Calculus of variations and optimal mass transportation: theory and applications

这一建议主要集中在两个主要方向，偏微分方程，重点是变分法，以及最优大众运输的理论和应用。*********偏微分方程（PDEs）由于与物理世界中的现象有关而引起广泛的兴趣。科学家和数学家已经积极地参与到由偏微分方程建模的无数问题的研究中。我主要对自然科学和微分几何中的非线性偏微分方程的研究感兴趣。除了开发新技术外，我还应用泛函和凸分析的工具来研究确定性和随机偏微分方程的定性性质。*********我也研究最优大众交通的理论和应用。最优质量输运是一门历史悠久的学科，有着极其丰富的理论和应用，它定义了概率分布之间的稳健度量。通过相关运输计划计算分布之间的最优位移使该理论在图像处理、癌症检测和统计机器学习等几个应用领域成为主流。这是一个充满活力和不断发展的领域，它带来了令人惊讶的新发现以及对经典结果的新解释。*********这个研究计划的目的是双重的。***首先，将变分微积分和变分不等式的技术与新建立的变分原理结合起来，研究和分析自然科学中出现的一大类随机和确定性偏微分方程的存在性和解的定性性质。******其次，考虑到标准和鞅多边际最优质量输运问题的解决方案能够提供信号强度和其他数据分布的精确模型，表征和仔细研究这些问题的解决方案。除了发展最优交通的数学理论外，我还将研究它在现实生活中的各种应用。*********我希望这项研究的结果在偏微分方程、几何测量理论、变分法、随机偏微分方程以及最优大众运输在其他领域的应用方面产生重大影响。

This proposal is focused on two primary directions, partial differential equations with an emphasis on the calculus of variations, and the theory and applications of optimal mass transportation.*********Partial differential equations (PDEs) are of widespread interest because of their connection with phenomena in the physical world. Scientists and mathematicians have become actively involved in the study of countless problems modeled by PDEs. I am mainly interested in the study of non-linear partial differential equations arising from natural sciences and differential geometry. In addition to developing new techniques, I apply tools from functional and convex analysis to study the qualitative properties of deterministic and stochastic partial differential equations.*********I also study the theory and application of optimal mass transportation. Optimal mass transport is a subject with a long history, with incredibly rich theory and applications, that defines robust metrics between probability distributions. The computation of optimal displacements between distributions through an associated transport plan makes this theory mainstream in several applicable fields including image processing, cancer detection, and statistical machine learning. It is a dynamic and growing area that has led to surprising new discoveries as well as novel interpretations of classical results.*********The purpose of this proposed research program is two-fold.***First, to combine techniques from the calculus of variations and variational inequalities with newly established variational principles to study and analyze the existence, and qualitative properties of solutions for a large class of stochastic and deterministic partial differential equations arising from the natural sciences.******Second, to characterize and scrutinize solutions of standard and martingale multi-marginal optimal mass transport problems given their abilities to provide accurate models for signal intensities and other data distributions. In addition to developing the mathematical theory of optimal transportation, I will also work on its various real-life applications.*********I expect the outcomes of this research to have significant impacts in the fields of partial differential equations, geometric measure theory, calculus of variations, stochastic PDEs, and applications of optimal mass transportation in other fields.