Most Download

Published in last 1 year | In last 2 years| In last 3 years| All| Most Downloaded in Recent Month | Most Downloaded in Recent Year|

In last 3 years

Please wait a minute...


For Selected: Download Citations EndNote Ris BibTeX Toggle Thumbnails

Select

A Weak Galerkin Harmonic Finite Element Method for Laplace Equation Ahmed Al-Taweel, Yinlin Dong, Saqib Hussain, Xiaoshen Wang Communications on Applied Mathematics and Computation    2021, 3 (3): 527-544.   DOI: 10.1007/s42967-020-00097-z Abstract （2497）      PDF       Knowledge map    Save In this article, a weak Galerkin finite element method for the Laplace equation using the harmonic polynomial space is proposed and analyzed. The idea of using the ${P_k}$-harmonic polynomial space instead of the full polynomial space ${P_k}$ is to use a much smaller number of basis functions to achieve the same accuracy when k ≥ 2. The optimal rate of convergence is derived in both H1 and L2 norms. Numerical experiments have been conducted to verify the theoretical error estimates. In addition, numerical comparisons of using the P2-harmonic polynomial space and using the standard P2 polynomial space are presented. Reference | Related Articles | Metrics | Comments（0）

Select

A Note on Numerical Algorithm for the Time-Caputo and Space-Riesz Fractional Difusion Equation Junhong Tian, Hengfei Ding Communications on Applied Mathematics and Computation    2021, 3 (4): 571-584.   DOI: 10.1007/s42967-021-00139-0 Abstract （6217）      PDF       Knowledge map    Save Recently, Zhang and Ding developed a novel fnite diference scheme for the timeCaputo and space-Riesz fractional difusion equation with the convergence order $\mathcal{O}\left(\tau^{2-\alpha}+h^{2}\right)$ in Zhang and Ding (Commun. Appl. Math. Comput. 2(1): 57–72, 2020). Unfortunately, they only gave the stability and convergence results for α ∈ (0, 1) and $\beta \in\left[\frac{7}{8}+\frac{\sqrt[3]{621+48 \sqrt{87}}}{24}+\frac{19}{8 \sqrt[3]{621+48 \sqrt{87}}}, 2\right]$. In this paper, using a new analysis method, we fnd that the original diference scheme is unconditionally stable and convergent with order $\mathcal{O}\left(\tau^{2-\alpha}+h^{2}\right)$ for all α ∈ (0, 1) and β ∈ (1, 2]. Finally, some numerical examples are given to verify the correctness of the results. Reference | Related Articles | Metrics | Comments（0）

Select

Learning the Mapping x ↦ $\sum\limits_{i = 1}^d {x_i^2}$: the Cost of Finding the Needle in a Haystack Jiefu Zhang, Leonardo Zepeda-Núñez, Yuan Yao, Lin Lin Communications on Applied Mathematics and Computation    2021, 3 (2): 313-335.   DOI: 10.1007/s42967-020-00078-2 Abstract （2773）      PDF       Knowledge map    Save The task of using the machine learning to approximate the mapping x ↦ $\sum\limits_{i = 1}^d {x_i^2}$ with xi ∈ [-1, 1] seems to be a trivial one. Given the knowledge of the separable structure of the function, one can design a sparse network to represent the function very accurately, or even exactly. When such structural information is not available, and we may only use a dense neural network, the optimization procedure to fnd the sparse network embedded in the dense network is similar to fnding the needle in a haystack, using a given number of samples of the function. We demonstrate that the cost (measured by sample complexity) of fnding the needle is directly related to the Barron norm of the function. While only a small number of samples are needed to train a sparse network, the dense network trained with the same number of samples exhibits large test loss and a large generalization gap. To control the size of the generalization gap, we fnd that the use of the explicit regularization becomes increasingly more important as d increases. The numerically observed sample complexity with explicit regularization scales as $\mathcal{O}$(d2.5), which is in fact better than the theoretically predicted sample complexity that scales as $\mathcal{O}$(d4). Without the explicit regularization (also called the implicit regularization), the numerically observed sample complexity is signifcantly higher and is close to $\mathcal{O}$(d4.5). Reference | Related Articles | Metrics | Comments（0）

Select

T-Jordan Canonical Form and T-Drazin Inverse Based on the T-Product Yun Miao, Liqun Qi, Yimin Wei Communications on Applied Mathematics and Computation    2021, 3 (2): 201-220.   DOI: 10.1007/s42967-019-00055-4 Abstract （6009）      PDF       Knowledge map    Save In this paper, we investigate the tensor similarity and propose the T-Jordan canonical form and its properties. The concepts of the T-minimal polynomial and the T-characteristic polynomial are proposed. As a special case, we present properties when two tensors commute based on the tensor T-product. We prove that the Cayley-Hamilton theorem also holds for tensor cases. Then, we focus on the tensor decompositions: T-polar, T-LU, T-QR and T-Schur decompositions of tensors are obtained. When an F-square tensor is not invertible with the T-product, we study the T-group inverse and the T-Drazin inverse which can be viewed as the extension of matrix cases. The expressions of the T-group and T-Drazin inverses are given by the T-Jordan canonical form. The polynomial form of the T-Drazin inverse is also proposed. In the last part, we give the T-core-nilpotent decomposition and show that the T-index and T-Drazin inverses can be given by a limit process. Reference | Related Articles | Metrics | Comments（0）

Select

Conservative Discontinuous Galerkin/Hermite Spectral Method for the Vlasov-Poisson System Francis Filbet, Tao Xiong Communications on Applied Mathematics and Computation    2022, 4 (1): 34-59.   DOI: 10.1007/s42967-020-00089-z Abstract （8966）      PDF       Knowledge map    Save We propose a class of conservative discontinuous Galerkin methods for the Vlasov-Poisson system written as a hyperbolic system using Hermite polynomials in the velocity variable. These schemes are designed to be systematically as accurate as one wants with provable conservation of mass and possibly total energy. Such properties in general are hard to achieve within other numerical method frameworks for simulating the Vlasov-Poisson system. The proposed scheme employs the discontinuous Galerkin discretization for both the Vlasov and the Poisson equations, resulting in a consistent description of the distribution function and the electric feld. Numerical simulations are performed to verify the order of the accuracy and conservation properties. Reference | Related Articles | Metrics | Comments（0）

Select

The Spectral Radii of Intersecting Uniform Hypergraphs Peng, Li Zhang, Xiao, Dong Zhang Communications on Applied Mathematics and Computation    2021, 3 (2): 243-256.   DOI: 10.1007/s42967-020-00073-7 Abstract （4781）      PDF       Knowledge map    Save The celebrated Erdős-Ko-Rado theorem states that given n ≥ 2k, every intersecting k-uniform hypergraph G on n vertices has at most $(_{k - 1}^{n - 1})$ edges. This paper states spectral versions of the Erdős-Ko-Rado theorem: let G be an intersecting k-uniform hypergraph on n vertices with n ≥ 2k. Then, the sharp upper bounds for the spectral radius of $\mathcal{A}$α(G) and $\mathcal{Q}$*(G) are presented, where $\mathcal{A}$α(G) = α$\mathcal{D}$(G) + (1 - α)$\mathcal{A}$(G) is a convex linear combination of the degree diagonal tensor $\mathcal{D}$(G) and the adjacency tensor $\mathcal{A}$(G) for 0 ≤ α < 1, and $\mathcal{Q}$*(G) is the incidence $\mathcal{Q}$-tensor, respectively. Furthermore, when n > 2k, the extremal hypergraphs which attain the sharp upper bounds are characterized. The proof mainly relies on the Perron-Frobenius theorem for nonnegative tensor and the property of the maximizing connected hypergraphs. Reference | Related Articles | Metrics | Comments（0）

Select

Uniform Subspace Correction Preconditioners for Discontinuous Galerkin Methods with hp-Refnement Will Pazner, Tzanio Kolev Communications on Applied Mathematics and Computation    2022, 4 (2): 697-727.   DOI: 10.1007/s42967-021-00136-3 Abstract （1500）      PDF       Knowledge map    Save In this paper, we develop subspace correction preconditioners for discontinuous Galerkin (DG) discretizations of elliptic problems with hp-refnement. These preconditioners are based on the decomposition of the DG fnite element space into a conforming subspace, and a set of small nonconforming edge spaces. The conforming subspace is preconditioned using a matrix-free low-order refned technique, which in this work, we extend to the hp-refnement context using a variational restriction approach. The condition number of the resulting linear system is independent of the granularity of the mesh h, and the degree of the polynomial approximation p. The method is amenable to use with meshes of any degree of irregularity and arbitrary distribution of polynomial degrees. Numerical examples are shown on several test cases involving adaptively and randomly refned meshes, using both the symmetric interior penalty method and the second method of Bassi and Rebay (BR2). Reference | Related Articles | Metrics | Comments（0）

Select

Superconvergent Interpolatory HDG Methods for Reaction Difusion Equations II: HHO-Inspired Methods Gang Chen, Bernardo Cockburn, John R. Singler, Yangwen Zhang Communications on Applied Mathematics and Computation    2022, 4 (2): 477-499.   DOI: 10.1007/s42967-021-00128-3 Abstract （1351）      PDF       Knowledge map    Save In Chen et al. (J. Sci. Comput. 81(3):2188-2212, 2019), we considered a superconvergent hybridizable discontinuous Galerkin (HDG) method, defned on simplicial meshes, for scalar reaction-difusion equations and showed how to defne an interpolatory version which maintained its convergence properties. The interpolatory approach uses a locally postprocessed approximate solution to evaluate the nonlinear term, and assembles all HDG matrices once before the time integration leading to a reduction in computational cost. The resulting method displays a superconvergent rate for the solution for polynomial degree k ≥ 1. In this work, we take advantage of the link found between the HDG and the hybrid high-order (HHO) methods, in Cockburn et al. (ESAIM Math. Model. Numer. Anal. 50(3):635-650, 2016) and extend this idea to the new, HHO-inspired HDG methods, defned on meshes made of general polyhedral elements, uncovered therein. For meshes made of shape-regular polyhedral elements afne-equivalent to a fnite number of reference elements, we prove that the resulting interpolatory HDG methods converge at the same rate as for the linear elliptic problems. Hence, we obtain superconvergent methods for k ≥ 0 by some methods. We thus maintain the superconvergence properties of the original methods. We present numerical results to illustrate the convergence theory. Reference | Related Articles | Metrics | Comments（0）

Select

High Order Semi-implicit Multistep Methods for Time-Dependent Partial Diferential Equations Giacomo Albi, Lorenzo Pareschi Communications on Applied Mathematics and Computation    2021, 3 (4): 701-718.   DOI: 10.1007/s42967-020-00110-5 Abstract （3402）      PDF       Knowledge map    Save We consider the construction of semi-implicit linear multistep methods that can be applied to time-dependent PDEs where the separation of scales in additive form, typically used in implicit-explicit (IMEX) methods, is not possible. As shown in Boscarino et al. (J. Sci. Comput. 68: 975–1001, 2016) for Runge-Kutta methods, these semi-implicit techniques give a great fexibility, and allow, in many cases, the construction of simple linearly implicit schemes with no need of iterative solvers. In this work, we develop a general setting for the construction of high order semi-implicit linear multistep methods and analyze their stability properties for a prototype linear advection-difusion equation and in the setting of strong stability preserving (SSP) methods. Our fndings are demonstrated on several examples, including nonlinear reaction-difusion and convection-difusion problems. Reference | Related Articles | Metrics | Comments（0）

Select

A Novel Staggered Semi-implicit Space-Time Discontinuous Galerkin Method for the Incompressible Navier-Stokes Equations F. L. Romeo, M. Dumbser, M. Tavelli Communications on Applied Mathematics and Computation    2021, 3 (4): 607-647.   DOI: 10.1007/s42967-020-00077-3 Abstract （3853）      PDF       Knowledge map    Save A new high-order accurate staggered semi-implicit space-time discontinuous Galerkin (DG) method is presented for the simulation of viscous incompressible fows on unstructured triangular grids in two space dimensions. The staggered DG scheme defnes the discrete pressure on the primal triangular mesh, while the discrete velocity is defned on a staggered edge-based dual quadrilateral mesh. In this paper, a new pair of equal-order-interpolation velocity-pressure fnite elements is proposed. On the primary triangular mesh (the pressure elements), the basis functions are piecewise polynomials of degree N and are allowed to jump on the boundaries of each triangle. On the dual mesh instead (the velocity elements), the basis functions consist in the union of piecewise polynomials of degree N on the two subtriangles that compose each quadrilateral and are allowed to jump only on the dual element boundaries, while they are continuous inside. In other words, the basis functions on the dual mesh are built by continuous fnite elements on the subtriangles. This choice allows the construction of an efcient, quadraturefree and memory saving algorithm. In our coupled space-time pressure correction formulation for the incompressible Navier-Stokes equations, the arbitrary high order of accuracy in time is achieved through the use of time-dependent test and basis functions, in combination with simple and efcient Picard iterations. Several numerical tests on classical benchmarks confrm that the proposed method outperforms existing staggered semi-implicit space-time DG schemes, not only from a computer memory point of view, but also concerning the computational time. Reference | Related Articles | Metrics | Comments（0）

Select

A Non-intrusive Correction Algorithm for Classifcation Problems with Corrupted Data Jun Hou, Tong Qin, Kailiang Wu, Dongbin Xiu Communications on Applied Mathematics and Computation    2021, 3 (2): 337-356.   DOI: 10.1007/s42967-020-00084-4 Abstract （14182）      PDF       Knowledge map    Save A novel correction algorithm is proposed for multi-class classifcation problems with corrupted training data. The algorithm is non-intrusive, in the sense that it post-processes a trained classifcation model by adding a correction procedure to the model prediction. The correction procedure can be coupled with any approximators, such as logistic regression, neural networks of various architectures, etc. When the training dataset is sufciently large, we theoretically prove (in the limiting case) and numerically show that the corrected models deliver correct classifcation results as if there is no corruption in the training data. For datasets of fnite size, the corrected models produce signifcantly better recovery results, compared to the models without the correction algorithm. All of the theoretical fndings in the paper are verifed by our numerical examples. Reference | Related Articles | Metrics | Comments（0）

Select

A Jacobi Spectral Collocation Method for Solving Fractional Integro-Differential Equations Qingqing Wu, Zhongshu Wu, Xiaoyan Zeng Communications on Applied Mathematics and Computation    2021, 3 (3): 509-526.   DOI: 10.1007/s42967-020-00099-x Abstract （2636）      PDF       Knowledge map    Save The aim of this paper is to obtain the numerical solutions of fractional Volterra integrodifferential equations by the Jacobi spectral collocation method using the Jacobi-Gauss collocation points. We convert the fractional order integro-differential equation into integral equation by fractional order integral, and transfer the integro equations into a system of linear equations by the Gausssian quadrature. We furthermore perform the convergence analysis and prove the spectral accuracy of the proposed method in L∞ norm. Two numerical examples demonstrate the high accuracy and fast convergence of the method at last. Reference | Related Articles | Metrics | Comments（0）

Select

Hermite-Discontinuous Galerkin Overset Grid Methods for the Scalar Wave Equation Oleksii Beznosov, Daniel Appel? Communications on Applied Mathematics and Computation    2021, 3 (3): 391-418.   DOI: 10.1007/s42967-020-00075-5 Abstract （9468）      PDF       Knowledge map    Save We present high order accurate numerical methods for the wave equation that combines efficient Hermite methods with geometrically flexible discontinuous Galerkin methods by using overset grids. Near boundaries we use thin boundary fitted curvilinear grids and in the volume we use Cartesian grids so that the computational complexity of the solvers approaches a structured Cartesian Hermite method. Unlike many other overset methods we do not need to add artificial dissipation but we find that the built-in dissipation of the Hermite and discontinuous Galerkin methods is sufficient to maintain the stability. By numerical experiments we demonstrate the stability, accuracy, efficiency, and the applicability of the methods to forward and inverse problems. Reference | Related Articles | Metrics | Comments（0）

Select

Energy-Based Discontinuous Galerkin Difference Methods for Second-Order Wave Equations Lu Zhang, Daniel Appelö, Thomas Hagstrom Communications on Applied Mathematics and Computation    2022, 4 (3): 855-879.   DOI: 10.1007/s42967-021-00149-y Abstract （1706）      PDF       Knowledge map    Save We combine the newly constructed Galerkin difference basis with the energy-based discontinuous Galerkin method for wave equations in second-order form. The approximation properties of the resulting method are excellent and the allowable time steps are large compared to traditional discontinuous Galerkin methods. The one drawback of the combined approach is the cost of inversion of the local mass matrix. We demonstrate that for constant coefficient problems on Cartesian meshes this bottleneck can be removed by the use of a modified Galerkin difference basis. For variable coefficients or non-Cartesian meshes this technique is not possible and we instead use the preconditioned conjugate gradient method to iteratively invert the mass matrices. With a careful choice of preconditioner we can demonstrate optimal complexity, albeit with a larger constant. Reference | Related Articles | Metrics | Comments（0）

Select

Von Neumann Stability Analysis of DG-Like and PNPM-Like Schemes for PDEs with Globally Curl-Preserving Evolution of Vector Fields Dinshaw S. Balsara, Roger Käppeli Communications on Applied Mathematics and Computation    2022, 4 (3): 945-985.   DOI: 10.1007/s42967-021-00166-x Abstract （1667）      PDF       Knowledge map    Save This paper examines a class of involution-constrained PDEs where some part of the PDE system evolves a vector field whose curl remains zero or grows in proportion to specified source terms. Such PDEs are referred to as curl-free or curl-preserving, respectively. They arise very frequently in equations for hyperelasticity and compressible multiphase flow, in certain formulations of general relativity and in the numerical solution of Schrödinger's equation. Experience has shown that if nothing special is done to account for the curl-preserving vector field, it can blow up in a finite amount of simulation time. In this paper, we catalogue a class of DG-like schemes for such PDEs. To retain the globally curl-free or curl-preserving constraints, the components of the vector field, as well as their higher moments, must be collocated at the edges of the mesh. They are updated using potentials collocated at the vertices of the mesh. The resulting schemes: (i) do not blow up even after very long integration times, (ii) do not need any special cleaning treatment, (iii) can operate with large explicit timesteps, (iv) do not require the solution of an elliptic system and (v) can be extended to higher orders using DG-like methods. The methods rely on a special curl-preserving reconstruction and they also rely on multidimensional upwinding. The Galerkin projection, highly crucial to the design of a DG method, is now conducted at the edges of the mesh and yields a weak form update that uses potentials obtained at the vertices of the mesh with the help of a multidimensional Riemann solver. A von Neumann stability analysis of the curl-preserving methods is conducted and the limiting CFL numbers of this entire family of methods are catalogued in this work. The stability analysis confirms that with the increasing order of accuracy, our novel curl-free methods have superlative phase accuracy while substantially reducing dissipation. We also show that PNPM-like methods, which only evolve the lower moments while reconstructing the higher moments, retain much of the excellent wave propagation characteristics of the DG-like methods while offering a much larger CFL number and lower computational complexity. The quadratic energy preservation of these methods is also shown to be excellent, especially at higher orders. The methods are also shown to be curl-preserving over long integration times. Reference | Related Articles | Metrics | Comments（0）

Select

Parallel Active Subspace Decomposition for Tensor Robust Principal Component Analysis Michael K. Ng, Xue-Zhong Wang Communications on Applied Mathematics and Computation    2021, 3 (2): 221-241.   DOI: 10.1007/s42967-020-00063-9 Abstract （7013）      PDF       Knowledge map    Save Tensor robust principal component analysis has received a substantial amount of attention in various felds. Most existing methods, normally relying on tensor nuclear norm minimization, need to pay an expensive computational cost due to multiple singular value decompositions at each iteration. To overcome the drawback, we propose a scalable and efcient method, named parallel active subspace decomposition, which divides the unfolding along each mode of the tensor into a columnwise orthonormal matrix (active subspace) and another small-size matrix in parallel. Such a transformation leads to a nonconvex optimization problem in which the scale of nuclear norm minimization is generally much smaller than that in the original problem. We solve the optimization problem by an alternating direction method of multipliers and show that the iterates can be convergent within the given stopping criterion and the convergent solution is close to the global optimum solution within the prescribed bound. Experimental results are given to demonstrate that the performance of the proposed model is better than the state-of-the-art methods. Reference | Related Articles | Metrics | Comments（0）

Select

Local Discontinuous Galerkin Methods for the abcd Nonlinear Boussinesq System Jiawei Sun, Shusen Xie, Yulong Xing Communications on Applied Mathematics and Computation    2022, 4 (2): 381-416.   DOI: 10.1007/s42967-021-00119-4 Abstract （2154）      PDF       Knowledge map    Save Boussinesq type equations have been widely studied to model the surface water wave. In this paper, we consider the abcd Boussinesq system which is a family of Boussinesq type equations including many well-known models such as the classical Boussinesq system, the BBM-BBM system, the Bona-Smith system, etc. We propose local discontinuous Galerkin (LDG) methods, with carefully chosen numerical fluxes, to numerically solve this abcd Boussinesq system. The main focus of this paper is to rigorously establish a priori error estimate of the proposed LDG methods for a wide range of the parameters a, b, c, d. Numerical experiments are shown to test the convergence rates, and to demonstrate that the proposed methods can simulate the head-on collision of traveling wave and finite time blow-up behavior well. Reference | Related Articles | Metrics | Comments（0）

Select

p-Multilevel Preconditioners for HHO Discretizations of the Stokes Equations with Static Condensation Lorenzo Botti, Daniele A. Di Pietro Communications on Applied Mathematics and Computation    2022, 4 (3): 783-822.   DOI: 10.1007/s42967-021-00142-5 Abstract （1769）      PDF       Knowledge map    Save We propose a p-multilevel preconditioner for hybrid high-order (HHO) discretizations of the Stokes equation, numerically assess its performance on two variants of the method, and compare with a classical discontinuous Galerkin scheme. An efficient implementation is proposed where coarse level operators are inherited using $ L^2 $-orthogonal projections defined over mesh faces and the restriction of the fine grid operators is performed recursively and matrix-free. Both h- and k-dependency are investigated tackling two- and three-dimensional problems on standard meshes and graded meshes. For the two HHO formulations, featuring discontinuous or hybrid pressure, we study how the combination of p-coarsening and static condensation influences the V-cycle iteration. In particular, two different static condensation procedures are considered for the discontinuous pressure HHO variant, resulting in global linear systems with a different number of unknowns and matrix non-zero entries. Interestingly, we show that the efficiency of the solution strategy might be impacted by static condensation options in the case of graded meshes. Reference | Related Articles | Metrics | Comments（0）

Select

Comparison of Semi-Lagrangian Discontinuous Galerkin Schemes for Linear and Nonlinear Transport Simulations Xiaofeng Cai, Wei Guo, Jing-Mei Qiu Communications on Applied Mathematics and Computation    2022, 4 (1): 3-33.   DOI: 10.1007/s42967-020-00088-0 Abstract （1384）      PDF       Knowledge map    Save Transport problems arise across diverse felds of science and engineering. Semi-Lagrangian (SL) discontinuous Galerkin (DG) methods are a class of high-order deterministic transport solvers that enjoy advantages of both the SL approach and the DG spatial discretization. In this paper, we review existing SLDG methods to date and compare numerically their performance. In particular, we make a comparison between the splitting and nonsplitting SLDG methods for multi-dimensional transport simulations. Through extensive numerical results, we ofer a practical guide for choosing optimal SLDG solvers for linear and nonlinear transport simulations. Reference | Related Articles | Metrics | Comments（0）

Select

CAMC Focused Section on Tensor Computation Delin Chu, Michael Ng, Liqun Qi, Qing-Wen Wang Communications on Applied Mathematics and Computation    2021, 3 (2): 199-199.   DOI: 10.1007/s42967-020-00113-2 Abstract （8015）      PDF       Knowledge map    Save Related Articles | Metrics | Comments（0）