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Krylov subspace methods and CG

The Krylov subspace

The set of basis vectors for the Krylov subspace $\mathcal{K}_k(A, b)$ are formed by repeated application of a matrix $A$ on a vector $b$

$\mathcal{K}_k(A, b) = \text{span}\{ b, Ab, A^2b, ..., A^{k-1}b \}$

Krylov subspace methods

Krylov subspace methods are an important family of iterative algorithms for solving $Ax=b$ . Lets suppose that $A$ is an $n \times n$ invertible matrix, and our only knowledge of $A$ is its matrix-vector product with an arbitrary vector $\mathbf{x}$ . Repeated application of $A$ , $n$ times on an initial vector $b$ gives us a sequence of vectors $\mathbf{b}, A \mathbf{b}, A^2 \mathbf{b}, ..., A^{n} \mathbf{b}$ . Since we are in $n$ -dimensional space, we are guaranteed that these $n+1$ vectors are linearly dependent, and therefore

$a_0 \mathbf{b} + a_1 A \mathbf{b} + ... + a_n A^n \mathbf{b} = 0$

for some set of coefficients $a_i$ . Let now pick the smallest index $k$ such that $a_k \ne 0$ , and multiply the above equation by $\frac{1}{a_k} A^{-k-1}$ , giving

$A^{-1} b = \frac{1}{a_k} ( a_{k+1} b + ... + a_n A^{n-k-1} b )$

This implies that $A^{-1} b$ can be found only via repeated application of $A$ , and also motivates the search for solutions from the Krylov subspace.

For each iteration $k$ of a Krylov subspace method, we choose the "best" linear combination of the Krylov basis vectors $\mathbf{b}, A\mathbf{b}, ... , A^{k−1} \mathbf{b}$ to form an improved solution $\mathbf{x}_k$ . Different methods give various definitions of "best", for example:

The residual $\mathbf{r}_k = \mathbf{b} − A\mathbf{x}_k$ is orthogonal to $\mathcal{K}_k$ (Conjugate Gradients).
The residual $\mathbf{r}_k$ has minimum norm for $\mathbf{x}_k$ in $\mathcal{K}_k$ (GMRES and MINRES).
$\mathbf{r}_k$ is orthogonal to a different space $\mathcal{K}_k(A^T)$ (BiConjugate Gradients).

Conjugate Gradient Method

Here we will give a brief summary of the CG method, for more details you can consult the text by Golub and Van Loan (Chapter 10).

The CG method is based on minimising the function

$\phi(x) = \frac{1}{2}x^T A x - x^T b$

If we set $x$ to the solution of $Ax =b$ , that is $x = A^{-1} b$ , then the value of $\phi(x)$ is at its minimum $\phi(A^{-1} b) = -b^T A^{-1} b / 2$ , showing that solving $Ax = b$ and minimising $\phi$ are equivalent.

At each iteration $k$ of CG we are concerned with the residual, defined as $r_k = b - A x_k$ . If the residual is nonzero, then at each step we wish to find a positive $\alpha$ such that $\phi(x_k + \alpha p_k) < \phi(x_k)$ , where $p_k$ is the search direction at each $k$ . For the classical steepest descent optimisation algorithm the search direction would be the residual $p_k = r_k$ , however, steepest descent can suffer from convergence problems, so instead we aim to find a set of search directions $p_k$ so that $p_k^T r_{k-1} \ne 0$ (i.e. at each step we are guaranteed to reduce $\phi$ ), and that the search directions are linearly independent. The latter condition guarantees that the method will converge in at most $n$ steps, where $n$ is the size of the square matrix $A$ .

It can be shown that the best set of search directions can be achieved by setting

$\begin{aligned} \beta_k &= \frac{-p^T_{k-1} A r_{k-1}}{p^T_{k-1} A p_{k-1}} \\ p_k &= r_{k-1} + \beta_k p_{k-1} \\ \alpha_k &= \frac{p^T_k r_{k-1}}{p^T_k A p_k} \end{aligned}$

This ensures that the search direction $\mathbf{p}_k$ is the closest vector to $\mathbf{r}_{k-1}$ that is also A-conjugate to $\mathbf{p}_1, ..., \mathbf{p}_{k-1}$ , i.e. $p^T_i A p_j=0$ for all $i \ne j$ , which gives the algorithm its name. After $k$ iterations the sequence of residuals $\mathbf{r}_i$ for $i=1..k$ form a set of mutually orthogonal vectors that span the Krylov subspace $\mathcal{K}_k$ .

Directly using the above equations in an iterative algorithm results in the standard CG algorithm. A more efficient algorithm can be derived from this by computing the residuals recursively via $r_k = r_{k-1} - \alpha_k A p_k$ , leading to the final algorithm given below (reproduced from Wikipedia):

Preconditioning

The CG method works well (i.e. converges quickly) if the condition number of the matrix $A$ is low. The condition number of a matrix gives a measure of how much the solution $x$ changes in response to a small change in the input $b$ , and is a property of the matrix $A$ itself, so can vary from problem to problem. In order to keep the number of iterations small for iterative solvers, it is therefore often necessary to use a preconditioner, which is a method of transforming what might be a difficult problem with a poorly conditioned $A$ , into a well conditioned problem that is easy to solve.

Consider the case of preconditioning for the CG methods, we start from the standard problem $A x = b$ , and we wish to solve an equivalent transformed problem given by

$\tilde{A} \tilde{x} = \tilde{b}$

where $\tilde{A} = C^{-1} A C^{-1}$ , $\tilde{x} = Cx$ , $\tilde{b} = C^{-1} b$ , and $C$ is a symmetric positive matrix.

We then simply apply the standard CG method as given above to this transformed problem. This leads to an algorithm which is then simplified by instead computing the transformed quantities $\tilde{p}_k = C p_k$ , $\tilde{x}_k = C x_k$ , and $\tilde{r}_k = C^{-1} r_k$ . Finally we define a matrix $M = C^2$ , which is known as the preconditioner, leading to the final preconditioned CG algorithm given below (reproduced and edited from Wikipedia):

$\mathbf{r}_0 := \mathbf{b} - \mathbf{A x}_0$
$\mathbf{z}_0 := \mathbf{M}^{-1} \mathbf{r}_0$
$\mathbf{p}_0 := \mathbf{z}_0$
$k := 0 \,$
repeat until $|| \mathbf{r}_k ||_2 < \epsilon ||\mathbf{b}||_2$
   $\alpha_k := \frac{\mathbf{r}_k^T \mathbf{z}_k}{ \mathbf{p}_k^T \mathbf{A p}_k }$
   $\mathbf{x}_{k+1} := \mathbf{x}_k + \alpha_k \mathbf{p}_k$
   $\mathbf{r}_{k+1} := \mathbf{r}_k - \alpha_k \mathbf{A p}_k$
   if $r_{k+1}$ is sufficiently small then exit loop end if
   $\mathbf{z}_{k+1} := \mathbf{M}^{-1} \mathbf{r}_{k+1}$
   $\beta_k := \frac{\mathbf{r}_{k+1}^T \mathbf{z}_{k+1}}{\mathbf{r}_k^T \mathbf{z}_k}$
   $\mathbf{p}_{k+1} := \mathbf{z}_{k+1} + \beta_k \mathbf{p}_k$
   $k := k + 1 \,$
end repeat

The key point to note here is that the preconditioner is used by inverting $M$ , so this matrix must be "easy" to solve in some fashion, and also result in a transformed problem with better conditioning.

Termination: The CG algorithm is normally run until convergence to a given tolerance which is based on the norm of the input vector $b$ . In the algorithm above we iterate until the residual norm is less than some fraction (set by the user) of the norm of $b$ .

What preconditioner to choose for a given problem is often highly problem-specific, but some useful general purpose preconditioners exist, such as the incomplete Cholesky preconditioner for preconditioned CG (see Chapter 10.3.2 of the Golub & Van Loan text given below). Chapter 3 of the Barrett et al. text, also cited below, contains descriptions of a few more commonly used preconditioners.

Krylov subspace methods and CG

The Krylov subspace

Krylov subspace methods

Conjugate Gradient Method

Preconditioning

Other Reading