蒙日-安培方程是一类二阶完全非线性偏微分方程，因其在微分几何、最优输运等领域中的重要应用而被广泛关注研究。发展高效的数值方法是求解蒙日-安培方程的有效途径之一，由于方程高度非线性，有效数值格式的设计和理论分析是一个极具挑战性的课题。申请人前期发展了一种基于单元片重构空间的数值方法，并应用于几类经典线性偏微分方程的数值求解，结果表明该方法具有极高的效率和灵活性，特别适合逼近高阶微分方程具有高正则性的解。本项目拟以申请人前期发展的新型数值方法为基础，试图将其应用于数值求解蒙日-安培方程，利用该方法的灵活性和适用于高阶问题的特点，期望发展出一种稳定高效的数值方法，使得数值求解蒙日-安培方程的研究获得一些实质的进步。

The Monge-Ampere equation is a fully nonlinear second-order partial differential equation of special kind, which has attracted pervasive interests due to its ever increasing importance in pure mathematical topics as differential geometry and in many application fields, for example optimal transportation and machine learning. This makes the numerical method for the Monge-Ampere equation a very hot area in the last decades. Though significant progress in developing numerical schemes for the Monge-Ampere equation has been made, it is still far from maturely solving it with numerical methods. Many problems remain open because of the full nonlinearity of this equation. Recently we have developed a novel numerical approach to approximate the solution of PDEs using a brand-new discontinuous space with patch reconstructed basis functions, and the approach is adopted to construct numerical schemes for several classical PDEs successfully. Our analysis and numerical results demonstrate great efficiency and remarkable flexibility of the new methods, particularly when it is used in problems with solutions of high regularity, which is one of the most important attributes of the Monge-Ampere equation. The object in this proposal is to develop a new numerical method to solve the Monge-Ampere equation based on the new proposed approximation space. We are expected to utilize these unique advantages of the new space to develop a new numerical method with much improved efficiency and robustness for the Monge-Ampere equation.