ADVANCED APPLIED MATHEMATICS

Module Nonlinear Optimization:

Elements of Operations Research,

Models and algorithms for Linear Programming problems.

Module Numerical methods for differential equations:

Elements of Numerical analysis,

Elements of Analysis on ordinary differential equations,

Elements of Analysis on partial differential equations

The course is partitioned into two modules: Nonlinear Optimization (6 credits) and Numerical methods for differential equations (6 credits)

Module Nonlinear Optimization.

Convex sets and Convex functions. 

Unconstrained minimization: Optimality conditions. Generic minimization scheme. Descent methods, Gradient descent method, Newton’s method, Quasi.-Newton’s method. 

Constrained Programming: Lagrange dual function, Optimality conditions, Equality constrained minimization problems, Sequential quadratic programming

Module Numerical methods for differential equations.

Methods for the numerical approximation of ordinary differential equations: shooting method, finite difference method, finite elements methods.

Methods for the numerical approximation of partial differential equations: finite difference method for elliptic hyperbolic and parabolic equations, finite elements methods for elliptic equations.

Module Nonlinear Optimization.

D1.1 Know optimality conditions for unconstrained optimization

D1.2 Know optimality conditions for constrained optimization. 

D1.3 Illustrate the main features of algorithms for unconstrained, equality constrained and inequality constrained optimization problems 

D2.1 Classify and compare various models and algorithms for nonlinear optimization problems 

D2.2 Utilize specific software for solving nonlinear optimization problems

D3.1 Choose between different algorithms the most suited for solving specific nonlinear optimization problems

D3.2 Evaluate advantages and disadvantages of individual algorithms for nonlinear optimization

D4.1 Discuss the main aspects (theoretical and algorithmc) in nonlinear optimization

D5.1 Deepen through personal study, the most recent aspects of Nonlinear Programming

D5.2 Study independently the most recent algorithmic developments in the area.

Module Numerical methods for differential equations.

D1.1 Know the fundamental concepts on the numerical approximation of ordinary differential equations

D1.2 Know the fundamental concepts on the numerical approximation of partial differential equations

D2.1 Produce algorithms for the numerical solution differential equations 

D2.2 Carry out matlab codes for the implementation of approximation methods for differential equations

D3.1 Evaluate the features of the various approximation methods in order to solve efficiently a given differential equation problem

D4.1 Express rigorously the features of the various approximation methods and of the results computed by these methods

D5.1 Learn, by personal study, new approximation techniques for differential equations.

D5.2 Generalize the approximation methods studied in the course for the solution of other differential equation problems.