摘要：In this paper,for the regularized Hermitian and skew-Hermitian splitting(RHSS)preconditioner introduced by Bai and Benzi(BIT Numer Math 57:287–311,2017)for the solution of saddle-point linear systems,we analyze the spectral properties of the preconditioned matrix when the regularization matrix is a special Hermitian positive semidefinite matrix which depends on certain parameters.We accurately describe the numbers of eigenvalues clustered at(0,0)and(2,0),if the iteration parameter is close to 0.An estimate about the condition number of the corresponding eigenvector matrix,which partly determines the convergence rate of the RHSS-preconditioned Krylov subspace method,is also studied in this work.
摘要：Bai et al.proposed the multistep Rayleigh quotient iteration(MRQI)as well as its inexact variant(IMRQI)in a recent work(Comput.Math.Appl.77:2396–2406,2019).These methods can be used to effectively compute an eigenpair of a Hermitian matrix.The convergence theorems of these methods were established under two conditions imposed on the initial guesses for the target eigenvalue and eigenvector.In this paper,we show that these two conditions can be merged into a relaxed one,so the convergence conditions in these theorems can be weakened,and the resulting convergence theorems are applicable to a broad class of matrices.In addition,we give detailed discussions about the new convergence condition and the corresponding estimates of the convergence errors,leading to rigorous convergence theories for both the MRQI and the IMRQI.
摘要：In this paper,by means of constructing the linear complementarity problems into the corresponding absolute value equation,we raise an iteration method,called as the nonlinear lopsided HSS-like modulus-based matrix splitting iteration method,for solving the linear complementarity problems whose coefficient matrix in R^(n×n)is large sparse and positive definite.From the convergence analysis,it is appreciable to see that the proposed method will converge to its accurate solution under appropriate conditions.Numerical examples demonstrate that the presented method precede to other methods in practical implementation.
摘要：In this paper,we study the nonlinear matrix equation X-A^(H)X^(-1)A=Q,where A,Q∈C^(n×n),Q is a Hermitian positive definite matrix and X∈C^(n×n)is an unknown matrix.We prove that the equation always has a unique Hermitian positive definite solution.We present two structure-preserving-doubling like algorithms to find the Hermitian positive definite solution of the equation,and the convergence theories are established.Finally,we show the effectiveness of the algorithms by numerical experiments.
摘要：For two-dimensional(2D)time fractional diffusion equations,we construct a numerical method based on a local discontinuous Galerkin(LDG)method in space and a finite differ-ence scheme in time.We investigate the numerical stability and convergence of the method for both rectangular and triangular meshes and show that the method is unconditionally stable.Numerical results indicate the effectiveness and accuracy of the method and con-firm the analysis.
摘要：Anomalous diffusion is a phenomenon that cannot be modeled accurately by second-order diffusion equations,but is better described by fractional diffusion models.The nonlocal nature of the fractional diffusion operators makes substantially more difficult the mathemati-cal analysis of these models and the establishment of suitable numerical schemes.This paper proposes and analyzes the first finite difference method for solving variable-coefficient one-dimensional(steady state)fractional differential equations(DEs)with two-sided fractional derivatives(FDs).The proposed scheme combines first-order forward and backward Euler methods for approximating the left-sided FD when the right-sided FD is approximated by two consecutive applications of the first-order backward Euler method.Our scheme reduces to the standard second-order central difference in the absence of FDs.The existence and uniqueness of the numerical solution are proved,and truncation errors of order h are demonstrated(h denotes the maximum space step size).The numerical tests illustrate the global 0(h)accu-racy,except for nonsmooth cases which,as expected,have deteriorated convergence rates.
摘要：The main aim of this paper is to analyze the numerical method based upon the spectral element technique for the numerical solution of the fractional advection-diffusion equa-tion.The time variable has been discretized by a second-order finite difference procedure.The stability and the convergence of the semi-discrete formula have been proven.Then,the spatial variable of the main PDEs is approximated by the spectral element method.The convergence order of the fully discrete scheme is studied.The basis functions of the spectral element method are based upon a class of Legendre polynomials.The numerical experiments confirm the theoretical results.
摘要：The H-tensor is a new developed concept in tensor analysis and it is an extension of the M-tensor.In this paper,we present some criteria for identifying nonsingular H-tensors and give two numerical examples.
摘要：In this article,some high-order local discontinuous Galerkin(LDG)schemes based on some second-order θ approximation formulas in time are presented to solve a two-dimen-sional nonlinear fractional diffusion equation.The unconditional stability of the LDG scheme is proved,and an a priori error estimate with O(h^(k+1)+At^(2))is derived,where k≥0 denotes the index of the basis function.Extensive numerical results with Q^(k)(k=0,1,2,3)elements are provided to confirm our theoretical results,which also show that the second-order convergence rate in time is not impacted by the changed parameter θ.
摘要：In this article,discrete variants of several results from vector calculus are studied for clas-sical finite difference summation by parts operators in two and three space dimensions.It is shown that existence theorems for scalar/vector potentials of irrotational/solenoidal vector fields cannot hold discretely because of grid oscillations,which are characterised explicitly.This results in a non-vanishing remainder associated with grid oscillations in the discrete Helmholtz Hodge decomposition.Nevertheless,iterative numerical methods based on an interpretation of the Helmholtz Hodge decomposition via orthogonal projections are pro-posed and applied successfully.In numerical experiments,the discrete remainder vanishes and the potentials converge with the same order of accuracy as usual in other first-order partial differential equations.Motivated by the successful application of the Helmholtz Hodge decomposition in theoretical plasma physics,applications to the discrete analysis of magnetohydrodynamic(MHD) wave modes are presented and discussed.
摘要：We propose an explicit,single-step discontinuous Galerkin method on moving grids using the arbitrary Lagrangian-Eulerian approach for one-dimensional Euler equations.The grid is moved with the local fluid velocity modified by some smoothing,which is found to con-siderably reduce the numerical dissipation introduced by Riemann solvers.The scheme preserves constant states for any mesh motion and we also study its positivity preservation property.Local grid refinement and coarsening are performed to maintain the mesh qual-ity and avoid the appearance of very small or large cells.Second,higher order methods are developed and several test cases are provided to demonstrate the accuracy of the proposed scheme.
摘要：Tensors are multi-arrays with more than two indices.In the last decade or so,many con-cepts and results in matrix theory-some of which are nontrivial-have been extended to tensors and have a wide range of applications.The present focused section entitled"tensor computation"in Communications on Applied Mathematics and Computation(CAMC)covers tensors in inversion,principal component analysis,spectral hyperspectral theory,decompositions,and rank structure.We do hope this focused section can offer fresh stimuli for the community of tensor computation and theory to promote and develop cutting-edge research in this important field.
摘要：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.
摘要：Tensor robust principal component analysis has received a substantial amount of attention in various fields.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 efficient 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.
摘要：The celebrated Erdos-Ko-Rado theorem states that given n≥2k,every intersecting k-uni-n-1 form hypergraph G on n vertices has at most(n-1 k-1)edges.This paper states spectral versions of the Erd6s-_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 Aα(G)and 2*(G)are presented,where Aα(G)=αD(G)+(1-α).A(G)is a convex linear combination of the degree diagonal tensor D(G)and the adjacency tensor A(G)for 0≤α<1,and Q^(*)(G)is the incidence 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.
摘要：In this article,two new algorithms are presented that convert a given data tensor train into either a Tucker decomposition with orthogonal matrix factors or a multi-scale entanglement renormalization ansatz(MERA).The Tucker core tensor is never explicitly computed but stored as a tensor train instead,resulting in both computationally and storage efficient algorithms.Both the multilinear Tucker-ranks as well as the MERA-ranks are automatically determined by the algorithm for a given upper bound on the relative approximation error.In addition,an iterative algorithm with low computational complexity based on solving an orthogonal Procrustes problem is proposed for the first time to retrieve optimal rank-lowering disentangler tensors,which are a crucial component in the construction of a low-rank MERA.Numerical experiments demonstrate the effectiveness of the proposed algorithms together with the potential storage benefit of a low-rank MERA over a tensor train.
摘要：The purpose of this note is to give a linear algebra algorithm to find out if a rank of a given tensor over a field F is at most k over the algebraic closure of F,where K is a given positive integer.We estimate the arithmetic complexity of our algorithm.
摘要：This focused section,edited by Wei Cai,Weinan E,Jan Hesthaven,Dongbin Xiu and Le-xing Ying,presents several recent developments in improving deep neural networks(DNNs)for scientific computing and classification applications.Original research works on DNNs are included on the following topics:drop-activations as regularization,non-intrusive corrections for classifier,complexity of finding sparse representations for high dimensional functions,and discovery of phase field models for multiphase material studies.
摘要：Overfitting frequently occurs in deep learning.In this paper,we propose a novel regularization method called drop-activation to reduce overfitting and improve generalization.The key idea is to drop nonlinear activation functions by setting them to be identity functions randomly during training time.During testing,we use a deterministic network with a new activation function to encode the average effect of dropping activations randomly.Our theoretical analyses support the regularization effect of drop-activation as implicit parameter reduction and verify its capability to be used together with batch normalization(Iolfe and Szegedy in Batch normalization:accelerating deep network training by reducing internal covariate shift.arXiv:1502.03167,2015).The experimental results on CIFAR10,CIFAR100,SVHN,EMNIST,and ImageNet show that drop-activation generally improves the performance of popular neural network architectures for the image classification task.Furthermore,as a regularizer drop-activation can be used in harmony with standard training and regularization techniques such as batch normalization and AutoAugment(Cubuk et al.in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition,pp.113-123,2019).The code is available at https://github.com/LeungSamWai/Drop-Activ ation.
摘要：The task of using the machine learning to approximate the mapping x→Σi=1^d xi^(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 find the sparse network embedded in the dense network is similar to finding the needle in a haystack,using a given number of samples of the function.We demonstrate that the cost(measured by sample complexity)of finding 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 find that the use of the explicit regularization becomes increasingly more important as d increases.The numerically observed sample complexity with explicit regularization scales as G(d^(2.5)),which is in fact better than the theoretically predicted sample complexity that scales as 0(d^(4)).Without the explicit regularization(also called the implicit regularization),the numerically observed sample complexity is significantly higher and is close to 0(d^(4.5)).
摘要：A novel correction algorithm is proposed for multi-class classification problems with corrupted training data.The algorithm is non-intrusive,in the sense that it post-processes a trained classification 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 sufficiently large,we theoretically prove(in the limiting case)and numerically show that the corrected models deliver correct classification results as if there is no corruption in the training data.For datasets of finite size,the corrected models produce significantly better recovery results,compared to the models without the correction algorithm.All of the theoretical findings in the paper are verified by our numerical examples.
摘要：In this paper,we introduce a new deep learning framework for discovering the phase-field models from existing image data.The new framework embraces the approximation power of physics informed neural networks(PINNs)and the computational efficiency of the pseudo-spectral methods,which we named pseudo-spectral PINN or SPINN.Unlike the baseline PINN,the pseudo-spectral PINN has several advantages.First of all,it requires less training data.A minimum of two temporal snapshots with uniform spatial resolution would be adequate.Secondly,it is computationally efficient,as the pseudo-spectral method is used for spatial discretization.Thirdly,it requires less trainable parameters compared with the baseline PINN,which significantly simplifies the training process and potentially assures fewer local minima or saddle points.We illustrate the effectiveness of pseudo-spectral PINN through several numerical examples.The newly proposed pseudo-spectral PINN is rather general,and it can be readily applied to discover other FDE-based models from image data.
摘要：Regularization methods have been substantially applied in image restoration due to the ill-posedness of the image restoration problem.Different assumptions or priors on images are applied in the construction of image regularization methods.In recent years,matrix low-rank approximation has been successfully introduced in the image denoising problem and significant denoising effects have been achieved.Low-rank matrix minimization is an NP-hard problem and it is often replaced with the matrix’s weighted nuclear norm minimization(WNNM).The assumption that an image contains an extensive amount of self-similarity is the basis for the construction of the matrix low-rank approximation-based image denoising method.In this paper,we develop a model for image restoration using the sum of block matching matrices’weighted nuclear norm to be the regularization term in the cost function.An alternating iterative algorithm is designed to solve the proposed model and the convergence analyses of the algorithm are also presented.Numerical experiments show that the proposed method can recover the images much better than the existing regularization methods in terms of both recovered quantities and visual qualities.
摘要：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.
摘要：In this paper,we present a modulus-based multisplitting iteration method based on multisplitting of the system matrix for a class of weakly nonlinear complementarity problem.And we prove the convergence of the method when the system matrix is an H_(+)-matrix.Finally,we give two numerical examples.
摘要：The aim of this paper is to develop a fully discrete local discontinuous Galerkin method to solve a class of variable-order fractional diffusion problems.The scheme is discretized by a weighted-shifted Grünwald formula in the temporal discretization and a local discontinuous Galerkin method in the spatial direction.The stability and the L^(2)-convergence of the scheme are proved for all variable-orderα(t)∈(0,1).The proposed method is of accuracy-order O(τ^(3)+h^(k+1)),whereτ,h,and k are the temporal step size,the spatial step size,and the degree of piecewise P^(k)polynomials,respectively.Some numerical tests are provided to illustrate the accuracy and the capability of the scheme.
摘要：We introduce adaptive moving mesh central-upwind schemes for one-and two-dimensional hyperbolic systems of conservation and balance laws.The proposed methods consist of three steps.First,the solution is evolved by solving the studied system by the second-order semi-discrete central-upwind scheme on either the one-dimensional nonuniform grid or the two-dimensional structured quadrilateral mesh.When the evolution step is complete,the grid points are redistributed according to the moving mesh differential equation.Finally,the evolved solution is projected onto the new mesh in a conservative manner.The resulting adaptive moving mesh methods are applied to the one-and two-dimensional Euler equations of gas dynamics and granular hydrodynamics systems.Our numerical results demonstrate that in both cases,the adaptive moving mesh central-upwind schemes outperform their uniform mesh counterparts.
摘要：In this article,novel smoothness indicators are presented for calculating the nonlinear weights of the weighted essentially non-oscillatory scheme to approximate the viscosity numerical solutions of Hamilton-Jacobi equations.These novel smoothness indicators are constructed from the derivatives of reconstructed polynomials over each sub-stencil.The constructed smoothness indicators measure the arc-length of the reconstructed polynomials so that the new nonlinear weights could get less absolute truncation error and give a high-resolution numerical solution.Extensive numerical tests are conducted and presented to show the performance capability and the numerical accuracy of the proposed scheme with the comparison to the classical WENO scheme.
摘要：The purpose of this paper is to study the oscillation of second-order half-linear neutral differential equations with advanced argument of the form(r(t)((y(t)+p(t)y(τ(t)))'')^(α))''+q(t)yα(σ(t))=0,t≥t_(0),when∫^(∞)r^(−1/α)(s)ds<∞.We obtain sufficient conditions for the oscillation of the studied equations by the inequality principle and the Riccati transformation.An example is provided to illustrate the results.
摘要：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.
应用数学与计算数学学报(英文)的期刊信息
曾用名：应用数学与计算数学学报
创刊时间：1986
地区：CN
语言：中文
热门主题：FRACTIONAL
NONLOCAL
DISCONTINUOUS
GALERKIN
METHODS
APPROXIMATION
SPACE
DERIVATIVE
CONVERGENCE
LAPLACIAN
