Faculty of Mathematics, Mechanics and Computer Science

Southern Federal University

Faculty of mathematics, mechanics and computer science

Milchakova str. 8a, Rostov-on-Don, 344090
Phone: (863) 2975 111;
Fax: (863) 2975 113;

SYLLABUS

1-semester course

Inverse Problems, Identification and Optimization

for Masters Program Computational Mechanics and Biomechanics.

4 ECTS Credits

Aim of the course

The mathematical modeling in mechanics and biomechanics is typically faced with direct and inverse problems. An inverse problem is a general framework that is used to convert observed measurements into information about a physical object or system that we are interested in. The reason is that many parameters are unknown a priori, and after some measurements have been done they may be restored as a result of a certain inverse problem solution. This theory is thoroughly connected with identification and optimization problems. All of them require specific mathematical techniques, which are the basic goal of the present lecture course.

Also this course analyzed the program coding techniques in C++ for the efficient numerical solution of problems in science and engineering.

Techniques, Skills, etc.

After completing the course, the students are expected to be able to:

·  distinguish inverse and direct problems;

·  compare different approaches to the inverse problems, identification and optimization;

·  choose and implement a suitable approach for a given problem;

·  ascertain basic properties of continuous and discrete problems and analyze their correlations;

·  understand and analyze the stability of a method applied;

·  develop their own C++ code to solve typical inverse, identification and optimization problems;

·  present coherent arguments to answer questions both orally and in writing.

Teaching

The following methods and forms of study are used in the course:

·  Lectures

·  Labs

·  Problem sets

·  Program coding

·  Self-study

·  Use of different reference books and Internet resources

At the end of the course the students are supposed to do problem sets and write a report, make an oral presentation and participate in discussion. Upon the successful completion, the students will gain 4 credits.

Course content

№
NO / Subject / Form of Lesson / Duration (hrs) / Date
1.  / Introduction. The concept of direct and inverse problems in mathematical physics, some examples. / Lecture / 1
2.  / Explicit solutions to some simple inverse problems. / Lecture / 1
3.  / The concept of stable and unstable problems in mathematical physics. / Lecture / 1
4.  / The Hadamard and Tikhonov stability – commons and differences. / Lecture
Self-study / 1
1
5.  / Integral equations of the 1-st kind as an unstable problem. / Lecture / 2
6.  / Tikhonov theorem and classes of correctness on compact sets. / Laboratory,
Self-study / 2
4
7.  / Iteration methods for unstable problems. / Lecture / 4
8.  / Steepest-descent method and its variety: convergence and stability in inverse problems. / Lecture
Self-study / 8
2
9.  / Application of iteration methods to solve integral equations of the 1-st kind. / Laboratory,
Self-study / 4
6
10.  / Application of iteration methods to inverse problems, which require calculation of derivatives to functions given approximately. / Lecture / 2
11.  / Discretization and quantization of image. Mathematical representation of digital images. Space resolution and resolution by brightness. / Laboratory,
Self-study, / 5
8
12.  / Spatial improvement of images. The concept of histogram. Averaging method. Logarithmic and power-function in contrast ratio improvement. Piecewise linear transformation. The tactics of brightness segment cut. / Lecture
Self-study / 2
2
13.  / Spatial filtration. The concept of window (mask). Masks for averaging and median filtration methods. Improvement of clearness. Filters with derivatives. / Lecture
Self-study / 2
2
14.  / Image segmentation. Discontinuity recognition, isolated dots. Lines recognition. / Lecture
Self-study / 2
2
15.  / Edge recognition by using first- and second- order derivatives. Using gradient and Laplacian operators. / Lecture
Self-study / 4
2
16.  / Optimization free of derivative usage. The gradient methods. Steepest-descent and conjugate gradient methods. / Laboratory,
Self-study / 6
8
17.  / Optimization methods using second-order derivatives. Newton-Kantorovich method. Method of Levenberg and Marcward. / Lecture
Self-study / 2
4
18.  / Stochastic methods. The basics of genetic algorithms. / Lecture
Self-study / 2
8
19.  / Summarizing presentation / Colloquium

Requirements

During the session students are required to

·  attend class lectures;

·  be prepared to laboratory hours;

·  attend the laboratory;

·  write a report;

·  represent the report results in oral presentation at the colloquium;

·  be prepared to participate in final course discussion.

Grade determination

·  Class participation - 30%

·  Laboratory work - 40%

·  Written report and its presentation – 20%

·  Participation in discussion – 10%

Literature

Core

1.  A.N.Tikhonov, V.Ya.Arseniin. Solutions of ill-posed problems. Wiley: NY, 1977.

2.  A.A.Goruynov, A.V.Saskovets. Inverse scattering problems in acoustics. Moscow: Moscow State University Press, 1989 (in Russian).

3.  P.E.Gill, W.Murray, M.H.Wright. Practical Optimization. – Academic Press: London, 1981.

4.  R.Gonzalez, R.Woods. Digital Image Processing. – Addison-Wesley: Reading, MA, 1992.

Additional

B. K. P. Horn, Robot Vision. – MIT Press: Cambridge, MA, 1985.

R.Bates, M.McDonnell. Image Restoration and Reconstruction. – Clarendon Press: NY, 1986.

Internet Resources

1. Visual C++

2. www.microsoft.com

3. Maple

4. Microsoft Windows XP, Windows 7 (licensed)

5. Microsoft Office 2003, 2010 (licensed)

6. Internet training rooms of the Faculty