This paper develops a class of general one-step discretization methods for solving the index-1 stochastic delay differential-algebraic equations. The existence and uniqueness theorem of strong solutions of index-1 equations is given. A strong convergence criterion of the methods is derived, which is applicable to a series of one-step stochastic numerical methods. Some specific numerical methods, such as the Euler-Maruyama method, stochastic  $θ$-methods, split-step $θ$-methods are proposed, and their strong convergence results are given. Numerical experiments further illustrate the theoretical results.

For the stationary Wigner equation with inflow boundary conditions, the numerical convergence with respect to the velocity mesh size are deteriorated due to the singularity at velocity zero. In this paper, using the fact that the solution of the stationary Wigner equation is subject to an integral constraint, we prove that the Wigner equation can be written into a form with a bounded operator $\mathcal{B}[V]$, which is equivalent to the operator $\mathcal{A}[V] = Θ[V]/v$ in the original Wigner equation under some conditions. Then the discrete operators discretizing $\mathcal{B}[V]$ are proved to be uniformly bounded with respect to the mesh size. Based on the theoretical findings, a singularity-free numerical method is proposed. Numerical results are provided to show our improved numerical scheme performs much better in numerical convergence than the original scheme based on discretizing $\mathcal{A}[V]$.

Numerous C⁰ discontinuous Galerkin (DG) schemes for the Kirchhoff plate bending problem are extended to solve a plate frictional contact problem, which is a fourth-order elliptic variational inequality of the second kind. This variational inequality contains a non-differentiable term due to the frictional contact. We prove that these C⁰ DG methods are consistent and stable, and derive optimal order error estimates for the quadratic element. A numerical example is presented to show the performance of the C⁰ DG methods; and the numerical convergence orders confirm the theoretical prediction.

Ultrasound imaging technique is one of the most non-invasive, practically harmless to the human body, accurate, cost effective and real-time techniques in medical diagnosis. However, ultrasound images suffer from the so-called speckle noise because of the imaging principle. The speckle noise reduces the quality and visibility of ultrasound images, thereby decreasing overall reliability of the images and interfering with the clinical diagnosis. In this paper, we propose a novel variational model under a combination of total variation regularization and Weberized total variation regularization and prove the existence and uniqueness of the minimizer for the variational problem. In order to efficiently solve the associated Euler-Lagrange equation consisting of nonlinear partial differential equation, we apply a finite difference method and develop several numerical techniques for solving the resulting discrete system. Numerical experiments on various synthetic and real ultrasound images not only confirm that our improved model is effective, but also it can provide significant improvement over evaluated models. Moreover, they also show that our proposed multigrid method has great potential applications to medical ultrasound imaging technique in delivering fast, accurate, and visually pleasing restoration results.

In this paper, we present a block Lanczos method for solving the large-scale CDT subproblem. During the algorithm, the original CDT subproblem is projected to a small-scale one, and then some classical methods are employed to solve this small-scale CDT subproblem to get a solution, which can be used to derive an approximate solution of the original CDT subproblem. Theoretical analysis of the error bounds for both the optimal value and the optimal solution is also proposed. Numerical experiments are carried out, and it is demonstrated that the block Lanczos method is effective and can achieve high accuracy for large-scale CDT subproblems.

In this paper, we present the composite rectangle rule for the computation of Hadamard finite-part integrals in boundary element methods with the hypersingular kernel 1/(x−s)² and we obtain the asymptotic expansion of error function of the middle rectangle rule. Based on the asymptotic expansion, two extrapolation algorithms are presented and their convergence rates are proved, which are the same as the Euler-Maclaurin expansions of classical middle rectangle rule approximations. At last, some numerical results are also illustrated to confirm the theoretical results and show the efficiency of the algorithms.

Based on the preconditioned modified Hermitian and skew-Hermitian splitting (PMHSS) iteration method for the complex symmetric linear system, two improved iterative methods, namely, the modified PMHSS (MPMHSS) method and the double modified PMHSS (DMPMHSS) method, are proposed in this paper. The spectral radii of the iteration matrices of two methods are given. We show that by choosing an appropriate parameter, MPMHSS could speed up the convergence on PMHSS. The DMPMHSS method is a four-step alternating iteration that is developed upon the two-step alternating iteration of MPMHSS. We discuss the choice of the parameters and establish the convergence of DMPMHSS. In particular, we give an analysis of the spectral radius of PMHSS and DMPMHSS at the parameter free situation, and we show that DMPMHSS converges faster than PMHSS in most cases. Our numerical experiments show these points.