TOPICS IN NUMERICAL ANALYSIS
The course is designed for students in Mathematics and Physics, but it is conceived also for students with a different background, but with an interest in scientific computing. The basic notions from analysis and geometry needed are Taylor expansions, linear combinations and bases in a finite dimensional vector space, vector and function norms. From the previous course in numerical analysis, we will use algorithms for the solution of linear systems of equations and the notion of condition number. A basic experience in programming is also helpful, but not mandatory.
The aim of this class is the study of the main algorithms of classical numerical analysis for the solution of problems in approximation theory and differential calculus. We will approach both the theoretical aspects and the computational issues of the problems we will consider.
Course content
1. Approximation of functions: Polynomial interpolation, Hermite interpolation, Least square interpolation, Piecewise polynomial interpolation, Extension to the two-dimensional case
2. Numerical integration: Quadrature on an interval, Quadrature rules based on polynomial interpolation and, Composite formulas, Adaptive methods, Monte Carlo quadrature and dimensional curse
3. Approximation and integration with orthogonal polynomials: Families of orthogonal polynomials, Gaussian integration, Integration on unbounded intervals with Hermite polynomials, Trigonometric interpolation and Discrete Fourier transform
4. Integration of ordinary differential equations: A brief review tools from analysis, Elementary methods and Issues on implementation, Analysis of one step methods, Stability, Runge-Kutta methods, Multistep methods, Adaptivity, Stiff problems and implicit methods, IMEX Runge-Kutta methods
Exam
The exam consists of two parts, which can also take place in the same day. In a first part, the student presents a brief report, which can also be carried out by a group of 2 or 3 people, regarding a project on a topic addressed during the course. Usually the project is computational, and the student must demonstrate a certain level of autonomy.
The second part is an oral exam on the contents of the course.
R. Leveque, “Finite volume methods for hyperbolic problems”, Cambridge. Notes and slides from the teacher
Borrowed from
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