72548 - Algebra and Numerical Analysis (2nd cycle)

Learning outcomes

Part A:

A first course in Numerical Analysis. Covers the basic techniques of the subject and provides a foundation for the efficient numerical solution of problems in science and engineering. Numerical methods to solve linear and nonlinear systems , numerical derivative, numerical integration, interpolation, approximation. A brief introduction to inverse problems and regularization techniques.

Part B:

This second part of the course presents numerical methods for the solution of problems modeled by both Ordinary Differential Equations (ODE) and Partial Differential Equations (PDE). The course discusses their analysis, applications, and computation of the solution (by first discretizing the equation, bringing it into a finite-dimensional subspace by a finite element method, or a finite difference method , and finally reducing the problem to the solution of an algebraic equations)

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.

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.

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.

Final discussion about the project and the theoretical part B.

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