Lecture 12: Great Iterative Methods¶
Previous lecture¶
- Gaussian elimination and graphs in more details
- Concept of iterative methods for linear systems:
- Richardson iteration and its convergence
- Chebyshev iteration
Missed experiment from the previous lecture¶
- Compare Richardson and Chebyshev convergence
- In theory, for $A = A^* > 0$
- Richardson method gives linear convergence rate with $q = \frac{\lambda_{\max} - \lambda_{\min}}{\lambda_{\max} + \lambda_{\min}} = \frac{\mathrm{cond}(A) - 1}{\mathrm{cond}(A)+1}$
- Chebyshev method gives linear convergence rate, but with $q = \frac{\sqrt{\mathrm{cond}(A)}-1}{\sqrt{\mathrm{cond}(A)}+1}$
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
plt.rc("text", usetex=True)
import scipy as sp
import scipy.sparse
import scipy.sparse.linalg as spla
import scipy
from scipy.sparse import csc_matrix
n = 16
ex = np.ones(n);
A = sp.sparse.spdiags(np.vstack((-ex, 2*ex, -ex)), [-1, 0, 1], n, n, 'csr');
rhs = np.ones(n)
ev1, vec = spla.eigsh(A, k=2, which='LA')
ev2, vec = spla.eigsh(A, k=2, which='SA')
lam_max = ev1[0]
lam_min = ev2[0]
tau_opt = 2.0/(lam_max + lam_min)
fig, ax = plt.subplots()
plt.close(fig)
niters = 100
x = np.zeros(n)
res_richardson = []
for i in range(niters):
rr = A.dot(x) - rhs
x = x - tau_opt * rr
res_richardson.append(np.linalg.norm(rr))
#Convergence of an ordinary Richardson (with optimal parameter)
plt.semilogy(res_richardson)
plt.xlabel("Number of iterations, $k$", fontsize=20)
plt.ylabel("Residual norm, $\|Ax_k - b\|_2$", fontsize=20)
plt.xticks(fontsize=20)
plt.yticks(fontsize=20)
plt.show()
niters = 64
roots = [np.cos((np.pi * (2 * i + 1)) / (2 * niters)) for i in range(niters)]
taus = [(lam_max + lam_min - (lam_min - lam_max) * r) / 2 for r in roots]
x = np.zeros(n)
r = A.dot(x) - rhs
res_cheb = [np.linalg.norm(r)]
# Implementation may be non-optimal if number of iterations is not power of two
def good_shuffle(idx):
if len(idx) == 1:
return idx
else:
new_len = int(np.ceil((len(idx) / 2)))
new_idx = good_shuffle(idx[:new_len])
res_perm = []
perm_count = 0
for i in new_idx:
res_perm.append(i)
perm_count += 1
if perm_count == len(idx):
break
res_perm.append(len(idx) + 1 - i)
perm_count += 1
if perm_count == len(idx):
break
return res_perm
good_perm = good_shuffle([i for i in range(1, niters+1)])
# good_perm = [i for i in range(niters, 0, -1)]
# good_perm = np.random.permutation([i for i in range(1, niters+1)])
for i in range(niters):
x = x - 1.0/taus[good_perm[i] - 1] * r
r = A.dot(x) - rhs
res_cheb.append(np.linalg.norm(r))
plt.semilogy(res_richardson, label="Richardson")
plt.semilogy(res_cheb, label="Chebyshev")
plt.legend(fontsize=20)
plt.xlabel("Number of iterations, $k$", fontsize=20)
plt.ylabel("Residual norm, $\|Ax_k - b\|_2$", fontsize=20)
plt.xticks(fontsize=20)
_ = plt.yticks(fontsize=20)
What happened with great Chebyshev iterations?¶
Today lecture¶
- Lanczos and Arnoldi orthogonalization of Krylov subspaces, optimality result for Krylov subspaces
- Main iterative methods: conjugate gradient, GMRES, etc
- Convergence estimates
Solution of linear systems and minimization of functionals¶
Instead of solving a linear system, we can minimize the residual:
$$R(x) = \Vert A x - f \Vert^2_2.$$
The condition $\nabla R(x) = 0$ gives
$$A^* A x = A^* f,$$
thus it has squared condition number, so direct minimization of the residual by standard optimization methods is rarely used.
For the symmetric positive definite case there is a much simpler functional.
Energy functional¶
Let $A = A^* > 0$, then the following functional
$$\Phi(x) = (Ax, x) - 2(f, x)$$
is called energy functional.
Properties of energy functional¶
- It is strictly convex (check!)
$$ \Phi(\alpha x + (1 - \alpha)y) < \alpha \Phi(x) + (1 - \alpha) \Phi(y)$$
Since it is strictly convex, it has unique local minimum, which is also global
Its global minimum $x_*$ satisfies
$$A x_* = f.$$
Indeed,
$$\nabla \Phi = 2(Ax - f).$$
and the first order optimality condition $\nabla \Phi (x_*) = 0$ yields
$$A x_* = f.$$
Approximation of the solution by a subspace¶
Given a linear $M$-dimensional subspace $\{y_1, \dots, y_M\}$, we want to find an approximate solution in this basis, i.e.
$$A x \approx f, \quad x = \sum_{k=1}^M c_k y_k,$$
where $c$ is the vector of coefficients.
In the symmetric positive definite case we need to minimize
$$(Ax, x) - 2(f, x)$$
subject to $$x = Y c,$$
where $Y=[y_1,\dots,y_M]$ is $n \times M$ and vector $c$ has length $M$.
Using the representation of $x$, we have the following minimization for $c$:
$$\widehat{\Phi}(c) = (A Y c, Y c) - 2(f, Y c) = (Y^* A Y c, c) - 2(Y^* f, c).$$
Note that this is the same functional, but for the Galerkin projection of $A$
$$Y^* A Y c = Y^* f,$$
which is an $M \times M$ linear system with symmetric positive definite matrix if $Y$ has full column rank.
But how to choose $Y$?
Selection of the subspace¶
In the Krylov subspace we generate the whole subspace from a single vector $r_0 = f - Ax_0$:
$$y_0\equiv k_0 = r_0, \quad y_1\equiv k_1 = A r_0, \quad y_2\equiv k_2 = A^2 r_0, \ldots, \quad y_{M-1}\equiv k_{M-1} = A^{M-1} r_0.$$
This gives the Krylov subpace of the $M$-th order
$$\mathcal{K}_M(A, r_0) = \mathrm{Span}(r_0, Ar_0, \ldots, A^{M-1} r_0).$$
- It is known to be quasi-optimal space given only matrix-vector product operation.
- Key reference here is "On the numerical solution of equation by which are determined in technical problems the frequencies of small vibrations of material systems", A. N. Krylov, 1931, text in russian
Solution $x_*$ lies in the Krylov subspace: $x_* \in \mathcal{K}_n(A, f)$¶
- According to Cayley–Hamilton theorem: $p(A) = 0$, where $p(\lambda) = \det(A - \lambda I)$
- $p(A)f = A^nf + a_1A^{n-1}f + \ldots + a_{n-1}Af + a_n f = 0$
- $A^{-1}p(A)f = A^{n-1}f + a_1A^{n-2}f + \ldots + a_{n-1}f + a_nA^{-1}f = 0$
- $x_* = A^{-1}f = -\frac{1}{a_n}(A^{n-1}f + a_1A^{n-2}f + \ldots + a_{n-1}f)$
- Thus, $x_* \in \mathcal{K}_n(A, f)$
Ill-conditioned of the natural basis¶
The natural basis in the Krylov subspace is very ill-conditioned, since
$$k_i = A^i r_0 \rightarrow \lambda_\max^i v,$$
where $v$ is the eigenvector, corresponding to the maximal eigenvalue of $A$,
i.e. $k_i$ become more and more collinear for large $i$.
import numpy as np
import matplotlib.pyplot as plt
import scipy.sparse as spsp
%matplotlib inline
n = 100
ex = np.ones(n);
A = spsp.spdiags(np.vstack((-ex, 2*ex, -ex)), [-1, 0, 1], n, n, 'csr');
f = np.ones(n)
x0 = np.random.randn(n)
subspace_order = 10
krylov_vectors = np.zeros((n, subspace_order))
krylov_vectors[:, 0] = f - A.dot(x0)
for i in range(1, subspace_order):
krylov_vectors[:, i] = A.dot(krylov_vectors[:, i-1])
s = np.linalg.svd(krylov_vectors, compute_uv=False)
print("Condition number = {}".format(s.max() / s.min()))
Solution: Compute orthogonal basis in the Krylov subspace.
Good basis in a Krylov subspace¶
In order to have stability, we first orthogonalize the vectors from the Krylov subspace using Gram-Schmidt orthogonalization process (or, QR-factorization).
$$K_j = \begin{bmatrix} r_0 & Ar_0 & A^2 r_0 & \ldots & A^{j-1} r_0\end{bmatrix} = Q_j R_j, $$
and the solution will be approximated as $$x \approx x_0 + Q_j c.$$
Short way to Arnoldi relation¶
Statement. The Krylov matrix $K_j$ satisfies an important recurrent relation (called Arnoldi relation)
$$A Q_j = Q_j H_j + h_{j, j-1} q_j e^{\top}_{j-1},$$
where $H_j$ is upper Hessenberg, and $Q_{j+1} = [q_0,\dots,q_j]$ has orthogonal columns that spans columns of $K_{j+1}$.
Let us prove it (consider $j = 3$ for simplicity):
$$A \begin{bmatrix} k_0 & k_1 & k_2 \end{bmatrix} = \begin{bmatrix} k_1 & k_2 & k_3 \end{bmatrix} = \begin{bmatrix} k_0 & k_1 & k_2 \end{bmatrix} \begin{bmatrix} 0 & 0 & \alpha_0 \\ 1 & 0 & \alpha_1 \\ 0 & 1 & \alpha_2 \\ \end{bmatrix} + \begin{bmatrix} 0 & 0 & k_3 - \alpha_0 k_0 - \alpha_1 k_1 - \alpha_2 k_2 \end{bmatrix}, $$
where $\alpha_s$ will be selected later. Denote $\widehat{k}_3 = k_3 - \alpha_0 k_0 - \alpha_1 k_1 - \alpha_2 k_2$.
In the matrix form,
$$A K_3 = K_3 Z + \widehat k_3 e^{\top}_2,$$
where $Z$ is the lower shift matrix with the last column $(\alpha_0,\alpha_1,\alpha_2)^T$, and $e_2$ is the last column of the identity matrix.
Let $$K_3 = Q_3 R_3$$ be the QR-factorization. Then,
$$A Q_3 R_3 = Q_3 R_3 Z + \widehat{k}_3 e^{\top}_2,$$
$$ A Q_3 = Q_3 R_3 Z R_3^{-1} + \widehat{k}_3 e^{\top}_2 R_3^{-1}.$$
Note that
$$e^{\top}_2 R_3^{-1} = \begin{bmatrix} 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} * & * & * \\ 0 & * & * \\ 0 & 0 & * \end{bmatrix} = \gamma e^{\top}_2,$$ and
$$R_3 Z R_3^{-1} = \begin{bmatrix} * & * & * \\* & * & * \\ 0 & * & * \\ \end{bmatrix},$$
in the general case it will be an upper Hessenberg matrix $H$, i.e. a matrix that
$$H_{ij} = 0, \quad \mbox{if } i > j + 1.$$
(Almost) Arnoldi relation¶
Let $Q_j$ be the orthogonal basis in the Krylov subspace, then we have almost the Arnoldi relation
$$A Q_j = Q_j H_j + \gamma\widehat{k}_j e^{\top}_{j-1},$$
where $H_j$ is an upper Hessenberg matrix, and
$$\widehat{k}_j = k_j - \sum_{s=0}^{j-1} \alpha_s k_s.$$
We select $\alpha_s$ in such a way that
$$Q^*_j \widehat{k}_j = 0.$$
Then, $\widehat{k}_j = h_{j, j-1} q_j,$ where $q_j$ is the last column of $Q_{j+1}$.
Arnoldi relation: final formula¶
We have
$$A Q_j = Q_j H_j + h_{j, j-1} q_j e^{\top}_{j-1}.$$
This is the crucial formula for the efficient generation of such subspaces.
For non-symmetric case, it is just modified Gram-Schmidt.
For the symmetric case, we have a much simpler form (Lanczos process).
Lanczos process¶
If $A = A^*$, then
$$Q^*_j A Q_j = H_j, $$
thus $H_j$ is hermitian, and thus it is tridiagonal, $H_j = T_j$.
This gives a short-term recurrence relation to generate the Arnoldi vectors $q_j$ without full orthogonalization.
Lanczos process (2)¶
$$ A Q_j = Q_j T_j + t_{j, j-1} q_j e^{\top}_{j-1}.$$
In order to get $q_j$, we need to compute just the last column of
$$t_{j, j-1} q_j = (A Q_j - Q_j T_j) e_{j-1} = A q_{j-1} - t_{j-1, j-1} q_{j-1} - t_{j-2, j-1} q_{j-2}. $$
The coefficients $\alpha_j = t_{j-1, j-1}$ and $\beta_j = t_{j-2, j-1}$ can be recovered from orthogonality constraints
$(q_j, q_{j-1}) = 0, \quad (q_j, q_{j-2}) = 0$
All the other constraints will be satisfied automatically!!
And we only need to store two vectors to get the new one.
From direct Lanczos method to the conjugate gradient¶
We can now get from the Lanczos recurrence to the famous conjugate gradient method.
We have for $A = A^* > 0$
$$A Q_j = Q_j T_j + T_{j, j-1} q_j.$$
Recall that when we minimize energy functional in basis $Y$ we get a system $Y^* A Y c = Y^* f,$. Here $Y = Q_j$, so the approximate solution of $Ax \approx f$ with $x_j = x_0 + Q_j c_j$ can be found by solving a small system
$$Q^*_j A Q_j c_j = T_j c_j = Q^*_j r_0 .$$
Since $f$ is the first Krylov subspace, then Note!!! (recall what the first column in $Q_j$ is)
$$Q^*_j r_0 = \Vert r_0 \Vert_2^2 e_0 = \gamma e_0.$$
We have a tridiagonal system of equations for $c$:
$$T_j c_j = \gamma e_0$$
and $x_j = Q_j c_j$.
We could stop at this point, but we want short recurrent formulas instead of solving linear system with matrix $T_j$ at each step.
Derivation of the following update formulas is not required on the oral exam!
Since $A$ is positive definite, $T_j$ is also positive definite, and it allows an LU decomposition
$T_j = L_j U_j$, where $L_j$ is a bidiagonal matrix with ones on the diagonal, $U_j$ is a upper bidiagonal matrix.
We need to define one subdiagonal in $L$ (with elements $c_1, \ldots, c_{j-1}$), main diagonal of $U_j$ (with elements $d_0, \ldots, d_{j-1}$ and superdiagonal of $U_j$ (with elements $b_1, \ldots, b_{j-1}$).
They have convenient recurrences:
$$c_i = b_i/d_{i-1}, \quad d_i = \begin{cases} a_1, & \mbox{if } i = 0, \\ a_i - c_i b_i, & \mbox{if } i > 0. \end{cases}$$
- For the solution we have
$$x_j = Q_j T^{-1}_j \gamma e_0 = \gamma Q_j (L_j U_j)^{-1} e_0 = \gamma Q_j U^{-1}_j L^{-1}_j e_0.$$
- We introduce two new quantities:
$$P_j = Q_j U^{-1}_j, \quad z_j = \gamma L^{-1}_j e_0.$$
- Now we have the following equation for $x_j$:
$$ x_j = P_j z_j$$
- Due to the recurrence relations, we have
$$P_j = \begin{bmatrix} P_{j-1} & p_j \end{bmatrix}, $$
and
$$z_j = \begin{bmatrix} z_{j-1} \\ \xi_{j} \end{bmatrix}.$$
- For $p_j$ and $\xi_j$ we have short-term recurrence relations (due to bidiagonal structure)
$$p_j = \frac{1}{d_j}\left(q_j - b_j p_{j-1} \right), \quad \xi_j = -c_j \xi_{j-1}.$$
- Thus, we arrive at short-term recurrence for $x_j$:
$$x_j = P_j z_j = P_{j-1} z_{j-1} + \xi_j p_j = x_{j-1} + \xi_j p_j.$$
and $q_j$ are found from the Lanczos relation (see slides above).
- This method for solving linear systems is called a direct Lanczos method. It is closely related to the conjugate gradient method.
Direct Lanczos method¶
We have the direct Lanczos method, where we store
$$p_{j-1}, q_j, x_{j-1}$$ to get a new estimate of $x_j$.
The main problem is with $q_j$: we have the three-term recurrence, but in the floating point arithmetic the orthogonality is can be lost, leading to numerical errors.
Let us do some demo.
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
import scipy as sp
import scipy.sparse as spsp
from scipy.sparse import csc_matrix
n = 128
ex = np.ones(n);
A = spsp.spdiags(np.vstack((ex, -2*ex, ex)), [-1, 0, 1], n, n, 'csr');
rhs = np.ones(n)
nit = 64
q1 = rhs/np.linalg.norm(rhs)
q2 = A.dot(q1)
q2 = q2 - np.dot(q2, q1)*q1
q2 = q2/np.linalg.norm(q2)
qall = [q1, q2]
for i in range(nit):
qnew = A.dot(qall[-1])
qnew = qnew - np.dot(qnew, qall[-1])*qall[-1]
qnew = qnew/np.linalg.norm(qnew)
qnew = qnew - np.dot(qnew, qall[-2])*qall[-2]
qnew = qnew/np.linalg.norm(qnew)
qall.append(qnew)
qall_mat = np.vstack(qall).T
print(np.linalg.norm(qall_mat.T.dot(qall_mat) - np.eye(qall_mat.shape[1])))
Conjugate gradient method¶
Instead of $q_j$ (last vector in the modified Gram-Schmidt process), it is more convenient to work with the residual
$$r_j = f - A x_j.$$
The resulting recurrency has the form
$x_j = x_{j-1} + \alpha_{j-1} p_{j-1}$
$r_j = r_{j-1} - \alpha_{j-1} A p_{j-1}$
$p_j = r_j + \beta_j p_{j-1}$.
Hence the name conjugate gradient: to the gradient $r_j$ we add a conjugate direction $p_j$.
We have orthogonality of residuals (check!):
$$(r_i, r_j) = 0, \quad i \ne j$$
and A-orthogonality of conjugate directions (check!):
$$ (A p_i, p_j) = 0,$$
which can be checked from the definition.
The equations for $\alpha_j$ and $\beta_j$ can be now defined explicitly from these two properties.
CG final formulas¶
We have $(r_{j}, r_{j-1}) = 0 = (r_{j-1} - \alpha_{j-1} A r_{j-1}, r_{j-1})$,
thus
$$\alpha_{j-1} = \frac{(r_{j-1}, r_{j-1})}{(A r_{j-1}, r_{j-1})}.$$
In the similar way, we have
$$\beta_{j-1} = \frac{(r_j, r_j)}{(r_{j-1}, r_{j-1})}.$$
Recall that
$x_j = x_{j-1} + \alpha_{j-1} p_{j-1}$
$r_j = r_{j-1} - \alpha_{j-1} A p_{j-1}$
$p_j = r_j + \beta_j p_{j-1}$.
Only one matrix-by-vector product per iteration.
CG derivation overview¶
- Want to find $x_*$ in Krylov subspace
- But natural basis is ill-conditioned, therefore we need orthogonalization
- Derive recurrent equation for sequential orthogonalization of the Krylov subspace basis
- Arnoldi process for non-symmetric matrix
- Lanczos process for symmetrix matrix
- Clever re-writing of these formulas gives short recurrence
Some history¶
More details here: https://www.siam.org/meetings/la09/talks/oleary.pdf
When Hestenes worked on conjugate bases in 1936, he was advised by a Harvard professor that it was too obvious for publication
- CG doesn’t work on slide rules.
- CG has little advantage over Gauss elimination for computation with calculators.
- CG is not well suited for a room of human "computers" – too much data exchange.
Properties of the CG method¶
We need to store 3 vectors.
Since it generates $A$-orthogonal sequence $p_1, \ldots, p_N$, aften $n$ steps it should stop (i.e., $p_{N+1} = 0$.)
In practice it does not have this property in finite precision, thus after its invention in 1952 by Hestens and Stiefel it was labeled unstable.
In fact, it is a brilliant iterative method.
$A$-optimality¶
Energy functional can be written as
$$(Ax, x) - 2(f, x) = (A (x - x_*), (x - x_*)) - (Ax _*, x_*),$$
where $A x_* = f$. Up to a constant factor,
$$ (A(x - x_*), (x -x_*)) = \Vert x - x_* \Vert^2_A$$
is the A-norm of the error.
Convergence¶
The CG method computes $x_k$ that minimizes the energy functional over the Krylov subspace, i.e. $x_k = p(A)f$, where $p$ is a polynomial of degree $k+1$, so
$$\Vert x_k - x_* \Vert_A = \inf\limits_{p} \Vert \left(p(A) - A^{-1}\right) f \Vert_A. $$
Using eigendecomposition of $A$ we have
$$A = U \Lambda U^*, \quad g = U^* f,$$ and
$\Vert x - x_* \Vert^2_A = \displaystyle{\inf_p} \Vert \left(p(\Lambda) - \Lambda^{-1}\right) g \Vert_\Lambda^2 = \displaystyle{\inf_p} \displaystyle{\sum_{i=1}^n} \frac{(\lambda_i p(\lambda_i) - 1)^2 g^2_i}{\lambda_i} = \displaystyle{\inf_{q, q(0) = 1}} \displaystyle{\sum_{i=1}^n} \frac{q(\lambda_i)^2 g^2_i}{\lambda_i} $
Selection of the optimal $q$ depends on the eigenvalue distribution.
Absolute and relative error¶
We have $$\Vert x - x_* \Vert^2_A \leq \sum_{i=1}^n \frac{g^2_i}{\lambda_i} \inf_{q, q(0)=1} \max_{j} q({\lambda_j})^2$$
The first term is just $$\sum_{i=1}^n \frac{g^2_i}{\lambda_i} = (A^{-1} f, f) = \Vert x_* \Vert^2_A.$$
And we have relative error bound
$$\frac{\Vert x - x_* \Vert_A }{\Vert x_* \Vert_A} \leq \inf_{q, q(0)=1} \max_{j} |q({\lambda_j})|,$$
so if matrix has only 2 different eigenvalues, then there exists a polynomial of degree 2 such that $q({\lambda_1}) =q({\lambda_2})=0$, so in this case CG converges in 2 iterations.
If eigenvalues are clustered and there are $l$ outliers, then after first $\mathcal{O}(l)$ iterations CG will converge as if there are no outliers (and hence the effective condition number is smaller).
The intuition behind this fact is that after $\mathcal{O}(l)$ iterations the polynomial has degree more than $l$ and thus is able to zero $l$ outliers.
Let us find another useful upper-bound estimate of convergence. Since
$$ \inf_{q, q(0)=1} \max_{j} |q({\lambda_j})| \leq \inf_{q, q(0)=1} \max_{\lambda\in[\lambda_\min,\lambda_\max]} |q({\lambda})| $$
The last term is just the same as for the Chebyshev acceleration, thus the same upper convergence bound holds:
$$\frac{\Vert x_k - x_* \Vert_A }{\Vert x_* \Vert_A} \leq \gamma \left( \frac{\sqrt{\mathrm{cond}(A)}-1}{\sqrt{\mathrm{cond}(A)}+1}\right)^k.$$
Finite termination & clusters¶
- If $A$ has only $m$ different eigenvalues, CG converges in $m$ iterations (proof in the blackboard).
- If $A$ has $m$ "clusters" of eigenvalues, CG converges cluster-by-cluster.
As a result, better convergence than Chebyshev acceleration, but slightly higher cost per iteration.
Summary¶
CG is the method of choice for symmetric positive definite systems:
- $\mathcal{O}(n)$ memory
- Square root of condition number in the estimates
- Automatic ignoring of the outliers/clusters
- $A$-optimality property
Non-linear conjugate gradient method¶
- CG minimizes the energy functional, which is quadratic in $x$
- CG formulas were used as starting point in developing methods to minimize arbitrary convex function
- Most popular CG extensions (so-called non-linear CG method) are
- Hestenes-Stiefel method
- Polak-Ribiere method - original paper in French
- Fletcher–Reeves method
Non-symmetric systems and the generalized minimal residual method (GMRES) (Y. Saad, M. Schultz, 1986)¶
Before we discussed symmetric positive definite systems. What happens if $A$ is non-symmetric?
We can still orthogonalize the Krylov subspace using Arnoldi process, and get
$$A Q_j = Q_j H_j + h_{j,j-1}q_j e^{\top}_{j-1}.$$
Let us rewrite the latter expression as
$$ A Q_j = Q_j H_j + h_{j,j-1}q_j e^{\top}_{j-1} = Q_{j+1} \widetilde H_j, \quad \widetilde H_j = \begin{bmatrix} h_{0,0} & h_{0,1} & \dots & h_{0,j-2} & h_{0,j-1} \\ h_{1,0} & h_{1,1} & \dots & h_{1,j-2} & h_{1,j-1} \\ 0& h_{2,2} & \dots & h_{2,j-2} & h_{2,j-1} \\ 0& 0 & \ddots & \vdots & \vdots \\ 0& 0 & & h_{j,j-1} & h_{j-1,j-1} \\ 0& 0 & \dots & 0 & h_{j,j-1}\end{bmatrix}$$
Then, if we need to minimize the residual over the Krylov subspace, we have
$$x_j = x_0 + Q_j c_j $$
and $x_j$ has to be selected as
$$ \Vert A x_j - f \Vert_2 = \Vert A Q_j c_j - r_0 \Vert_2 \rightarrow \min_{c_j}.$$
Using the Arnoldi recursion, we have
$$ \Vert Q_{j+1} \widetilde H_j c_j - r_0 \Vert_2 \rightarrow \min_{c_j}.$$
Using the orthogonal invariance under multiplication by unitary matrix, we get
$$ \Vert \widetilde H_j c_j - \gamma e_0 \Vert_2 \rightarrow \min_{c_j},$$
where we have used that $Q^*_{j+1} r_0 = \gamma e_0.$
This is just a linear least squares with $(j+1)$ equations and $j$ unknowns.
The matrix is also upper Hessenberg, thus its QR factorization can be computed in a very cheap way.
This allows the computation of $c_j$. This method is called GMRES (generalized minimal residual)
Summary of the GMRES¶
- Minimizes the residual directly
- No normal equations
- Memory grows with the number of iterations as $\mathcal{O}(j^2)$, so restarts typically implemented (just start GMRES from the new initial guess).
import scipy.sparse.linalg as la
from scipy.sparse import csc_matrix, csr_matrix
import numpy as np
import matplotlib.pyplot as plt
import time
%matplotlib inline
n = 150
ex = np.ones(n);
lp1 = sp.sparse.spdiags(np.vstack((ex, -2*ex, ex)), [-1, 0, 1], n, n, 'csr');
e = sp.sparse.eye(n)
A = sp.sparse.kron(lp1, e) + sp.sparse.kron(e, lp1)
A = csr_matrix(A)
rhs = np.ones(n * n)
plt.figure(figsize=(10, 5))
f, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 5))
for restart in [5, 40, 200]:
hist = []
def callback(rk):
hist.append(np.linalg.norm(rk) / np.linalg.norm(rhs))
st = time.time()
sol = la.gmres(A, rhs, x0=np.zeros(n*n), maxiter=200, restart=restart, callback=callback, tol=1e-16)
current_time = time.time() - st
ax1.semilogy(np.array(hist), label='rst={}'.format(restart))
ax2.semilogy([current_time * i / len(hist) for i in range(len(hist))], np.array(hist), label='rst={}'.format(restart))
ax1.legend(loc='best')
ax2.legend(loc='best')
ax1.set_xlabel("Number of outer iterations", fontsize=20)
ax2.set_xlabel("Time, sec", fontsize=20)
ax1.set_ylabel(r"$\frac{||r_k||_2}{||rhs||_2}$", fontsize=20)
ax2.set_ylabel(r"$\frac{||r_k||_2}{||rhs||_2}$", fontsize=20)
plt.sca(ax1)
plt.yticks(fontsize=20)
plt.sca(ax2)
plt.yticks(fontsize=20)
f.tight_layout()
plt.show()
import scipy.sparse.linalg as la
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
# Example from http://www.caam.rice.edu/~embree/39961.pdf
A = np.array([[1, 1, 1],
[0, 1, 3],
[0, 0, 1]]
)
rhs = np.array([2, -4, 1])
x0 = np.zeros(3)
for restart in [1, 2, 3]:
hist = []
def callback(rk):
hist.append(np.linalg.norm(rk)/np.linalg.norm(rhs))
_ = la.gmres(A, rhs, x0=x0, maxiter=20, restart=restart, callback=callback)
plt.semilogy(np.array(hist), label='rst={}'.format(restart))
plt.legend()
plt.xlabel("Number of outer iterations", fontsize=20)
plt.ylabel(r"$\frac{||r_k||_2}{||rhs||_2}$", fontsize=20)
plt.xticks(fontsize=20)
plt.yticks(fontsize=20)
plt.tight_layout()
Next lecture¶
- Iterative methods continued (BiCG, Minres, QMR), preconditioners.