Mathematical Sciences: Sampling Solutions of Ill-Posed Problems

本项目将发展求解病态积分和算子方程的数值方法。这些方法是基于索博列夫空间核子空间中函数抽样定理的一种新的一般方法。各种类型的误差（截断、混叠、抖动和幅度）将被研究。介绍了正则Mikusinski算子的采样理论以及不适定问题的小波采样。从雷达技术到石油采收等领域的应用，推动了该项目的发展。这些研究是诸如扫描和断层扫描等应用软件开发基础的一部分

This project will develop numerical methods for the solution of ill-posed integral and operator equations. The methods are based on a new general approach to sampling theorems for functions in reproducing kernel subspaces of a Sobolev space. Various types of errors (truncation, aliasing, jitter, and amplitude) will be studied. Sampling theory for regular Mikusinski operators as well as wavelet sampling of ill- posed problems will be initiated. This project is motivated by several applications arising in subjects ranging from radar technology to oil recovery. These studies are part of the foundation of software development in applications such as scanning and tomography