- Code: DT0312
- Unit Coordinator: Antonio Cicone
- ECTS Credits: 6
- Semester: 2
- Year: 1
- Campus: University of L'Aquila
- Language: English
- Aims:
The Aim of this course is to provide the student with knowledge of Numerical Linear Algebra and Numerical Optimisation and ability to analyze theoretical properties and design mathematical software based on the proposed schemes.
On successful completion of this module, the student should
- have profound knowledge and understanding of the most relevant numerical methods for Numerical Linear Algebra and Numerical Optimisation and the design of accurate and eucient mathematical software;
- demonstrate skills in choosing the most suitable method in relation to the problem to be solved and ability to provide theoretical analysis and mathematical software based on the proposed schemes;
- demonstrate capacity to read and understand other texts on the related topics.
- Content:
MATRIX FACTORIZATIONS
LU decomposition, Cholesky decomposition. Singular value decomposition and applications (image processing, recommender systems). QR decomposition and least squares. Householder triangularization. Conditioning and stability in the case of linear systems.EIGENVALUE PROBLEMS
Approximation of the spectral radius. Power method and its variants. Reduction to Hessemberg form. Rayleigh quotient, inverse iteration. QR algorithm with and without shift. Jacobi method. Givens-Householder algorithm. Google PageRank.ITERATIVE METHODS FOR LINEAR SYSTEMS
Overview of iterative methods. Arnold iterations, Krylov iterations. GMRES. Lanczos method. Conjugate gradient. Preconditioners. Preconditioned conjugate gradient.NUMERICAL OPTIMISATION
Continuous versus discrete optimization. Constrained and unconstrained optimization. Global and local optimization. Overview of optimization algorithms. Convexity.
Line search methods. Convergence of line search methods. Rate of convergence. Steepest descent, quasi-Newton methods. Step-length selection algorithms. Trust region methods. Cauchy point and related algorithms. Dogleg method. Global convergence. Algorithms based on nearly exact solutions. Conjugate gradient methods. Basic properties. Rate of convergence. Preconditioning. Nonlinear conjugate gradient methods: Fletcher-Reeves method, Polak-Ribiere method. - Pre-requisites:
Basic analysis, basic Numerical Analysis and Linear Algebra. Basic
Probability Theory. - Reading list:
Quarteroni,Sacco,Saleri, Numerical Mathematics, Springer-Verlag, 2007
- Code: DT0639
- Unit Coordinator: Lothar Nannen
- Programme: InterMaths
- ECTS Credits: 6
- Semester: 2
- Year: 1
- Campus: Vienna University of Technology
- Language: English
- Aims:
Being able to check if the numerical solution of an ordinary differential equation makes sense. Furthermore, depending on the inherent structure of the differential equation, selecting the appropriate numerical integrator.
- Content:
Initial and boundary value problems for ordinary and partial differential equations, one-step and multi-step methods, adaptivity, structure perserving integrators, introduction to numerical methods for partial differential equations.
- Pre-requisites:
Basic lectures of analysis, linear algebra and numerical analysis.
- Reading list:
Lecture notes will be provided
- Code: DT0640
- Unit Coordinator: Markus Faustmann
- Programme: InterMaths
- ECTS Credits: 7
- Semester: 2
- Year: 1
- Campus: Vienna University of Technology
- Language: English
- Aims:
Being able to solve stationary partial differential equations numerically, analyse the quality of numerical solutions, select proper methods and implement them in a computer program.
- Content:
- Variational formulations
- Sobolev spaces, H(div), H(curl)
- Finite element spaces (h, p, hp)
- Mixed formulations,
- Discontinuous Galerkin Methods,
- Time-dependent problems - Pre-requisites:
Applied Mathematics Foundations
- Reading list:
- Lecture notes,
- Dietrich Braess: Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, Cambridge University Press, 2007
- Cleas Johnson: Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge Univ. Press, 1987, Dover 2009
- Susanne Brenner & Ridgway Scott: The Mathematical Theory of Finite Elements, Springer 2008
- Alexandre Ern & Jean-Luc Guermond: Theory and Practice of Finite Elements, Springer, 2010
- Unit Coordinator: dr hab. inż. Edyta Hetmaniok, prof. PŚ; prof. dr hab. inż. Damian Słota
- ECTS Credits: 4
- Year: 2
- Campus: Silesian University of Technology
- Language: English
- Aims:
The aim of the course is to get knowledge on numerical methods and technics of solving the problems possible to meet in engineering applications with implementation by using mathematical software.
- Content:
1. Elements of the error theory.
2. Approximate solutions of the nonlinear equations and their systems.
3. Solving linear equations.
4. Approximate methods of determining the eigenvalues and eigenvectors.
5. Interpolation.
6. Approximation.
7. Numerical integration.
8. Approximate methods of solving the initial and boundary problems.
9. Approximate methods of solving integral equations.
