Relaxation methods for solving the tensor equation arising from the higher‐order Markov chains

In this paper, we propose several relaxation algorithms for solving the tensor equation arising from the higher‐order Markov chain and the multilinear PageRank. The semi‐symmetrization technique on the original equation is also employed to modify the proposed algorithms. The convergence analysis is given for the proposed algorithms. It is shown that the new algorithms are more efficient than the existing ones by some numerical experiments when relaxation parameters are chosen suitably.     

Termination criteria for inexact fixed‐point schemes

We analyze inexact fixed‐point iterations where the generating function contains an inexact solve of an equation system to answer the question of how tolerances for the inner solves influence the iteration error of the outer fixed‐point iteration. Important applications are the Picard iteration and partitioned fluid‐structure interaction. For the analysis, the iteration is modeled as a perturbed fixed‐point iteration, and existing analysis is extended to the nested case x = F ( S ( x )). We prove that if the iteration converges, it converges to the exact solution irrespective of the tolerance in the inner systems, provided that a nonstandard relative termination criterion is employed, whereas standard relative and absolute criteria do not have this property. Numerical results demonstrate the effectiveness of the approach with the nonstandard termination criterion. Copyright © 2015 John Wiley & Sons, Ltd.     

Convergence of a transition probability tensor of a higher‐order Markov chain to the stationary probability vector

In this paper, first we introduce a new tensor product for a transition probability tensor originating from a higher‐order Markov chain. Subsequently, some properties of the new tensor product are explained, and its relationship with the stationary probability vector is studied. Also, similarity between results obtained by this new product and the first‐order case is shown. Furthermore, we prove the convergence of a transition probability tensor to the stationary probability vector. Finally, we show how to achieve a stationary probability vector with some numerical examples and make some comparison between the proposed method and another existing method for obtaining stationary probability vectors. Copyright © 2016 John Wiley & Sons, Ltd.     

Shifted power method for computing tensor H‐eigenpairs

In this paper, we propose a shifted symmetric higher‐order power method for computing the H‐eigenpairs of a real symmetric even‐order tensor. The local convergence of the method is proved. In addition, by utilizing the fixed‐point analysis, we can characterize exactly which H‐eigenpairs can be found and which cannot be found by the method. Numerical examples are presented to illustrate the performance of the method. Copyright © 2016 John Wiley & Sons, Ltd.     

Interior‐point methods and preconditioning for PDE‐constrained optimization problems involving sparsity terms

Partial differential equation (PDE)–constrained optimization problems with control or state constraints are challenging from an analytical and numerical perspective. The combination of these constraints with a sparsity‐promoting L¹ term within the objective function requires sophisticated optimization methods. We propose the use of an interior‐point scheme applied to a smoothed reformulation of the discretized problem and illustrate that such a scheme exhibits robust performance with respect to parameter changes. To increase the potency of this method, we introduce fast and efficient preconditioners that enable us to solve problems from a number of PDE applications in low iteration numbers and CPU times, even when the parameters involved are altered dramatically.     

Uzawa methods for a class of block three‐by‐three saddle‐point problems

In this work, we consider numerical methods for solving a class of block three‐by‐three saddle‐point problems, which arise from finite element methods for solving time‐dependent Maxwell equations and some other applications. The direct extension of the Uzawa method for solving this block three‐by‐three saddle‐point problem requires the exact solution of a symmetric indefinite system of linear equations at each step. To avoid heavy computations at each step, we propose an inexact Uzawa method, which solves the symmetric indefinite linear system in some inexact way. Under suitable assumptions, we show that the inexact Uzawa method converges to the unique solution of the saddle‐point problem within the approximation level. Two special algorithms are customized for the inexact Uzawa method combining the splitting iteration method and a preconditioning technique, respectively. Numerical experiments are presented, which demonstrated the usefulness of the inexact Uzawa method and the two customized algorithms.     

Generalized skew‐Hermitian triangular splitting iteration methods for saddle‐point linear systems

A generalized skew‐Hermitian triangular splitting iteration method is presented for solving non‐Hermitian linear systems with strong skew‐Hermitian parts. We study the convergence of the generalized skew‐Hermitian triangular splitting iteration methods for non‐Hermitian positive definite linear systems, as well as spectrum distribution of the preconditioned matrix with respect to the preconditioner induced from the generalized skew‐Hermitian triangular splitting. Then the generalized skew‐Hermitian triangular splitting iteration method is applied to non‐Hermitian positive semidefinite saddle‐point linear systems, and we prove its convergence under suitable restrictions on the iteration parameters. By specially choosing the values of the iteration parameters, we obtain a few of the existing iteration methods in the literature. Numerical results show that the generalized skew‐Hermitian triangular splitting iteration methods are effective for solving non‐Hermitian saddle‐point linear systems with strong skew‐Hermitian parts. Copyright © 2013 John Wiley & Sons, Ltd.     

Optimal parameters in the HSS‐like methods for saddle‐point problems

For the Hermitian and skew‐Hermitian splitting iteration method and its accelerated variant for solving the large sparse saddle‐point problems, we compute their quasi‐optimal iteration parameters and the corresponding quasi‐optimal convergence factors for the more practical but more difficult case that the (1, 1)‐block of the saddle‐point matrix is not algebraically equivalent to the identity matrix. In addition, the algebraic behaviors and the clustering properties of the eigenvalues of the preconditioned matrices with respect to these two iterations are investigated in detail, and the formulas for computing good iteration parameters are given under certain principle for optimizing the distribution of the eigenvalues. Copyright © 2008 John Wiley & Sons, Ltd.     

Graded Galerkin methods for the high‐order convection‐diffusion problem

We develop a Galerkin method using the Hermite spline on an admissible graded mesh for solving the high‐order singular perturbation problem of the convection‐diffusion type. We identify a special function class to which the solution of the convection‐diffusion problem belongs and characterize the approximation order of the Hermite spline for such a function class. The approximation order is then used to establish the optimal order of uniform convergence for the Galerkin method. Numerical results are presented to confirm the theoretical estimate.© 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

On the perturbation of rank‐one symmetric tensors

The problem of symmetric rank‐one approximation of symmetric tensors is important in independent components analysis, also known as blind source separation, as well as polynomial optimization. We derive several perturbative results that are relevant to the well‐posedness of recovering rank‐one structure from approximately‐rank‐one symmetric tensors. We also specialize the analysis of the shifted symmetric higher‐order power method, an algorithm for computing symmetric tensor eigenvectors, to approximately‐rank‐one symmetric tensors. Copyright © 2012 John Wiley & Sons, Ltd.     

On a new family of high‐order iterative methods for the matrix pth root

The main goal of this paper is to approximate the principal pth root of a matrix by using a family of high‐order iterative methods. We analyse the semi‐local convergence and the speed of convergence of these methods. Concerning stability, it is well known that even the simplified Newton method is unstable. Despite it, we present stable versions of our family of algorithms. We test numerically the methods: we check the numerical robustness and stability by considering matrices that are close to be singular and badly conditioned. We find algorithms of the family with better numerical behavior than the Newton and the Halley methods. These two algorithms are basically the iterative methods proposed in the literature to solve this problem. Copyright © 2015 John Wiley & Sons, Ltd.     

High‐resolution relaxation scheme for the two‐dimensional Riemann problems in gas dynamics

We present a new relaxation method for the numerical approximation of the two‐dimensional Riemann problems in gas dynamics. The novel feature of the technique proposed here is that it does not require either a Riemann solver or a characteristics decomposition. The high resolution of the method is achieved by using a third‐order reconstruction for the space discretization and a third‐order TVD Runge‐Kutta scheme for the time integration. Numerical experiments, using several configurations of Riemann problems in gas dynamics, are included to confirm the high resolution of the new relaxation scheme. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Uniform convergence of Galerkin‐multigrid methods for nonconforming finite elements for second‐order problems with less than full elliptic regularity

In this article we prove uniform convergence estimates for the recently developed Galerkin‐multigrid methods for nonconforming finite elements for second‐order problems with less than full elliptic regularity. These multigrid methods are defined in terms of the “Galerkin approach,” where quadratic forms over coarse grids are constructed using the quadratic form on the finest grid and iterated coarse‐to‐fine intergrid transfer operators. Previously, uniform estimates were obtained for problems with full elliptic regularity, whereas these estimates are derived with less than full elliptic regularity here. Applications to the nonconforming P₁, rotated Q₁, and Wilson finite elements are analyzed. The result applies to the mixed method based on finite elements that are equivalent to these nonconforming elements. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 203–217, 2002; DOI 10.1002/num.10004     

Eigenvalue problems for nonlinear third‐order m‐point p‐Laplacian dynamic equations on time scales

This work deals with the existence and uniqueness of a nontrivial solution for the third‐order p‐Laplacian m‐point eigenvalue problems on time scales. We find several sufficient conditions of the existence and uniqueness of nontrivial solution of eigenvalue problems when λ is in some interval. The proofs are based on the nonlinear alternative of Leray–Schauder. To illustrate the results, some examples are included. Copyright © 2014 John Wiley & Sons, Ltd.     

Spectral collocation methods for nonlinear weakly singular Volterra integro‐differential equations

In this paper, a spectral collocation approximation is proposed for neutral and nonlinear weakly singular Volterra integro‐differential equations (VIDEs) with non‐smooth solutions. We use some suitable variable transformations to change the original equation into a new equation, so that the solution of the resulting equation possesses better regularity, and the the Jacobi orthogonal polynomial theory can be applied conveniently. Under reasonable assumptions on the nonlinearity, we carry out a rigorous error analysis in L^∞ norm and weighted L² norm. To perform the numerical simulations, some test examples (linear and nonlinear) are considered with nonsmooth solutions, and numerical results are presented. Further more, the comparative study of the proposed methods with some existing numerical methods is provided.     

Fast tensor method for summation of long‐range potentials on 3D lattices with defects

In this paper, we present a method for fast summation of long‐range potentials on 3D lattices with multiple defects and having non‐rectangular geometries, based on rank‐structured tensor representations. This is a significant generalization of our recent technique for the grid‐based summation of electrostatic potentials on the rectangular L × L × L lattices by using the canonical tensor decompositions and yielding the O(L) computational complexity instead of O(L³) by traditional approaches. The resulting lattice sum is calculated as a Tucker or canonical representation whose directional vectors are assembled by the 1D summation of the generating vectors for the shifted reference tensor, once precomputed on large N × N × N representation grid in a 3D bounding box. The tensor numerical treatment of defects is performed in an algebraic way by simple summation of tensors in the canonical or Tucker formats. To diminish the considerable increase in the tensor rank of the resulting potential sum, the ?‐rank reduction procedure is applied based on the generalized reduced higher‐order SVD scheme. For the reduced higher‐order SVD approximation to a sum of canonical/Tucker tensors, we prove the stable error bounds in the relative norm in terms of discarded singular values of the side matrices. The required storage scales linearly in the 1D grid‐size, O(N), while the numerical cost is estimated by O(NL). The approach applies to a general class of kernel functions including those for the Newton, Slater, Yukawa, Lennard‐Jones, and dipole‐dipole interactions. Numerical tests confirm the efficiency of the presented tensor summation method; we demonstrate that a sum of millions of Newton kernels on a 3D lattice with defects/impurities can be computed in seconds in Matlab implementation. The tensor approach is advantageous in further functional calculus with the lattice potential sums represented on a 3D grid, like integration or differentiation, using tensor arithmetics of 1D complexity. Copyright © 2015 John Wiley & Sons, Ltd.     

Homoclinic solutions for nonlinear general fourth‐order differential equations

This work provides sufficient conditions for the existence of homoclinic solutions of fourth‐order nonlinear ordinary differential equations. Using Green's functions, we formulate a new modified integral equation that is equivalent to the original nonlinear equation. In an adequate function space, the corresponding nonlinear integral operator is compact, and it is proved an existence result by Schauder's fixed point theorem. Copyright © 2017 John Wiley & Sons, Ltd.     

A layer adaptive B‐spline collocation method for singularly perturbed one‐dimensional parabolic problem with a boundary turning point

In this article, we develop a parameter uniform numerical method for a class of singularly perturbed parabolic equations with a multiple boundary turning point on a rectangular domain. The coefficient of the first derivative with respect to x is given by the formula a₀(x, t)x^p, where a₀(x, t) ≥ α > 0 and the parameter p ∈ [1,∞) takes the arbitrary value. For small values of the parameter ε, the solution of this particular class of problem exhibits the parabolic boundary layer in a neighborhood of the boundary x = 0 of the domain. We use the implicit Euler method to discretize the temporal variable on uniform mesh and a B‐spline collocation method defined on piecewise uniform Shishkin mesh to discretize the spatial variable. Asymptotic bounds for the derivatives of the solution are established by decomposing the solution into smooth and singular component. These bounds are applied in the convergence analysis of the proposed scheme on Shishkin mesh. The resulting method is boundary layer resolving and has been shown almost second‐order accurate in space and first‐order accurate in time. It is also shown that the proposed method is uniformly convergent with respect to the singular perturbation parameter ε. Some numerical results are given to confirm the predicted theory and comparison of numerical results made with a scheme consisting of a standard upwind finite difference operator on a piecewise uniform Shishkin mesh. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1143–1164, 2011     

Application of collocation method for solving a parabolic‐hyperbolic free boundary problem which models the growth of tumor with drug application

In this article, we want to solve a free boundary problem which models tumor growth with drug application. This problem includes five time dependent partial differential equations. The tumor considered in this model consists of three kinds of cells, proliferative cells, quiescent cells, and dead cells. Three different first‐order hyperbolic equations are given that describe the evolution of cells and other two second‐order parabolic equations describe the diffusion of nutrient and drug concentration. We solve the problem using the collocation method. Then, we prove stability and convergence of method. Also, some examples are considered to show the efficiency of method. Copyright © 2016 John Wiley & Sons, Ltd.     

Semilocal and global convergence of the Newton‐HSS method for systems of nonlinear equations

Newton‐HSS methods, which are variants of inexact Newton methods different from the Newton–Krylov methods, have been shown to be competitive methods for solving large sparse systems of nonlinear equations with positive‐definite Jacobian matrices (J. Comp. Math. 2010; 28 :235–260). In that paper, only local convergence was proved. In this paper, we prove a Kantorovich‐type semilocal convergence. Then we introduce Newton‐HSS methods with a backtracking strategy and analyse their global convergence. Finally, these globally convergent Newton‐HSS methods are shown to work well on several typical examples using different forcing terms to stop the inner iterations. Copyright © 2010 John Wiley & Sons, Ltd.