An improved GPBi-CG algorithm suitable for distributed parallel computing

An improved generalized product-type bi-conjugate gradient (GPBi-CG) method (IGPBi-CG method, in brief) for solving large sparse linear systems with unsymmetrical coefficient matrices is proposed for distributed parallel environments. The method reduces three global synchronization points to two by reconstructing GPBi-CG method and the communication time required for the inner product can be efficiently overlapped with useful computation. The cost is only slightly increased computation time, which can be ignored compared with the reduction of communication time. Performance and isoefficiency analysis show that the IGPBi-CG method has better parallelism and scalability than the GPBi-CG method. Numerical experiments show that the scalability can be improved by a factor greater than 1.5 and the improvement in parallel communication performance approaches 33.3˙%.     

IMinpert： An Incomplete Minimum Perturbation Algorithm for Large Unsymmetric Linear Systems

This paper gives the truncated version of the Minpert method:the incomplete minimum perturbation algorithm(IMinpert).It is based on an incomplete orthogonal- ization of the Krylov vectors in question,and gives a quasi-minimum backward error solution over the Krylov subspace.In order to make the practical implementation of IMinpert easy and convenient,we give another approximate version of the IMinpert method:A-IMinpert.Theoretical properties of the latter algorithm are discussed.Nu- merical experiments are reported to show the proposed method is effective in practice and is competitive with the Minpert algorithm.     

The concept of special inner products for deriving new conjugate gradient-like solvers for non-symmetric sparse linear systems

Yserentant's and Bramble/Pasciak/Xu's hierarchical preconditioners only require the hierarchical node connection of a finite element grid. So they could be applied to nonsymmetric FE-systems too, having a symmetric preconditioner. We investigate the question whether this fact could be useful in the CG-like iterative methods. The main key is the consideration of an appropriate choice of the inner product defined in the N-vector space. It is shown, how the inner product influences the formulas of the methods, the rate of convergence and some other properties.     

An improved Wei–Yao–Liu nonlinear conjugate gradient method for optimization computation

In this paper, we take a little modification to the Wei–Yao–Liu nonlinear conjugate gradient method proposed by Wei et al. [Z. Wei, S. Yao, L. Liu, The convergence properties of some new conjugate gradient methods, Appl. Math. Comput. 183 (2006) 1341–1350] such that the modified method possesses better convergence properties. In fact, we prove that the modified method satisfies sufficient descent condition with greater parameter in the strong Wolfe line search and converges globally for nonconvex minimization. We also extend these results to the Hestenes–Stiefel method and prove that the modified HS method is globally convergent for nonconvex functions with the standard Wolfe conditions. Numerical results are reported by using some test problems in the CUTE library.     

Parallel algebraic hybrid solvers for large 3D convection-diffusion problems

In this paper we study the parallel scalability of variants of an algebraic additive Schwarz preconditioner for the solution of large three dimensional convection diffusion problems in a non-overlapping domain decomposition framework. To alleviate the computational cost, both in terms of memory and floating-point complexity, we investigate variants based on a sparse approximation or on mixed 32- and 64-bit calculation. The robustness and the scalability of the preconditioners are investigated through extensive parallel experiments on up to 2,000 processors. Their efficiency from a numerical and parallel performance view point are reported. This research activity was partially supported within the framework of the ANR-CIS project Solstice (ANR-06-CIS6- 010).     

GPBi‐CGstab(L): A Lanczos‐type product method unifying Bi‐CGstab(L) and GPBi‐CG

Lanczos‐type product methods (LTPMs), in which the residuals are defined by the product of stabilizing polynomials and the Bi‐CG residuals, are effective iterative solvers for large sparse nonsymmetric linear systems. Bi‐CGstab(L) and GPBi‐CG are popular LTPMs and can be viewed as two different generalizations of other typical methods, such as CGS, Bi‐CGSTAB, and Bi‐CGStab2. Bi‐CGstab(L) uses stabilizing polynomials of degree L, while GPBi‐CG uses polynomials given by a three‐term recurrence (or equivalently, a coupled two‐term recurrence) modeled after the Lanczos residual polynomials. Therefore, Bi‐CGstab(L) and GPBi‐CG have different aspects of generalization as a framework of LTPMs. In the present paper, we propose novel stabilizing polynomials, which combine the above two types of polynomials. The resulting method is referred to as GPBi‐CGstab(L). Numerical experiments demonstrate that our presented method is more effective than conventional LTPMs.     

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

For singular linear systems A x=b, ORTHOMIN(2) is known theoretically to attain the minimum residual min _xR nb–A x₂ 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).     

A hybrid method for computing Lyapunov exponents

In this paper we propose a numerical method for computing all Lyapunov coefficients of a discrete time dynamical system by spatial integration. The method extends an approach of Aston and Dellnitz (Comput Methods Appl Mech Eng 170:223–237, 1999) who use a box approximation of an underlying ergodic measure and compute the first Lyapunov exponent from a spatial average of the norms of the Jacobian for the iterated map. In the hybrid method proposed here, we combine this approach with classical QR-oriented methods by integrating suitable R-factors with respect to the invariant measure. In this way we obtain approximate values for all Lyapunov exponents. Assuming somewhat stronger conditions than those of Oseledec’ multiplicative theorem, these values satisfy an error expansion that allows to accelerate convergence through extrapolation. W.-J. Beyn and A. Lust was supported by CRC 701 ‘Spectral Analysis and Topological Methods in Mathematics’. The paper is mainly based on the PhD thesis [27] of A. Lust.     

An efficient gradient method with approximate optimal stepsize for the strictly convex quadratic minimization problem

In this paper, a new type of stepsize, approximate optimal stepsize, for gradient method is introduced to interpret the Barzilai–Borwein (BB) method, and an efficient gradient method with an approximate optimal stepsize for the strictly convex quadratic minimization problem is presented. Based on a multi-step quasi-Newton condition, we construct a new quadratic approximation model to generate an approximate optimal stepsize. We then use the two well-known BB stepsizes to truncate it for improving numerical effects and treat the resulted approximate optimal stepsize as the new stepsize for gradient method. We establish the global convergence and R-linear convergence of the proposed method. Numerical results show that the proposed method outperforms some well-known gradient methods.     

On the parallel GSAOR method for block diagonally dominant matrices

In this paper we consider the parallel generalized SAOR iterative method based on the generalized AOR iterative method presented by James for solving large nonsingular system. We obtain some convergence theorems for the case when coefficient matrix is a block diagonally dominant matrix or a generalized block diagonal dominant matrix. A numerical example is given to illustrate to our results.     

A hybrid numerical-asymptotic boundary integral method for high-frequency acoustic scattering

We propose a new robust method for the computation of scattering of high-frequency acoustic plane waves by smooth convex objects in 2D. We formulate this problem by the direct boundary integral method, using the classical combined potential approach. By exploiting the known asymptotics of the solution, we devise particular expansions, valid in various zones of the boundary, which express the solution of the integral equation as a product of explicit oscillatory functions and more slowly varying unknown amplitudes. The amplitudes are approximated by polynomials (of minimum degree d) in each zone using a Galerkin scheme. We prove that the underlying bilinear form is continuous in L ₂, with a continuity constant that grows mildly in the wavenumber k. We also show that the bilinear form is uniformly L ₂-coercive, independent of k, for all k sufficiently large. (The latter result depends on rather delicate Fourier analysis and is restricted in 2D to circular domains, but it also applies to spheres in higher dimensions.) Using these results and the asymptotic expansion of the solution, we prove superalgebraic convergence of our numerical method as d → ∞ for fixed k. We also prove that, as k → ∞, d has to increase only very modestly to maintain a fixed error bound (d ∼ k ^1/9 is a typical behaviour). Numerical experiments show that the method suffers minimal loss of accuracy as k →∞, for a fixed number of degrees of freedom. Numerical solutions with a relative error of about 10⁻⁵ are obtained on domains of size for k up to 800 using about 60 degrees of freedom.     

非线性系统的精确解

Based on the computerized symbolic, a new generalized tanh functions method is used for constructing exact travelling wave solutions of nonlinear partial differential equations （PDES） in a unified way. The main idea of our method is to take full advantage of an auxiliary ordinary differential equation which has more new solutions. At the same time, we present a more general transformation, which is a generalized method for finding more types of travelling wave solutions of nonlinear evolution equations （NLEEs）. More new exact travelling wave solutions to two nonlinear systems are explicitly obtained.     

New periodic and soliton solutions by application of Exp-function method for nonlinear evolution equations

In this letter, the Kaup–Kupershmidt, (2+1)$(2+1)$-dimensional Potential Kadomtsev–Petviashvili (shortly PKP) equations are presented and the Exp-function method is employed to compute an approximation to the solution of nonlinear differential equations governing the problem. It has been attempted to show the capabilities and wide-range applications of the Exp-function method. This method can be used as an alternative to obtain analytic and approximate solution of different types of differential equations applied in engineering mathematics.