# Using a Priori Information for Constructing Regularizing Algorithms

## Document Type

Presentation Abstract

## Presentation Date

11-21-2003

## Abstract

Many problems of science, technology and engineering are posed in the form of operator equation of the first kind with operator and right part approximately known. Often such problems turn out to be ill-posed. It means that they may have no solutions, or may have non-unique solution, or/and these solutions may be unstable. Usually, non-existence and non-uniqueness can be overcome by searching some ''generalized'' solutions, the last is left to be unstable. So for solving such problems is necessary to use the special methods - regularizing algorithms.

The theory of solving linear and nonlinear ill-posed problems is advanced greatly today (see for example [1, 2]). A general scheme for constructing regularizing algorithms on the base of Tikhonov variational approach is considered in [2].

It is very well known that ill-posed problems have unpleasant properties even in the cases when there exist stable methods (regularizing algorithms) of their solution. So at first it is recommended to study all a priori information, to find all physical constraints, which may make it possible to construct a well-posed mathematical model of the physical phenomena.

Computational programs for linear ill-posed problems with a priori information (monotonicity, convexity, known number of extremes, sourcewise representation of unknown solution, etc.) could be found in [1] and other author's publications, and could be generalized for nonlinear problems also. If the constraints are not sufficient to make a problem well-posed, then it is necessary to use all these constraints but we must also know error level of experimental data. As examples of successful applications of these regularizing algorithms to practical problems we consider inverse problems of vibrational spectroscopy and electron microscopy.

References

- Tikhonov, A.N., Goncharsky, A.V., Stepanov, V.V. and Yagola, A.G., Numerical Methods for the Solution of Ill-Posed Problems. Kluwer Academic Publ., Dordrecht, (1995).
- Tikhonov, A.N., Leonov, A.S. and Yagola, A.G., Nonlinear Ill-Posed Problems. Chapman and Hall, London, (1995).

## Recommended Citation

Yagola, Anatoly, "Using a Priori Information for Constructing Regularizing Algorithms" (2003). *Colloquia of the Department of Mathematical Sciences*. 154.

https://scholarworks.umt.edu/mathcolloquia/154

## Additional Details

Friday, 21 November 2003

4:10 p.m. in Skaggs 117