Image restoration is a common problem in visual process. In this paper, a modified minimization model is presented, which combines the L1 and L2 fidelity terms with a combined quadratic L2 and TV regularizer just as the regularizer of Cai et al. (2013). The combined regularizer has the priorities of preserving desirable edges and ensuring several kinds of noises can be removed clearly. Split-Bregman algorithm is effi- ciently employed to solve this model and convergence analysis is also discussed. Moreover, we extend the proposed model and algorithm for image restoration involving blurry images and color images. Experimental results show that our proposed model and algorithm have good performance both in visual and ISNR values for different kinds of blurs and noises including mixed noise.

The immersed boundary method has evolved into one of the most useful computational methods in studying fluid structure interaction. On the other hand, the immersed boundary method is also known to suffer from a severe timestep stability restriction when using an explicit time discretization. In this paper, we propose several efficient semi-implicit schemes to remove this stiffness from the immersed boundary method for the two-dimensional Stokes flow. First, we obtain a novel unconditionally stable semi-implicit discretization for the immersed boundary problem. Using this unconditionally stable discretization as a building block, we derive several efficient semi-implicit schemes for the immersed boundary problem by applying the small scale decomposition to this unconditionally stable discretization. Our stability analysis and extensive numerical experiments show that our semi-implicit schemes offer much better stability property than the explicit scheme. Unlike other implicit or semi-implicit schemes proposed in the literature, our semi-implicit schemes can be solved explicitly in the spectral space. Thus the computational cost of our semi-implicit schemes is comparable to that of an explicit scheme, but with a much better stability property.

he incompressibility constraint makes Navier-Stokes equations difficult. A reformulation to a better posed problem is needed before solving it numerically. The sequential regularization method (SRM) is a reformulation which combines the penalty method with a stabilization method in the context of constrained dynamical systems and has the benefit of both methods. In the paper, we study the existence and uniqueness for the solution of the SRM and provide a simple proof of the convergence of the solution of the SRM to the solution of the Navier-Stokes equations. We also give error estimates for the time discretized SRM formulation.

Jiang et al. (Transportation Research Part B, 2011, 45(2), 343-363) proposed a predictive continuum dynamic user-optimal (PDUO-C) model to investigate the dynamic characteristics of traffic flow and the corresponding route-choice behavior of travelers. The modeled region is a dense urban city, which is arbitrary in shape and has a single central business district (CBD). However, we argue that the model is not well posed due to an inconsistency in the route-choice strategy. To overcome this inconsistency, we revisit the PDUO-C problem, and construct an improved path-choice strategy. The improved model consists of a conservation law to govern the density, in which the flow direction is determined by the improved path-choice strategy, and a Hamilton-Jacobi equation to compute the total travel cost. The simultaneous satisfaction of both equations can be treated as a fixed-point problem. A self-adaptive method of successive averages (MSA) is proposed to solve this fixed-point problem. This method can automatically determine the optimal MSA step size using the least squares approach. Numerical examples are used to demonstrate the effectiveness of the model and the solution algorithm.

This paper proposes and analyzes arbitrarily high-order discontinuous Galerkin (DG) and finite volume methods which provably preserve the positivity of density and pressure for the ideal magnetohydrodynamics (MHD) on general meshes. Unified auxiliary theories are built for rigorously analyzing the positivity-preserving (PP) property of numerical MHD schemes with a Harten–Lax–van Leer (HLL) type flux on polytopal meshes in any space dimension. The main challenges overcome here include establishing certain relation between the PP property and a discrete divergence of magnetic field on general meshes, and estimating proper wave speeds in the HLL flux to ensure the PP property. In the 1D case, we prove that the standard DG and finite volume methods with the proposed HLL flux are PP, under a condition accessible by a PP limiter. For the multidimensional conservative MHD system, the standard DG methods with a PP limiter are not PP in general, due to the effect of unavoidable divergence error in the magnetic field. We construct provably PP high-order DG and finite volume schemes by proper discretization of the symmetrizable MHD system, with two divergence-controlling techniques: the locally divergence-free elements and suitably discretized Godunov–Powell source term. The former technique leads to zero divergence within each cell, while the latter controls the divergence error across cell interfaces. Our analysis reveals in theory that a coupling of these two techniques is very important for positivity preservation, as they exactly contribute the discrete divergence terms which are absent in standard multidimensional DG schemes but crucial for ensuring the PP property. Several numerical tests further confirm the PP property and the effectiveness of the proposed PP schemes. Unlike the conservative MHD system, the exact smooth solutions of the symmetrizable MHD system are proved to retain the positivity even if the divergence-free condition is not satisfied. Our analysis and findings further the understanding, at both discrete and continuous levels, of the relation between the PP property and the divergence-free constraint.