Large and Sparse Matrix Problems (ENM and GU)

7.5 credit points

January - March 2012.


Examiner and Lecturer: Ivar Gustafsson


- See you at the course start on Monday January 16, time 10.00 - 12.45 in room MVH11 at Mathematical Sciences building.

Some words about the course

Large and sparse matrix problems arise for instance in numerical approximation of differential equations, network problems and optimisation. In this course we study numerical techniques for solution of systems of linear equations and eigenvalue problems with this type of matrices.
For systems of linear equations we present two classes of methods: iterative and direct. Among the iterative methods we study basic stationary methods like Jacobi and SOR methods, orthogonalising methods like conjugate gradients and multigrid methods. The direct methods are based on Gaussian elimination with different renumberings of the unknowns in order to keep computing time and memory requirement small.
The eigenvalue routines presented are based on Lanczos and Arnoldi algorithms with or without spectral transformations. When solving the homework assignments (inlämningsuppgifter) and experimental assignments (laborationer), the students get experiences in implementation and evaluation of different algorithms for sparse problems.
For some learning outcomes of this course, see outcomes.

Course Literature

Teaching and Learning Methods

Lectures, five homework assignments (solved individually) and five computer exercises (performed in groups of two students).
The homework assignments give bonus credit points for the examination, totally at most five times one points.
Passed computer exercises will be graded with grades 3, 4 or 5, based on quantity and/or quality.

Lectures

Mondays 10-12 in room MVH11 (Mathematical Sciences building)
Wednesdays 13-15 in MVH11.

Outline

The following sections in the textbook will be considered.
Chapter 6, sections 1 - 6, 9.
Chapter 7: sections 1 - 4
The following questions in the textbook are recommended:
Chapter 6: 1, 2, 3, 4, 11, 12, 13, 14, 15
Chapter 7: 1, 2, 3
Some additional questions will be given along the lectures.

Detailed program on line

January 16

INTRODUCTION.
Learning outcomes
Outline of the course, problems and methods.
Available software on the web.
Large, sparse problems in applications
The model test problem: Poisson's equation in one and two dimensions.

January 18

FIRST PART OF THE COURSE: Iterative methods for systems of linear equations, Chapter 6 in the book.
Discussion on HA1 and CE1.
Basic iterative methods, Jacobi´s and Gauss-Seidel methods.
The spectral radius and convergence,
Lemma 6.4, Lemma 6.5, and Thm 6.1.
Rate of convergence
Good properties of basic iterative methods

January 23

Ordering of unknowns, red-black ordering
Irreducibility, strongly connected graph
Convergence criterion for Jacobi's and Gauss-Seidel methods, Thm 6.3.
SOR, successive overrelaxation method
Symmetrizable iterative methods
Convergence rate of SOR, property(A), consistently ordering
Important convergence results: Thm 6.6 and Thm 6.7

January 25

Convergence acceleration, section 6.5.6.
Semi-iterative methods, polynomial acceleration
Chebyshev polynomials for minimizing the spectral radius.
Stopping criteria
Krylov methods, section 6.6
Arnoldi's and Lanczos methods.

January 30

Solving Ax=b with Krylov methods.
Several "best" approximations.
The Conjugate Gradient method.
Gradient methods as search methods.
Choice of search directions and steplength.
The rate of convergence of the CG method.
Preconditioning.
Incomplete Cholesky factorizations.

February 1

Modified incomplete Cholesky factorization
Discussion on CE2.
Multigrid methods
Solution (smoothing), restriction and interpolation operators
Multigrid v-cycle, full multigrid.
Computational complexity for multigrid.
Multigrid for the 1D poisson problem

February 6

Convergence rate of multigrid for the 1D Poisson problem.
SECOND PART OF THE COURSE: Direct methods for large, sparse matrices, lecture notes.
Data structures for sparse matrices.
Compressed row and compressed diagonal storage.
Bandmatrices,
Theorem 1: No fill-in outside the band in non-pivoting Gaussian elimination.

February 13

General sparse matrices, envelope,
Theorem 2: no fill-in outside the envelope in non-pivoting Gaussian elimination.
More on compressed row and column storages.
Dynamic and static sparse structures, cholinc resp. (m)ic(d).
Orderings for sparsity, for minimizing fill-in
Graph theory for ordering
Reverse Cuthill-McKee ordering
Elimination graphs
Discussion om CE3

February 15

More on elimination graphs, Theorem 3.
The minimum degree ordering
Markowitz algorithm
Nested dissection orderings
MATLAB tools for sparse orderings and graphs.
Symbolic and numerical factorization

February 20

Numerical methods for large sparse least-squares problems, a survey (not to be examined).
Discussion on HA4 and CE4
THIRD PART OF THE COURSE: Methods for large sparse eigenproblems, chapter 7 in the text book.
Introduction, the power method, invers iteration with shift.
Relation to the Krylov space, Ritz vectors and Ritz values.
Krylov methods for symmetric eigenproblems.

February 22

Rayleigh-Ritz method
Lanczos-Raylegh-Ritz method for symmetric problems.
Theorem 7.1 with proof
Theorem 7.2 with proof

February 27

I solved three exercises as training for the examination:
- Exercise 5 in lecture notes on solving systems with iterative methods
- Solution of Question 7.2
- Solution of Question 7.3
Discussion on Question 7.3 related to CE5.
The Lanczos method in exact arithmetic.
Misconvergence:
(i) small component of q_1 in the direction of an eigenvector.
(ii) almost multiple eigenvalues.
The Lanczos method in floating point arithmetic:
+ can compute behind the theoretical break-down.
- loss of orthogonality.

February 29

Reorthogonalization, Selective reorthogonalization, Thm 7.3 (without proof).
Discussion om HA5 and CE5.
I proved the statement in Ex 6.9.
I solved an exercise on generalized eigenproblems.

Summary of the course

Part 1: Iterative methods for large, sparse systems of equations, Demmel chapter 6

Testexampel:
Poisson's equation, five-point-matrix, Kronecker product.
Basic iterative methods:
Fixpointiteration, splitting, Jacobi's method, Gauss-Seidel's method, SOR method.
Iteration matrix, norm and spectral radius.
Irreducibility, diagonal dominans.
Convergence rate, condition number.
Consistently ordering, property(A), red-black ordering.
Converegence acceleration.
Krylov techniques:
Krylov space.
Lanczos and Arnoldi's methods, Conjugate Gradient techniques.
Preconditioning, incomplete factorization.
Multigrid method:
V-cycle, full multigrid.
Three operators: smoother, interpolation anf restriction
Multigrid for Poisson's problem in 1D, idea of convergence rate.

Part 2: Direct methods for large, sparse systems of linear equations, lecture notes.

Gaussian elimination and reordering.
Bandmatrices.
Envelope minimization
fill-in minimization.
Sparse storage techniques.
Ordering methods for sparsity: RCM, minimum degree
Nested dissection for 2D Poisson's problem
Graph theory, elimination graphs.
Symbolic and numerical factorization.
Dynamic and static sparse structures.

Part 3: Iterative methods for large, sparse eigenproblems, Demmel chapter 7.

Krylov technique (again).
Lanczos-Rayleigh-Ritz method.
Lanczos method in exact arithmetic, reorthogonalization, misconvergence.
Lanczos method in floating point arithmetic.
Selective reorthogonalization.

Material for the course

Examination

Date for the written examination: Friday March 9, 13.30 - 17.30, room MVH12.

Means of assistance at the examination: none.
The total number of points in the written examination is 25 plus possibly 5 bonus points. The number of attained points on the written examination and the computer exercises are added and the final grade for the course is based on this sum.
Preliminary grades CTH: 28p for grade 3, 35p for grade 4 and 42p for grade 5.
Preliminary grades GU: 28p for grade G and 38p for grade VG.

Consultation

Thursday March 8, 10-12 in my office.

Examination: March 9 2012.
As expected, all passed the examination.

Last year's examination: March 17 2011.