- Code: DT0307
- Unit Coordinator: Raffaele D'Ambrosio, Carmen Scalone
- Programme: RealMaths
- ECTS Credits: 6
- Semester: 2
- Year: 1
- Campus: University of L'Aquila
- Language: English
- Delivery: In-class
- Aims:
Learning objectives.
The aim of the course is to provide knowledge and skills in the field of discretization of differential problems and skills in the analysis of theoretical properties and in the design of mathematical software based on the proposed numerical schemes.
Learning outcomes.
At the end of the course, each learner should:
- know the most relevant numerical methods for the approximation of differential equations, together with the aspects related to their implementation in an accurate and efficient mathematical software;
- understand and be able to explain notions, theses and proofs in the field of numerical mathematics for differential problems;
- be able to apply their knowledge to solve numerically, using a programming environment, differential problems;
- be able to read and understand other texts on related topics. - Content:
WELL POSEDNESS OF THE PROBLEM
Hadamard well-posed differential problems. Picard-Lindelof iterations, analysis of their convergence, long-term analysis. Existence and uniqueness of solutions. Continuous dependence on initial data and vector fields. Gronwall lemma. Examples of differential problems of interest in real applications.DIFFERENCE EQUATIONS
Linear difference equations. Space of solutions and its basis. Casorati matrix. General solution of homogeneous and inhomogeneous equations. Characteristic equations.ONE-STEP DISCRETIZATIONS
Discretization of the domain, numerical grids. Euler method. Consistency, zero-stability and convergence. Local truncation error. Trapezoidal method. Fixed point iterations for implicit schemes.LINEAR MULTISTEP METHODS
Multistep discretizations, starting procedure. First and second characteristic polynomials. Implicit methods and convergence analysis of fixed point iterations. Local truncation error. Linear difference operator. Consistency. Zero-stability and root condition. Convergence and order of convergence. Lax equivalence theorem. Methods arising from quadrature formulae. Adams-Bashforth and Adams-Moulton schemes. Dahlquist barriers.RUNGE-KUTTA METHODS
Definition and classification of Runge-Kutta methods. Order conditions. Butcher theory of order: B-trees, square bracket representation, order, symmetry and density; elementary differentials; elementary weights. B-series. Convergence analysis. Butcher barriers. Implicit methods: Gauss, Radau and Lobatto methods.LINEAR STABILITY ANALYSIS AND STIFF PROBLEMS
Dahlquist test problem. Linear stability analysis for linear multistep and Runge-Kutta methods. Regions and intervals of absolute stability. Stepsize restrictions. Boundary locus. A-stability, L-stability, A(alpha)-stability. Rational approximations of the exponential, Padé approximations. Stiff problems, stiffness ratio. A-priori Prothero-Robinson analysis.VARIABLE STEPSIZE IMPLEMENTATION
Predictor-corrector schemes. Error estimates. Stepsize control strategy: computation of the optimal stepsize, PI and PID controller. Newton iterations for implicit Runge-Kutta methods.GEOMETRIC NUMERICAL INTEGRATION
Numerical conservation of contractivity for dissipative problems. One-sided Lipschitz. Logarithmic norm. G-stability, B-stability, algebraic stability. Differential problems with first integrals. Conservation of linear invariants. Conservation of quadratic invariants. Symplectic Runge-Kutta methods. Hamiltonian problems: symplecticity of the flow, preservation of the Hamiltonian. Long-time behavior of symplectic methods to Hamiltonian problems. Modified differential equations. Benettin-Giorgilli theorem. - Pre-requisites:
Basic Numerical Analysis and differential equations.
- Reading list:
- R. D'Ambrosio, Numerical Approximation of Ordinary Differential Problems - From Deterministic to Stochastic Numerical Methods, Springer (2023).
- E. Hairer, C. Lubich, G. Wanner, Geometric Numerical Integration, Springer (2006).
- 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
