arXiv subjects
Publications and source records attributed to Xiaoping Xie.
This paper analyzes a discretization of a stochastic parabolic optimal control problem, where the diffusion term contains the control variable. With rough data, the convergence of the discretization is derived. In addition, a Monte-Carlo method is presented.
This paper studies the convergence of three temporal semi-discretizations for a backward semilinear stochastic evolution equation. For general terminal value and general coefficient with Lipschitz continuity, the convergence of the first two temporal semi-discretizations is established, and an explicit convergence rate is derived for the third temporal semi-discretization. The third temporal semi-discretization is applied to a general stochastic linear quadratic control problem, and the convergence of a temporally semi-discrete approximation to the optimal control is established.
Mixed finite element methods are considered for a ferrofluid flow model with magnetization paralleled to the magnetic field. The ferrofluid model is a coupled system of the Maxwell equations and the incompressible Navier-Stokes equations. By skillfully introducing some new variables, the model is rewritten as several decoupled subsystems that can be solved independently. Mixed finite element formulations are given to discretize the decoupled systems with proper finite element spaces. Existence and uniqueness of the mixed finite element solutions are shown, and optimal order error estimates are obtained under some reasonable assumptions. Numerical experiments confirm the theoretical results.
We propose an unfitted finite element method for numerically solving the time-harmonic Maxwell equations on a smooth domain. The model problem involves a Lagrangian multiplier to relax the divergence constraint of the vector unknown. The embedded boundary of the domain is allowed to cut through the background mesh arbitrarily. The unfitted scheme is based on a mixed interior penalty formulation, where Nitsche penalty method is applied to enforce the boundary condition in a weak sense, and a penalty stabilization technique is adopted based on a local direct extension operator to ensure the stability for cut elements. We prove the inf-sup stability and obtain optimal convergence rates under the energy norm and the $L^2$ norm for both the vector unknown and the Lagrangian multiplier. Numerical examples in both two and three dimensions are presented to illustrate the accuracy of the method.
This paper studies the convergence of a spatial semi-discretization for a backward semilinear stochastic parabolic equation. The filtration is general, and the spatial semi-discretization uses the standard continuous piecewise linear element method. Firstly, higher regularity of the solution to the continuous equation is derived. Secondly, the first-order spatial accuracy is derived for the spatial semi-discretization. Thirdly, an application of the theoretical result to a stochastic linear quadratic control problem is presented.
In this paper, we develop a class of mixed finite element methods for the ferrofluid flow model proposed by Shliomis [Soviet Physics JETP, 1972]. We show that the energy stability of the weak solutions to the model is preserved exactly for both the semi- and fully discrete finite element solutions. Furthermore, we prove the existence and uniqueness of the discrete solutions and derive optimal error estimates for both the the semi- and fully discrete schemes. Numerical experiments confirm the theoretical results.
This paper analyzes a class of globally divergence-free (and therefore pressure-robust) hybridizable discontinuous Galerkin (HDG) finite element methods for stationary Navier-Stokes equations. The methods use the $\mathcal{P}_{k}/\mathcal{P}_{k-1}$ $(k\geq1)$ discontinuous finite element combination for the velocity and pressure approximations in the interior of elements, and piecewise $\mathcal{P}_k/\mathcal{P}_{k}$ for the trace approximations of the velocity and pressure on the inter-element boundaries. It is shown that the uniqueness condition for the discrete solution is guaranteed by that for the continuous solution together with a sufficiently small mesh size. Based on the derived discrete HDG Sobolev embedding properties, optimal error estimates are obtained. Numerical experiments are performed to verify the theoretical analysis.
This paper considers weak Galerkin finite element approximations for a quasistatic Maxwell viscoelastic model. The spatial discretization uses piecewise polynomials of degree $k \ (k\geq 1)$ for the stress approximation, degree $k+1$ for the velocity approximation, and degree $k$ for the numerical trace of velocity on the inter-element boundaries. The temporal discretization in the fully discrete method adopts a backward Euler difference scheme. We show the existence and uniqueness of the semi-discrete and fully discrete solutions, and derive optimal a priori error estimates. Numerical examples are provided to support the theoretical analysis.
This paper is devoted to the numerical analysis of a piecewise constant discontinuous Galerkin method for time fractional subdiffusion problems. The regularity of weak solution is firstly established by using variational approach and Mittag-Leffler function. Then several optimal error estimates are derived with low regularity data. Finally, numerical experiments are conducted to verify the theoretical results.
This paper considers the temporal discretization of an inverse problem subject to a time fractional diffusion equation. Firstly, the convergence of the L1 scheme is established with an arbitrary sectorial operator of spectral angle $< π/2 $, that is the resolvent set of this operator contains $ \{z\in\mathbb C\setminus\{0\}:\ |\operatorname{Arg} z|< θ\}$ for some $ π/2 < θ< π$. The relationship between the time fractional order $α\in (0, 1)$ and the constants in the error estimates is precisely characterized, revealing that the L1 scheme is robust as $ α$ approaches $ 1 $. Then an inverse problem of a fractional diffusion equation is analyzed, and the convergence analysis of a temporal discretization of this inverse problem is given. Finally, numerical results are provided to confirm the theoretical results.
We propose and analyze an unfitted finite element method for solving elliptic problems on domains with curved boundaries and interfaces. The approximation space on the whole domain is obtained by the direct extension of the finite element space defined on interior elements, in the sense that there is no degree of freedom locating in boundary/interface elements. The boundary/jump conditions are imposed in a weak sense in the scheme. The method is shown to be stable without any mesh adjustment or any special stabilization. Optimal convergence rates under the $L^2$ norm and the energy norm are derived. Numerical results in both two and three dimensions are presented to illustrate the accuracy and the robustness of the method.
Semi-discrete and fully discrete mixed finite element methods are considered for Maxwell-model-based problems of wave propagation in linear viscoelastic solid. This mixed finite element framework allows the use of a large class of existing mixed conforming finite elements for elasticity in the spatial discretization. In the fully discrete scheme, a Crank-Nicolson scheme is adopted for the approximation of the temporal derivatives of stress and velocity variables. Error estimates of the semi-discrete and fully discrete schemes, as well as an unconditional stability result for the fully discrete scheme, are derived. Numerical experiments are provided to verify the theoretical results.
This paper is devoted to the error analysis of a time-spectral algorithm for fractional diffusion problems of order $α$ ($0 < α< 1$). The solution regularity in the Sobolev space is revisited, and new regularity results in the Besov space are established. A time-spectral algorithm is developed which adopts a standard spectral method and a conforming linear finite element method for temporal and spatial discretizations, respectively. Optimal error estimates are derived with nonsmooth data. Particularly, a sharp temporal convergence rate $1+2α$ is shown theoretically and numerically.
How to effectively remove the noise while preserving the image structure features is a challenging issue in the field of image denoising. In recent years, fractional PDE based methods have attracted more and more research efforts due to the ability to balance the noise removal and the preservation of image edges and textures. Among the existing fractional PDE algorithms, there are only a few using spatial fractional order derivatives, and all the fractional derivatives involved are one-sided derivatives. In this paper, an efficient feature-preserving fractional PDE algorithm is proposed for image denoising based on a nonlinear spatial-fractional anisotropic diffusion equation. Two-sided Grumwald-Letnikov fractional derivatives were used in the PDE model which are suitable to depict the local self-similarity of images. The Short Memory Principle is employed to simplify the approximation scheme. Experimental results show that the proposed method is of a satisfactory performance, i.e. it keeps a remarkable balance between noise removal and feature preserving, and has an extremely high structural retention property.
Deep learning based methods have achieved the state-of-the-art performance in image denoising. In this paper, a deep learning based denoising method is proposed and a module called fusion block is introduced in the convolutional neural network. For this so-called Noise Fusion Convolutional Neural Network (NFCNN), there are two branches in its multi-stage architecture. One branch aims to predict the latent clean image, while the other one predicts the residual image. A fusion block is contained between every two stages by taking the predicted clean image and the predicted residual image as a part of inputs, and it outputs a fused result to the next stage. NFCNN has an attractive texture preserving ability because of the fusion block. To train NFCNN, a stage-wise supervised training strategy is adopted to avoid the vanishing gradient and exploding gradient problems. Experimental results show that NFCNN is able to perform competitive denoising results when compared with some state-of-the-art algorithms.
An interface/boundary-unfitted eXtended hybridizable discontinuous Galerkin (X-HDG) method of arbitrary order is proposed for linear elasticity interface problems on unfitted meshes with respect to the interface and domain boundary. The method uses piecewise polynomials of degrees $k\ (>= 1)$ and $k-1$ respectively for the displacement and stress approximations in the interior of elements inside the subdomains separated by the interface, and piecewise polynomials of degree $k$ for the numerical traces of the displacement on the inter-element boundaries inside the subdomains and on the interface/boundary of the domain. Optimal error estimates in $L^2$-norm for the stress and displacement are derived. Finally, numerical experiments confirm the theoretical results and show that the method also applies to the case of crack-tip domain.
This paper proposes semi-discrete and fully discrete hybridizable discontinuous Galerkin (HDG) methods for the Burgers' equation in two and three dimensions. In the spatial discretization, we use piecewise polynomials of degrees $ k \ (k \geq 1), k-1$ and $ l \ (l=k-1; k) $ to approximate the scalar function, flux variable and the interface trace of scalar function, respectively. In the full discretization method, we apply a backward Euler scheme for the temporal discretization. Optimal a priori error estimates are derived. Numerical experiments are presented to support the theoretical results.
This paper proposes an interface/boundary-unfitted eXtended hybridizable discontinuous Galerkin (X-HDG) method for Darcy-Stokes-Brinkman interface problems in two and three dimensions. The method uses piecewise linear polynomials for the velocity approximation and piecewise constants for both the velocity gradient and pressure approximations in the interior of elements inside the subdomains separated by the interface, uses piecewise constants for the numerical traces of velocity on the inter-element boundaries inside the subdomains, and uses piecewise constants or linear polynomials for the numerical traces of velocity on the interface. Optimal error estimates are derived for the interface-unfitted X-HDG scheme. Numerical experiments are provided to verify the theoretical results and the robustness of the proposed method.