Math Courses
# Advanced topics in numerical analysis

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Europe/Rome

Ex-ISEF/Building-Room D (GSSI)
### Ex-ISEF/Building-Room D

#### GSSI

20

Description

**Lecturers**

Nicola Guglielmi, nicola.guglielmi@gssi.it

Francesco Tudisco, francesco.tudisco@gssi.it

**Timetable and workload**

Lectures: 60 hours

Final project assignment: 20 hours

**Course description and outcomes**

This course is an introduction to modern numerical analysis and numerical optimization. The primary objective of the course is to develop a graduate-level understanding of computational mathematics and skills to solve a range of real-world mathematical problems on a computer by implementing advanced numerical algorithms using a scientific computing language (such as MATLAB or Julia).

**Course requirements**

Calculus and basic linear algebra and numerical analysis. Previous programming experience in any language may help.

**Course content**

The course will cover the following topics

PRELIMINARIES (NG)

- Quadrature
- I.1. Quadrature formulas, order conditions.
- I.2. Error analysis
- I.3. Superconvergence and orthogonal polynomials
- I.4. Gauss quadrature.

- Initial Value Problems for ODEs
- II.1. One-step (Runge-Kutta) methods
- II.2. Stiff problems
- II.3. Collocation methods.

PRELIMINARIES (FT)

- Eigenvalues and variational characterization(s)
- (Nonlinear) Cheeger inequality and projection operators
- Power method and subspace iterations
- Basic iterative solvers for linear systems

MAIN TOPICS (NG)

- Elements of numerical optimization
- IV.1. The main methods of convex optimization.
- IV.2. Univariate optimization. Unimodular functions.

- Gradient methods
- V.1. Gradient systems.
- V.2. Classical gradient methods.
- V.3. Conjugate direction methods.

- Constrained optimization
- VI.1. Penalty methods.
- VI.2. Projection methods.

- VII. Eigenvalue optimization
- VII.1. Variational properties of eigenvalues.
- VII.2. Structured spectral optimization.

MAIN TOPICS (FT)

- GMRES and Krylov subspace methods for eigenvalues
- Quasi-Newton methods for nonlinear systems and optimization
- Linear Multistep and Nesterov acceleration
- Nonlinear fixed point iterations and monotone operators
- Stochastic gradient descent

**Books of reference**

- E. Hairer, G. Wanner, S. P. Nørsett;
*Solving Ordinary Differential Equations I* - R. Fletcher,
*Practical methods of optimization.*Second edition. - W. Gautschi,
*Numerical analysis. An introduction.* - J. Stoer and R. Bulirsch,
*Introduction to numerical analysis.*Third edition - Y. Saad;
*Iterative methods for Sparse Linear Systems*(Free Online Version) - Y. Saad;
*Numerical methods for Large Eigenvalue Problems*(Free Online Version) - E.K. Ryu, S. Boyd;
*A premier on monotone operator methods*(Free Oline Version) - J. Nocedal, S. J. Wright;
*Numerical optimization*

**Examination and grading**

Students will be evaluated on the basis of a written exam and computational assessment to be taken at the end of the course. Both tests are graded based on the ECTS grading scale.