二维及更高维椭圆Monge-Ampère方程粘性解的收敛有限差分求解器
Convergent finite difference solvers for viscosity solutions of the elliptic Monge-Ampère equation in dimensions two and higher
AI总结：
针对椭圆Monge-Ampère方程，提出一种宽模板单调有限差分格式，利用Barles-Souganidis理论证明收敛到唯一粘性解，并采用阻尼牛顿法求解，通过系统方法确定迭代初值，在二维和三维数值实验中展示了方法的精度和速度。
摘要：
椭圆Monge-Ampère方程是一个完全非线性偏微分方程，起源于几何曲面理论，并已应用于动力气象学、弹性力学、几何光学、图像处理和图像配准。解可能是奇异的，此时标准数值方法失效。需要新颖的求解方法以实现稳定性和收敛到弱（粘性）解。在本文中，我们为Monge-Ampère方程构建了一种宽模板有限差分离散化。该格式是单调的，因此Barles-Souganidis理论允许我们证明该格式的解收敛到方程的唯一粘性解。该格式的解通过阻尼牛顿法求得。我们证明了牛顿法的收敛性，并提供了一种系统的方法来确定牛顿迭代的起始点。给出了二维和三维的计算结果，展示了该方法在多个精确解上的速度和精度，这些解的正则性范围从光滑到不可微。
英文摘要：
The elliptic Monge-Ampère equation is a fully nonlinear Partial Differential Equation that originated in geometric surface theory and has been applied in dynamic meteorology, elasticity, geometric optics, image processing and image registration. Solutions can be singular, in which case standard numerical approaches fail. Novel solution methods are required for stability and convergence to the weak (viscosity) solution. In this article we build a wide stencil finite difference discretization for the \MA equation. The scheme is monotone, so the Barles-Souganidis theory allows us to prove that the solution of the scheme converges to the unique viscosity solution of the equation. Solutions of the scheme are found using a damped Newton's method. We prove convergence of Newton's method and provide a systematic method to determine a starting point for the Newton iteration. Computational results are presented in two and three dimensions, which demonstrates the speed and accuracy of the method on a number of exact solutions, which range in regularity from smooth to non-differentiable.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
