Ing Ind - Inf (Mag.)(ord. 270) - BV (477) ENERGY ENGINEERING - INGEGNERIA ENERGETICA

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093595 - ADVANCED MATHEMATICAL METHODS (C.I.)

Obiettivi dell'insegnamento

The course is made of two parts, as follows.

Functional analysis and numerics for PDEs.

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.

Numerical methods for optimization.

The course offers an introduction to linear and discrete optimization. It covers the use of mathematical programming as a means to solve real-world decision-making problems. The course is centered upon presenting optimization models and solution algorithms that fit a wide a variety of applications in engineering, with special attention towards energy applications. The course is accompanied by computer laboratory sessions, which use AMPL modeling language and state-of-the art commercial solvers to tackle optimization problems arising in a variety of fields of application.

Risultati di apprendimento attesi

Functional analysis and numerics for PDEs.

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.

Numerical methods for optimization.

Students will be able to model real-world decision-making problems with linear and discrete mathematical programming models, with a particular interest to energetic ones. The course will provide the students all the necessary modelling techniques. Students will learn the AMPL modelling language, and will thus be able to solve the modelled problems whit commercial solvers. Thanks to this course, students will also be able to use exact algorithms for both linear and discrete problems, and will have some basic knowledge on heuristic algorithms. Moreover, the course will teach the students how to consider the most important problems defined on graphs and networks and to select the correct algortihm to solve them.

Argomenti trattati

Functional analysis and numerics for PDEs.

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.

Numerical methods for optimization.

1. Introduction

Presenting applications of linear and discrete optimization. Introducing the steps required for formulating optimization problems (decision variables, objective function, constraints). Modeling techniques for Linear Programming and Integer Linear Programming.

2. Graph and network optimization

Optimum spanning trees. Maximum flows: Ford-Fulkerson algorithm and maximum flow-minimum cut theorem. Minimum cost flows.

3. Linear Programming (LP)

LP models. Geometric aspects (vertices of the feasible region) and algebraic aspects (basic feasible solutions). Fundamental properties of linear programming. The Simplex method. LP duality theory: pair of dual problems, weak and strong duality, complementary slackness conditions. Economic interpretation. Sensitivity analysis. Special cases: assignment and transportation problems.

4. Integer Linear Programming (ILP)

ILP models for, among others, scheduling and production planning problems. Relaxation: linear, combinatorial and Lagrangian. Exact methods: Branch & Bound algorithm and cutting plane methods.

5. Heuristics

Local search techniques, metaheuristics applied to scheduling problems and production planning problems. Tabu search.

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

Functional analysis and numerics for PDEs.

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.

Numerical methods for optimization.

The course is held with ex cathedra lessons and hands-on training.

The final evaluation of learning will be obtained through a written test, composed by numerical problems on the main topics of the course. No mid-term tests will take place.

The test evaluates the student's ability for modelling real-world decision making problems, and their ability to translate them into AMPL modelling language. The test will evaluate the student's knowledge on exact algorihtms for solving linear and discrete problems and their ability of chosign the correct algorithm for specific problems, with a particular attention to problems on graphs and networks.

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