Report of CIMPA School Inverse problem:theory and applications 5 to 14 May 2014 Erbil, Kurdistan-region, IRAQ

Transcription

1 Report of CIMPA School Inverse problem:theory and applications 5 to 14 May 2014 Erbil, Kurdistan-region, IRAQ Abdeljalil Nachaoui 1 Fatima Aboud 2 October 3, Laboratoire de Mathématiques Jean Leray, Université de Nantes, France 2 Department of Mathematics, College of Science, University of Diyala, Irak

2 Introduction This school is a continuation of the activities in Kurdistan since 2008 supported by the CIMPA and the French Embassy in Bagdad. The school was preceded in September 2011 by an intensive course on numerical methods for approximation of partial differential equations (24 students and researchers has participated in this workshop, these participants come from Kurdistan and different region of Iraq) and in May 2012 by a workshop about free boundary problems and a series of seminars about the application of numerical method with practice examples (these workshops and seminars took place in two different Universities in Kurdistan-Iraq with a participation of about 32 Iraqi students and researchers). Some basic courses about inverse problemes were taken place for the periode from 25 to 31 October These courses are a way to bring together researchers from Iraq and to do a preparation for our CIMPA school. Opening ceromony started by some talks by the the representer of the President of the University of Salahaddin and the University of Diyala, the deans of the college of sciences in these Universities, the councillor cultural of the Franch embassy in Erbil and the chef of commuttee of orginization of the school. The opening ceromony finished by a presentation of CIMPA activities in Iraq since 2009 given by Fatima ABOUD. The program of the school consists of 6 hours and 30 minutes of courses and communication by day. The day was loaded by the scientific program so almost every evening the participants and the lecturers have the diner together, which was a rest moments for every one after a long day of differnts courses. The local orginisors with some local participants have proposed some visits for the foreigns participants to the histrical place of Erbil, such as the Citadel and the bazar in center of Erbil. The Citadel of Erbil is the historical city centre of Erbil in Iraq, it has been claimed that the site is the oldest continuously inhabited town in the world dates to the 5th millennium BC, and possibly earlier. A social program orginized by the local orginisors of Salahaddin University, in Friday 9 May, the program contains a visit to the Shanadar cave (which is located in the bradost series of mountains of Erbil, neanderthals has been lived their before 70,000-50,000 BC. 0.1 Goel of the school This school will give possibility to young mathematicians from Iraq and others countries who do research in partial differential equations and Numerical Analysis, to be initiated to inverse problem technique which is one of the ways to handle original problem in such a way that it can be easily solved. The school will provide impetus to study and research in these areas. It may be remarked that inverse problem techniques are not as popular as it should be in this region. The aim was to introduce students and non-experts in the field of inverse problems in order to build momentum around this topic that covers many areas of research ranging from modeling to simulation through mathematical analysis of partial differential equations and ordinary and delay differential equations as well as the development of new solution algorithms. The scientific program of 1

3 this school included lectures, seminars and slots reserved for participants who wish to present their work. The objective of this last possibility was to know the domain of research of the participants or their initial training, to encourage those who cannot afford to go exhibit in international conferences. Presentations courses and seminars were prepared as courses for students and not as seminars for colleagues. That is, the lecturer have prepared them courses with an objective of initiation, and stimulus to research activity, and therefore start with reminders and detailed behavior of many examples of motivation and application. Covered Topics Boundary inverse problems: free boundary problems, interface problems, shape optimization. Completion of data problems, methods of optimization and iterative reconstruction methods Inverse problems and adapted spectral techniques. Inverse problems governed by delay differential equations: theoretical and numerical methods Scientific Committee F. Aliev (Director of Institute of Applied Mathematics Baku State University, Azebaijan) A. Nachaoui (Université de Nantes, France), T. Tadumdaze (Institute of Applied Mathematics, Tbilisi State University, Tbilisi,Georgia), O. Veliev (Dogus university, Turkey), A. Zeghal (Université Sultan Moulay Sliman, Morocco) Local organizing committee Chair : Dr Herish Omer, College of Sciences, department of Mathematics, University of Salahaddin Dr Rostam Karim Saeed, College of Sciences, department of Mathematics, University of Salahaddin :rostamkarim@yahoo.com Dr Ibrahim Othman, College of Sciences, department of Mathematics, University of Salahaddin Dr Ammar Seddiq Mahmood, department of Mathematics, University of Musol asmahmood65@yahoo.fr Dr Raheam Al-Saphory, department of Mathematics, University of Tikrit saphory@hotmail.com Sponsors The sponsors of our school was the University of Salahaddin, the University of Diyala, CIMPA and the University of Nantes, see the appendex. CIMPA: US dollars University of Diyala: US dollars University of Salaheddin: US dollars 2

4 Universite de Nantes: US dollars with a total of despenses : US dollars Participation There were 3 participants from Morrocco, 6 participants from Pakistan, 124 participants from Iraq (100 from Kurdistan region and 24 from Iraqi Universities out Kurdistan). Unfurtuntly, there were 2 participants from India who were not able to participate for the problem of visa and 6 participants from Iran who also can not participate for the raison of not having the permission of them institutions. Scientific Courses Inverse problems: Identification and reconstruction. A. Nachaoui : France. Prof. A. Nachaoui is member of editorial board of many international journals and published several research papers on inverse problems. He has also organized two International Conferences on inverse problems and he was in scientific comity of three others. He has also organized many workshops on this field in Morocco, Azerbaijan Turkey, and Kurdistan-Iraq. Abstracts: Mathematical models of phenomena encountered in science and engineering are usually formulated as a set of differential or integral equations. For a full description of the phenomena, coefficients of governing equations, the geometry of a body, initial and boundary conditions should be specified. In this case the problems are referred to as direct problems which they have been well examined for linear, nonlinear and multidimensional cases with regular and irregular geometries, including the question of solution uniqueness and stability. Once, one of those parameters or conditions just mentioned is either unknown or not fully specified, the problem is classified as an inverse problem. In the present course, we are interested by the resolution of a class of nonlinear inverse boundary value problems which describes numerous applications in many areas of science and engineering. We present a large class of techniques developed over the past several decades including a wide range of numerical optimization techniques with a strong focus on regularizing iterative methods. We show convergence results for these methods and discuss technical numerical implementation using finite and boundary element methods Boundary Inverse Problems and Shape Optimization. A. Chakib, Morocco. A. Chakib, Professor at university of Sultan moulay Sliman Beni Mellal Morocco, is a specialist in shape optimization. Abstracts: Many shape optimization problems can be seen in the larger framework of optimal control problems: indeed an admissible shape plays the 3

5 role of as admissible control, and the corresponding state variable is usually the solution of PDE on control domain. This series of lectures will be mainly focused on interface problems, these problems are reformulate as shape optimisation problems where one has to deform and modify the admissible shapes in order to comply with a given cost that needs to be minimized. A few theorems prove the existence of solutions under further geometrical, topological or regularity constraints. The classical domain variation method will be presented and numerical approximations will be constructed. Outline of the course: Basic principles of shape optimization. Domain variation and Existence theory. Gradient of a functional and Shape derivative. Technical numerical issue Methods of self-adaptive mesh based on a posteriori estimates. A Bergam Morocco Mrs. A. Bergam, professor at University Abdelmalik Assaadi Tetouan Morocco, is a specialist in a posteriori estimators for linear and non linear problems. Abstracts: Solutions of inverse problems describing natural phenomena often have singularities in regions of the studied domain. Precision calculations around these areas, sources of gross errors, are necessary to achieve an acceptable description of the phenomenon. Techniques for mesh adaptation are to refine locally the source regions of gross errors, and solve the problem with a sufficient number of points which allows a good improvement in accuracy. We are interested in this course to a self-adaptive mesh based on a posteriori estimates, which are to increase the error, according to the quadratic sum of certain quantities explicitly calculated, and called: local indicators of error. We show how these error indicators permit not only to perform the identification of source regions of gross errors but also with a choice of a strategy of refinement, the construction of a criterion for mesh adaptation. As example we will develop an analysis of an inverse problem of complete data, approached by finite elements. Inverse Problems: From Regularization Theory to Probabilistic and Bayesian Inference, A. Djafari, France Ali Mohammad-Djafari is research Director at CNRS in L2S (umr 8506 Supelec- CNRS-Univ Paris Sud), France. Outline of contents of course: Inverse problems examples Classification of Inversion methods: Analytical, Parametric and Non Parametric Algebraic methods Regularization theory Bayesian inference for inverse problems Full Bayesian with hyperparameter estimation Two main steps in Bayesian approach: Prior modeling and Bayesian computation Priors which enforce sparsity Heavy tailed: Double Exponential, Generalized Gaussian,... Mixture models: Mixture of Gaussians, Student-t,... Gauss-Markov-Potts models 4

6 Computational tools: MCMC and Variational Bayesian Approximation (VBA) Some results and applications X ray Computed Tomography, Microwave and Ultrasound imaging,... Control/ Fictitious Domain Method for Solving Optimal Shape Design Problems, M. Nachaoui, Morocco. M. Nachaoui is an assistant professor of the University of the Sultan Moulay Ismail Morocco. Abstract: This series of lectures concerns a method for numerical realization of optimal shape design problems based on the combination of a fictitious domain and an optimal control approach. This approach enables us to perform all calculations on fixed domain with a fixed grid. Conclusion and recommendation In the following we give some positive and negative points and the futurs projects produced by this school. some positive and negative points One of the most positive points the discusion between the participants and the lecturer and between the participants after each course. Specialy the discussion between the participants when they return back to hotel in the evening, a mini-course about the use of LaTex was done by Mostafa Johri (marocan participant) to some of the local participants who wished to know how to use LaTex and even how to install it. Not all the local participants were able to stay all the period of the school. Some of the local participants were not able to have the authorazation of them Universities, specially in the case that were more than one participante from the same department. The Futur projects CIMPA workshop represent by Prof Djafari for the next year. Project of CIMPA research school which proposed by Prof Ali Mohammed Djafari and Fatima Mohammad Aboud to take place in Erbil-Kurdistan region-iraq. Intensive master course which will be given by Prof Chakib for the students of the department of mathematics-university of Salahaddin. 5

7 Some links Fatima ABOUD workshop CIMPA M NACHAOUI May pdf Nachaoui CIMPAWorkshop2013.pdf 6