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 topics Preparation guidelines for presentations Introductory round A. Potschka Direct methods for sparse linear systems – 2

Organizational matters ◮ 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 A. Potschka Direct methods for sparse linear systems – 3

Grading criteria ◮ 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 A. Potschka Direct methods for sparse linear systems – 4

Sparse matrices ◮ 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 ( n 2 ) ◮ Requires special data structures ◮ Sparsity pattern connected to graphs ◮ Arise in many application problems A. Potschka Direct methods for sparse linear systems – 5

Applications: Networks Source: ❤tt♣✿✴✴✇✇✇✳❝✐s❡✳✉❢❧✳❡❞✉✴r❡s❡❛r❝❤✴s♣❛rs❡✴♠❛tr✐❝❡s✴ A. Potschka Direct methods for sparse linear systems – 6

Applications: Circuits Source: ❤tt♣✿✴✴✇✇✇✳❝✐s❡✳✉❢❧✳❡❞✉✴r❡s❡❛r❝❤✴s♣❛rs❡✴♠❛tr✐❝❡s✴ A. Potschka Direct methods for sparse linear systems – 7

Applications: Linear programming Source: ❤tt♣✿✴✴✇✇✇✳❝✐s❡✳✉❢❧✳❡❞✉✴r❡s❡❛r❝❤✴s♣❛rs❡✴♠❛tr✐❝❡s✴ A. Potschka Direct methods for sparse linear systems – 8

Applications: Nonlinear programming Source: ❤tt♣✿✴✴✇✇✇✳❝✐s❡✳✉❢❧✳❡❞✉✴r❡s❡❛r❝❤✴s♣❛rs❡✴♠❛tr✐❝❡s✴ A. Potschka Direct methods for sparse linear systems – 9

Applications: Partial differential equations Source: ❤tt♣✿✴✴✇✇✇✳❝✐s❡✳✉❢❧✳❡❞✉✴r❡s❡❛r❝❤✴s♣❛rs❡✴♠❛tr✐❝❡s✴ A. Potschka Direct methods for sparse linear systems – 10

Linear equations Ax = b Solution alternatives: ◮ Direct methods 1. Decomposition: A = LU , A = QR , A = LL T 2. Forward/backward substitution ◮ 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 ) A. Potschka Direct methods for sparse linear systems – 11

Topic: Crash course in graph theory 1 R. Diestel. Graph theory . 4th ed. Graduate texts in mathematics. Springer, 2012. 2 T.A. Davis. Direct methods for sparse linear systems . Vol. 2. Fundamentals of Algorithms. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 2006, pp. 4–6. A. Potschka Direct methods for sparse linear systems – 12

Topic: Basics of sparse matrices ◮ Memory formats ◮ Matrix modifications and arithmetic ◮ Solution of triangular systems 3 R. Diestel. Graph theory . 4th ed. Graduate texts in mathematics. Springer, 2012. 4 T.A. Davis. Direct methods for sparse linear systems . Vol. 2. Fundamentals of Algorithms. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 2006, pp. 7–35. A. Potschka Direct methods for sparse linear systems – 13

Topic: Cholesky decomposition ◮ A symmetric positive definite ◮ A = LL T 5 T.A. Davis. Direct methods for sparse linear systems . Vol. 2. Fundamentals of Algorithms. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 2006, pp. 37–67. A. Potschka Direct methods for sparse linear systems – 14

Topic: Orthogonal decomposition ◮ A = QR , Q T Q = I ◮ Householder reflectors ◮ Givens rotations 6 T.A. Davis. Direct methods for sparse linear systems . Vol. 2. Fundamentals of Algorithms. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 2006, pp. 69–82. 7 G.H. Golub and C.F . van Loan. Matrix Computations . 3rd ed. Baltimore: Johns Hopkins University Press, 1996, pp. 206–247. A. Potschka Direct methods for sparse linear systems – 15

Topic: LU decomposition ◮ A = LU ◮ UMFPACK: Matlab \ 8 T.A. Davis. Direct methods for sparse linear systems . Vol. 2. Fundamentals of Algorithms. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 2006, pp. 83–94. 9 T.A. Davis. “Algorithm 832: UMFPACK – an unsymmetric-pattern multifrontal method with a column pre-ordering strategy”. In: ACM Trans. Math. Software 30 (2004), pp. 196–199. A. Potschka Direct methods for sparse linear systems – 16

Topic: Minimum degree ordering ◮ Preserving sparsity of matrix factors ◮ Minimum degree ordering 10 T.A. Davis. Direct methods for sparse linear systems . Vol. 2. Fundamentals of Algorithms. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 2006, pp. 99–112. A. Potschka Direct methods for sparse linear systems – 17

Topic: Maximum matching ◮ Preserving sparsity of matrix factors ◮ Maximum matching ◮ Dulmage–Mendelsohn decomposition 11 T.A. Davis. Direct methods for sparse linear systems . Vol. 2. Fundamentals of Algorithms. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 2006, pp. 112–126. A. Potschka Direct methods for sparse linear systems – 18

Topic: Profile reduction, nested dissection, solution ◮ Preserving sparsity of matrix factors ◮ Bandwidth and profile reduction ◮ Nested dissection ◮ Solution of decomposed systems 12 T.A. Davis. Direct methods for sparse linear systems . Vol. 2. Fundamentals of Algorithms. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 2006, pp. 127–139. A. Potschka Direct methods for sparse linear systems – 19

Topic: Implicit LU decomposition ◮ Decomposition with possibility of updates 13 R. Fletcher. “Approximation Theory and Optimization. Tributes to M.J.D. Powell”. In: ed. by M.D. Buhmann and A. Iserles. Cambridge University Press, 1997. Chap. Dense factors of sparse matrices, pp. 145–166. A. Potschka Direct methods for sparse linear systems – 20

List of topics 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 A. Potschka Direct methods for sparse linear systems – 21

Preparation guidelines for presentations ◮ 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 A. Potschka Direct methods for sparse linear systems – 22

Introductory round ◮ Name ◮ Country ◮ Semester ◮ Study program ◮ Possible topics for seminar presentation A. Potschka Direct methods for sparse linear systems – 23