An efficient iterative method based on two-stage splitting methods to solve weakly nonlinear systems

[EN] In this paper, an iterative method for solving large, sparse systems of weakly nonlinear equations is presented. This method is based on Hermitian/skew-Hermitian splitting (HSS) scheme. Under suitable assumptions, we establish the convergence theorem for this method. In addition, it is shown that any faster and less time-consuming two-stage splitting method that satisfies the convergence theorem can be replaced instead of the HSS inner iterations. Numerical results, such as CPU time, show the robustness of our new method. This method is easy, fast and convenient with an accurate solution.The third and fourth authors have been partially supported by the Spanish Ministerio de Ciencia, Innovacion y Universidades PGC2018-095896-B-C22 and Generalitat Valenciana PROMETEO/2016/089.Amiri, A.; Darvishi, MT.; Cordero Barbero, A.; Torregrosa Sánchez, JR. (2019). An efficient iterative method based on two-stage splitting methods to solve weakly nonlinear systems. Mathematics. 7(9):1-17. https://doi.org/10.3390/math7090815S1177

A fast algorithm to solve systems of nonlinear equations

[EN] A new HSS-based algorithm for solving systems of nonlinear equations is presented and its semilocal convergence is proved. Spectral properties of the new method are investigated. Performance profile for the new scheme is computed and compared with HSS algorithm. Besides, by a numerical example in which a two-dimensional nonlinear convection diffusion equation is solved, we compare the new method and the Newton-HSS method. Numerical results show that the new scheme solves the problem faster than the NewtonHSS scheme in terms of CPU -time and number of iterations. Moreover, the application of the new method is found to be fast, reliable, flexible, accurate, and has small CPU time.This research was partially supported by Ministerio de Economia y Competitividad, Spain under grants MTM2014-52016-C2-2-P and Generalitat Valenciana, Spain PROMETEO/2016/089.Amiri, A.; Cordero Barbero, A.; Darvishi, M.; Torregrosa Sánchez, JR. (2019). A fast algorithm to solve systems of nonlinear equations. Journal of Computational and Applied Mathematics. 354:242-258. https://doi.org/10.1016/j.cam.2018.03.048S24225835

Trigonometric transform splitting methods for real symmetric Toeplitz systems

In this paper, we study efficient iterative methods for real symmetric Toeplitz systems based on the trigonometric transformation splitting (TTS) of the real symmetric Toeplitz matrix A. Theoretical analyses show that if the generating function f of the n × n Toeplitz matrix A is a real positive even function, then the TTS iterative methods converge to the unique solution of the linear system of equations for sufficient large n. Moreover, we derive an upper bound of the contraction factor of the TTS iteration which is dependent solely on the spectra of the two TTS matrices involved. Different from the CSCS iterative method in Ng (2003) in which all operations counts concern complex operations when the DFTs are employed, even if the Toeplitz matrix A is real and symmetric, our method only involves real arithmetics when the DCTs and DSTs are used. The numerical experiments show that our method works better than CSCS iterative method and much better than the positive definite and skew-symmetric splitting (PSS) iterative method in Bai et al. (2005) and the symmetric Gauss–Seidel (SGS) iterative method.National Natural Science Foundation of China under Grant No. 11371075info:eu-repo/semantics/publishedVersio

Generalized complex geometry

Generalized complex geometry, as developed by Hitchin, contains complex and symplectic geometry as its extremal special cases. In this thesis, we explore novel phenomena exhibited by this geometry, such as the natural action of a B-field. We provide new examples, including some on manifolds admitting no known complex or symplectic structure. We prove a generalized Darboux theorem which yields a local normal form for the geometry. We show that there is an elliptic deformation theory and establish the existence of a Kuranishi moduli space. We then define the concept of a generalized Kahler manifold. We prove that generalized Kahler geometry is equivalent to a bi-Hermitian geometry with torsion first discovered by physicists. We then use this result to solve an outstanding problem in 4-dimensional bi-Hermitian geometry: we prove that there exists a Riemannian metric on the complex projective plane which admits exactly two distinct Hermitian complex structures with equal orientation. Finally, we introduce the concept of generalized complex submanifold, and show that such sub-objects correspond to D-branes in the topological A- and B-models of string theory.Comment: Oxford University DPhil thesis, 107 page