ELEMENTS OF NUMERICAL METHODS FOR DATA REPRESENTATION AND SIMULATION
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- Versione italiana
- Academic year
- 2021/2022
- Teacher
- GIACOMO DIMARCO
- Credits
- 6
- Curriculum
- GENERALE
- Didactic period
- Primo Semestre
- SSD
- MAT/08
Training objectives
- The course aims to provide the basics of mathematical modeling through differential equations and the basis of the Euclidean numerical approximation of data and functions.
Furthermore, several tools for the development of numerical methods for solving such models will be provided and methods for the calculation of least squares approximants, both in the continuous and in the discrete case will be discussed.
The course includes a part of hours dedicated to laboratory activity, in which the Matlab Optimization Toolbox is also used.
The main knowledge provided by the course will be:
- basic concepts for numerical modeling of partial differential linear differential problems: elliptic equations, parabolic equations, hyperbolic equations, diffusion and transport equations, conservation laws, balance laws;
- general concepts about the approximation problem in functional spaces, both in the continuous and in the discrete case;
- methods for solving numerical differential equations: finite elements and finite differences. Analysis of the convergence and stability of the main methods. Algorithmic and computer implementation aspects in multi dimension;
- some of the main algorithms for the solution of linear and non-linear problems for the approximation in Euclidean norm, both in the continuous and in the discrete case;
- methods for solving linear differential equations using Euclidean approximants. Prerequisites
- Good knowledge of differential calculus, applied mathematics, and mathematical analysis is recommended to follow the course. Basic skills in the Matlab programming language are highly recommended.
Course programme
- The course includes a total of 48 hours of lessons, divided into two parts: a part of a total of 24 hours on ordinary differential equations and partial differential equations, a part of a total of 24 hours on the Euclidean approximation.
* First part on differential equations
In this part, 16 hours will be devoted to theory and the remaining 8 hours to lab activity with Matlab. The topics of this part will be:
1. Introduction to partial differential equations: definitions, classifications and first examples.
2. Diffusion equations: finite difference methods, spectral methods. Applications to heat transfer problem in one and two spatial dimensions.
3. Elliptic equations: finite difference methods, finite elements methods, spectral methods. Applications to structural mechanics in one and two spatial dimensions.
4. Linear hyperbolic equations: finite volume methods. Applications to diffusion of pollutants in the air in one and two spatial dimensions.
* Second part on Euclidean approximation
About 21 hours will be dedicated to the theoretical aspects and the remainder to exercises with Matlab. The hourly division (indicated in brackets) may vary, even significantly, according to the difficulties encountered by the students in the various parts of the program. The covered topics will be:
1. Introduction to the fundamental concepts of Numerical Approximation; selection criteria of the approximant. (1)
2. Linear approximation problem: formulation of the continuous case and the discrete case in Lp spaces. Fundamental theorem of linear approximation in Lp spaces. (2)
3. Ordinary least squares approximation (OLS), continuous case: generalized polynomial of best approximation; normal equations; orthogonal and orthonormal algebraic polynomials; generalized Fourier series; Bessel inequality, Parseval identity,. A quick mention to Gram-Schmidt procedure, convergence in Euclidean norm, Legendre polynomials, Clenshaw-Curtis formula, Fourier series (3)
4. OLS approximation, discrete case: normal equations, existence and uniqueness of the solution; outline of the statistical acceptability of the model, geometric interpretation; Moore-Penrose pseudo-inverse; solution of the normal equations system and its conditioning; maximum rank and incomplete rank cases; orthogonal polynomials in the discrete case; SVD solution of the linear problem; quick mention to statistical interpretation and the Gauss-Markoff Theorem. (7)
5. Total least squares (TLS): separation of singular values, solution by SVD, uniqueness; relationship with OLS, conditioning, sensitivity; the orthogonal distance linear regression problem. (3)
6. Introduction to the non-linear least squares problem (NLS): concepts, first order methods: Gauss-Newton (G-N) and relaxed G-N; descent property of the G-N direction. (2)
7. Lab.: approximation by trigonometric polynomials and FFT; TLS and the orthogonal distance linear regression problem; introduction to the Optimization toolbox; geometric fit; SVD applications. (6) Didactic methods
- The course is divided into 35 hours frontal/streaming teachingc on the theoretical aspects and 13 hours of lab sessions using Matlab.
Learning assessment procedures
- The exam aims to verify that the students acquired an adequate level of knowledge of the course topics, from both the theoretical as well as the laboratory point of view. The exam will consist of a practical part and a theoretical part (oral only).
The practical part consists of a project chosen by the student within a list proposed by the teachers. The project must be implemented using Matlab and the student will produce a short report on that.
The theoretical part includes a discussion about the project, supplemented by questions that may span over all the course topics.
The final score is given by the sum, thresholded at 30 points, of the project score and oral score. Each of the two evaluations cannot exceed 17 points. Reference texts
- For the first part:
1) Modellistica Numerica per Problemi Differenziali. A. Quarteroni. Springer, 2008.
2) Finite Volume Methods for Hyperbolic Problems. R. J. LeVeque. Cambridge University Press 2002.
3) Numerical Solution of Partial Differential Equations. An Introduction. 2nd Edition. K. W. Morton, D. F. Mayers, Cambridge University Press, 2005
4) E. F. Toro. Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer, 2009.
5) E. Godlewski, P.A. Raviart. Numerical Approximation of Hyperbolic Systems of Conservation Laws, Springer, 1996.
Per the second part:
- Åke Björck, "Numerical Methods for Least Squares Problems", SIAM, 1996. ISBN-13: 978-0-898713-60-2. ISBN-10: 0-89871-360-9.
- Matlab Optimization Toolbox User's Guide
- Appunti del docente
