Direct methods for sparse linear systems Seminar Summer semester - - PowerPoint PPT Presentation
Direct methods for sparse linear systems Seminar Summer semester 2017 Andreas Potschka Heidelberg University April 19, 2017 A. Potschka Direct methods for sparse linear systems 1 Overview Organizational matters Introduction List of
Seminar Summer semester 2017 Andreas Potschka Heidelberg University April 19, 2017
Direct methods for sparse linear systems – 1
Organizational matters Introduction List of topics Preparation guidelines for presentations Introductory round
Direct methods for sparse linear systems – 2
◮ Wednesdays, 14–16 Uhr ◮ Kickoff: April 19 ◮ Location: INF 205, SR1 ◮ Target group: MSc
◮ Mathematics ◮ Scientific computing ◮ Computer science
◮ Language: English ◮ One presentation per session (45–75 min plus discussion) ◮ Credit Points: 6 CP ◮ Prerequisites: Presentation, regular attendance
Direct methods for sparse linear systems – 3
◮ Quality of contents
◮ Mathematical precision ◮ Focus on the essential aspects, adapted to audience ◮ Clear structure
◮ Presentation style
◮ Comprehensible pronounciation ◮ Adequate tempo of presentation ◮ Responsiveness to questions from the audience
◮ Presentation technique
◮ Choice: Black board, PowerPoint, L
A
T EXbeamer, etc.
◮ Readable, well-structured, meaningful black board and slides ◮ Focus on one message per slide ◮ Nominal style instead of full sentences ◮ Avoid clutter ◮ Handout
Direct methods for sparse linear systems – 4
◮ Matrices with many zero entries ◮ Simple examples: 0 matrix, identity matrix, band matrices ◮ Memory requirement for sparse n × n matrix: O(n) instead of O(n2) ◮ Requires special data structures ◮ Sparsity pattern connected to graphs ◮ Arise in many application problems
Direct methods for sparse linear systems – 5
Source: ❤tt♣✿✴✴✇✇✇✳❝✐s❡✳✉❢❧✳❡❞✉✴r❡s❡❛r❝❤✴s♣❛rs❡✴♠❛tr✐❝❡s✴
Direct methods for sparse linear systems – 6
Source: ❤tt♣✿✴✴✇✇✇✳❝✐s❡✳✉❢❧✳❡❞✉✴r❡s❡❛r❝❤✴s♣❛rs❡✴♠❛tr✐❝❡s✴
Direct methods for sparse linear systems – 7
Source: ❤tt♣✿✴✴✇✇✇✳❝✐s❡✳✉❢❧✳❡❞✉✴r❡s❡❛r❝❤✴s♣❛rs❡✴♠❛tr✐❝❡s✴
Direct methods for sparse linear systems – 8
Source: ❤tt♣✿✴✴✇✇✇✳❝✐s❡✳✉❢❧✳❡❞✉✴r❡s❡❛r❝❤✴s♣❛rs❡✴♠❛tr✐❝❡s✴
Direct methods for sparse linear systems – 9
Source: ❤tt♣✿✴✴✇✇✇✳❝✐s❡✳✉❢❧✳❡❞✉✴r❡s❡❛r❝❤✴s♣❛rs❡✴♠❛tr✐❝❡s✴
Direct methods for sparse linear systems – 10
Solution alternatives:
◮ Direct methods
◮ To minimize fill-in: Analyze and permute ◮ Alternative: Iterative methods
fixed-point solvers, Krylov subspace methods, multi-grid, . . . (Seminar Iterative methods for sparse linear systems)
Direct methods for sparse linear systems – 11
2T.A. Davis. Direct methods for sparse linear systems. Vol. 2. Fundamentals of
Direct methods for sparse linear systems – 12
◮ Memory formats ◮ Matrix modifications and arithmetic ◮ Solution of triangular systems
4T.A. Davis. Direct methods for sparse linear systems. Vol. 2. Fundamentals of
Direct methods for sparse linear systems – 13
◮ A symmetric positive definite ◮ A = LLT
5T.A. Davis. Direct methods for sparse linear systems. Vol. 2. Fundamentals of
Direct methods for sparse linear systems – 14
◮ A = QR, QTQ = I ◮ Householder reflectors ◮ Givens rotations
6T.A. Davis. Direct methods for sparse linear systems. Vol. 2. Fundamentals of
7G.H. Golub and C.F
. van Loan. Matrix Computations. 3rd ed. Baltimore: Johns Hopkins University Press, 1996, pp. 206–247.
Direct methods for sparse linear systems – 15
◮ A = LU ◮ UMFPACK: Matlab \
8T.A. Davis. Direct methods for sparse linear systems. Vol. 2. Fundamentals of
9T.A. Davis. “Algorithm 832: UMFPACK – an unsymmetric-pattern multifrontal method
with a column pre-ordering strategy”. In: ACM Trans. Math. Software 30 (2004),
Direct methods for sparse linear systems – 16
◮ Preserving sparsity of matrix factors ◮ Minimum degree ordering
10T.A. Davis. Direct methods for sparse linear systems. Vol. 2. Fundamentals of
Direct methods for sparse linear systems – 17
◮ Preserving sparsity of matrix factors ◮ Maximum matching ◮ Dulmage–Mendelsohn decomposition
11T.A. Davis. Direct methods for sparse linear systems. Vol. 2. Fundamentals of
Direct methods for sparse linear systems – 18
◮ Preserving sparsity of matrix factors ◮ Bandwidth and profile reduction ◮ Nested dissection ◮ Solution of decomposed systems
12T.A. Davis. Direct methods for sparse linear systems. Vol. 2. Fundamentals of
Direct methods for sparse linear systems – 19
◮ Decomposition with possibility of updates
factors of sparse matrices, pp. 145–166.
Direct methods for sparse linear systems – 20
Nr Date Topic Name 1 10.05.2017 Crash course in graph theory 2 17.05.2017 Basics of sparse matrices 3 24.05.2017 Cholesky decomposition 4 31.05.2017 Orthogonal decomposition 5 07.06.2017 LU decomposition 6 21.06.2017 Minimum degree ordering 7 28.06.2017 Maximum matching 8 05.07.2017 Profile reduction, nested dissection, solution 9 12.07.2017 Implicit LU decomposition
Direct methods for sparse linear systems – 21
◮ Who is my audience?
Imagine one or two concrete persons!
◮ How much time do I have? ◮ Structure: Overview, main part, summary ◮ One week before presentation:
Meet me to discuss slides/black board
◮ Your presentation is more than your slides
Deliver at least one, better two exercise presentations
Direct methods for sparse linear systems – 22
◮ Name ◮ Country ◮ Semester ◮ Study program ◮ Possible topics for seminar presentation
Direct methods for sparse linear systems – 23