Orthogonal Residual Methods Research Articles

An Efficient Strategy for Solving Structural Nonlinear Equations by Combining the Orthogonal Residual Method and Normal Flow Technique

This paper presents a new procedure for solving structural nonlinear problems by combining the orthogonal residual method (ORM) and normal flow technique (NFT). The perpendicularity condition to the Davidenko flow, introduced by the NFT, which must be satisfied during the iterative process, overcome the difficulties, i.e. the poor convergence and inefficiency of the ORM close to the limit points, particularly the displacement limit points (snap-back behavior). Basically, the idea of the proposed strategy is to adjust the load parameter, which is treated as a variable in the nonlinear incremental-iterative solution process, assuming that the unbalanced forces (residual forces) must be orthogonal to the incremental displacements. This constraint is used together with the NFT perpendicularity condition. The proposed procedure is tested, and its efficiency is corroborated through the analyses of slender shallow and nonshallow arches and an L-frame since they exhibit highly nonlinear behaviors under certain loading conditions. It is concluded that the proposed procedure can overcome the numerical instability problems in the neighborhood of critical points when using only the conventional OR process, and the procedure compares favorably with the arc-length method, minimum residual displacement method, and generalized displacement control method.

Deflated and augmented global Krylov subspace methods for the matrix equations

Global Krylov subspace methods are among the most efficient algorithms to solve matrix equation AX=B. Deflation and augmentation techniques are used to accelerate the convergence of Krylov subspace methods. There are two different approaches for deflated and augmented methods: an augmentation space is applied explicitly in every step, or the global method is used for solving a projected problem and then a correction step is applied at the end. In this paper, we present a framework of deflation and augmentation approaches for accelerating the convergence of the global methods for the solution of nonsingular linear matrix equations AX=B. Then, we define deflated and augmented global algorithms. Also, we analyze the deflated and augmented global minimal residual and global orthogonal residual methods. Finally, we present numerical examples to illustrate the effectiveness of different versions of the new algorithms.

A hybrid variant of the BiCOR method for a nonsymmetric linear system with a complex spectrum

A biconjugate A -orthogonal residual method based on the biconjugate A -orthonormalization procedure has been proposed and denoted as BiCOR by Jing, Huang, et al. [Y.-F. Jing, T.-Z. Huang, Y. Zhang, L. Li, G.-H. Cheng, Z.-G. Ren, Y. Duan, T. Sogabe, and B. Carpentieri, Lanczos-type variants of the COCR method for complex nonsymmetric linear systems, J. Comput. Phys. 228 (2009) 6376–6394]. Here, a hybrid variant of BiCOR is presented to avoid generating the transpose of the system matrix and to optimize the convergence of a nonsymmetric linear system with a complex spectrum.

Convergence properties of some block Krylov subspace methods for multiple linear systems

In the present paper, we give some convergence results of the global minimal residual methods and the global orthogonal residual methods for multiple linear systems. Using the Schur complement formulae and a new matrix product, we give expressions of the approximate solutions and the corresponding residuals. We also derive some useful relations between the norm of the residuals.

Analysis of the convergence of the minimal and the orthogonal residual methods

We consider two Krylov subspace methods for solving linear systems, which are the minimal residual method and the orthogonal residual method. These two methods are studied without referring to any particular implementations. By using the Petrov–Galerkin condition, we describe the residual norms of these two methods in terms of Krylov vectors, and the relationship between there two norms. We define the Ritz singular values, and prove that the convergence of these two methods is governed by the convergence of the Ritz singular values.

By how much can residual minimization accelerate the convergence of orthogonal residual methods

We capitalize upon the known relationship between pairs of orthogonal and minimal residual methods (or, biorthogonal and quasi-minimal residual methods) in order to estimate how much smaller the residuals or quasi-residuals of the minimizing methods can be compared to those of the corresponding Galerkin or Petrov–Galerkin method. Examples of such pairs are the conjugate gradient (CG) and the conjugate residual (CR) methods, the full orthogonalization method (FOM) and the generalized minimal residual (GMRES) method, the CGNE and BiCG versions of applying CG to the normal equations, as well as the biconjugate gradient (BiCG) and the quasi-minimal residual (QMR) methods. Also the pairs consisting of the (bi)conjugate gradient squared (CGS) and the transpose-free QMR (TFQMR) methods can be added to this list if the residuals at half-steps are included, and further examples can be created easily. The analysis is more generally applicable to the minimal residual (MR) and quasi-minimal residual (QMR) smoothing processes, which are known to provide the transition from the results of the first method of such a pair to those of the second one. By an interpretation of these smoothing processes in coordinate space we deepen the understanding of some of the underlying relationships and introduce a unifying framework for minimal residual and quasi-minimal residual smoothing. This framework includes the general notion of QMR-type methods.

A dual orthogonality procedure for non-linear finite element equations

In the orthogonal residual procedure for solution of non-linear finite element equations, the load is adjusted in each equilibrium iteration to satisfy an orthogonality condition to the current displacement increment. It is shown here that the quasi-Newton formulation of the orthogonal residual method consists of a simple one-term correction of the displacement subincrement, and that this correction leads to orthogonality between the corrected displacement subincrement and the current increment of the internal force vector, thus defining a dual orthogonality algorithm. It is demonstrated how the algorithm can be implemented to combine a single update of the stiffness matrix for each load increment in normal circumstances with full updates locally if increasing stiffness is encountered. The algorithm is illustrated by examples.