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.
2
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.
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Course content
№
Subject
NO
1.
Introduction. The
Form of
Duration
Lesson
(hrs)
Lecture
1
Explicit solutions to Lecture
1
concept of direct and
inverse problems in
mathematical physics,
some examples.
2.
some simple inverse
problems.
3.
The concept of
Lecture
1
Lecture
1
Self-study
1
Lecture
2
Laboratory,
2
Self-study
4
Lecture
4
Lecture
8
Self-study
2
stable and unstable
problems in mathematical
physics.
4.
The Hadamard and
Tikhonov stability –
commons and differences.
5.
Integral equations
of the 1-st kind as an
unstable problem.
6.
Tikhonov theorem
and classes of correctness
on compact sets.
7.
Iteration methods
for unstable problems.
8.
Steepest-descent
method and its variety:
convergence and stability
Date
4
in inverse problems.
9.
Application of
Laboratory,
4
iteration methods to solve
Self-study
6
Lecture
2
Laboratory,
5
Self-study,
8
Lecture
2
Self-study
2
Lecture
2
Self-study
2
integral equations of the 1st kind.
10.
Application of
iteration methods to
inverse problems, which
require calculation of
derivatives to functions
given approximately.
11.
Discretization and
quantization of image.
Mathematical
representation of digital
images. Space resolution
and resolution by
brightness.
12.
Spatial
improvement of images.
The concept of histogram.
Averaging method.
Logarithmic and powerfunction in contrast ratio
improvement. Piecewise
linear transformation. The
tactics of brightness
segment cut.
13.
Spatial filtration.
The concept of window
5
(mask). Masks for
averaging and median
filtration methods.
Improvement of clearness.
Filters with derivatives.
14.
Image
segmentation.
Lecture
2
Self-study
2
Discontinuity recognition,
isolated dots. Lines
recognition.
15.
Edge recognition by Lecture
using first- and second-
Self-study
4
2
order derivatives. Using
gradient and Laplacian
operators.
16.
Optimization free of Laboratory,
derivative usage. The
6
Self-study
8
Lecture
2
Self-study
4
Lecture
2
Self-study
8
gradient methods.
Steepest-descent and
conjugate gradient
methods.
17.
Optimization
methods using secondorder derivatives. NewtonKantorovich method.
Method of Levenberg and
Marcward.
18.
Stochastic methods.
The basics of genetic
algorithms.
6
19.
Summarizing
Colloquium
presentation
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,
2.
A.A.Goruynov, A.V.Saskovets. Inverse scattering problems in acoustics.
1977.
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.
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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