Shao-Liang Zhang

GPBi-CG: Generalized Product-type Methods Based on Bi-CG for Solving Nonsymmetric Linear Systems

Recently Bi-CGSTAB as a variant of Bi-CG has been proposed for solving nonsymmetric linear systems, and its attractive convergence behavior has been confirmed in many numerical experiments. Bi-CGSTAB can be characterized by its residual polynomial which consists of the product of the residual polynomial of Bi-CG with other polynomials generated from two-term recurrence relations. In this paper, we propose a unified way to generalize a class of product-type methods whose residual polynomials can be factored by the residual polynomial of Bi-CG and other polynomials with standard three-term recurrence relations. Such product-type methods which are based on Bi-CG can be regarded as generalizations of Bi-CGSTAB. From the unified way, the well-known variants of the product-type methods, like CGS, Bi-CGSTAB, Bi-CGSTAB2, are reacquired again.

Linear algebraic calculation of the Green’s function for large-scale electronic structure theory

A linear algebraic method named the shifted conjugate-orthogonal conjugate-gradient method is introduced for large-scale electronic structure calculation. The method gives an iterative solver algorithm of the Greens function and the density matrix without calculating eigenstates. The problem is reduced to independent linear equations at many energy points and the calculation is actually carried out only for a single energy point. The method is robust against the round-off error and the calculation can reach the machine accuracy. With the observation of residual vectors, the accuracy can be controlled, microscopically, independently for each element of the Greens function, and dynamically, at each step in dynamical simulations. The method is applied to both a semiconductor and a metal.

Necessary and sufficient conditions for the convergence of Orthomin(k) on singular and inconsistent linear systems

Summary. We consider the convergence of Orthomin(k) on singular and inconsistent linear systems. Criteria for the breakdown of Orthomin(k) are discussed and analyzed. Moreover, necessary and sufficient conditions for the convergence of Orthomin(k) for any right hand side are given, and a rate of convergence is provided as well. Finally, numerical experiments are shown to confirm the convergence theorem.

A block IDR(s) method for nonsymmetric linear systems with multiple right-hand sides

The IDR(s) based on the induced dimension reduction (IDR) theorem, is a new class of efficient algorithms for large nonsymmetric linear systems. IDR(1) is mathematically equivalent to BiCGStab at the even IDR(1) residuals, and IDR(s) with s>1 is competitive with most Bi-CG based methods. For these reasons, we extend the IDR(s) to solve large nonsymmetric linear systems with multiple right-hand sides. In this paper, a variant of the IDR theorem is given at first, then the block IDR(s), an extension of IDR(s) based on the variant IDR(s) theorem, is proposed. By analysis, the upper bound on the number of matrix-vector products of block IDR(s) is the same as that of the IDR(s) for a single right-hand side in generic case, i.e., the total number of matrix-vector products of IDR(s) may be m times that of of block IDR(s), where m is the number of right-hand sides. Numerical experiments are presented to show the effectiveness of our proposed method.

A Variant of the ORTHOMIN(2) Method for Singular Linear Systems

For singular linear systems Ax=b, ORTHOMIN(2) is known theoretically to attain the minimum residual min x∈Rn‖b−Ax‖2 under a certain condition. However, in the actual computation with finite precision arithmetic, the residual is often observed to be reduced further than the theoretically expected level. Therefore, we propose a variant of ORTHOMIN(2), which is mathematically equivalent to the original ORTHOMIN(2) method, but uses recurrence formulas that are different from those of ORTHOMIN(2); they contain alternative expressions for the auxiliary vector and the recurrence coefficients. Although our implementation has the same computational costs as ORTHOMIN(2), numerical experiments on singular systems show that our implementation is more accurate and less affected by rounding errors than ORTHOMIN(2).

IDR(s) for solving shifted nonsymmetric linear systems

The IDR(s) method by Sonneveld and van Gijzen (2008) has recently received tremendous attention since it is effective for solving nonsymmetric linear systems. In this paper, we generalize this method to solve shifted nonsymmetric linear systems. When solving this kind of problem by existing shifted Krylov subspace methods, we know one just needs to generate one basis of the Krylov subspaces due to the shift-invariance property of Krylov subspaces. Thus the computation cost required by the basis generation of all shifted linear systems, in terms of matrix-vector products, can be reduced. For the IDR(s) method, we find that there also exists a shift-invariance property of the Sonneveld subspaces. This inspires us to develop a shifted version of the IDR(s) method for solving the shifted linear systems.

Parse-matrix evolution for symbolic regression

Data-driven model is highly desirable for industrial data analysis in case the experimental model structure is unknown or wrong, or the concerned system has changed. Symbolic regression is a useful method to construct the data-driven model (regression equation). Existing algorithms for symbolic regression such as genetic programming and grammatical evolution are difficult to use due to their special target programming language (i.e., LISP) or additional function parsing process. In this paper, a new evolutionary algorithm, parse-matrix evolution (PME), for symbolic regression is proposed. A chromosome in PME is a parse-matrix with integer entries. The mapping process from the chromosome to the regression equation is based on a mapping table. PME can easily be implemented in any programming language and free to control. Furthermore, it does not need any additional function parsing process. Numerical results show that PME can solve the symbolic regression problems effectively.

A class of product-type Krylov-subspace methods for solving nonsymmetric linear systems

Abstract In the present paper, a class of product-type Krylov-subspace methods for solving nonsymmetric linear systems is discussed. A characteristic of this class is the relationship r n =H n (A) r n BCG where r n is the residual vector corresponding to the nth iterate x n , and r n BCG is the nth residual generated in bi-conjugate gradient method. The polynomial Hn is chosen to speed up and stabilize convergence, while satisfying standard three-term recurrence relations. These product-type methods can be regarded as unifications and generalizations of CGS and Bi-CGSTAB.

Solution of generalized shifted linear systems with complex symmetric matrices

We develop the shifted COCG method [R. Takayama, T. Hoshi, T. Sogabe, S.-L. Zhang, T. Fujiwara, Linear algebraic calculation of Greens function for large-scale electronic structure theory, Phys. Rev. B 73 (165108) (2006) 1-9] and the shifted WQMR method [T. Sogabe, T. Hoshi, S.-L. Zhang, T. Fujiwara, On a weighted quasi-residual minimization strategy of the QMR method for solving complex symmetric shifted linear systems, Electron. Trans. Numer. Anal. 31 (2008) 126-140] for solving generalized shifted linear systems with complex symmetric matrices that arise from the electronic structure theory. The complex symmetric Lanczos process with a suitable bilinear form plays an important role in the development of the methods. The numerical examples indicate that the methods are highly attractive when the inner linear systems can efficiently be solved.

Some modifications of low-dimensional simplex evolution and their convergence

Low-dimensional simplex evolution (LDSE) is a real-coded evolutionary algorithm for global optimization. In this paper, we introduce three techniques to improve its performance: low-dimensional reproduction (LDR), normal struggle (NS) and variable dimension (VD). LDR tries to preserve the elite by keeping some of its (randomly chosen) components. LDR can also help the offspring individuals to escape from the hyperplane determined by their parents. NS tries to enhance its local search capability by allowing unlucky individual search around the best vertex of m-simplex. VD tries to draw lessons from recent failure by making further exploitation on its most promising sub-facet. Numerical results show that these techniques can improve the efficiency and reliability of LDSE considerably. The convergence properties are then analysed by finite Markov chains. It shows that the original LDSE might fail to converge, but modified LDSE with the above three techniques will converge for any initial population. To evaluate the convergence speed of modified LDSE, an estimation of its first passage time (of reaching the global minimum) is provided.