Computational Krylov‐based methods for large‐scale differential Sylvester matrix problems

In the present paper, we propose Krylov‐based methods for solving large‐scale differential Sylvester matrix equations having a low‐rank constant term. We present two new approaches for solving such differential matrix equations. The first approach is based on the integral expression of the exact solution and a Krylov method for the computation of the exponential of a matrix times a block of vectors. In the second approach, we first project the initial problem onto a block (or extended block) Krylov subspace and get a low‐dimensional differential Sylvester matrix equation. The latter problem is then solved by some integration numerical methods such as the backward differentiation formula or Rosenbrock method, and the obtained solution is used to build the low‐rank approximate solution of the original problem. We give some new theoretical results such as a simple expression of the residual norm and upper bounds for the norm of the error. Some numerical experiments are given in order to compare the two approaches.     

Finite‐region stability and finite‐region boundedness for 2D Roesser models

In this paper, we establish finite‐region stability (FRS) and finite‐region boundedness analysis methods to investigate the transient behavior of discrete two‐dimensional Roesser models. First, by building special recursive formulas, a sufficient FRS condition is built via solvable linear matrix inequalities constraints. Next, by designing state feedback controllers, the finite‐region stabilization issue is analyzed for the corresponding two‐dimensional closed‐loop system. Similar to FRS analysis, the finite‐region boundedness problem is addressed for Roesser models with exogenous disturbances and corresponding criteria, and linear matrix inequalities conditions are reported. To conclude the paper, we provide numerical examples to confirm the validity of the proposed methods. Copyright © 2016 John Wiley & Sons, Ltd.     

On order‐reducible sinc discretizations and block‐diagonal preconditioning methods for linear third‐order ordinary differential equations

By introducing a variable substitution, we transform the two‐point boundary value problem of a third‐order ordinary differential equation into a system of two second‐order ordinary differential equations (ODEs). We discretize this order‐reduced system of ODEs by both sinc‐collocation and sinc‐Galerkin methods, and average these two discretized linear systems to obtain the target system of linear equations. We prove that the discrete solution resulting from the linear system converges exponentially to the true solution of the order‐reduced system of ODEs. The coefficient matrix of the linear system is of block two‐by‐two structure, and each of its blocks is a combination of Toeplitz and diagonal matrices. Because of its algebraic properties and matrix structures, the linear system can be effectively solved by Krylov subspace iteration methods such as GMRES preconditioned by block‐diagonal matrices. We demonstrate that the eigenvalues of certain approximation to the preconditioned matrix are uniformly bounded within a rectangle on the complex plane independent of the size of the discretized linear system, and we use numerical examples to illustrate the feasibility and effectiveness of this new approach. Copyright © 2013 John Wiley & Sons, Ltd.     

Modified Hermitian and skew‐Hermitian splitting methods for non‐Hermitian positive‐definite linear systems

To further study the Hermitian and non‐Hermitian splitting methods for a non‐Hermitian and positive‐definite matrix, we introduce a so‐called lopsided Hermitian and skew‐Hermitian splitting and then establish a class of lopsided Hermitian/skew‐Hermitian (LHSS) methods to solve the non‐Hermitian and positive‐definite systems of linear equations. These methods include a two‐step LHSS iteration and its inexact version, the inexact Hermitian/skew‐Hermitian (ILHSS) iteration, which employs some Krylov subspace methods as its inner process. We theoretically prove that the LHSS method converges to the unique solution of the linear system for a loose restriction on the parameter α. Moreover, the contraction factor of the LHSS iteration is derived. The presented numerical examples illustrate the effectiveness of both LHSS and ILHSS iterations. Copyright © 2007 John Wiley & Sons, Ltd.     

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.     

Modulus‐based multigrid methods for linear complementarity problems

By employing modulus‐based matrix splitting iteration methods as smoothers, we establish modulus‐based multigrid methods for solving large sparse linear complementarity problems. The local Fourier analysis is used to quantitatively predict the asymptotic convergence factor of this class of multigrid methods. Numerical results indicate that the modulus‐based multigrid methods of the W‐cycle can achieve optimality in terms of both convergence factor and computing time, and their asymptotic convergence factors can be predicted perfectly by the local Fourier analysis of the corresponding modulus‐based two‐grid methods.     

Unified multipliers‐free theory of dual‐primal domain decomposition methods

The concept of dual‐primal methods can be formulated in a manner that incorporates, as a subclass, the non preconditioned case. Using such a generalized concept, in this article without recourse to “Lagrange multipliers,” we introduce an all‐inclusive unified theory of nonoverlapping domain decomposition methods (DDMs). One‐level methods, such as Schur‐complement and one‐level FETI, as well as two‐level methods, such as Neumann‐Neumann and preconditioned FETI, are incorporated in a unified manner. Different choices of the dual subspaces yield the different dual‐primal preconditioners reported in the literature. In this unified theory, the procedures are carried out directly on the matrices, independently of the differential equations that originated them. This feature reduces considerably the code‐development effort required for their implementation and permit, for example, transforming 2D codes into 3D codes easily. Another source of this simplification is the introduction of two projection‐matrices, generalizations of the average and jump of a function, which possess superior computational properties. In particular, on the basis of numerical results reported there, we claim that our jump matrix is the optimal choice of the B operator of the FETI methods. A new formula for the Steklov‐Poincaré operator, at the discrete level, is also introduced. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Quasi‐Toeplitz splitting iteration methods for unsteady space‐fractional diffusion equations

We construct a class of quasi‐Toeplitz splitting iteration methods to solve the two‐sided unsteady space‐fractional diffusion equations with variable coefficients. By making full use of the structural characteristics of the coefficient matrix, the method only requires computational costs of O(n log n) with n denoting the number of degrees of freedom. We develop an appropriate circulant matrix to replace the Toeplitz matrix as a preconditioner. We discuss the spectral properties of the quasi‐circulant splitting preconditioned matrix. Numerical comparisons with existing approaches show that the present method is both effective and efficient when being used as matrix splitting preconditioners for Krylov subspace iteration methods.     

Adaptive iterative regularization schemes for two‐dimensional integral‐algebraic systems

The main idea of this paper is to utilize the adaptive iterative schemes based on regularization techniques for moderately ill‐posed problems that are obtained by a system of linear two‐dimensional Volterra integral equations with a singular matrix in the leading part. These problems may arise in the modeling of certain heat conduction processes as well as in the dynamic simulation packages such as compressible flow through a plant piping network. Owing to the ill‐posed nature of the first kind Volterra equation that appears in the system, we will focus on the two families of regularization algorithms, ie, the Landweber and Lavrentiev type methods, where we treat both the exact and perturbed data. Our aim is to work directly with the original Volterra equations without any kind of reduction. Two fast iterative algorithms with reasonable computational complexity are developed. Numerical experiments on a few test problems are used to illustrate the validity and efficiency of the proposed iterative methods in comparison with the classical regularization methods.     

Modulus‐based synchronous multisplitting iteration methods for linear complementarity problems

By an equivalent reformulation of the linear complementarity problem into a system of fixed‐point equations, we construct modulus‐based synchronous multisplitting iteration methods based on multiple splittings of the system matrix. These iteration methods are suitable to high‐speed parallel multiprocessor systems and include the multisplitting relaxation methods such as Jacobi, Gauss–Seidel, successive overrelaxation, and accelerated overrelaxation of the modulus type as special cases. We establish the convergence theory of these modulus‐based synchronous multisplitting iteration methods and their relaxed variants when the system matrix is an H ₊ ‐matrix. Numerical results show that these new iteration methods can achieve high parallel computational efficiency in actual implementations. Copyright © 2012 John Wiley & Sons, Ltd.     

Solutions of symmetry‐constrained least‐squares problems

In this paper, two new matrix‐form iterative methods are presented to solve the least‐squares problem: and matrix nearness problem: where matrices and are given; ??₁ and ??₂ are the set of constraint matrices, such as symmetric, skew symmetric, bisymmetric and centrosymmetric matrices sets and S_XY is the solution pair set of the minimum residual problem. These new matrix‐form iterative methods have also faster convergence rate and higher accuracy than the matrix‐form iterative methods proposed by Peng and Peng (Numer. Linear Algebra Appl. 2006; 13 : 473–485) for solving the linear matrix equation AXB+CYD=E. Paige's algorithms, which are based on the bidiagonalization procedure of Golub and Kahan, are used as the framework for deriving these new matrix‐form iterative methods. Some numerical examples illustrate the efficiency of the new matrix‐form iterative methods. Copyright © 2008 John Wiley & Sons, Ltd.     

The tensor t‐function: A definition for functions of third‐order tensors

A definition for functions of multidimensional arrays is presented. The definition is valid for third‐order tensors in the tensor t‐product formalism, which regards third‐order tensors as block circulant matrices. The tensor function definition is shown to have similar properties as standard matrix function definitions in fundamental scenarios. To demonstrate the definition's potential in applications, the notion of network communicability is generalized to third‐order tensors and computed for a small‐scale example via block Krylov subspace methods for matrix functions. A complexity analysis for these methods in the context of tensors is also provided.     

A unitary joint diagonalization algorithm for nonsymmetric higher‐order tensors based on Givens‐like rotations

Based on Givens‐like rotations, we present a unitary joint diagonalization algorithm for a set of nonsymmetric higher‐order tensors. Each unitary rotation matrix only depends on one unknown parameter which can be analytically obtained in an independent way following a reasonable assumption and a complex derivative technique. It can serve for the canonical polyadic decomposition of a higher‐order tensor with orthogonal factors. Furthermore, based on cross‐high‐order cumulants of observed signals, we show that the proposed algorithm can be applied to solve the joint blind source separation problem. The simulation results reveal that the proposed algorithm has a competitive performance compared with those of several existing related methods.     

Modified modulus‐based matrix splitting iteration methods for linear complementarity problems

For solving the large sparse linear complementarity problems, we establish modified modulus‐based matrix splitting iteration methods and present the convergence analysis when the system matrices are H₊‐matrices. The optima of parameters involved under some scopes are also analyzed. Numerical results show that in computing efficiency, our new methods are superior to classical modulus‐based matrix splitting iteration methods under suitable conditions. Copyright © 2015 John Wiley & Sons, Ltd.     

Subspace‐by‐subspace preconditioners for structured linear systems

We consider the iterative solution of symmetric positive‐definite linear systems whose coefficient matrix may be expressed as the outer product of low‐rank terms. We derive suitable preconditioners for such systems, and demonstrate their effectiveness on a number of test examples. We also consider combining these methods with existing techniques to cope with the commonly‐occuring case where the coefficient matrix is the linear sum of elements, some of which are of very low rank. Copyright © 1999 John Wiley & Sons, Ltd.     

Compress‐and‐restart block Krylov subspace methods for Sylvester matrix equations

Block Krylov subspace methods (KSMs) comprise building blocks in many state‐of‐the‐art solvers for large‐scale matrix equations as they arise, for example, from the discretization of partial differential equations. While extended and rational block Krylov subspace methods provide a major reduction in iteration counts over polynomial block KSMs, they also require reliable solvers for the coefficient matrices, and these solvers are often iterative methods themselves. It is not hard to devise scenarios in which the available memory, and consequently the dimension of the Krylov subspace, is limited. In such scenarios for linear systems and eigenvalue problems, restarting is a well‐explored technique for mitigating memory constraints. In this work, such restarting techniques are applied to polynomial KSMs for matrix equations with a compression step to control the growing rank of the residual. An error analysis is also performed, leading to heuristics for dynamically adjusting the basis size in each restart cycle. A panel of numerical experiments demonstrates the effectiveness of the new method with respect to extended block KSMs.     

A memory‐efficient finite volume method for advection‐diffusion‐reaction systems with nonsmooth sources

We present a parallel matrix‐free implicit finite volume scheme for the solution of unsteady three‐dimensional advection‐diffusion‐reaction equations with smooth and Dirac‐Delta source terms. The scheme is formally second order in space and a Newton–Krylov method is employed for the appearing nonlinear systems in the implicit time integration. The matrix‐vector product required is hardcoded without any approximations, obtaining a matrix‐free method that needs little storage and is well‐suited for parallel implementation. We describe the matrix‐free implementation of the method in detail and give numerical evidence of its second‐order convergence in the presence of smooth source terms. For nonsmooth source terms, the convergence order drops to one half. Furthermore, we demonstrate the method's applicability for the long‐time simulation of calcium flow in heart cells and show its parallel scaling. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq31: 143–167, 2015     

Single‐cell compact finite‐difference discretization of order two and four for multidimensional triharmonic problems

In this article, we discuss finite‐difference methods of order two and four for the solution of two‐and three‐dimensional triharmonic equations, where the values of u,(?²u/?n²) and (?⁴u/?n⁴) are prescribed on the boundary. For 2D case, we use 9‐ and for 3D case, we use 19‐ uniform grid points and a single computational cell. We introduce new ideas to handle the boundary conditions and do not require to discretize the boundary conditions at the boundary. The Laplacian and the biharmonic of the solution are obtained as byproduct of the methods. The resulting matrix system is solved by using the appropriate block iterative methods. Computational results are provided to demonstrate the fourth‐order accuracy of the proposed methods. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Block‐triangular preconditioners for PDE‐constrained optimization

In this paper we investigate the possibility of using a block‐triangular preconditioner for saddle point problems arising in PDE‐constrained optimization. In particular, we focus on a conjugate gradient‐type method introduced by Bramble and Pasciak that uses self‐adjointness of the preconditioned system in a non‐standard inner product. We show when the Chebyshev semi‐iteration is used as a preconditioner for the relevant matrix blocks involving the finite element mass matrix that the main drawback of the Bramble–Pasciak method—the appropriate scaling of the preconditioners—is easily overcome. We present an eigenvalue analysis for the block‐triangular preconditioners that gives convergence bounds in the non‐standard inner product and illustrates their competitiveness on a number of computed examples. Copyright © 2010 John Wiley & Sons, Ltd.     

Understanding Saul'yev‐type unconditionally stable schemes from exponential splitting

Saul'yev‐type asymmetric schemes have been widely used in solving diffusion and advection equations. In this work, we show that Saul'yev‐type schemes can be derived from the exponential splitting of the semidiscretized equation which fundamentally explains their unconditional stability. Furthermore, we show that optimal schemes are obtained by forcing each scheme's amplification factor to match that of the exact amplification factor. A new second‐order explicit scheme is found for solving the advection equation with the identical amplification factor as the implicit Crank–Nicolson algorithm. Other new schemes for solving the advection–diffusion equation are also derived.© 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1961–1983, 2014