complex ones. Here we provide a way of constructing the PML for nonlocal models, which decays the waves exponentially impinging in the layer and makes reflections at the truncated boundary very tiny. To numerically solve the nonlocal PML problem, we design the asymptotically compatible (AC) scheme for spatially nonlocal operator by combining Talbot's contour, and a Verlet-type scheme for time evolution. The accuracy and effectiveness of our approach are illustrated by various numerical examples.

math.NA↗

arXiv2022

Sharp error estimate of variable time-step IMEX BDF2 scheme for parabolic integro-differential equations with initial singularity arising in finance

The recently developed technique of DOC kernels has been a great success in the stability and convergence analysis for BDF2 scheme with variable time steps. However, such an analysis technique seems not directly applicable to problems with initial singularity. In the numerical simulations of solutions with initial singularity, variable time-steps schemes like the graded mesh are always adopted to achieve the optimal convergence, whose first adjacent time-step ratio may become pretty large so that the acquired restriction is not satisfied. In this paper, we revisit the variable time-step implicit-explicit two-step backward differentiation formula (IMEX BDF2) scheme presented in [W. Wang, Y. Chen and H. Fang, \emph{SIAM J. Numer. Anal.}, 57 (2019), pp. 1289-1317] to compute the partial integro-differential equations (PIDEs) with initial singularity. We obtain the sharp error estimate under a mild restriction condition of adjacent time-step ratios $r_{k}: =τ_{k}/τ_{k-1} \; (k\geq 3) < r_{\max} = 4.8645 $ and a much mild requirement on the first ratio, i.e., $r_2>0$. This leads to the validation of our analysis of the variable time-step IMEX BDF2 scheme when the initial singularity is dealt by a simple strategy, i.e., the graded mesh $t_k=T(k/N)^γ$. In this situation, the convergence of order $\mathcal{O}(N^{-\min\{2,γα\}})$ is achieved with $N$ and $α$ respectively representing the total mesh points and indicating the regularity of the exact solution. This is, the optical convergence will be achieved by taking $γ_{\text{opt}}=2/α$. Numerical examples are provided to demonstrate our theoretical analysis.

math.NA↗

arXiv2022

Unconditionally optimal error estimate of a linearized variable-time-step BDF2 scheme for nonlinear parabolic equations

In this paper we consider a linearized variable-time-step two-step backward differentiation formula (BDF2) scheme for solving nonlinear parabolic equations. The scheme is constructed by using the variable time-step BDF2 for the linear term and a Newton linearized method for the nonlinear term in time combining with a Galerkin finite element method (FEM) in space. We prove the unconditionally optimal error estimate of the proposed scheme under mild restrictions on the ratio of adjacent time-steps, i.e. $0<r_k < r_{\max} \approx 4.8645$ and on the maximum time step. The proof involves the discrete orthogonal convolution (DOC) and discrete complementary convolution (DCC) kernels, and the error splitting approach. In addition, our analysis also shows that the first level solution $u^1$ obtained by BDF1 (i.e. backward Euler scheme) does not cause the loss of global accuracy of second order. Numerical examples are provided to demonstrate our theoretical results.

math.NA↗

arXiv2021

Variational Autoencoder with CCA for Audio-Visual Cross-Modal Retrieval

Cross-modal retrieval is to utilize one modality as a query to retrieve data from another modality, which has become a popular topic in information retrieval, machine learning, and database. How to effectively measure the similarity between different modality data is the major challenge of cross-modal retrieval. Although several reasearch works have calculated the correlation between different modality data via learning a common subspace representation, the encoder's ability to extract features from multi-modal information is not satisfactory. In this paper, we present a novel variational autoencoder (VAE) architecture for audio-visual cross-modal retrieval, by learning paired audio-visual correlation embedding and category correlation embedding as constraints to reinforce the mutuality of audio-visual information. On the one hand, audio encoder and visual encoder separately encode audio data and visual data into two different latent spaces. Further, two mutual latent spaces are respectively constructed by canonical correlation analysis (CCA). On the other hand, probabilistic modeling methods is used to deal with possible noise and missing information in the data. Additionally, in this way, the cross-modal discrepancy from intra-modal and inter-modal information are simultaneously eliminated in the joint embedding subspace. We conduct extensive experiments over two benchmark datasets. The experimental outcomes exhibit that the proposed architecture is effective in learning audio-visual correlation and is appreciably better than the existing cross-modal retrieval methods.

cs.IR↗

arXiv2021

DL-AMP and DBTO: An Automatic Merge Planning and Trajectory Optimization and Its Application in Autonomous Driving

This paper presents an automatic merging algorithm for autonomous driving vehicles, which decouples the specific motion planning problem into a Dual-Layer Automatic Merge Planning (DL_AMP) and a Descent-Based Trajectory Optimization (DBTO). This work leads to great improvements in finding the best merge opportunity, lateral and longitudinal merge planning and control, trajectory postprocessing and driving comfort.

cs.RO↗

arXiv2021

Numerical solution of a one-dimensional nonlocal Helmholtz equation by Perfectly Matched Layers

We consider the computation of a nonlocal Helmholtz equation by using Perfectly Matched Layer (PML). We first derive the nonlocal PML equation by extending PML modifications from the local operator to the nonlocal operator of integral form. We then give stability estimates of some weighted average value of the nonlocal Helmholtz solution and prove that (i) the weighted average value of the nonlocal PML solution decays exponentially in PML layers in one case; (ii) in the other case, the weighted average value of the nonlocal Helmholtz solution itself decays exponentially outside some domain. Particularly for a typical kernel function $γ_1(s)=\frac12 e^{-| s|}$, we obtain the Green's function of the nonlocal Helmholtz equation, and use the Green's function to further prove that (i) the nonlocal PML solution decays exponentially in PML layers in one case; (ii) in the other case, the nonlocal Helmholtz solution itself decays exponentially outside some domain. Based on our theoretical analysis, the truncated nonlocal problems are discussed and an asymptotic compatibility scheme is also introduced to solve the resulting truncated problems. Finally, numerical examples are provided to verify the effectiveness and validation of our nonlocal PML strategy and theoretical findings.

math.NA↗

arXiv2021

Simultaneous Imaging Achiral and Chiral Domains beyond Diffraction Limit by Structured-illumination Microscopy

Modern optical microscopy methods have been advanced to provide super resolution at high imaging speed, but not chirality discriminative. We recently proposed chiral structured-illumination microscopy (SIM) method to image chiral fluorescent domains at sub-wavelength resolution. Chiral SIM is suitable for imaging chiral domains at sub-wavelength resolution but loses the high spatial frequency of the achiral ones. In order to obtain the full picture of all fluorescent domains at sub-wavelength resolution, we propose an advanced version of chiral SIM, termed double SIM, which enables simultaneous imaging of achiral and chiral domains at sub-wavelength resolution. In double SIM, the illumination field must be spatially structured both in the intensity and optical chirality so that moiré effects can be concurrently generated on the achiral and chiral fluorescent domains of a sample. This allows down-modulating the high spatial frequency of both domains at the same time and thus provides sub-wavelength details after image reconstruction. To generate the illumination field with concurrently structured intensity and optical chirality, we propose interfering two coherent circularly polarized light with the same handedness at the sample plane. We introduce the working principle of double SIM and theoretically demonstrate the feasibility of this method using different kinds of synthetic samples.

physics.optics↗

arXiv2021

Perfectly Matched Layers for nonlocal Helmholtz equations II: multi-dimensional cases

Perfectly matched layers (PMLs) are formulated and applied to numerically solve nonlocal Helmholtz equations in one and two dimensions. In one dimension, we present the PML modifications for the nonlocal Helmholtz equation with general kernels and theoretically show its effectiveness in some sense. In two dimensions, we give the PML modifications in both Cartesian coordinates and polar coordinates. Based on the PML modifications, nonlocal Helmholtz equations are truncated in one and two dimensional spaces, and asymptotic compatibility schemes are introduced to discretize the resulting truncated problems. Finally, numerical examples are provided to study the "numerical reflections" by PMLs and demonstrate the effectiveness and validation of our nonlocal PML strategy.

math.NA↗

arXiv2021

Plasmonic elliptical nanoholes for chiroptical analysis and enantioselective optical trapping

A simple yet effective achiral platform using elliptical nanoholes for chiroptical analysis is demonstrated. Under linearly polarized excitation, an elliptical nanohole in a thin gold film can generate a localized chiral optical field for chiroptical analysis and simultaneously serve as a near-field optical trap to capture dielectric and plasmonic nanospheres. In particular, the trapping potential is enantioselective for dielectric nanospheres, i.e., the hole traps or repels the dielectric nanoparticles depending on the sample chirality. For plasmonic nanospheres, the trapping potential well is much deeper than that for dielectric particles, rendering the enantioselectivity less pronounced. This platform is suitable for chiral analysis with nanoparticle-based solid-state extraction and pre-concentration. Compared to plasmonic chiroptical sensing using chiral structures or circularly polarized light, elliptical nanoholes are a simple and effective platform, which is expected to have a relatively low background because chiroptical noise from the structure or chiral species outside the nanohole is minimized. The usage of linearly polarized excitation also makes the platform easily compatible with a commercial optical microscope.

physics.optics↗

arXiv2021

Sharp pointwise-in-time error estimate of L1 scheme for nonlinear subdiffusion equations

An essential feature of the subdiffusion equations with the $α$-order time fractional derivative is the weak singularity at the initial time. The weak regularity of the solution is usually characterized by a regularity parameter $σ\in (0,1)\cup(1,2)$. Under this general regularity assumption, we here obtain the pointwise-in-time error estimate of the widely used L1 scheme for nonlinear subdiffusion equations. To the end, we present a refined discrete fractional-type Grönwall inequality and a rigorous analysis for the truncation errors. Numerical experiments are provided to demonstrate the effectiveness of our theoretical analysis.

math.NA↗

arXiv2021

Frequency Principle in Deep Learning Beyond Gradient-descent-based Training

Frequency perspective recently makes progress in understanding deep learning. It has been widely verified in both empirical and theoretical studies that deep neural networks (DNNs) often fit the target function from low to high frequency, namely Frequency Principle (F-Principle). F-Principle sheds light on the strength and the weakness of DNNs and inspires a series of subsequent works, including theoretical studies, empirical studies and the design of efficient DNN structures etc. Previous works examine the F-Principle in gradient-descent-based training. It remains unclear whether gradient-descent-based training is a necessary condition for the F-Principle. In this paper, we show that the F-Principle exists stably in the training process of DNNs with non-gradient-descent-based training, including optimization algorithms with gradient information, such as conjugate gradient and BFGS, and algorithms without gradient information, such as Powell's method and Particle Swarm Optimization. These empirical studies show the universality of the F-Principle and provide hints for further study of F-Principle.

cs.LG↗

arXiv2020

Implicit bias with Ritz-Galerkin method in understanding deep learning for solving PDEs

This paper aims at studying the difference between Ritz-Galerkin (R-G) method and deep neural network (DNN) method in solving partial differential equations (PDEs) to better understand deep learning. To this end, we consider solving a particular Poisson problem, where the information of the right-hand side of the equation f is only available at n sample points, that is, f is known at finite sample points. Through both theoretical and numerical studies, we show that solution of the R-G method converges to a piecewise linear function for the one dimensional (1D) problem or functions of lower regularity for high dimensional problems. With the same setting, DNNs however learn a relative smooth solution regardless of the dimension, this is, DNNs implicitly bias towards functions with more low-frequency components among all functions that can fit the equation at available data points. This bias is explained by the recent study of frequency principle (Xu et al., (2019) [17] and Zhang et al., (2019) [11, 19]). In addition to the similarity between the traditional numerical methods and DNNs in the approximation perspective, our work shows that the implicit bias in the learning process, which is different from traditional numerical methods, could help better understand the characteristics of DNNs.

math.NA↗

arXiv2020

A priori generalization error for two-layer ReLU neural network through minimum norm solution

We focus on estimating \emph{a priori} generalization error of two-layer ReLU neural networks (NNs) trained by mean squared error, which only depends on initial parameters and the target function, through the following research line. We first estimate \emph{a priori} generalization error of finite-width two-layer ReLU NN with constraint of minimal norm solution, which is proved by \cite{zhang2019type} to be an equivalent solution of a linearized (w.r.t. parameter) finite-width two-layer NN. As the width goes to infinity, the linearized NN converges to the NN in Neural Tangent Kernel (NTK) regime \citep{jacot2018neural}. Thus, we can derive the \emph{a priori} generalization error of two-layer ReLU NN in NTK regime. The distance between NN in a NTK regime and a finite-width NN with gradient training is estimated by \cite{arora2019exact}. Based on the results in \cite{arora2019exact}, our work proves an \emph{a priori} generalization error bound of two-layer ReLU NNs. This estimate uses the intrinsic implicit bias of the minimum norm solution without requiring extra regularity in the loss function. This \emph{a priori} estimate also implies that NN does not suffer from curse of dimensionality, and a small generalization error can be achieved without requiring exponentially large number of neurons. In addition the research line proposed in this paper can also be used to study other properties of the finite-width network, such as the posterior generalization error.

cs.LG↗