Generalized minimal residual method

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In mathematics, the generalized minimal residual method (usually abbreviated GMRES) is an iterative method for the numerical solution of a system of linear equations. The method approximates the solution by the vector in a Krylov subspace with minimal residual. The Arnoldi iteration is used to find this vector.

The GMRES method was developed by Yousef Saad and Martin H. Schultz in 1986.^[1]

1 The method
2 Convergence
3 Extensions of the method
4 Comparison with other solvers
5 Implementation
- 5.1 Solving the least squares problem
- 5.2 Pseudocode
6 Notes
7 References

[edit] The method

Denote the system of linear equations which needs to be solved by

$Ax = b. \,$

The matrix A is assumed to be invertible of size m-by-m. Furthermore, it is assumed that b is normalized, i.e., ||b|| = 1 (in this article, ||·|| denotes the Euclidean norm).

The nth Krylov subspace for this problem is

$K_n = \operatorname{span} \, \{ b, Ab, A^2b, \ldots, A^{n-1}b \}. \,$

GMRES approximates the exact solution of Ax = b by the vector x_n ∈ K_n that minimizes the norm of the residual Ax_n − b.

The vectors b, Ab, …, Aⁿ⁻¹b are almost linearly dependent, so instead of this basis, the Arnoldi iteration is used to find orthonormal vectors

$q_1, q_2, \ldots, q_n \,$

which form a basis for K_n. Hence, the vector x_n ∈ K_n can be written as x_n = Q_ny_n with y_n ∈ Rⁿ, where Q_n is the m-by-n matrix formed by q₁, …, q_n.

The Arnoldi process also produces an (n+1)-by-n upper Hessenberg matrix $\tilde{H}_n$ with

$AQ_n = Q_{n+1} \tilde{H}_n. \,$

Because $Q n$ is orthogonal, we have

$\| Ax_n - b \| = \| \tilde{H}_ny_n - e_1 \|, \,$

where

$e_1 = (1,0,0,\ldots,0) \,$

is the first vector in the standard basis of Rⁿ⁺¹. Hence, $x n$ can be found by minimizing the norm of the residual

$r_n = \tilde{H}_n y_n - e_1.$

This is a linear least squares problem of size n.

This yields the GMRES method. At every step of the iteration:

do one step of the Arnoldi method;
find the $y n$ which minimizes ||r_n||;
compute $x n = Q n y n$ ;
repeat if the residual is not yet small enough.

At every iteration, a matrix-vector product Aq_n must be computed. This costs about 2m² floating-point operations for general dense matrices of size m, but the cost can decrease to O(m) for sparse matrices. In addition to the matrix-vector product, O(n m) floating-point operations must be computed at the nth iteration.

[edit] Convergence

The nth iterate minimizes the residual in the Krylov subspace K_n. Since every subspace is contained in the next subspace, the residual decreases monotonically. After m iterations, where m is the size of the matrix A, the Krylov space K_m is the whole of R^m and hence the GMRES method arrives at the exact solution. However, the idea is that after a small number of iterations (relative to m), the vector x_n is already a good approximation to the exact solution.

This does not happen in general. Indeed, a theorem of Greenbaum, Pták and Strakoš states that for every monotonically decreasing sequence a₁, …, a_m−1, a_m = 0, one can find a matrix A such that the ||r_n|| = a_n for all n, where r_n is the residual defined above. In particular, it is possible to find a matrix for which the residual stays constant for m − 1 iterations, and only drops to zero at the last iteration.

In practice, though, GMRES often performs well. This can be proven in specific situations. If A is positive definite, then

$\|r_n\| \leq \left( 1-\frac{\lambda_{\mathrm{min}}(A^\top + A)}{2 \lambda_{\mathrm{max}}(A^T + A)} \right)^{n/2} \|r_0\|,$

where $λ min$ and $λ max$ denote the smallest and largest eigenvalue, respectively.

If A is symmetric and positive definite, then we even have

$\|r_n\| \leq \left( \frac{\kappa_2^2(A)-1}{\kappa_2^2(A)} \right)^{n/2} \|r_0\|.$

where $κ 2 (A)$ denotes the condition number of A in the Euclidean norm.

In the general case, where A is not positive definite, we have

$\|r_n\| \le \inf_{p \in P_n} \|p_n(A)\| \le \kappa_2(V) \inf_{p \in P_n} \max_{\lambda \in \sigma(A)} |p(\lambda)|, \,$

where P_n denotes the set of polynomials of degree at most n with p(0) = 1, V is the matrix appearing in the spectral decomposition of A, and σ(A) is the spectrum of A. Roughly speaking, this says that fast convergence occurs when the eigenvalues of A are clustered away from the origin and A is not too far from normality.^[2]

All these inequalities bound only the residuals instead of the actual error, that is, the distance between the current iterate x_n and the exact solution.

[edit] Extensions of the method

Like other iterative methods, GMRES is usually combined with a preconditioning method in order to speed up convergence.

The cost of the iterations grow like O(n²), where n is the iteration number. Therefore, the method is sometimes restarted after a number, say k, of iterations, with x_k as initial guess. The resulting method is called GMRES(k).

[edit] Comparison with other solvers

The Arnoldi iteration reduces to the Lanczos iteration for symmetric matrices. The corresponding Krylov subspace method is the minimal residual method (MinRes) of Paige and Saunders. Unlike the unsymmetric case, the MinRes method is given by a three-term recurrence relation. It can be shown that there is no Krylov subspace method for general matrices, which is given by a short recurrence relation and yet minimizes the norms of the residuals, as GMRES does.

Another class of methods builds on the unsymmetric Lanczos iteration, in particular the BiCG method. These use a three-term recurrence relation, but they do not attain the minimum residual, and hence the residual does not decrease monotonically for these methods. Convergence is not even guaranteed.

The third class is formed by methods like CGS and BiCGSTAB. These also work with a three-term recurrence relation (hence, without optimality) and they can even terminate prematurely without achieving convergence. The idea behind these methods is to choose the generating polynoms of the iteration sequence suitably.

None of these three classes is the best for all matrices; there are always examples in which one class outperforms the other. Therefore, multiple solvers are tried in practice to see which one is the best for a given problem.

[edit] Implementation

[edit] Solving the least squares problem

One part of the GMRES method is to find the vector $y n$ which minimizes

$\| \tilde{H}_n y_n - e_1 \|. \,$

This can be done by computing a QR decomposition: find an (n+1)-by-(n+1) orthogonal matrix Ω_n and an (n+1)-by-n upper triangular matrix $\tilde{R}_n$ such that

$\Omega_n \tilde{H}_n = \tilde{R}_n.$

The triangular matrix has one more row than it has columns, so its bottom row consists of zero. Hence, it can be decomposed as

$\tilde{R}_n = \begin{bmatrix} R_n \\ 0 \end{bmatrix},$

where $R n$ is an n-by-n (thus square) triangular matrix.

The QR decomposition can be updated cheaply from one iteration to the next, because the Hessenberg matrices differ only by a row of zeros and a column:

$\tilde{H}_{n+1} = \begin{bmatrix} \tilde{H}_n & h_n \\ 0 & h_{n+1,n} \end{bmatrix},$

where h_n = (h_1n, … h_nn)^T. This implies that premultiplying the Hessenberg matrix with Ω_n, augmented with a zero row and column, yields almost a triangular matrix:

$\begin{bmatrix} \Omega_n & 0 \\ 0 & 0 \end{bmatrix} \tilde{H}_{n+1} = \begin{bmatrix} R_n & r_k \\ 0 & \rho \\ 0 & \sigma \end{bmatrix}$

This would be triangular if σ is zero. To remedy this, one needs the Givens rotation

$G_n = \begin{bmatrix} I_{n-1} & 0 & 0 \\ 0 & c_n & s_n \\ 0 & -s_n & c_n \end{bmatrix}$

where

$c_n = \frac{\rho}{\sqrt{\rho^2+\sigma^2}} \quad\mbox{and}\quad s_n = \frac{\sigma}{\sqrt{\rho^2+\sigma^2}}.$

With this Givens rotation, we form

$\Omega_{n+1} = G_n \begin{bmatrix} \Omega_n & 0 \\ 0 & 0 \end{bmatrix}.$

Indeed,

$\Omega_{n+1} \tilde{H}_{n+1} = \begin{bmatrix} R_n & r_n \\ 0 & r_{nn} \\ 0 & 0 \end{bmatrix} \quad\text{with}\quad r_{nn} = \sqrt{\rho^2+\sigma^2}$

is a triangular matrix.

Given the QR decomposition, the minimization problem is easily solved by noting that

$\| \tilde{H}_n y_n - e_1 \| = \| \Omega_n (\tilde{H}_n y_n - e_1) \| = \| \tilde{R}_n y_n - \Omega_n e_1 \|.$

Denoting the vector $Ω n e 1$ by

$\tilde{g}_n = \begin{bmatrix} g_n \\ \gamma_n \end{bmatrix}$

with g_n ∈ Rⁿ and γ_n ∈ R, this is

$\| \tilde{H}_n y_n - e_1 \| = \| \tilde{R}_n y_n - \Omega_n e_1 \| = \left\| \begin{bmatrix} R_n \\ 0 \end{bmatrix} y - \begin{bmatrix} g_n \\ \gamma_n \end{bmatrix} \right\|.$

The vector y that minimizes this expression is given by

$y_n = R_n^{-1} g_n.$

Again, the vectors $g n$ are easy to update.^[3]

[edit] Pseudocode

Inputs: A, b, x₀.

$r_0 \leftarrow b - Ax_0 \,$
if $r 0 = 0$ then

return

x 0

end if
$v_1 = \frac{r_0}{|r_0|_2} \,$
$γ 1 = | r 0 | 2$
for j ← 1, …, n do

for i ← 1, …, j do

$h_{ij} \leftarrow v_i^\top Av_j \,$

end for

$w_j \leftarrow Av_j - \sum_{i=1}^j h_{ij} v_i \,$

$h_{j+1,j} \leftarrow |w_j|_2 \,$

for i ← 1, …, j − 1 do

$\begin{pmatrix} h_{ij} \\ h_{i+1,j} \end{pmatrix} \leftarrow \begin{pmatrix} c_{i+1} & s_{i+1} \\ s_{i+1} & -c_{i+1} \end{pmatrix} \begin{pmatrix} h_{ij} \\ h_{i+1,j} \end{pmatrix}$

end for

$\beta \leftarrow \sqrt{h_{jj}^2 + h_{j+1,j}^2} \,$

$s_{j+1} \leftarrow \frac{h_{j+1,j}}{\beta} \,$

$c_{j+1} \leftarrow \frac{h_{jj}}{\beta} \,$

$h_{jj} \leftarrow \beta \,$

$\gamma_{j+1} \leftarrow s_{j+1} \gamma_j \,$

$\gamma_j \leftarrow c_{j+1} \gamma_j \,$ .

if $\gamma_{j+1} \neq 0$ then

$v_{j+1} \leftarrow \frac{w_j}{h_{j+1,j}} \,$

else

for i ← j, …, 1 do

$\alpha_i \leftarrow \frac{1}{h_{ii}} \left( \gamma_i - \sum_{k=i+1}^j h_{ik} \alpha_k \right) \,$

end for

$x \leftarrow x_0 + \sum_{i=1}^j \alpha_i v_i \,$

return x

end if

end for

[edit] Notes

^ Saad and Schultz
^ Trefethen & Bau, Thm 35.2
^ Stoer and Bulirsch, §8.7.2

[edit] References

A. Meister, Numerik linearer Gleichungssysteme, 2nd edition, Vieweg 2005, ISBN 978-3-528-13135-7.
Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd edition, Society for Industrial and Applied Mathematics, 2003. ISBN 978-0-89871-534-7.
Y. Saad and M.H. Schultz, "GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems", SIAM J. Sci. Stat. Comput., 7:856-869, 1986. DOI:10.1137/0907058.
J. Stoer and R. Bulirsch, Introduction to numerical analysis, 3rd edition, Springer, New York, 2002. ISBN 978-0-387-95452-3.
Lloyd N. Trefethen and David Bau, III, Numerical Linear Algebra, Society for Industrial and Applied Mathematics, 1997. ISBN 978-0-89871-361-9.