Symmetric variational methods

假设有人给你一份脏了的影印本，你需要对它进行润色。人眼非常善于猜测缺失的曲线片段应该如何连接起来。现在假设你有数百个这样的副本。电脑能做到吗？问题是我们可以在不知道怎么做的情况下做这些事情。必须用计算机自己的语言准确地告诉计算机该做什么。这意味着我们需要用数学来解决曲线补全的问题。这个项目是对这个问题和类似问题的一个贡献，这些问题可以用同样的数学术语来描述。基本上，我们尝试了许多不同的完井，然后问，哪一个看起来最好？表述“最佳外观”的含义是一个问题，找到描述解决方案的方法是另一个问题。成功的关键是使解决方法类似于我们已经知道和喜爱的数学。

Suppose you are given a dirty photocopy and you need to touch it up.The human eye is very good at guessing how missing pieces of curvesshould be joined up. Now suppose you have hundreds of such copies.Can a computer do it? The problem is that we can do these thingswithout knowing how we do it. A computer has to be told exactlywhat to do, in the computer's own language. This means we needto solve the problem of curve completion using mathematics.This project is a contribution to this and similar problemswhich can be described in the same mathematical terms.Basically, we try lots of different completions and ask,which looks best? Formulating what it means to ``look best is one problem, finding a way to describe the solutionis another. The key to success is to make the solutionmethod resemble mathematics we already know and love.