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Published in last 1 year |  In last 2 years |  In last 3 years |  All Please wait a minute... For Selected: Download Citations EndNote Ris BibTeX Toggle Thumbnails 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 （14183）      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 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 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 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） 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 （7014）      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 Preface to Focused Section on Efcient HighcOrder Time Discretization Methods for Partial Diferential Equations Sebastiano Boscarino, Giuseppe Izzo, Lorenzo Pareschi, Giovanni Russo, Chi-Wang Shu Communications on Applied Mathematics and Computation    2021, 3 (4): 605-605.   DOI: 10.1007/s42967-021-00164-z Abstract （6846）      PDF       Knowledge map    Save 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 （6218）      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 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 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 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 Linear-Quadratic Optimal Control Problems for Mean-Field Stochastic Differential Equation with Lévy Process Hong Xiong, Maoning Tang, Qingxin Meng Communications on Applied Mathematics and Computation    2022, 4 (4): 1386-1415.   DOI: 10.1007/s42967-021-00181-y Abstract （3472）      PDF       Knowledge map    Save This paper investigates a linear-quadratic mean-field stochastic optimal control problem under both positive definite case and indefinite case where the controlled systems are mean-field stochastic differential equations driven by a Brownian motion and Teugels martingales associated with Lévy processes. In either case, we obtain the optimality system for the optimal controls in open-loop form, and by means of a decoupling technique, we obtain the optimal controls in closed-loop form which can be represented by two Riccati differential equations. Moreover, the solvability of the optimality system and the Riccati equations are also obtained under both positive definite case and indefinite case. 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 （3403）      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 Perturbation Analysis for t-Product-Based Tensor Inverse, Moore-Penrose Inverse and Tensor System Zhengbang Cao, Pengpeng Xie Communications on Applied Mathematics and Computation    2022, 4 (4): 1441-1456.   DOI: 10.1007/s42967-022-00186-1 Abstract （3360）      PDF       Knowledge map    Save This paper establishes some perturbation analysis for the tensor inverse, the tensor Moore-Penrose inverse, and the tensor system based on the t-product. In the settings of structured perturbations, we generalize the Sherman-Morrison-Woodbury (SMW) formula to the t-product tensor scenarios. The SMW formula can be used to perform the sensitivity analysis for a multilinear system of equations. Reference | Related Articles | Metrics | Comments（0） Select Space-Fractional Diffusion with Variable Order and Diffusivity:Discretization and Direct Solution Strategies Hasnaa Alzahrani, George Turkiyyah, Omar Knio, David Keyes Communications on Applied Mathematics and Computation    2022, 4 (4): 1416-1440.   DOI: 10.1007/s42967-021-00184-9 Abstract （3341）      PDF       Knowledge map    Save We consider the multidimensional space-fractional diffusion equations with spatially varying diffusivity and fractional order. Significant computational challenges are encountered when solving these equations due to the kernel singularity in the fractional integral operator and the resulting dense discretized operators, which quickly become prohibitively expensive to handle because of their memory and arithmetic complexities. In this work, we present a singularity-aware discretization scheme that regularizes the singular integrals through a singularity subtraction technique adapted to the spatial variability of diffusivity and fractional order. This regularization strategy is conveniently formulated as a sparse matrix correction that is added to the dense operator, and is applicable to different formulations of fractional diffusion equations. We also present a block low rank representation to handle the dense matrix representations, by exploiting the ability to approximate blocks of the resulting formally dense matrix by low rank factorizations. A Cholesky factorization solver operates directly on this representation using the low rank blocks as its atomic computational tiles, and achieves high performance on multicore hardware. Numerical results show that the singularity treatment is robust, substantially reduces discretization errors, and attains the first-order convergence rate allowed by the regularity of the solutions. They also show that considerable savings are obtained in storage (O(N1.5)) and computational cost (O(N2)) compared to dense factorizations. This translates to orders-of-magnitude savings in memory and time on multidimensional problems, and shows that the proposed methods offer practical tools for tackling large nonlocal fractional diffusion simulations. Reference | Related Articles | Metrics | Comments（0） Select Modeling Fast Diffusion Processes in Time Integration of Stiff Stochastic Differential Equations Xiaoying Han, Habib N. Najm Communications on Applied Mathematics and Computation    2022, 4 (4): 1457-1493.   DOI: 10.1007/s42967-022-00188-z Abstract （3333）      PDF       Knowledge map    Save Numerical algorithms for stiff stochastic differential equations are developed using linear approximations of the fast diffusion processes, under the assumption of decoupling between fast and slow processes. Three numerical schemes are proposed, all of which are based on the linearized formulation albeit with different degrees of approximation. The schemes are of comparable complexity to the classical explicit Euler-Maruyama scheme but can achieve better accuracy at larger time steps in stiff systems. Convergence analysis is conducted for one of the schemes, that shows it to have a strong convergence order of 1/2 and a weak convergence order of 1. Approximations arriving at the other two schemes are discussed. Numerical experiments are carried out to examine the convergence of the schemes proposed on model problems. Reference | Related Articles | Metrics | Comments（0） Select Two Energy-Preserving Compact Finite Difference Schemes for the Nonlinear Fourth-Order Wave Equation Xiaoyi Liu, Tingchun Wang, Shilong Jin, Qiaoqiao Xu Communications on Applied Mathematics and Computation    2022, 4 (4): 1509-1530.   DOI: 10.1007/s42967-022-00193-2 Abstract （3322）      PDF       Knowledge map    Save In this paper, two fourth-order compact finite difference schemes are derived to solve the nonlinear fourth-order wave equation which can be viewed as a generalized model from the nonlinear beam equation. Differing from the existing compact finite difference schemes which preserve the total energy in a recursive sense, the new schemes are proved to perfectly preserve the total energy in the discrete sense. By using the standard energy method and the cut-off function technique, the optimal error estimates of the numerical solutions are established, and the convergence rates are of O(h4 + τ2) with mesh-size h and time-step τ. In order to improve the computational efficiency, an iterative algorithm is proposed as the outer solver and the double sweep method for pentadiagonal linear algebraic equations is introduced as the inner solver to solve the nonlinear difference schemes at each time step. The convergence of the iterative algorithm is also rigorously analyzed. Several numerical results are carried out to test the error estimates and conservative properties. Reference | Related Articles | Metrics | Comments（0） Select Dynamical Soliton Wave Structures of One-Dimensional Lie Subalgebras via Group-Invariant Solutions of a Higher-Dimensional Soliton Equation with Various Applications in Ocean Physics and Mechatronics Engineering Oke Davies Adeyemo, Chaudry Masood Khalique Communications on Applied Mathematics and Computation    2022, 4 (4): 1531-1582.   DOI: 10.1007/s42967-022-00195-0 Abstract （3275）      PDF       Knowledge map    Save Having realized various significant roles that higher-dimensional nonlinear partial differential equations (NLPDEs) play in engineering, we analytically investigate in this paper, a higher-dimensional soliton equation, with applications particularly in ocean physics and mechatronics (electrical electronics and mechanical) engineering. Infinitesimal generators of Lie point symmetries of the equation are computed using Lie group analysis of differential equations. In addition, we construct commutation as well as Lie adjoint representation tables for the nine-dimensional Lie algebra achieved. Further, a one-dimensional optimal system of Lie subalgebras is also presented for the soliton equation. This consequently enables us to generate abundant group-invariant solutions through the reduction of the understudy equation into various ordinary differential equations (ODEs). On solving the achieved nonlinear differential equations, we secure various solitonic solutions. In consequence, these solutions containing diverse mathematical functions furnish copious shapes of dynamical wave structures, ranging from periodic, kink and kink-shaped nanopteron, soliton (bright and dark) to breather waves with extensive wave collisions depicted. We physically interpreted the resulting soliton solutions by imploring graphical depictions in three dimensions, two dimensions and density plots. Moreover, the gained group-invariant solutions involved several arbitrary functions, thus exhibiting rich physical structures. We also implore the power series technique to solve part of the complicated differential equations and give valid comments on their results. Later, we outline some applications of our results in ocean physics and mechatronics engineering. Reference | Related Articles | Metrics | Comments（0） Select Dual Quaternions and Dual Quaternion Vectors Liqun Qi, Chen Ling, Hong Yan Communications on Applied Mathematics and Computation    2022, 4 (4): 1494-1508.   DOI: 10.1007/s42967-022-00189-y Abstract （3258）      PDF       Knowledge map    Save We introduce a total order and an absolute value function for dual numbers. The absolute value function of dual numbers takes dual number values, and has properties similar to those of the absolute value function of real numbers. We define the magnitude of a dual quaternion, as a dual number. Based upon these, we extend 1-norm, ∞-norm, and 2-norm to dual quaternion vectors. Reference | Related Articles | Metrics | Comments（0） Select Parametric Regression Approach for Gompertz Survival Times with Competing Risks H. Rehman, N. Chandra Communications on Applied Mathematics and Computation    2022, 4 (4): 1175-1190.   DOI: 10.1007/s42967-021-00154-1 Abstract （3180）      PDF       Knowledge map    Save Regression models play a vital role in the study of data regarding survival of subjects. The Cox proportional hazards model for regression analysis has been frequently used in survival modelling. In survival studies, it is also possible that survival time may occur with multiple occurrences of event or competing risks. The situation of competing risks arises when there are more than one mutually exclusive causes of death (or failure) for the person (or subject). In this paper, we developed a parametric regression model using Gompertz distribution via the Cox proportional hazards model with competing risks. We discussed point and interval estimation of unknown parameters and cumulative cause-specific hazard function with maximum-likelihood method and Bayesian method of estimation. The Bayes estimates are obtained based on non-informative priors and symmetric as well as asymmetric loss functions. To observe the finite sample behaviour of the proposed model under both estimation procedures, we carried out a Monte Carlo simulation analysis. To demonstrate our methodology, we also included real data analysis. Reference | Related Articles | Metrics | Comments（0） Select Reconstruction of a Heat Equation from One Point Observations H. Al Attas, A. Boumenir Communications on Applied Mathematics and Computation    2022, 4 (4): 1280-1292.   DOI: 10.1007/s42967-021-00174-x Abstract （3055）      PDF       Knowledge map    Save We are concerned with the reconstruction of the heat sink coefficient in a one-dimensional heat equation from the observations of solutions at the same point. This direct method which is based on spectral estimation and asymptotics techniques provides a fast algorithm and also an alternative to the Gelfand-Levitan theory or minimization procedures. Reference | Related Articles | Metrics | Comments（0）