In this work we introduce a new unconditionally convergent explicit Tree-Grid Method for solving stochastic control problems with one space and one time dimension or equivalently, the corresponding Hamilton-Jacobi-Bellman equation. We prove the convergence of the method and outline the relationships to other numerical methods. The case of vanishing diffusion is treated by introducing an artificial diffusion term. We illustrate the superiority of our method to the standardly used implicit finite difference method on two numerical examples from finance. 

In this paper, Nodal discontinuous Galerkin method is presented to approximate Time-domain Lorentz model equations in meta-materials. The upwind flux is chosen in spatial discrete scheme. Low-storage five-stage fourth-order explicit Runge-Kutta method is employed in time discrete scheme. An error estimate of accuracy $\mathcal{O}(τ^4+h^n)$ is proved under the $L^2$-norm, specially $\mathcal{O}(τ^4+h^{n+1})$ can be obtained. Numerical experiments for transverse electric (TE) case and transverse magnetic (TM) case are demonstrated to verify the stability and the efficiency of the method in low and high wave frequency.

Variational methods are an important class of methods for general image restoration. Boosting technique has been shown capable of improving many image denoising algorithms. This paper discusses a boosting technique for general variational image restoration methods. It broadens the applications of boosting techniques to a wide range of image restoration problems, including not only denoising but also deblurring and inpainting. In particular, we combine the recent SOS technique with dynamic parameter to variational methods. The dynamic regularization parameter is motivated by Meyer's analysis on the ROF model. In each iteration of the boosting scheme, the variational model is solved by augmented Lagrangian method. The convergence analysis of the boosting process is shown in a special case of total variation image denoising with a "disk" input data. We have implemented our boosting technique for several image restoration problems such as denoising, inpainting and deblurring. The numerical results demonstrate promising improvement over standard variational restoration models such as total variation based models and higher order variational model as total generalized variation.

In this paper, based on the finite element formulation, we focus on multiple solutions and their evolution with time for a laminar flow in a permeable channel with expanding or contracting walls. Both Newtonian fluid and micropolar fluid are considered. For the Newtonian fluid model, we find that the profile of the unique solution in the case of injection remains the same for long time, which indicates that the solution may be stable. On the other hand, in the case of large suction, the profile of multiple solutions changes in time, which suggests that the multiple solutions may be unstable. Similar behaviors and conclusions are observed for the micropolar fluid model under different boundary parameters.

Numerical computation plays an important role in the study of differential equations with time-delay, because a simple and explicit analytic solution is usually un available. Time-stepping methods based on discretizing the temporal derivative with some step-size $∆t$ are the main tools for this task. To get accurate numerical solutions, in many cases it is necessary to require $∆t < τ$ and this will be a rather unwelcome restriction when $τ$, the quantity of time-delay, is small. In this paper, we propose a method for a class of time-delay problems, which is completely meshless. The idea lies in representing the solution by its Laplace inverse transform along a carefully designed contour in the complex plane and then approximating the contour integral by the Filon-Clenshaw-Curtis (FCC) quadrature in a few fast growing subintervals. The computations of the solution for all time points of interest are naturally parallelizable and for each time point the implementations of the FCC quadrature in all subintervals are also parallelizable. For each time point and each subinterval, the FCC quadrature can be implemented by fast Fourier transform. Numerical results are given to check the efficiency of the proposed method.

We propose the two-step modulus-based matrix splitting iteration methods for a class of nonlinear complementarity problems. The corresponding convergence theory is established when the system matrix is an $H_+$-matrix. Theoretical analysis gives the choice of parameter matrix involved based on the $H$-compatible splitting of the system matrix. Moreover, in actual implementation, the choices of iterative parameters for two-step modulus-based accelerated overrelaxation methods are studied. Numerical experiments show that the method is efficient and further verify the convergence theorems.

The purpose of this paper is to derive the generalized conjugate residual (GCR) algorithm for finding the least squares solution on a class of Sylvester matrix equations. We prove that if the system is inconsistent, the least squares solution can be obtained with infinite iterative steps in the absence of round-off errors. Furthermore, we provide a method for choosing the initial matrix to obtain the minimum norm least squares solutionof the problem. Finally, we give some numerical examples to illustrate the performance of GCR algorithm.

We present the local convergence analysis of a fifth order Traub-Steffensen-Chebyshev-like composition for solving nonlinear equations in Banach spaces. In earlier studies, hypotheses on the Fréchet derivative up to the fifth order of the operator under consideration are used to prove the convergence order of the method although only divided differences of order one appear in the method. That restricts the applicability of the method. In this paper, we extended the applicability of the fifth order Traub-Steffensen-Chebyshev-like composition without using hypotheses on the derivatives of the operator involved. Our convergence conditions are weaker than the conditions used in earlier studies. Numerical examples where earlier results cannot apply to solve equations but our results can apply are also given in this study.

This paper is concerned with a piecewise smooth rational quasi-interpolation with algebraic accuracy of degree $(n+1)$ to approximate the scattered data in $\mathbb{R}^3$. We firstly use the modified Taylor expansion to expand the mean value coordinates interpolation with algebraic accuracy of degree one to one with algebraic accuracy of degree $(n+1)$. Then, based on the triangulation of the scattered nodes in $\mathbb{R}^2$, on each triangle a rational quasi-interpolation function is constructed. The constructed rational quasi-interpolation is a linear combination of three different expanded mean value coordinates interpolations and it has algebraic accuracy of degree $(n+1)$. By comparing accuracy, stability, and efficiency with the $C^1$-Tri-interpolation method of Goodman [16] and the MQ Shepard method, it is observed that our method has some computational advantages.

The primal-dual hybrid gradient method is a classic way to tackle saddle-point problems. However, its convergence is not guaranteed in general. Some restrictions on the step size parameters, e.g., $τσ  ≤ 1/‖A^TA‖$, are imposed to guarantee the convergence. In this paper, a new convergent method with no restriction on parameters is proposed. Hence the expensive calculation of $‖A^TA‖$ is avoided. This method produces a predictor like other primal-dual methods but in a parallel fashion, which has the potential to speed up the method. This new iteration is then updated by a simple correction to guarantee the convergence. Moreover, the parameters are adjusted dynamically to enhance the efficiency as well as the robustness of the method. The generated sequence monotonically converges to the solution set. A worst-case $\mathcal{O}(1/t)$ convergence rate in ergodic sense is also established under mild assumptions. The numerical efficiency of the proposed method is verified by applications in LASSO problem and Steiner tree problem.