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Methods in Scientific Computing (DD2365), 7.5hp, Spring 2017
Course goals
The goal of the course is to present general and efficient numerical methods and algorithms for basic models of computational science, in particular particle models, ordinary differential equations (ODE) and partial differential equations (PDE). Research challenges in the field are highlighted, e.g. with respect to parallel and distributed computing.
Course activites
- 16 lectures
- 8 laboratory sessions
Examination
- Written examination, 4.5hp (Grade: A, B, C, D, E, FX, F)
- Written exam based on the lecture notes of the course, and the lab modules Lab1-Lab6.
- Exam maximum 50p: 40p (A), 35p (B), 30p (C), 25p (D), 20p (E), <20p (F)
- Lab modules Lab1-Lab6, 3.0hp (P/F)
- Lab modules: each LabX consists of a set of mandatory problems to complete (P/F) + a set of optional problems that give bonus points for the written exam: max 2p/LabX (max 12p in total).
- Deadline for lab reports: March 14 (After this date no bonus points will be awarded.)
- Lab reports for each module should be submitted online.
Teachers
Johan Hoffman
Office hours: Tuesdays 9:00-10:00 (Office 4432, Lindstedtsvägen 5)
Johan Jansson
Contact person for Lab2-Lab5.
Tania Bakhos
Contact person for Lab1 and Lab6.
Literature
Lecture notes distributed in time for the lectures.
Extra reading
[LIN] Trefethen, Bau, "Numerical Linear Algebra", SIAM, (ISBN 978-0-898713-61-9) [SIAM]
[CDE] Eriksson, Estep, Hansbo, Johnson, "Computational Differential Equations", Studentlitteratur, (ISBN 91-44-49311-8), 1996. [Bokus] [Studentlitteratur] [Kårbokhandeln]
NOTE: In some electronic versions of the CDE book, the Chapter 13 "Calculus of Several Variables" may be missing. For the problem sets the correct chapters should be: Chapter 8 "Two-Point Boundary Value Problems", Chapter 15: "The Poisson Equation", Chapter 21: "The Power of Abstraction".
Lab modules
Lab1: Iterative methods for solving linear systems
[Lab1 - PM]
[Lab1 - part 1 (Python)]
[Lab1 - part 2 (Python)]
[Lab1 - part 1 (Jupyter)]
[Lab1 - part 2 (Jupyter)]
[The Conjugate Gradient Method - Shewchuk]
Lab2: ODE time stepping
[Lab2 - PM and Jupyter source]
Lab3: FEM assembly
Lab4: PDE and FEM in 1D/2D
Lab5: Adaptive FEM
Lab6: Optimization
Software
SciPy - Python-based ecosystem of open-source software for mathematics, science, and engineering
FEniCS project - computing platform for partial differential equations (PDE)
Week plan
Week 1
- Lecture 1: Vector spaces
[Lecture notes 1] - Lecture 2: Matrix algebra
[Lecture notes 2] - Lecture 3: Linear systems of equations - direct methods
[Lecture notes 3]
Week 2
- Lecture 4: Eigenvalues and eigenvectors
[Lecture notes 4] - Lecture 5: Iterative methods
[Lecture notes 5] - Lab 1: Iterative methods for solving linear systems
Week 3
- Lecture 6: Nonlinear equations - Newton's method
[Lecture notes 6] - Lecture 7: ODE - time stepping IVP
[Lecture notes 7] - Lab 2: ODE time stepping
Week 4
- Lecture 8: Course review
- Lecture 9: Function approximation - piecewise polynomials, interpolation, LS/L2-projection
- Lecture 10: FEM for 1D BVP model problems
Week 5
- Lecture 11: FEM algorithms - quadrature, mesh, reference element, assembly algorithm
- Lecture 12: PDE - FEM for 2D/3D BVP model problems
- Lecture 13: PDE - semi-discretization of IBVP
Week 6
- Lecture 14: Optimization - Adaptive FEM
- Lecture 15: Optimization - Parameter estimation
- Lecture 16: Course review
Week 7
- Lab 3: FEM Assembly
- Lab 4: PDE and FEM in 1D/2D
- Lab 5: Adaptive FEM
Week 8
- Lab 6: Optimization
- Lab 7: Lab review
- Lab 8: Lab review
Week 9
Written exam