We use a positive parameter to develop a dierentiable perturbed reconstruction model to solve the L1–L2 magnetic resonance image (MRI) reconstruction problem. We use Bregman iterative formulation to solve the dierentiable perturbed L1–L2 model, and lagged diusivity xed-point iteration to solve the minimization problem in the Bregman iteration. Two Fourier transforms and an inverse Fourier transform are used to accelerate L1–L2 MRI reconstruction. Real MR images are used to test the method in numerical experiments. The results demonstrate that the proposed method is very ecient for L1–L2MRI reconstruction.

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.

Following our previous work \cite{HLY}, we introduce the notions of partial seed homomorphisms and partial ideal rooted cluster morphisms. Related to the theory of Green's equivalences, the isomorphism classes of sub-rooted cluster algebras of a rooted cluster algebra are corresponded one-by-one to the regular $\mathcal D$-classes of the semigroup consisting of partial seed endomorphisms of the initial seed. Moreover, for a rooted cluster algebra from a Riemannian surface, they are also corresponded to the isomorphism classes of the so-called paunched surfaces.

Convergence of a second-order shock-capturing scheme for the system of isentropic gas dynamics with $ {L^\infty }$ initial data is established by analyzing the entropy dissipation measures. This scheme is modified from the classical MUSCL scheme to treat the vacuum problem in gas fluids and to capture local entropy near shock waves. Convergence of this scheme for the piston problem is also discussed.

Difference approximations of hyperbolic partial differential equations with highly oscillatory coefficients and initial values are studied. Analysis of strong and weak convergence is carried out in the practically interesting case when the discretization step sizes are essentially independent of the oscillatory wave lengths. © 1993 John Wiley & Sons, Inc.