We are interested in front propagation problems in the presence of obstacles. We extend a previous work (Bokanowski, Cheng and Shu 09}), to propose a simple and direct discontinuous Galerkin (DG) method adapted to such front propagation problems. We follow the formulation of Bokanowski et al. (Bokanowski_Forcadel_Zidani_2010), leading to a level set formulation driven by $\min(u_t + H(x,\nabla u), u-g(x))=0$, where $g(x)$ is an obstacle function. The DG scheme is motivated by the variational formulation when the Hamiltonian $H$ is a linear function of $\nabla u$, corresponding to linear convection problems in presence of obstacles. The scheme is then generalized to nonlinear equations, written in an explicit form. Stability analysis are performed for the linear case with Euler forward, a Heun scheme and a Runge-Kutta third order time discretization using the technique proposed in Zhang and Shu 2010. Several numerical examples are provided to demonstrate the robustness of the method. Finally, a narrow band approach is considered in order to reduce the computational cost.

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In this paper, we study the rate of convergence of the cyclic projection algorithm applied to finitely many basic semialgebraic convex sets. We establish an explicit convergence rate estimate which relies on the maximum degree of the polynomials that generate the basic semialgebraic convex sets and the dimension of the underlying space. We achieve our results by exploiting the algebraic structure of the basic semialgebraic convex sets.

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.

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