Diffusive limit with geometric correction of unsteady neutron transport equation
Lei Wu
Kinetic & Related Models 2017, 10(4): 1163-1203 doi: 10.3934/krm.2017045

We consider the diffusive limit of an unsteady neutron transportequation in a two-dimensional plate with one-speed velocity. We show the solution can be approximated by the sum of interior solution, initial layer, and boundary layer with geometric correction. Also, we construct a counterexample to the classical theory in [1] which states the behavior of solution near boundary can be described by the Knudsen layer derived from the Milne problem.

keywords: Compatibility condition $\epsilon$-Milne problem Knudsen layer geometric correction diffusive limit
A new spectral method for $l_1$-regularized minimization
Lei Wu Zhe Sun
Inverse Problems & Imaging 2015, 9(1): 257-272 doi: 10.3934/ipi.2015.9.257
In this paper, we propose an iterative method for solving the $\ell_1$-regularized minimization problem $\min_{x\in\mathbb{R}^n} f(x)+\rho^T |x|$, which has great applications in the areas of compressive sensing. The construction of our method consists of two main steps: (1) to reformulate an $\ell_1$-problem into a nonsmooth equation $h^\tau(x)={\bf 0}$, where ${\bf 0}\in \mathbb{R}^n$ is the zero vector; and (2) to use $-h^\tau(x)$ as a search direction. The proposed method can be regarded as spectral residual method because we use the residual as a search direction at each iteration. Under mild conditions, we establish the global convergence of the method with a nonmonotone line search. Some numerical experiments are performed to compare the behavior of the proposed method with three other methods. The results positively support the proposed method.
keywords: spectral residual method convex optimization Compressed sensing global convergence. $l_1$-regularized

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