A New Stabilized Finite Element Method for Solving Transient Navier-Stokes Equations with High Reynolds Number
Chunmei Xie & Minfu Feng
10.4208/jcm.1810-m2018-0096
J. Comp. Math., 38 (2020), pp. 395-416.
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In this paper, we present a new stabilized finite element method for transient Navier-Stokes equations with high Reynolds number based on the projection of the velocity and pressure. We use Taylor-Hood elements and the equal order elements in space and second order difference in time to get the fully discrete scheme. The scheme is proven to possess the absolute stability and the optimal error estimates. Numerical experiments show that our method is effective for transient Navier-Stokes equations with high Reynolds number and the results are in good agreement with the value of subgrid-scale eddy viscosity methods, Petro-Galerkin finite element method and streamline diffusion method.
Image Denoising via Time-Delay Regularization Coupled Nonlinear Diffusion Equations
Qianting Ma
A novel nonlinear anisotropic diffusion model is proposed for image denoising which can be viewed as a novel regularized model that preserves the cherished features of Perona-Malik to some extent. It is characterized by a local dependence in the diffusivity which manifests itself through the presence of $p(x)$-Laplacian and time-delay regularization. The proposed model offers a new nonlinear anisotropic diffusion which makes it possible to effectively enhance the denoising capability and preserve the details while avoiding artifacts. Accordingly, the restored image is very clear and becomes more distinguishable. By Galerkin's method, we establish the well-posedness in the weak setting. Numerical experiments illustrate that the proposed model appears to be overwhelmingly competitive in restoring the images corrupted by Gaussian noise.
Enhanced Block-Sparse Signal Recovery Performance via Truncated $ℓ_2/ℓ_{1−2}$ Minimization
Weichao Kong, Jianjun Wang, Wendong Wang & Feng Zhang
In this paper, we investigate truncated $ℓ_2/ℓ_{1−2}$ minimization and its associated alternating direction method of multipliers (ADMM) algorithm for recovering the block sparse signals. Based on the block restricted isometry property (Block-RIP), a theoretical analysis is presented to guarantee the validity of proposed method. Our theoretical results not only show a less error upper bound, but also promote the former recovery condition of truncated ℓ1−2 method for sparse signal recovery. Besides, the algorithm has been compared with some state-of-the-art algorithms and numerical experiments have shown excellent performances on recovering the block sparse signals.
Efficient Linear Schemes with Unconditional Energy Stability for the Phase Field Model of Solid-State Dewetting Problems
Jie Chen, Zhengkang He, Shuyu Sun, Shimin Guo & Zhangxin Chen
In this paper, we study linearly first and second order in time, uniquely solvable and unconditionally energy stable numerical schemes to approximate the phase field model of solid-state dewetting problems based on the novel "scalar auxiliary variable" (SAV) approach, a new developed efficient and accurate method for a large class of gradient flows. The schemes are based on the first order Euler method and the second order backward differential formulas (BDF2) for time discretization, and finite element methods for space discretization. The proposed schemes are proved to be unconditionally stable and the discrete equations are uniquely solvable for all time steps. Various numerical experiments are presented to validate the stability and accuracy of the proposed schemes.
A Fully Discrete Implicit-Explicit Finite Element Method for Solving the FitzHugh-Nagumo Model
Li Cai, Ye Sun, Feifei Jing, Yiqiang Li, Xiaoqin Shen & Yufeng Nie
This work develops a fully discrete implicit-explicit finite element scheme for a parabolic-ordinary system with a nonlinear reaction term which is known as the FitzHugh-Nagumo model from physiology. The first-order backward Euler discretization for the time derivative, and an implicit-explicit discretization for the nonlinear reaction term are employed for the model, with a simple linearization technique used to make the process of solving equations more efficient. The stability and convergence of the fully discrete implicit-explicit finite element method are proved, which shows that the FitzHugh-Nagumo model is accurately solved and the trajectory of potential transmission is obtained. The numerical results are also reported to verify the convergence results and the stability of the proposed method.
Convergence Analysis of Parareal Algorithm Based on Milstein Scheme for Stochastic Differential Equations
Liying Zhang, Jing Wang, Weien Zhou, Landong Liu & Li Zhang
In this paper, we propose a parareal algorithm for stochastic differential equations (SDEs), which proceeds as a two-level temporal parallelizable integrator with the Milstein scheme as the coarse propagator and the exact solution as the fine propagator. The convergence order of the proposed algorithm is analyzed under some regular assumptions. Finally, numerical experiments are dedicated to illustrating the convergence and the convergence order with respect to the iteration number $k$, which show the efficiency of the proposed method.
ReLU Deep Neural Networks and Linear Finite Elements
Juncai He, Lin Li, Jinchao Xu & Chunyue Zheng
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In this paper, we investigate the relationship between deep neural networks (DNN) with rectified linear unit (ReLU) function as the activation function and continuous piecewise linear (CPWL) functions, especially CPWL functions from the simplicial linear finite element method (FEM). We first consider the special case of FEM. By exploring the DNN representation of its nodal basis functions, we present a ReLU DNN representation of CPWL in FEM. We theoretically establish that at least $2$ hidden layers are needed in a ReLU DNN to represent any linear finite element functions in $\Omega \subseteq \mathbb{R}^d$ when $d\ge2$. Consequently, for $d=2,3$ which are often encountered in scientific and engineering computing, the minimal number of two hidden layers are necessary and sufficient for any CPWL function to be represented by a ReLU DNN. Then we include a detailed account on how a general CPWL in $\mathbb R^d$ can be represented by a ReLU DNN with at most $\lceil\log_2(d+1)\rceil$ hidden layers and we also give an estimation of the number of neurons in DNN that are needed in such a representation. Furthermore, using the relationship between DNN and FEM, we theoretically argue that a special class of DNN models with low bit-width are still expected to have an adequate representation power in applications. Finally, as a proof of concept, we present some numerical results for using ReLU DNNs to solve a two-point boundary problem to demonstrate the potential of applying DNN for numerical solution of partial differential equations.
A Stochastic Moving Balls Approximation Method over a Smooth Inequality Constraint
Leiwu Zhang
We consider the problem of minimizing the average of a large number of smooth component functions over one smooth inequality constraint. We propose and analyze a stochastic Moving Balls Approximation (SMBA) method. Like stochastic gradient (SG) methods, the SMBA method's iteration cost is independent of the number of component functions and by exploiting the smoothness of the constraint function, our method can be easily implemented. Theoretical and computational properties of SMBA are studied, and convergence results are established. Numerical experiments indicate that our algorithm dramatically outperforms the existing Moving Balls Approximation algorithm (MBA) for the structure of our problem.