$L^1$ based optimization is widely used in image denoising, machine learning and related applications. One of the main features of such approach is that it naturally provide a sparse structure in the numerical solutions. In this paper, we study an $L^1$ based mixed DG method for second-order elliptic equations in the non-divergence form. The elliptic PDE in nondivergence form arises in the linearization of fully nonlinear PDEs. Due to the nature of the equations, classical finite element methods based on variational forms can not be employed directly. In this work, we propose a new optimization scheme coupling the classical DG framework with recently developed $L^1$ optimization technique. Convergence analysis in both energy norm and $L^{\infty}$ norm are obtained under weak regularity assumption. Such $L^1$ models are nondifferentiable and therefore invalidate traditional gradient methods. Therefore all existing gradient based solvers are no longer feasible under this setting. To overcome this difficulty, we characterize solutions of $L^1$ optimization as fixed-points of proximity equations and utilize matrix splitting technique to obtain a class of fixed-point proximity algorithms with convergence analysis. Various numerical examples are displayed to illustrate the numerical solution has sparse structure with careful choice of the bases of the finite dimensional spaces. Numerical examples in both smooth and nonsmooth settings are provided to validate the theoretical results.

翻译：基于$L^1$的优化方法广泛应用于图像去噪、机器学习及相关领域。这类方法的主要特点之一是能够自然地为数值解提供稀疏结构。本文针对非散度形式的二阶椭圆方程，研究了一种基于$L^1$的混合DG方法。非散度形式椭圆偏微分方程源于完全非线性偏微分方程的线性化。由于方程本身的特性，基于变分形式的经典有限元方法无法直接适用。本文提出了一种新的优化方案，将经典DG框架与最新发展的$L^1$优化技术相结合。在弱正则性假设下，分别获得了能量范数和$L^{\infty}$范数下的收敛性分析。此类$L^1$模型不可微，因此传统梯度方法失效。在此设定下，所有现有的基于梯度的求解器均不再适用。为克服这一困难，我们将$L^1$优化的解表征为邻近方程的固定点，并利用矩阵分裂技术获得一类具有收敛性分析的固定点邻近算法。通过多种数值算例展示，在有限维空间基函数合理选取下，数值解具有稀疏结构。同时提供了光滑和非光滑设定下的数值算例以验证理论结果。