标题： 莫涅-阿普拉斯方程：各种形式和数值方法 显示英文标题

标题： The Monge-Ampere equation: various forms and numerical methods

摘要： 我们提出了Monge-Ampere方程的三种新形式，该方程被用于图像处理以及原始宇宙中质量传输的重建。 本文的核心是我们的傅里叶积分形式，我们建立了核的正性和尖锐界性质。 这是在奇数维空间中开发一种新的数值求解空间周期性Monge-Ampere问题方法的基础。 通过一个宇宙学类型的测试问题展示了收敛性，在该问题中，每个局部物体中的物质分布为高斯分布，而Monge-Ampere方程的右边是这些分布的总和。 显示英文摘要

摘要： We present three novel forms of the Monge-Ampere equation, which is used, e.g., in image processing and in reconstruction of mass transportation in the primordial Universe. The central role in this paper is played by our Fourier integral form, for which we establish positivity and sharp bound properties of the kernels. This is the basis for the development of a new method for solving numerically the space-periodic Monge-Ampere problem in an odd-dimensional space. Convergence is illustrated for a test problem of cosmological type, in which a Gaussian distribution of matter is assumed in each localised object, and the right-hand side of the Monge-Ampere equation is a sum of such distributions.