GMRES

The most important use of the Arnoldi iteration is to solve the square linear system $\mathbf{A}\mathbf{x}=\mathbf{b}$, resulting in a well-known algorithm called GMRES.^[1]

In Demo 8.4.1, we attempted to replace the linear system $\mathbf{A}\mathbf{x}=\mathbf{b}$ by the lower-dimensional approximation

where $\mathbf{K}_m$ is the Krylov matrix generated using $\mathbf{A}$ and the seed vector $\mathbf{b}$. This method was unstable due to the poor conditioning of $\mathbf{K}_m$, which is a numerically poor basis for $\mathcal{K}_m$. Instead, we can use the orthonormal basis $\mathbf{Q}_m$ generated by the Arnoldi iteration.

8.5.1Derivation¶

If we set $\mathbf{x}=\mathbf{Q}_m\mathbf{z}$ in the residual $\mathbf{b} - \mathbf{A}\mathbf{x}$, we obtain a way to minimize the norm of the residual over $\mathcal{K}_m$:

From the fundamental Arnoldi identity (8.4.10), this is equivalent to

Note that $\mathbf{q}_1$ is a unit multiple of $\mathbf{b}$, so $\mathbf{b} = \|\mathbf{b}\| \mathbf{Q}_{m+1}\mathbf{e}_1$. Thus, (8.5.3) becomes

The least-squares problems (8.5.2), (8.5.3), and (8.5.4) are all $n\times m$. But observe that for any $\mathbf{w}\in\mathbb{C}^{m+1}$,

The first norm in that equation is on $\mathbb{C}^n$, while the last is on the much smaller space $\mathbb{C}^{m+1}$. Hence, the least-squares problem (8.5.4) is equivalent to

which is of size $(m+1)\times m$. We call the solution of this minimization $\mathbf{z}_m$, and then $\mathbf{x}_m=\mathbf{Q}_m \mathbf{z}_m$ is the $m$th approximation to the solution of $\mathbf{A}\mathbf{x}=\mathbf{b}$.

In exact arithmetic, GMRES should get the exact solution when $m=n$, but the goal is to reduce the residual enough to stop at some $m \ll n$.

A basic implementation of GMRES is given in Function 8.5.2.

8.5.2Convergence¶

In Demo 8.4.1, we tried to implement the GMRES idea using the Krylov matrix. After some initial reduction in the norm of the residual, improvement stopped due to the poor conditioning of that matrix. We return to the linear system in that example now to show that a stable implementation based on the Arnoldi iteration continues to improve the residual nearly to machine precision.

Thanks to Theorem 8.4.1, minimization of $\|\mathbf{b}-\mathbf{A}\mathbf{x}\|$ over $\mathcal{K}_{m+1}$ includes minimization over the subset $\mathcal{K}_m$. Hence:

Unfortunately, making other conclusive statements about the convergence of GMRES is not easy. Demo 8.5.1 shows the cleanest behavior: essentially, linear convergence down to the range of machine epsilon. But it is possible for the convergence to go through phases of sublinear and superlinear convergence as well. There is a strong dependence on the eigenvalues of the matrix, a fact we address with more precision and detail in the next section.

8.5.3Restarting¶

One of the practical challenges in GMRES is that as the dimension of the Krylov subspace grows, the number of new entries to be found in $\mathbf{H}_m$ and the total number of basis vectors grow, too. Consequently, both the work and the storage requirements are quadratic in $m$, which can become intolerable in some applications. For this reason, GMRES is often used with restarting.

Suppose $\hat{\mathbf{x}}$ is an approximate solution of $\mathbf{A}\mathbf{x}=\mathbf{b}$. Then, if we set $\mathbf{x}=\mathbf{u}+\hat{\mathbf{x}}$, we have $\mathbf{A}(\mathbf{u}+\hat{\mathbf{x}}) = \mathbf{b}$, or $\mathbf{A}\mathbf{u} = \mathbf{b} - \mathbf{A}\hat{\mathbf{x}}$. The conclusion is that if we get an approximate solution and compute its residual $\mathbf{r}=\mathbf{b} - \mathbf{A}\hat{\mathbf{x}}$, then we need only to solve $\mathbf{A}\mathbf{u} = \mathbf{r}$ in order to get a correction to $\hat{\mathbf{x}}$.^[3]

Restarting guarantees a fixed upper bound on the per-iteration cost of GMRES. However, this benefit comes at a price. Even though restarting preserves progress made in previous iterations, the Krylov space information is discarded and the residual minimization process starts again over low-dimensional spaces. That can significantly retard or even stagnate the convergence.

Restarting creates a tradeoff between the number of iterations and the speed per iteration. It’s essentially impossible in general to predict the ideal restart location in any given problem, so one goes by experience and hopes for the best.

There are other ways to avoid the growth in computational effort as the GMRES/Arnoldi iteration proceeds. Three of the more popular variations are abbreviated CGS, BiCGSTAB, and QMR. We do not describe them in this book.

8.5.4Exercises¶

1. ✍ (See also Exercise 8.4.1.) Consider the linear system with

   $$\mathbf{A}=\displaystyle \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{bmatrix}, \qquad \mathbf{b}=\mathbf{e}_1.$$

   (8.5.7)

   (a) Find the exact solution by inspection.

   (b) Find the GMRES approximate solutions $\mathbf{x}_m$ for $m=1,2,3,4$. (You should not try to do the Arnoldi iteration by hand. Instead, apply the result in (8.4.4) to get a tractable least-squares problem.)

2. ✍ (Continuation of Exercise 8.4.3.) Show that if $\mathbf{x}_m\in\mathcal{K}_m$, then the residual $\mathbf{b}-\mathbf{A}\mathbf{x}_m$ is equal to $q(\mathbf{A})\mathbf{b}$, where $q$ is a polynomial of degree at most $m$ and $q(0)=1$. (This fact is a key one for many convergence results.)

3. ✍ Explain why GMRES, in exact arithmetic, converges to the true solution in $n$ iterations for an $n\times n$ matrix if $\operatorname{rank}(\mathbf{K}_n)=n$. (Hint: Consider how the algorithm is defined from first principles.)

4. ⌨ Let $\mathbf{A}$ be the $n\times n$ tridiagonal matrix

   $$\begin{bmatrix} -4 & 1 & & & \\ 1 & -4 & 1 & & \\ & \ddots & \ddots & \ddots & \\ & & 1 & -4 & 1 \\ & & & 1 & -4 \end{bmatrix}$$

   (8.5.8)

   and let the $n$-vector $\mathbf{b}$ have elements $b_i=i/n$. For $n=8,16,32,64$, run Function 8.5.2 for $m=n/2$ iterations. On one semi-log graph, plot $\|\mathbf{r}_k\|/\|\mathbf{b}\|$ for all the cases. How does the convergence rate of GMRES seem to depend on $n$?

5. ⌨ In this exercise you will see the strong effect the eigenvalues of the matrix may have on GMRES convergence. Let

   $$\mathbf{B}= \begin{bmatrix} 1 & & & \\ & 2 & & \\ & & \ddots & \\ & & & 100 \end{bmatrix},$$

   (8.5.9)

   let $\mathbf{I}$ be a $100\times 100$ identity, and let $\mathbf{Z}$ be a $100\times 100$ matrix of zeros. Also let $\mathbf{b}$ be a $200\times 1$ vector of ones. You will use GMRES with restarts, as in Demo 8.5.2 (i.e., not the book’s version of gmres).

   (a) Let $\mathbf{A} = \begin{bmatrix} \mathbf{B} & \mathbf{I} \\ \mathbf{Z} & \mathbf{B} \end{bmatrix}.$ What are its eigenvalues (no computer required here)? Apply gmres with tolerance 10^-10 for 100 iterations without restarts, and plot the residual convergence.

   (b) Repeat part (a) with restarts every 20 iterations.

   (c) Now let $\mathbf{A} = \begin{bmatrix} \mathbf{B} & \mathbf{I} \\ \mathbf{Z} & -\mathbf{B} \end{bmatrix}.$ What are its eigenvalues? Repeat part (a). Which matrix is more difficult for GMRES? (Note: Even though this matrix is triangular, GMRES has no way of exploiting that fact.)

6. ⌨ (Continuation of Exercise 8.3.5.) We again consider the $n^2\times n^2$ sparse matrix defined by FNC.poisson(n). The solution of $\mathbf{A}\mathbf{x}=\mathbf{b}$ may be interpreted as the deflection of a lumped membrane in response to a load represented by $\mathbf{b}$.

   (a) For $n=10,15,20,25$, let $\mathbf{b}$ be the vector of $n^2$ ones and apply Function 8.5.2 for 50 iterations. On one semi-log graph, plot the four convergence curves $\|\mathbf{r}_m\|/\|\mathbf{b}\|$.

   (b) For the case $n=25$ make a surface plot of x after reshaping it to a 25×25 matrix. It should look physically plausible (though upside-down for a weighted membrane).