Ing Ind - Inf (Mag.)(ord. 270) - BV (479) MANAGEMENT ENGINEERING - INGEGNERIA GESTIONALE

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055894 - NUMERICAL APPROXIMATION OF MATHEMATICAL MODELS AND APPLICATIONS

Obiettivi dell'insegnamento

The primary aim of this course is to give an introduction to some basic aspects in numerical modeling of problems relevant for Management Engineering, such as decision-making applications and business strategies. Laboratory sessions and team projects carried out by small groups of students during the course complement the theoretical lectures and provide the students with a practical “hands-on” training.

The secondary (but not less important) aim of the course is to train students to use a widely spread programming language as Matlab for algorithm development and numerical computations. Students will also learn to apply numerical models to mathematical problems stemming from real world applications and critically interpreting the results.

Blended learning and flipped class innovative teaching methods will be exploited through the course to allow students to improve specific communication and soft skills such as problem solving, critical thinking and teamwork skills. As a matter of fact, students are required to work in team on a real project whose results will be shown to the class by the end of the course.

Risultati di apprendimento attesi

The course will allow the students to:

- develop and critically apply a deep knowledge of numerical approximation of mathematical models for solving differential problems stemming from real world applications in Management Engineering, such as decision-making and production applications, supply chain management and logistics, and business and finance strategies, independently of they own field of specialization;

- critical reasoning and interpreting the results obtained in the light of the theory through the implementation of the algorithms using the Matlab software;

- know, understand and be able to use critically numerical methods for differential problems, as well as evaluate their properties and implement them via computer codes;

The team-project will allow students to:

- improve higher-order thinking skills (critical thinking, problem solving, communicative and working group skills);

- summarize and present the results achieved during the analysis and implementation activities

It is expected a critical knowledge and understanding and a fully comprehensive ability to distinguish the different situations and make reasoned choices and justify the procedures followed.

Argomenti trattati

- Principles of numerical mathematics. Basic principle of mathematical and numerical modelling: sources of errors in computational models, well-posedness and condition number of a mathematical model, the concept of stability of numerical methods.

- Some classical differential models in Management Engineering. The Malthusian, Logistic and Leslies populations growth models; the Phillips’ stabilization model and the Samuelson’s multiplier-accelerator model for macroeconomic systems; price evolution models in market economy; demand-supply dynamics models; the Solow-Swan model of long-run economic growth; the Black-Scholes model.

- Numerical methods for ordinary differential equations. The Cauchy problem: main results of existence and uniqueness of the solution, Liapunov stability. One-step numerical methods: forward and backward Euler methods, Crank-Nicolson and Heun methods. Analysis of one-step methods: consistency, zero-stability and convergence; absolute stability, regions of absolute stability. Methods for the solution of first-order systems of differential equations: the theta-method. Methods for the solution of second-order differential equations (hints). Applications to models for macroeconomic, price evolution and demand-supply dynamics, and long-run economic growth.

- Numerical methods for partial differential equations. Classification and examples of PDEs.The Poisson equation: main properties and well-posedness of the weak formulation. The Galerkin finite element discretization of the Poisson equation. Theoretical results of consistency, stability and convergence of the method. Introduction to the heat equation: weak formulation, the semi-discrete and fully discrete approximations and theoretical properties. Applications to the Black-Scholes model.

-Hints on optimization and optimal control problems. Linear programming: the simplex method. Definition of optimal control problems, some examples of optimal control problems (maximisation of the stream of profit, plan of the production to minimize the total costs).

Lab sessions. The methods presented during the lectures will be numerically investigated during the laboratory sessions based on employing the software Matlab.

Team project. The project represents an integral part of the course. Each project deals with a real-world problem, stemming from a practical application in the field of management engineering. The objective of the project is to train students in developing the following higher-order thinking skills: critical thinking, problem solving, communicative and working group skills. Projects will be assigned at the beginning of the semester and the results will be presented at the whole class at the end of the course.

Prerequisiti

Calculus, linear algebra and some programming experience.

Modalità di valutazione

The exam consists of a written test and of an oral presentation of the team project. To pass the exam the student must pass each part of the exam with a score greater than or equal to 18/30; the final score is then obtained as the weighted average of the two scores, with weights respectively equal to 0.6 for the written test and 0.4 for the team project evaluation.

Written test. The written exam takes place in the computer room. The exam covers all the theoretical and practical arguments considered during the lectures and lab sessions. Part of the questions and problems are solved numerically with Matlab. The questions focus on definitions, algorithms, theoretical results, and important examples. Light calculations may be needed. It is not allowed to use any form of course material.

Team project. Students are required to complete a team project focusing on a practical problem that involve modeling, numerical and computational aspects. Each team consists of approximately 2-4 students. The project is assigned at the beginning of the course and the results are presented at the end of the course. Team project presentations last 15 minutes plus 5 minutes for questions. The assessment of the project will be based on subject knowledge matters, oral delivery, quality of the work, demonstration of teamwork and individual contributions.

Bibliografia

Alfio Quarteroni, Fausto Saleri, Paola Gervasio, Scientific Computing with MATLAB and Octave, Editore: Springer, Anno edizione: 2014, ISBN: 978-3-642-45367-0
Sandro Salsa, Annamaria Squellati, Dynamical Systems and Optimal Control, Editore: Bocconi Univ Pr, Anno edizione: 2018, ISBN: 978-8885486522

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