二阶非散度型椭圆型方程的无梯度DG方法
A Non-gradient DG method for second-order Elliptic Equations in the Non-divergence Form
摘要：
基于$L^1$的优化算法广泛应用于图像去噪、机器学习等领域。这种方法的一个主要特点是它自然地提供了一个稀疏结构的数值解。本文研究了二阶非散度型椭圆型方程的基于$L^1$的混合DG方法。非发散形式的椭圆型偏微分方程是在完全非线性偏微分方程线性化过程中产生的。由于方程的性质，不能直接采用基于变分形式的经典有限元法。在这项工作中，我们提出了一个新的优化方案耦合的经典DG框架与最近开发的$L^1$优化技术。在弱正则性假设下，得到了能量范数和$L^{\infty}$范数下的收敛性分析.这种$L^1$模型是不可微的，因此使传统的梯度法无效。因此，所有现有的基于梯度的解算器在此设置下不再可行。为了克服这一困难，我们将$L^1$优化问题的解刻画为邻近方程的不动点，利用矩阵分裂技术得到了一类不动点邻近算法，并进行了收敛性分析.数值算例表明，通过选择有限维空间的基，数值解具有稀疏结构。在光滑和非光滑设置的数值例子来验证理论结果。
英文摘要：
$L^1$ 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.
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