不适定问题或称反问题的研究从20世纪末成为国际上的热点问题,也是现代数学家广为关注的研究领域。随着生产和科学技术的发展,离散不适定问题在自动控制、图像处理、地球物理等诸多领域都有广泛的应用。而反问题求解面临的一个本质性困难是不适定性,求解不适定问题的普遍方法是:用与原不适定问题相“邻近”的适定问题的解去逼近原问题的解,这种方法称为正则化方法。如何建立有效的正则化方法是反问题领域中不适定问题研究的重要内容。当前,最为流行的正则化方法是基于变分原理的Tikhonov正则化及其改进方法。The study of ill-posed problems or inverse problems has become a hot topic in the world since the end of the 20th century, and it is also a research field widely concerned by modern mathematicians. With the development of production and scientific technology, discrete ill-posed problems have been widely used in many fields, such as automatic control, image processing, geophysics, etc. An essential difficulty in solving the inverse problem is illness. The general method for solving ill-posed problems is to approximate the solution of the original problem with the solution of the well-posed problem “adjacent” to the original ill-posed problem. This method is called regularization method. How to establish an effective regularization method is an important content of ill-posed problems in the field of inverse problems. At present, the most popular regularization methods are Tikhonov regularization based on variational principle and its improved method.
