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Scheda Riassuntiva
Anno Accademico 2019/2020
Scuola Scuola di Ingegneria Industriale e dell'Informazione
Insegnamento 093595 - ADVANCED MATHEMATICAL METHODS (C.I.)
  • 093593 - FUNCTIONAL ANALYSIS AND NUMERICS FOR PDES
Docente Fumagalli Alessio
Cfu 5.00 Tipo insegnamento Modulo Di Corso Strutturato

Corso di Studi Codice Piano di Studio preventivamente approvato Da (compreso) A (escluso) Insegnamento
Ing Ind - Inf (Mag.)(ord. 270) - BV (477) ENERGY ENGINEERING - INGEGNERIA ENERGETICA*AZZZZ093595 - ADVANCED MATHEMATICAL METHODS (C.I.)

Obiettivi dell'insegnamento

Solving optimization problems subject to constraints given in terms of partial differential equations (PDEs) with additional constraints on the controls and/or states is one of the most challenging problems in the context of industrial, economical and engineering applications. For the treatment of such optimization problems the interaction of optimization techniques and numerical simulation plays a central role. The objective of the course is to introduce the mathematical field of optimization with PDE constraints and present different optimization strategies, tailored to specific problem, for the verification of the optimization algorithms.


Risultati di apprendimento attesi

Lessons, exercise classes and computer laboratories will allow the student who has passed the exam to know and understand:

a) basic concepts of functional analysis, with particular regards to the study of partial differential equations
b) main properties and well-posedness of the weak formulation of boundary value problems for partial differential equations
c) fundamentals of the finite element method and its application to the discretization of elliptic, parabolic and Stokes equations
d) fundamentals of unconstrained and constrained optimization problems
e) theoretical properties of the numerical methods considered
f) fundamentals of optimal control problems with partial differential equation constraints
g) the use of the software FreeFem++ for the implementation of the considered numerical methods.

Furthermore, the student will be able to:

a) implement numerical algorithms for the solution of relevant problems in engineering that require the use of the finite element method
b) implement numerical algorithms for the solution of relevant problems in engineering that require the use of optimization strategies
c) solve mathematical problems formulated in term of partial differential equations with the finite element method
d) introduce and use properly numerical methods for the approximation of constrained and unconstrained minimization problems
e) combine finite element discretization with numerical methods for optimization for the solution of optimal control problems with PDE constraints.

The student must demonstrate a critical and in-deep knowledge of the contents given in the course which is not limited to the plain exposition of definitions and results. Moreover, he/she should be able to solve exercises with a rigorous and logical approach in agreement with the theory.


Argomenti trattati

1. Elements of functional analysis for partial differential equations. Definitions, classification and examples of PDEs. Functional and bilinear forms. Sobolev spaces. Spaces of time dependent functions.

2. Parabolic problems. Introduction to the heat equation from the principles of thermodynamics. Poisson’s equation as a stationary heat problem. Weak formulation and Galerkin finite element discretization of the heat equation. General results of consistency, stability and convergence of the method.

3. Stokes and Navier-Stokes equations. Derivation of the constitutive laws in the case of a incompressible flow. Compatible boundary conditions. Weak formulation. Discretization by finite elements. Compatible ad incompatible finite element spaces. Stabilization techniques. The saddle point algebraic problem. Incompressible Navier-Stokes equation. The different treatment of the convective term: implicit, semi-explicit, fully explicit. Fixed point techniques for the approximation of the nonlinear term.

4. Fundamentals of Unconstrained Optimization. Introduction to line search methods: search directions. The Wolfe conditions. Sufficient Decrease. Convergence of Line Search Methods. Convergence of the Steepest Descent Method. Newton's Method. The BFGS method: Properties and implementation. Global convergence of the BFGS method.

5. Theory of Constrained Optimization. Local and global solutions. First order optimality conditions. Second order conditions. Quadratic programming. Direct Solution of the KKT System. Quadratic Penalty and Augmented Lagrangian Methods.

6. Optimal control with partial differential equation constraints. Definition of optimal control problems. A control problem for linear systems. Minimization of linear functionals. Some examples of optimal control problems.


Prerequisiti

Who is going to attend the course is expected to have rooted basis in mathematical analysis and linear algebra and some basic knowledge of numerical analysis.


Modalità di valutazione

The course will offer lectures, class exercises and computer labs. Computer labs will make use of the open-source code FreeFem++ for the solutions of 1) partial differential equation by finite element methods and 2) optimization problems by Newton, quasi-Newton and BFGS methods.

Course attendance is warmly suggested. Students’ skills and knowledge will be evaluated by a written test, composed by numerical problems on the main topics of the course and some open questions. In the written exam, the student is expected to:

a) discuss and present the numerical approximation of partial differential equation, with particular emphasis to elliptic, parabolic and Stokes problems, including weak formulations, finite element discretizations and algebraic representations of the discretized problem
b) discuss and present numerical optimization methods for the solution of unconstrained and constrained (with equality constraints) minimization problems.
c) implement in FreeFem++ suitable numerical algorithms for the solution of PDEs, optimization problems and optimal control problems with PDE constraints.

Written exams will be scheduled in the spots established by the academic calendar (2 in January-February, 2 in June-July and 1 in August-September). The exam is passed if one get a score greater than or equal to 18 over 33 points. No mid term exams are foreseen.


Bibliografia
Risorsa bibliografica obbligatoriaAlfio Quarteroni, Numerical Models for Differential Problems, Editore: Springer, ISBN: 978-88-470-1070-3
Risorsa bibliografica obbligatoriaLuca Formaggia, Fausto Saleri, Alessandro Veneziani, Solving Numerical PDEs: Problems, Applications, Exercises, Editore: Springer, ISBN: 978-88-470-2411-3
Risorsa bibliografica obbligatoriaJorge Nocedal Stephen J. Wright, Numerical Optimization, Editore: Springer, ISBN: 978-0387-30303-1

Software utilizzato
Software Info e download Virtual desktop
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PC studente
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Aule
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Altri corsi
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FreeFem++ Vedi sito produttore SI SI

Forme didattiche
Tipo Forma Didattica Ore di attività svolte in aula
(hh:mm)
Ore di studio autonome
(hh:mm)
Lezione
32:00
48:00
Esercitazione
7:00
10:30
Laboratorio Informatico
11:00
16:30
Laboratorio Sperimentale
0:00
0:00
Laboratorio Di Progetto
0:00
0:00
Totale 50:00 75:00

Informazioni in lingua inglese a supporto dell'internazionalizzazione
Insegnamento erogato in lingua Inglese
Disponibilità di materiale didattico/slides in lingua inglese
Disponibilità di libri di testo/bibliografia in lingua inglese
Possibilità di sostenere l'esame in lingua inglese
Disponibilità di supporto didattico in lingua inglese
schedaincarico v. 1.6.9 / 1.6.9
Area Servizi ICT
05/12/2021