Numerical Analysis and Optimization
Prerequisiti
The course is suitable for those students who have had a first introductory course on numerical methods but have not yet been exposed to systematic courses on Numerical Integration of ODEs and Optimization.
Programma
Part I, Numerical Methods for the integration of ODEs:
-Review of ODEs theory
-Runge-Kutta methods
-Multi-step methods
-Stiffness and A-stability
-Introduction to geometric numerical integration
-Conservation of first integrals and methods on manifolds
-Symmetric integration and reversibility
-Symplectic integration of Hamiltonian systems
-Geometric integration by multi-step methods
-Implementation of numerical methods
Part II, Methods of Numerical Optimization Unconstrained optimization:
-Gradient descent
-Newton and Quasi-Newton methods, Nesterov's method
-Globalization techniques
-Constrained optimization
-Method of Lagrange multipliers and augmented Lagrangian methods
-Interior point methods
-KKT systems
-The ADMM method
Obiettivi formativi
The goal of this course is to provide the students the basic tools of numerical integration of ODEs and of optimization (both constrained and unconstrained). The topics will be treated both from a theoretical point of view and with regard to algorithmic aspects.
Riferimenti bibliografici
-Arieh Iserles, A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, 2008
-Ernst Hairer, Gerhard Wanner, Christian Lubich, Geometric Numerical Integration, Springer, 2006
-J. Nocedal and S. Wright, Numerical Optimization, Springer, 1999
-Additional references will be provided in the course of the lectures