ИСТИНА

Multidimensional ill-posed problems Anatoly Yagola Department of Mathematics, Faculty of Physics, Lomonosov Moscow State University Moscow 119991, RUSSIA E-mail: yagola@physics.msu.ru It is very important now to develop methods of solving multidimensional ill-posed problems using regularization procedures and parallel computers. The main purpose of the talk is to show how 2D and 3D Fredholm integral equations of the 1st kind can be effectively solved. We will consider ill-posed problems on compact sets of convex functions [2] and functions convex along lines parallel to coordinate axes [3]. Recovery of magnetic target parameters from magnetic sensor measurements has attracted wide interests and found many practical applications. However, difficulties present in identifying the permanent magnetization due to the complications of magnetization distributions over the ship body, and errors and noises of measurement data degrade the accuracy and quality of the parameter identification. In this paper, we use a two step sequential solutions to solve the inversion problem. In the first step, a numerical model is built and used to determine the induced magnetization of the ship. In the second step, we solve a type of continuous magnetization inversion problem by solving 2D and 3D Fredholm integral equations of the 1st kind. We use parallel computing which allows solve the inverse problem with high accuracy. Tikhonov regularization has been applied in solving the inversion problems. The proposed methods have been validated using simulation data with added noises [4, 6]. 2D and 3D inverse problems also could be found in tomography [3] and electron microscopy [5]. We will demonstrate examples of applied problems and discuss methods of numerical solving. Acknowledgement This paper was supported by the Visby program and RFBR grants 11-01-00040–а and 12-01-91153-NSFC-a. References 1. V. Titarenko, A. Yagola. Linear ill-posed problems on sets of convex functions on two-dimensional sets. – J. of Inverse and Ill-Posed Problems, 2006, v. 14, No 7, pp. 735-750. 2. V. Titarenko, A. Yagola. Solution of ill-posed problems on sets of functions convex along all lines parallel to coordinate axes. - Journal of Inverse and Ill-posed Problems, 2008, Vol. 16, No. 8, pp. 805–824. 3. S. Titarenko, Philip J. Withers. A. Yagola. An analytic formula for ring artefact suppression in X-ray tomography. – Applied Mathematics Letters, v. 23, № 12, 2010, pp. 1489-1495. 4. D.V. Lukyanenko, A.G. Yagola, N.A. Evdokimova. Application of inversion methods in solving ill-posed problems for magnetic parameter identification of steel hull vessel. – Journal of Inverse and Ill-Posed Problems, 2011, v. 18, issue 9, pp. 1013–1029. 5. N.A. Koshev, N.A. Orlikovsky, E.I. Rau, A.G. Yagola. Solution of the inverse problem of restoring the signals from an electronic microscope in the backscattered electron mode on the class of bounded variation functions. - Numerical Methods and Programming, 2011, v. 12, pp. 362-367 (in Russian). 6. D.V. Luk’yanenko, A.G.Yagola. Application of multiprocessor systems for solving inverse problems leading to Fredholm integral equations of the first kind. – Proceedings of the Institute of Mathematics and Mechanics of the Ural branch of the Russian Academy of Sciences, 2012, v. 18, No 1, pp. 222-234 (in Russian).