1. Direct and Iterative Methods for Sparse Linear Systems
Shirley Moore
CPS5401 Fall 2015
svmoore.pbworks.com
December 1, 2015

2. Learning Objectives
Describe advantages and disadvantages of direct and iterative methods for solving sparse linear systems.
Apply appropriate method from a solver library to solve a particular sparse linear system, including both symmetric positive definite and nonsymmetric matrices.
Be able to find and make use of documentation on sparse solver libraries.

3. Direct vs. Iterative Methods
In a direct method, the matrix of the initial linear system is transformed or factorized into a simpler form, which can be solved easily. The exact solution is obtained in a finite number of arithmetic operations, if not considering numerical rounding errors.
Iterative methods compute a sequence of approximate solutions, which converges to the exact solution in the limit, i.e., in practice until a desired accuracy is obtained.

4. Direct vs. Iterative Methods (cont.)
Direct methods have been preferred to iterative methods for solving linear systems, mainly because of their simplicity and robustness.
However, the emergence of conjugate gradient methods and Krylov subspace iterations has provided an efficient alternative to direct solvers.
Nowadays, iterative methods are almost mandatory in complex applications, mainly because of memory and computational requirements that prohibit the use of direct methods.
Iterative methods usually involve a matrix-vector multiplication procedure that is cheap to compute on modern computer architectures.
When the matrix A is very large and is composed of a majority of nonzero elements, the LU factorization would contain many more nonzero coefficients than the matrix A itself.
Nonetheless, in some peculiar applications, very ill-conditioned matrices arise that may require a direct method for solving the problem at hand.

5. Direct Solvers for Sparse Linear Systems
Direct solvers for sparse matrices involve much more complicated algorithms than for dense matrices. The main complication is due to the need for efficient handling the fill-in in the factors L and U. A typical sparse solver consists of four distinct steps as opposed to two in the dense case:
An ordering step that reorders the rows and columns so that the factors suffer little fill, or so that the matrix has special structure such as block triangular form.
An analysis step or symbolic factorization that determines the nonzero structures of the factors and creates suitable data structures for the factors.
Numerical factorization that computes the L and U factors.
A solve step that performs forward and back substitution using the factors.

6. Direct Solver Packages
SuperLU
SuperLU for sequential machines
SuperLU_MT for shared memory parallel machines
SuperLU_DIST for distributed memory parallel machines
See survey of direct solvers by Xiaoye Li
See also research by Tim Davis

7. Iterative Methods
Use successive approximations to obtain more accurate solutions to a linear system
Suitable for large sparse linear systems
Stationary methods are older, simple to understand and implement, not as effective
Perform same operation each iteration
Nonstationary methods are more recent, harder to understand, highly effective
Have iteration-dependent coefficients
Typically use a transformation matrix called a preconditioner that improves convergence of the method
Reference: Templates book

8. Stationary Methods Jacobi Gauss-Seidel Successive Overrelaxation (SOR)
Based on solving for every variable locally with respect to the other variables
One iteration of the method corresponds to solving for every variable once.
Resulting method is easy to understand and implement, but convergence is slow.
Gauss-Seidel 
Like the Jacobi method, except that it uses updated values as soon as they are available
In general, if the Jacobi method converges, the Gauss-Seidel method will converge faster than the Jacobi method, though still relatively slowly.
Successive Overrelaxation (SOR)
Can be derived from the Gauss-Seidel method by introducing an extrapolation parameter ω
For the optimal choice of ω, SOR may converge faster than Gauss-Seidel by an order of magnitude.
Symmetric Successive Overrelaxation (SSOR)
No advantage over SOR as a stand-alone iterative method
However, it is useful as a preconditioner for nonstationary methods.

9. Nonstationary Methods
Conjugate Gradient (CG )
The conjugate gradient method derives its name from the fact that it generates a sequence of conjugate (or orthogonal) vectors. These vectors are the residuals of the iterates. They are also the gradients of a quadratic functional, the minimization of which is equivalent to solving the linear system. CG is an extremely effective method when the coefficient matrix is symmetric positive definite, since storage for only a limited number of vectors is required.
Minimum Residual (MINRES ) and Symmetric LQ (SYMMLQ )
These methods are computational alternatives for CG for coefficient matrices that are symmetric but possibly indefinite. SYMMLQ will generate the same solution iterates as CG if the coefficient matrix is symmetric positive definite.
Conjugate Gradient on the Normal Equations : CGNE  and CGNR
These methods are based on the application of the CG method to one of two forms of the normal equations  When the coefficient matrix is nonsymmetric and nonsingular, the normal equations matrices will be symmetric and positive definite, and hence CG can be applied. The convergence may be slow, since the spectrum of the normal equations matrices will be less favorable.

10. Nonstationary Methods (cont.)
Generalized Minimal Residual (GMRES )
The Generalized Minimal Residual method computes a sequence of orthogonal vectors (like MINRES), and combines these through a least-squares solve and update. However, unlike MINRES (and CG) it requires storing the whole sequence, so that a large amount of storage is needed. For this reason, restarted versions of this method are used. In restarted versions, computation and storage costs are limited by specifying a fixed number of vectors to be generated. This method is useful for general nonsymmetric matrices.
BiConjugate Gradient (BiCG )
The Biconjugate Gradient method generates two CG-like sequences of vectors, one based on a system with the original coefficient matrix A , and one on AT. Instead of orthogonalizing each sequence, they are made mutually orthogonal, or ``bi-orthogonal''. This method, like CG, uses limited storage. It is useful when the matrix is nonsymmetric and nonsingular; however, convergence may be irregular, and there is a possibility that the method will break down. BiCG requires a multiplication with the coefficient matrix and with its transpose at each iteration.
Quasi-Minimal Residual (QMR )
The Quasi-Minimal Residual method applies a least-squares solve and update to the BiCG residuals, thereby smoothing out the irregular convergence behavior of BiCG, which may lead to more reliable approximations. In full glory, it has a look ahead strategy built in that avoids the BiCG breakdown. Even without look ahead, QMR largely avoids the breakdown that can occur in BiCG. On the other hand, it does not effect a true minimization of either the error or the residual, and while it converges smoothly, it often does not improve on the BiCG in terms of the number of iteration steps.

11. Nonstationary Methods (cont.)
Conjugate Gradient Squared (CGS )
The Conjugate Gradient Squared method is a variant of BiCG that applies the updating operations for the A-sequence and the AT-sequences both to the same vectors. Ideally, this would double the convergence rate, but in practice convergence may be much more irregular than for BiCG, which may sometimes lead to unreliable results. A practical advantage is that the method does not need the multiplications with the transpose of the coefficient matrix.
Biconjugate Gradient Stabilized (Bi-CGSTAB )
The Biconjugate Gradient Stabilized method is a variant of BiCG, like CGS, but using different updates for the -sequence in order to obtain smoother convergence than CGS.
Chebyshev  Iteration
The Chebyshev Iteration recursively determines polynomials with coefficients chosen to minimize the norm of the residual in a min-max sense. The coefficient matrix must be positive definite and knowledge of the extremal eigenvalues is required. This method has the advantage of requiring no inner products.

12. Conjugate Gradient Method
Popular iterative method for solving large systems of sparse linear equations Ax=b
A is known, square, symmetric, positive-definite
Reference: Shewchuk

13. Quadratic Form Quadratic form
If A is symmetric and positive-definite, f(x) is minimized by the solution to Ax=b
A=AT

14. Method of Steepest Descent
Choose the direction in which f decreases the most: the direction opposite to f’(x(i))
Error
Error e(i) = x(i) – x is a vector that indicates how far we are from the solution.
residual r(i) = b – Ax(i) indicates how far we are from the correct value of b.
Think of the residual as the error transformed by A into the same space as b.
Think of the residual as the direction of steepest descent.
Residual
Residual as the direction of steepest descent

15. Line Search  = 0 line search chooses α to minimize f along a line
α minimizes f when the directional derivative is equal to zero
Setting this to zero, α should be chosen so that r(0) and f’x(1) are orthogonal
Bottom figure:f is minimized where the projection of the gradient onto the line is zero

16. Summary b – A b – A( ) Starting at [-2,-2] Converges at [2,-2]
Each gradient is orthogonal to the previous gradient.
Premultiply by –A and add b to eliminate one matrix-vector multiply

17. Convergence Analysis Two cases where one-step convergence is possible
e(i) is the eigenvector (of A)
All eigenvalues of A are the same

18. Convergence Analysis (cont)
In general, depends on condition number of A and the slope of e(0) (relative to coord. system of eigenvectors) at the starting point

19. Method of Conjugate Directions
Idea: Pick a set of n orthogonal directions. In each direction, take exactly one step. After n steps, we’ll be done  
Steepest descent can take steps in the same direction as earlier steps.
But can’t compute α(i) without knowing e(i+1) in which case we would already know the solution.
Solution: make the search direction A-orthogonal (also called A-conjugate) instead of orthogonal. New requirement is that d(i) and e(i+1) be A-orthogonal.
Algorithm should look like:
But we don’t know e(i) !

20. Conjugate Directions Make the search directions {d(i)}A-orthogonal
Def: A-orthogonal
Make the search directions {d(i)}A-orthogonal A
And require
Equivalent to finding the minimum point along the search direction d(i)
To see this, set the directional derivative equal to zero
and
A-orthogonal
Find minimum along d(i)

21. Conjugate Directions
To get stepsize
Algorithm Summary

22. Proof
Can this really solve x in n steps? [since we’ve changed orthogonal to A-orthogonal?]
Intuition:
Each step of conjugate directions eliminates one A-orthogonal component of error

23. represent error in basis of {d(i)}
Initial error
represent error in basis of {d(i)}
Initial error e(0) can be expressed as a sum of A-orthogonal components.
Each step of conjugate directions eliminates one of these components.
After n iterations, every component is cut away and e(n) = 0.
After n iterations, e(n) = 0

24. To find A-orthogonal directions
Conjugate Gram-Schmidt process
n linearly independent vectors
To construct d(i), take ui and subtract out any components that are not A-orthogonal to the previous d vectors
e.g. u1 is comprised of two components: u* which is A-orthogonal to d(0) and u+ which is parallel to d(0). After conjugation, only the A-orthogonal portion remains.
Difficulty: need to keep all the old search directions to construct the new one!

25. Conjugate Gradients
Conjugate directions where the search directions are constructed by conjugation of the residuals
Setting ui = r(i)
Reasons: residuals are independent, orthogonal to previous search directions, …

26. Method of Conjugate Gradients
Each new residual is orthogonal to all the previous residuals and search directions.
Each new search direction is constructed to be A-orthogonal to all the previous residuals and search directions.
d(2) is a linear combination of r(2) and d(1).
Krylov subspace
Krylov subspace is a subspace formed by repeatedly applying a matrix to a vector.

27. Reduction of Space Complexity
Conjugate Gradients is a misnomer: the gradients are not conjugate and the conjugate directions are not all gradients.
“Conjugated Gradients” would be more accurate.
Let

28. Convergence Analysis of Conjugate Gradients
With exact arithmetic, CG is complete after n iterations.
Accumulated roundoff error causes residual to gradually lose accuracy, and cancellation error causes search vectors to lose A-orthogonality.
CG is useful for problems so large that is not feasible to run even n iterations.
CG is quicker if there are duplicated eigenvalues.
CG converges more quickly when eigenvalues are clustered together.

29. Convergence Analysis of Conjugate Gradients (cont.)
See Appendix C3 of Shewchuk

30. Time and Space Complexity
Shewchuk, p. 38
How does this compare to the time and space complexity of Gaussian elimination?

31. Preconditioning

32. Transformed Preconditioned Conjugate Gradient Method

33. Untransformed Preconditioned Conjugate Gradient Method
Effectiveness of preconditioner determined by condition number of M^(-1)A.
For large-scale problems, CG should almost always be used with a preconditioner.

34. History of Conjugate Gradient Method
Conjugate Direction methods were probably first presented by Schmidt [14] in 1908, and were independently reinvented by Fox, Huskey, and Wilkinson [7] in 1948.
In the early fifties, the method of Conjugate Gradients was discovered independently by Hestenes [10] and Stiefel [15]; shortly thereafter, they jointly published what is considered the seminal reference on CG [11].
Convergence bounds for CG in terms of Chebyshev polynomials were developed by Kaniel [12]. A more thorough analysis of CG convergence is provided by van der Sluis and van der Vorst [16].
CG was popularized as an iterative method for large, sparse matrices by Reid [13] in 1971.
CG was generalized to nonlinear problems in 1964 by Fletcher and Reeves [6], based on work by Davidon [4] and Fletcher and Powell [5]. Convergence of nonlinear CG with inexact line searches was analyzed by Daniel [3].
A history and extensive annotated bibliography of CG to the mid-seventies is provided by Golub and O’Leary [9]. Most research since that time has focused on nonsymmetric systems.

35. Templates section on Conjugate Gradient Method

36. MATLAB Preconditioned Conjugate Gradients Method

37. Generalized Minimal Residual (GMRES) Method
Generalizes MINRES to unsymmetric systems
Templates book:

38. MATLAB documentation on GMRES

39. Iterative Method Packages
Included in PETSc
Sparse linear solvers
ITSOL and pARMS
Dongarra’s survey of freely available linear algebra software
Vendor libraries
IBM ESSL/PESSL
Cray-supported version of PETSc
Numerical Algorithms Group (NAG)