Optimal recovery in problems of solving linear integral equations with a priori information

AbstractIn this article, we consider an inverse problem for the integral equation of the convolution type in a multidimensional case. This problem is severely ill-posed. To deal with this problem, using a priori information (sourcewise representation) based on optimal recovery theory we propose a new method. The regularization and optimization properties of this method are proved. An optimal minimal a priori error of the problem is found. Moreover, a so-called optimal regularized approximate solution and its corresponding error estimation are considered. Efficiency and applicability of this method are demonstrated in a numerical example of the image deblurring problem with noisy data.

Download Full-text

COMBINED APPROXIMATION METHODS FOR SOLVING THE PROBLEMS OF GRAVITY AND MAGNETIC PROSPECTING

Геология и геофизика Юга России ◽

10.23671/vnc.2018.3.16544 ◽

2018 ◽

Author(s):

И.А. Керимов ◽

И.Э. Степанова ◽

Д.Н. Раевский

Keyword(s):

Inverse Problem ◽

A Priori ◽

Integral Representations ◽

Approximation Methods ◽

A Priori Information ◽

Mathematical Experiment ◽

Gravity And Magnetic ◽

Priori Information ◽

Linear Integral ◽

Combined Approximations

В статье исследуется взаимосвязь различных вариантов метода линейных интегральных представлений. Комбинированные аппроксимации рельефа и геопотенциальных полей позволяют осуществить более тонкую «настройку» метода при решении обратных задач геофизики и геоморфологии, а также наиболее полно учесть априорную информацию о высотных отметках и элементах аномальных полей. Приводится описание методики нахождения численного решения обратной задачи по поиску распределений эквивалентных по внешнему полю носителей масс. Обсуждаются результаты математического эксперимента The article investigates the interrelation of various variants of the method of linear integral representations. Combined approximations of the relief and geopotential fields allow for a more subtle «tuning» of the method in solving the inverse problems of geophysics and geomorphology, and also take into account a priori information about altitude marks and elements of anomalous fields. A description of the procedure for finding the numerical solution of the inverse problem in the search for distributions of mass carriers equivalent in the external field is given. The results of a mathematical experiment are discussed

Download Full-text