arXiv subjects

Xiaobing Feng

Publications and source records attributed to Xiaobing Feng.

At least 55 records · Page 3Linked to original sources

arXiv2020

Analysis of Fully Discrete Mixed Finite Element Methods for Time-dependent Stochastic Stokes Equations with Multiplicative Noise

This paper is concerned with fully discrete mixed finite element approximations of the time-dependent stochastic Stokes equations with multiplicative noise. A prototypical method, which comprises of the Euler-Maruyama scheme for time discretization and the Taylor-Hood mixed element for spatial discretization is studied in detail. Strong convergence with rates is established not only for the velocity approximation but also for the pressure approximation (in a time-averaged fashion). A stochastic inf-sup condition is established and used in a nonstandard way to obtain the error estimate for the pressure approximation in the time-averaged fashion. Numerical results are also provided to validate the theoretical results and to gauge the performance of the proposed fully discrete mixed finite methods.

math.NA↗

arXiv2019

A Narrow-stencil finite difference method for approximating viscosity solutions of fully nonlinear elliptic partial differential equations with applications to Hamilton-Jacobi-Bellman equations

This paper presents a new narrow-stencil finite difference method for approximating the viscosity solution of second order fully nonlinear elliptic partial differential equations including Hamilton-Jacobi-Bellman equations. The proposed finite difference method naturally extends the Lax-Friedrichs method for first order problems to second order problems by introducing a key stabilization and guiding term called a "numerical moment". The numerical moment uses the difference of two (central) Hessian approximations to resolve the potential low-regularity of viscosity solutions. It is proved that the proposed scheme is well posed (i.e, it has a unique solution) and stable in both the l-2 norm and the l-infinity norm. The highlight of the paper is to prove the convergence of the proposed scheme to the viscosity solution of the underlying fully nonlinear second order problem using a novel discrete comparison argument. This paper extends the one-dimensional analogous method of Feng, Kao, and Lewis to the higher-dimensional setting. Numerical tests are presented to gauge the performance of the proposed finite difference methods and to validate the convergence result of the paper.

math.NA↗

arXiv2019

$H^1$-norm error estimate for a nonstandard finite element approximation of second-order linear elliptic PDEs in non-divergence form

This paper establishes the optimal $H^1$-norm error estimate for a nonstandard finite element method for approximating $H^2$ strong solutions of second order linear elliptic PDEs in non-divergence form with continuous coefficients. To circumvent the difficulty of lacking an effective duality argument for this class of PDEs, a new analysis technique is introduced; the crux of it is to establish an $H^1$-norm stability estimate for the finite element approximation operator which mimics a similar estimate for the underlying PDE operator recently established by the authors and its proof is based on a freezing coefficient technique and a topological argument. Moreover, both the $H^1$-norm stability and error estimate also hold for the linear finite element method.

math.NA↗

arXiv2019

The Phase Field Method for Geometric Moving Interfaces and Their Numerical Approximations

This paper surveys recent numerical advances in the phase field method for geometric surface evolution and related geometric nonlinear partial differential equations (PDEs). Instead of describing technical details of various numerical methods and their analyses, the paper presents a holistic overview about the main ideas of phase field modeling, its mathematical foundation, and relationships between the phase field formalism and other mathematical formalisms for geometric moving interface problems, as well as the current state-of-the-art of numerical approximations of various phase field models with an emphasis on discussing the main ideas of numerical analysis techniques. The paper also reviews recent development on adaptive grid methods and various applications of the phase field modeling and their numerical methods in materials science, fluid mechanics, biology and image science.

math.NA↗

arXiv2019

Mass- and energy-conserved numerical schemes for nonlinear Schrödinger equations

In this paper, we propose a family of time-stepping schemes for approximating general nonlinear Schrödinger equations. The proposed schemes all satisfy both mass conservation and energy conservation. Truncation and dispersion error analyses are provided for each proposed scheme. Efficient fixed-point iterative solvers are also constructed to solve the resulting nonlinear discrete problems. As a byproduct, an efficient one-step implementation of the BDF schemes is obtained as well. Extensive numerical experiments are presented to demonstrate the convergence and the capability of capturing the blow-up phenomenon of the proposed schemes.

math.NA↗

arXiv2019

Fully Discrete Mixed Finite Element Methods for the Stochastic Cahn-Hilliard Equation with Gradient-type Multiplicative Noise

This paper develops and analyzes some fully discrete mixed finite element methods for the stochastic Cahn-Hilliard equation with gradient-type multiplicative noise that is white in time and correlated in space. The stochastic Cahn-Hilliard equation is formally derived as a phase field formulation of the stochastically perturbed Hele-Shaw flow. The main result of this paper is to prove strong convergence with optimal rates for the proposed mixed finite element methods. To overcome the difficulty caused by the low regularity in time of the solution to the stochastic Cahn-Hilliard equation, the Hölder continuity in time with respect to various norms for the stochastic PDE solution is established, and it plays a crucial role in the error analysis. Numerical experiments are also provided to validate the theoretical results and to study the impact of noise on the Hele-Shaw flow as well as the interplay of the geometric evolution and gradient-type noise.

math.NA↗

arXiv2019

Analysis of the Vanishing Moment Method and its Finite Element Approximations for Second-order Linear Elliptic PDEs in Non-divergence Form

This paper is concerned with continuous and discrete approximations of $W^{2,p}$ strong solutions of second-order linear elliptic partial differential equations (PDEs) in non-divergence form. The continuous approximation of these equations is achieved through the Vanishing Moment Method (VMM) which adds a small biharmonic term to the PDE. The structure of the new fourth-order PDE is a natural fit for Galerkin-type methods unlike the original second order equation since the highest order term is in divergence form. The well-posedness of the weak form of the perturbed fourth order equation is shown as well as error estimates for approximating the strong solution of the original second-order PDE. A $C^1$ finite element method is then proposed for the fourth order equation, and its existence and uniqueness of solutions as well as optimal error estimates in the $H^2$ norm are shown. Lastly, numerical tests are given to show the validity of the method.

math.NA↗

arXiv2019

Background subtraction on depth videos with convolutional neural networks

Background subtraction is a significant component of computer vision systems. It is widely used in video surveillance, object tracking, anomaly detection, etc. A new data source for background subtraction appeared as the emergence of low-cost depth sensors like Microsof t Kinect, Asus Xtion PRO, etc. In this paper, we propose a background subtraction approach on depth videos, which is based on convolutional neural networks (CNNs), called BGSNet-D (BackGround Subtraction neural Networks for Depth videos). The method can be used in color unavailable scenarios like poor lighting situations, and can also be applied to combine with existing RGB background subtraction methods. A preprocessing strategy is designed to reduce the influences incurred by noise from depth sensors. The experimental results on the SBM-RGBD dataset show that the proposed method outperforms existing methods on depth data.

cs.CV↗

arXiv2018

Auto-tuning Neural Network Quantization Framework for Collaborative Inference Between the Cloud and Edge

Recently, deep neural networks (DNNs) have been widely applied in mobile intelligent applications. The inference for the DNNs is usually performed in the cloud. However, it leads to a large overhead of transmitting data via wireless network. In this paper, we demonstrate the advantages of the cloud-edge collaborative inference with quantization. By analyzing the characteristics of layers in DNNs, an auto-tuning neural network quantization framework for collaborative inference is proposed. We study the effectiveness of mixed-precision collaborative inference of state-of-the-art DNNs by using ImageNet dataset. The experimental results show that our framework can generate reasonable network partitions and reduce the storage on mobile devices with trivial loss of accuracy.

cs.DC↗

arXiv2018

Strong convergence of a fully discrete finite element method for a class of semilinear stochastic partial differential equations with multiplicative noise

This paper develops and analyzes a fully discrete finite element method for a class of semilinear stochastic partial differential equations (SPDEs) with multiplicative noise. The nonlinearity in the diffusion term of the SPDEs is assumed to be globally Lipschitz and the nonlinearity in the drift term is only assumed to satisfy a one-side Lipschitz condition. The semilinear SPDEs considered in this paper is a direct generalization of the SODEs considered in [13]. There are several difficulties which need to be overcome for this generalization. First, obviously the spatial discretization, which does not appear in the SODE case, adds an extra layer of difficulty. It turns out a special discretization must be designed to guarantee certain properties for the numerical scheme and its stiffness matrix. In this paper we use a finite element interpolation technique to discretize the nonlinear drift term. Second, in order to prove the strong convergence of the proposed fully discrete finite element method, stability estimates for higher order moments of the $H^1$-seminorm of the numerical solution must be established, which are difficult and delicate. A judicious combination of the properties of the drift and diffusion terms and a nontrivial technique borrowed from [16] is used in this paper to achieve the goal. Finally, stability estimates for the second and higher order moments of the $L^2$-norm of the numerical solution is also difficult to obtain due to the fact that the mass matrix may not be diagonally dominant. This is done by utilizing the interpolation theory and the higher moment estimates for the $H^1$-seminorm of the numerical solution. After overcoming these difficulties, it is proved that the proposed fully discrete finite element method is convergent in strong norms with nearly optimal rates of convergence.

math.NA↗

arXiv2018

An efficient Monte Carlo interior penalty discontinuous Galerkin method for the time-harmonic Maxwell's equations with random coefficients

This paper develops an efficient Monte Carlo interior penalty discontinuous Galerkin method for electromagnetic wave propagation in random media. This method is based on a multi-modes expansion of the solution to the time-harmonic random Maxwell equations. It is shown that each mode function satisfies a Maxwell system with random sources defined recursively. An unconditionally stable IP-DG method is employed to discretize the nearly deterministic Maxwell system and the Monte Carlo method combined with an efficient acceleration strategy is proposed for computing the mode functions and the statistics of the electromagnetic wave. A complete error analysis is established for the proposed multi-modes Monte Carlo IP-DG method. It is proved that the proposed method converges with an optimal order for each of three levels of approximations. Numerical experiments are provided to validate the theoretical results and to gauge the performance of the proposed numerical method and approach.

math.NA↗

arXiv2018

A New Approach for 4DVar Data Assimilation

Four-dimensional variational data assimilation (4DVar) has become an increasingly important tool in data science with wide applications in many engineering and scientific fields such as geoscience1-12, biology13 and the financial industry14. The 4DVar seeks a solution that minimizes the departure from the background field and the mismatch between the forecast trajectory and the observations within an assimilation window. The current state-of-the-art 4DVar offers only two choices by using different forms of the forecast model: the strong- and weak-constrained 4DVar approaches15-16. The former ignores the model error and only corrects the initial condition error at the expense of reduced accuracy; while the latter accounts for both the initial and model errors and corrects them separately, which increases computational costs and uncertainty. To overcome these limitations, here we develop an integral correcting 4DVar (i4DVar) approach by treating all errors as a whole and correcting them simultaneously and indiscriminately. To achieve that, a novel exponentially decaying function is proposed to characterize the error evolution and correct it at each time step in the i4DVar. As a result, the i4DVar greatly enhances the capability of the strong-constrained 4DVar for correcting the model error while also overcomes the limitation of the weak-constrained 4DVar for being prohibitively expensive with added uncertainty. Numerical experiments with the Lorenz model show that the i4DVar significantly outperforms the existing 4DVar approaches. It has the potential to be applied in many scientific and engineering fields and industrial sectors in the big data era because of its ease of implementation and superior performance.

physics.data-an↗

arXiv2018

A Discontinuous Ritz Method for a Class of Calculus of Variations Problems

This paper develops an analogue (or counterpart) to discontinuous Galerkin (DG) methods for approximating a general class of calculus of variations problems. The proposed method, called the discontinuous Ritz (DR) method, constructs a numerical solution by minimizing a discrete energy over DG function spaces. The discrete energy includes standard penalization terms as well as the DG finite element (DG-FE) numerical derivatives developed recently by Feng, Lewis, and Neilan in [Feng2013]. It is proved that the proposed DR method converges and that the DG-FE numerical derivatives exhibit a compactness property which is desirable and crucial for applying the proposed DR method to problems with more complex energy functionals. Numerical tests are provided on the classical $p$-Laplace problem to gauge the performance of the proposed DR method.

math.NA↗

arXiv2018

Nonstandard local discontinuous Galerkin methods for fully nonlinear second order elliptic and parabolic equations in high dimensions

This paper is concerned with developing accurate and efficient numerical methods for fully nonlinear second order elliptic and parabolic partial differential equations (PDEs) in multiple spatial dimensions. It presents a general framework for constructing high order local discontinuous Galerkin (LDG) methods for approximating viscosity solutions of these fully nonlinear PDEs. The proposed LDG methods are natural extensions of a narrow-stencil finite difference framework recently proposed by the authors for approximating viscosity solutions. The idea of the methodology is to use multiple approximations of first and second order derivatives as a way to resolve the potential low regularity of the underlying viscosity solution. Consistency and generalized monotonicity properties are proposed that ensure the numerical operator approximates the differential operator. The resulting algebraic system has several linear equations coupled with only one nonlinear equation that is monotone in many of its arguments. The structure can be explored to design nonlinear solvers. This paper also presents and analyzes numerical results for several numerical test problems in two dimensions which are used to gauge the accuracy and efficiency of the proposed LDG methods.

math.NA↗

arXiv2016

An enhanced finite element method for a class of variational problems exhibiting the Lavrentiev gap phenomenon

This paper develops an enhanced finite element method for approximating a class of variational problems which exhibit the \textit{Lavrentiev gap phenomenon} in the sense that the minimum values of the energy functional have a nontrivial gap when the functional is minimized on spaces $W^{1,1}$ and $W^{1,\infty}$. To remedy the standard finite element method, which fails to converge for such variational problems, a simple and effective cut-off procedure is utilized to design the (enhanced finite element) discrete energy functional. In essence the proposed discrete energy functional curbs the gap phenomenon by capping the derivatives of its input on a scale of $O(h^{-α})$ (where $h$ denotes the mesh size) for some positive constant $α$. A sufficient condition is proposed for determining the problem-dependent parameter $\a$. Extensive 1-D and 2-D numerical experiment results are provided to show the convergence behavior and the performance of the proposed enhanced finite element method.

math.NA↗

arXiv2016

Convergent semi-Lagrangian methods for the Monge-Ampère equation on unstructured grids

This paper is concerned with developing and analyzing convergent semi-Lagrangian methods for the fully nonlinear elliptic Monge-Ampère equation on general triangular grids. This is done by establishing an equivalent (in the viscosity sense) Hamilton-Jacobi-Bellman formulation of the Monge-Ampère equation. A significant benefit of the reformulation is the removal of the convexity constraint from the admissible space as convexity becomes a built-in property of the new formulation. Moreover, this new approach allows one to tap the wealthy numerical methods, such as semi-Lagrangian schemes, for Hamilton-Jacobi-Bellman equations to solve Monge-Ampère type equations. It is proved that the considered numerical methods are monotone, pointwise consistent and uniformly stable. Consequently, its solutions converge uniformly to the unique convex viscosity solution of the Monge-Ampère Dirichlet problem. A super-linearly convergent Howard's algorithm, which is a Newton type method, is utilized as the nonlinear solver to take advantage of the monotonicity of the scheme. Numerical experiments are also presented to gauge the performance of the proposed numerical method and the nonlinear solver.

math.NA↗

arXiv2016

Interior Penalty Discontinuous Galerkin Methods for Second Order Linear Non-Divergence Form Elliptic PDEs

This paper develops interior penalty discontinuous Galerkin (IP-DG) methods to approximate $W^{2,p}$ strong solutions of second order linear elliptic partial differential equations (PDEs) in non-divergence form with continuous coefficients. The proposed IP-DG methods are closely related to the IP-DG methods for advection-diffusion equations, and they are easy to implement on existing standard IP-DG software platforms. It is proved that the proposed IP-DG methods have unique solutions and converge with optimal rate to the $W^{2,p}$ strong solution in a discrete $W^{2,p}$-norm. The crux of the analysis is to establish a DG discrete counterpart of the Calderon-Zygmund estimate and to adapt a freezing coefficient technique used for the PDE analysis at the discrete level. As a byproduct of our analysis, we also establish broken $W^{1,p}$-norm error estimates for IP-DG approximations of constant coefficient elliptic PDEs. Numerical experiments are provided to gauge the performance of the proposed IP-DG methods and to validate the theoretical convergence results.

math.NA↗

arXiv2016

An efficient Monte Carlo interior penalty discontinuous Galerkin method for elastic wave scattering in random media

This paper develops and analyzes an efficient Monte Carlo interior penalty discontinuous Galerkin (MCIP-DG) method for elastic wave scattering in random media. The method is constructed based on a multi-modes expansion of the solution of the governing random partial differential equations. It is proved that the mode functions satisfy a three-term recurrence system of partial differential equations (PDEs) which are nearly deterministic in the sense that the randomness only appears in the right-hand side source terms, not in the coefficients of the PDEs. Moreover, the same differential operator applies to all mode functions. A proven unconditionally stable and optimally convergent IP-DG method is used to discretize the deterministic PDE operator, an efficient numerical algorithm is proposed based on combining the Monte Carlo method and the IP-DG method with the $LU$ direct linear solver. It is shown that the algorithm converges optimally with respect to both the mesh size $h$ and the sampling number $M$, and practically its total computational complexity is only amount to solving very few deterministic elastic Helmholtz equations using the $LU$ direct linear solver. Numerically experiments are also presented to demonstrate the performance and key features of the proposed MCIP-DG method.

math.NA↗