Probabilistic numerical methods for Inverse Problems

这个项目关注的是函数空间的数值方法，可能包括反问题的应用。特别是，函数空间的概率方法为我们提供了一个数学框架来定义兴趣量和我们观察到的数据之间的关系。因为这些关系是概率的，所以它也使我们能够以概率的方式对量进行推断。反问题的一个很好的例子是通过查看卫星图像来确定结构的材料属性。从本质上讲，问题的定义非常模糊，例如，解决方案相对于数据不稳定或不是唯一的。对于这样的问题，我们需要找到定义和刻画函数空间的方法，并能够从噪声和有限的数据中学习。这一数学理论是解决这类不适定问题的方法不断发展的基础。

This project is concerned with numerical methods for function spaces that would possibly include applications to inverse problems. In particular, probabilistic methods for function spaces provide us with a mathematical framework to define relationships between quantities of interest and our observed data. Because the relationships are probabilistic, it also enables us to make inferences on the quantities in a probabilistic manner. A good example of an inverse problem is determining material properties of a structure by looking at a satellite image. In its nature, the problem is very ill-defined, e.g. the solutions are unstable with respect to the data or not unique. For such problems, we need to find ways to define and characterize function spaces and to be able to learn from noisy and limited data. This mathematical theory is fundamental for the ongoing development of methods to solve such ill-posed problems.