Optimal recovery and Extremum problems: methods and solutions

• Grant number: 13/15840-4
• Support Opportunities: Research Grants - Visiting Researcher Grant - International
• Start date: January 27, 2014
• End date: April 24, 2014
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Dimitar Kolev Dimitrov
• Grantee: Dimitar Kolev Dimitrov
• Visiting researcher: Konstantin Osipenko
• Visiting researcher institution: MATI-Russian State Technological University, Russia
• Host Institution: Instituto de Biociências, Letras e Ciências Exatas (IBILCE). Universidade Estadual Paulista (UNESP). Campus de São José do Rio Preto. São José do Rio Preto , SP, Brazil
• Associated research grant: 09/13832-9 - Orthogonal polynomials, special functions and applications, AP.TEM

Abstract

This research and teaching project develops a common approach to a number of optimization problems basing on the theory of optimal recovery. One of the central ideas of this approach, which originates from the famous work of Andrei Nikolaevich Kolmogorov, is, for an arbitrary given method at certain a priori given assumptions, to introduce a margin of solution error and then to find a method for which this error is minimal. Thus it becomes possible to compare different methods and as a result to find optimal methods. The Visitor will talk about new algorithms for solving optimization problems with inaccurate initial data. That will have a form of lectures or research seminars (involving PhD and Master students from IBILCE UNESP). After these lectures a broad scientific discussion and talking over are planned. In particular, we will discuss research plans related to the above topic where we expect to obtain optimal recovery methods addressing a number of problems in Mathematical Physics in case of continuous and discrete models, as well as some problems associated with recovery of functions given by inaccurate Radon transformations. The project develops a common approach to the construction of optimal algorithms for solving problems with inaccurate initial data. An essential part of that approach is related to the study of the case when the error of input data and of the metric optimizing the error are non-Euclidean. Specific tasks which will be used at constructing various algorithms are associated with the study of continuous and discrete models of Mathematical Physics, where the original data is set inaccurately. For example, for discrete models we will obtain a number of new finite-difference methods, adapted to the inaccurate input data. These methods will be applied to the problems of optimal recovery of functions by inaccurately known Radon transformation, which is widely used in tomography. (AU)