34794 - Numerical Analysis (2nd cycle)

Learning outcomes

At the end of the course, the student knows the numerical-mathematical aspects and the main numerical algorithms that allow to solve problems of interest in Engineering. In particular, the student knows basic numerical methods for solving linear and non-linear systems of large dimensions, interpolation, least squares approximation, integration and derivation. Basic numerical methods  for the regularization of ill-posed inverse problems. Given this basic knowledge, the main objective is to introduce students to numerical methods for differential equations with ordinary differential equations and partial differential equations with particular reference to finite difference and basic finite element schemes. The course includes a laboratory activity which is an integral part of it, the MATLAB software will be used.

Course contents

PART A:

1. [ANALISI] Basics of numerical computing: floating-point arithmetic, roundoff errors, algorithms, problem conditioning, numerical stability.

2. [ALGEBRA] Linear algebra: matrices, vector and matrix norm.

3. [ANALISI]Introduction to programming using MATLAB.

4. [ALGEBRA]Solving Linear Systems. Direct methods: LU factorization, pivoting, Gaussian elimination, Cholesky factorization.

5. [ALGEBRA]Solving Linear Systems. Itarative methods:Gauss-Seidel, Conjugated Gradients, Preconditioning.

6. [ANALISI]Numerical solution of nonlinear equations and systems: bisection method, Newton's method, secant, regula falsi.

7. [ANALISI]Polynomial interpolation and piecewise polynomial interpolation

8. [ALGEBRA]Polynomial approximation of data by least squares: normal equations, method based on QR factorization and SVD.

9. [ALGEBRA]Regularization methods for ill-posed problems

10. [ANALISI] Numerical integration: Newton Cotes quadrature formulas of simple and composite.

11. [ANALISI] Numerical differentiation

Part B:

1. Numerical Solution of Ordinary Differential Equations: One step methods; Control of error; Definition of the step; Multi-step methods; Predictor corrector method; Methods for Stiff Problems;

2. Boundary value problems;

3. Numerical Solution of Partial Differential Equations; Classification; domain of dependence of the first order equations; Finite difference methods for parabolic problems; transport equation, hyperbolic problems: Galerkin method for Parabolic Problems; Elliptic equations: finite difference method and finite element methods.

4. Introduction to PDETOOL and its use in the analysis of some models. 

Readings/Bibliography

Cleve Moler, Numerical Computing with MATLAB , Ed. SIAM, 2004.
Michael T. Heath, Scientific Computing: An Introductory Survey , 2nd ed., McGraw-Hill, 2002.

A.Quarteroni, F.Saleri, P.Gervasio, Scientific Computing with MATLAB and Octave, 2010

A. Quarteroni, Modellistica Numerica per problemi Differenziali , Springer, Ed. 4a, 2008.

A. Quarteroni, Numerical Models for Differential Problems (3rd Edition), Springer International Publishing, 2017.

Randall J. Leveque, Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems, 2007

Teaching methods

class hours and computational experiments in lab with the teacher.

Assessment methods

Part A: Final examination in lab.

Part B: Projects where the numerical methods are used in specific applications will be assigned throughout the course.

Teaching tools

Experience in Lab. is an essential part of the course. Matlab is used as problem solving environment, matrix-vector programming language, graphics.

Slides provided in the WEB site in the Platform iol.unibo.it

Office hours

See the website of Serena Morigi