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IPI

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.

KRM

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.

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