Teaching is organised as follows:  
Unit  Credits  Academic sector  Period  Academic staff 
NUMERICAL OPTIMIZATION  3  MAT/07MATHEMATICAL PHYSICS  I semestre 
Giacomo Albi

MODELLING SEMINAR  3  MAT/07MATHEMATICAL PHYSICS  I semestre 
Nicola Sansonetto

The aim of the first module is to deepen the knowledge and skills especially in the modern theory of dynamical systems and give the student a solid appreciation of the deep connections between mathematics and other scientific disciplines, both in terms of the mathematical problems that they inspire and the important role that mathematics plays in scientific research and industry. Mathematical software tools, and others, will be used to implement algorithms for the solution of the real world problems studied during the course. At the end of the course the student is expected to be able to complete professional and technical tasks of a high level in the context of mathematical modelling and computation, both working alone and in groups. In particular the student will be able to write a model of a real problem, to recognise the effective parameters and analyse the model and its possible implications. The second module wants to provide sufficient theoretical and numerical background for the optimal control of dynamical systems. Such problems will be developed by means of real application examples, and recent research studies. At the end of the course students will be able to decide which numerical method is suitable for the solution of some specific optimal control problems. He/She will be able to provide theoretical results on the controllability and stability of certain optimal control problem and numerical methods. He/She will be able to develop his/her own code, and capable choose the appropriate optimization method for each application shown during the course.
The course presents different differential models with application in biology, economics and robotics.
The analysis of these models will be enached by the study of theoretical aspects, and the development of several computational methods.
PART I: (Numerical Optimization)
* Linear and Nonlinear optimization, KKT conditions, gradient methods, quasiNewton and Newton methods. Convex optimization.
* Optimal control: direct methods (shooting, collocation), indirect methods (forwardbackward) and dynamic programming, ModelPredictive Control.
* Inverse problems and parameters estimation.
Examples and exercise with software ( Matlab/Octave and CVX).
PARTE II: (Modelling)
* Modelling of complex and multiagent systems (swarming, opinion formaton, Network, and (non)holonomic systems).
* Bifurcation analysis, and geometric control.
The programme is in accordance with the ECMI standards (European Consortium for Mathematics in Industry, https://ecmiindmath.org/)
The student is expected to demonstrate the ability to mathematically formalize and solve models used in several scientific discipline, using, adapting and developing the models and advanced methods discussed during the lectures. To that end the final evaluation will consist in a written and oral exam.
Written exam: One question/exercise for each part of the course (Part I and Part II), the solution will possible require the use of computer.
Oral exam: Subject of student choice and discussion of the written exam with questions.
The subject of student choice can be substituted with the development of a smallproject to be decided together with the teacher.
Reference books  
Author  Title  Publisher  Year  ISBN  Note 
L. T. Biegler  Nonlinear Programming  SIAM  2010  
Nocedal, Jorge, Stephen Wright  Numerical optimization  Springer Science & Business Media  2006  
Betts, J.  Practical Methods for Optimal Control and Estimation Using Nonlinear Programming  SIAM  2010 
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