We consider the solution of the rational matrix equations, or generalized algebraic Riccati equations with rational terms, arising in stochastic optimal control in continuous- and discrete-time. Fixed-point iteration and (modified) Newton's methods will be considered. In particular, the convergence results of a new modified Newton's method, for both continuous- and discrete-time rational Riccati equations, will be presented.

Recently there have two different effective methods proposed by Kanzow et al. in (Kanzow, 2001 [8]) and (Kanzow and Petra, 2004 [9]), respectively, which commonly use the FischerBurmeister (FB) function to recast the mixed complementarity problem (MCP) as a constrained minimization problem and a nonlinear system of equations, respectively. They all remark that their algorithms may be improved if the FB function is replaced by other NCP functions. Accordingly, in this paper, we employ the generalized FischerBurmeister (GFB) where the 2-norm in the FB function is relaxed to a general p-norm (p> 1) for the two methods and investigate how much the improvement is by changing the parameter p as well as which method is influenced more when we do so, by the performance profiles of iterations and function evaluations for the two methods with different p on MCPLIB collection.

In this paper, we present a staggered discontinuous Galerkin method for the approximation of the incompressible NavierStokes equations. Our new method combines the advantages of discontinuous Galerkin methods and staggered meshes, and results in many good properties, namely local and global conservations, optimal convergence and superconvergence through the use of a local postprocessing technique. Another key feature is that our method provides a skew-symmetric discretization of the convection term, with the aim of giving a better conservation property compared with existing discretizations. We will present extensive numerical results, including Kovasznay flow, Taylor vortex flow, lid-driven cavity flow, parallel plate flow and channel expansion flow, to show the performance of the method.

We present a novel modification to the well-known infomax algorithm of blind source separation. Under natural gradient descent, the infomax algorithm converges to a stationary point of a limiting ordinary differential equation. However, due to the presence of saddle points or local minima of the corresponding likelihood function, the algorithm may be trapped around these bad stationary points for a long time, especially if the initial data are near them. To speed up convergence, we propose to add a sequence of random perturbations to the infomax algorithm to shake the iterating sequence so that it is captured by a path descending to a more stable stationary point. We analyze the convergence of the randomly perturbed algorithm, and illustrate its fast convergence through numerical examples on blind demixing of stochastic signals. The examples have analytical structures so that saddle points or local minima of

We propose to incorporate a weighted difference of anisotropic and isotropic total variation (TV) norms into a relaxed formulation of the two phase Mumford-Shah (MS) model for image segmentation. We show results exceeding those obtained by the MS model when using the standard TV norm to regularize partition boundaries. In particular, examples illustrating the qualitative differences between the proposed model and the standard MS one are shown. A fast numerical method is introduced to minimize the proposed model utilizing the difference-of-convex algorithm (DCA) and the primal dual hybrid gradient (PDHG) method.