To provide advanced mathematical tools that will be used throughout the rest of the program.
Course Prerequisites
Complex numbers: cartesian, polar and exponential representations; properties and operations. Linear algebra: matrix operations (sum, multiplication, determinant); eigenvalues and eigenvectors. Calculus tools: Taylor expansion, integration methods (integration by substitution and by parts).
Teaching Methods
Frontal lessons on the blackboard. Class notes (handouts) will be given for each one of the 4 topics of the course.
Assessment Methods
Written exam. The exam consists of a certain number of exercises to be solved, using the techniques covered in class. Examples of previously assigned exams will be made available to students. The use of calculators, books, and/or notes is not permitted during the exam.
Texts
• Optimization of N-variate functions (Ch. 1): J. Nocedal, S.Wright. “Numerical Optimization”, Springer; • Ordinary Differential Equations (Ch. 2): G. Teschl. “Ordinary Differential Equations and Dynamical Systems”, American Mathematical Society. • Function approximation, transforms (Ch. 3): D. Kammler. “A First Course in Fourier Analysis”, Cambridge University Press. • Partial Differential Equations (Ch. 4): S. Salsa, “Partial Differential Equations in Action: From Modelling to Theory”, Springer.
Contents
1. Optimization of N-variate functions. Free and constrained optimization of N-variate functions. Lagrange multipliers and KKT conditions. 2. Ordinary Differential Equations (ODEs). Scalar ODEs and system of ODEs. Analytic solutions of linear systems of ODEs (exponential matrix). Study of the harmonic oscillator (damped and with external force). Equilibria of linear and non-linear systems (linearization). 3. Function approximation and Fourier. Space of square-summable functions, orthonormal bases and Parseval’s identity, Fourier and Legendre expansions, interpolation and least squares approximation. Fourier transform, Dirac’s delta. 4. Introduction to Partial Differential Equations (PDEs). Classification, boundary & initial conditions, model equations (Laplace equation, Heat equation, Wave equation), basic analytic methods (separation of variables, Fourier), numerical methods (finite differences).
