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In this paper, high dimensional two-sided space fractional diffusion equations, derived from the fractional Fick’s law, and with monotonic variable diffusion coefficients, are solved by alternating direction implicit method. Each linear system corresponding to each spatial direction thus resulted is solved by Krylov subspace method. The method is accelerated by applying an approximate inverse preconditioner, where under certain conditions we showed that the normalized preconditioned matrix is equal to a sum of identity matrix, a matrix with small norm, and a matrix with low rank, such that the preconditioned Krylov subspace method converges superlinearly. We also briefly present some fast algorithms whose computational cost for solving the linear systems is O ( n log n ) , where n is the matrix size. The results are illustrated by some numerical examples. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we incorporate a nonmonotone technique with the new proposed adaptive trust region radius (Shi and Guo, 2008)  in order to propose a new nonmonotone trust region method with an adaptive radius for unconstrained optimization. Both the nonmonotone techniques and adaptive trust region radius strategies can improve the trust region methods in the sense of global convergence. The global convergence to first and second order critical points together with local superlinear and quadratic convergence of the new method under some suitable conditions. Numerical results show that the new method is very efficient and robustness for unconstrained optimization problems. [Copyright &y& Elsevier]

Abstract: In this paper, a kind of nonlinear optimization problems with nonlinear inequality constraints are discussed, and a new SQP feasible descent algorithm for solving the problems is presented. At each iteration of the new algorithm, a convex quadratic program (QP) which always has feasible solution is solved and a master direction is obtained, then, an improved (feasible descent) direction is yielded by updating the master direction with an explicit formula, and in order to avoid the Maratos effect, a height-order correction direction is computed by another explicit formula of the master direction and the improved direction. The new algorithm is proved to be globally convergent and superlinearly convergent under mild conditions without the strict complementarity. Furthermore, the quadratic convergence rate of the algorithm is obtained when the twice derivatives of the objective function and constrained functions are adopted. Finally, some numerical tests are reported. [Copyright &y& Elsevier]

In this paper, we combine the new trust region subproblem proposed in [1] with the nonmonotone technique to propose a new algorithm for unconstrained optimization—the nonmonotone adaptive trust region method. The local and global convergence properties of the nonmonotone adaptive trust region method are proved. Its efficiency is tested by numerical results. [Copyright &y& Elsevier]