2023-10-18 Krylov and preconditioning#

Last time#

• Classical iterative methods

• Concept of preconditioning

Today#

• Krylov methods (focus on GMRES)

• PETSc experiments

• Simple preconditioners

using Plots
default(linewidth=3)
using LinearAlgebra
using SparseArrays

function my_spy(A)
cmax = norm(vec(A), Inf)
s = max(1, ceil(120 / size(A, 1)))
spy(A, marker=(:square, s), c=:diverging_rainbow_bgymr_45_85_c67_n256, clims=(-cmax, cmax))
end

h = 2 / (n + 1)
rows = Vector{Int64}()
cols = Vector{Int64}()
vals = Vector{Float64}()
idx((i, j),) = (i-1)*n + j
in_domain((i, j),) = 1 <= i <= n && 1 <= j <= n
stencil_advect = [-wind[1], -wind[2], 0, wind[1], wind[2]] / h
stencil_diffuse = [-1, -1, 4, -1, -1] * kappa / h^2
for i in 1:n
for j in 1:n
neighbors = [(i-1, j), (i, j-1), (i, j), (i+1, j), (i, j+1)]
end
end
sparse(rows, cols, vals)
end

advdiff_matrix (generic function with 1 method)


Krylov subspaces#

All matrix iterations work with approximations in a Krylov subspace, which has the form

$K_n = \big[ b \big| Ab \big| A^2 b \big| \dotsm \big| A^{n-1} b \big] .$

This matrix is horribly ill-conditioned and cannot stably be computed as written. Instead, we seek an orthogonal basis $$Q_n$$ that spans the same space as $$K_n$$. We could write this as a factorization

$K_n = Q_n R_n$

where the first column $$q_0 = b / \lVert b \rVert$$. The $$R_n$$ is unnecessary and hopelessly ill-conditioned, so a slightly different procedure is used.

Arnoldi iteration#

The Arnoldi iteration applies orthogonal similarity transformations to reduce $$A$$ to Hessenberg form, starting from a vector $$q_0 = b$$,

$A = Q H Q^* .$

Let’s multiply on the right by $$Q$$ and examine the first $$n$$ columns,

$A Q_n = Q_{n+1} H_n$

where $$H_n$$ is an $$(n+1) \times n$$ Hessenberg matrix.

Aside: Conditioning of Arnoldi process#

The Arnoldi process is well-conditioned because $$Q$$ is orthogonal and

\begin{align}\begin{aligned} \lVert H_n \rVert \le \lVert Q_{n+1}^* \rVert \lVert A \rVert \lVert Q_n \rVert \le \lVert A \rVert .\\For a lower bound, we have\\ \sigma_{\min}(A)^2 \le x^* A^* A x \end{aligned}\end{align}

for all $$x$$ of norm 1. It must also be true for any $$x = Q_n y$$ where $$\lVert y\rVert = 1$$, thus

$\sigma_{\min}(A)^2 \le y^* Q_n^* A^* A Q_n y = y^* H_n^* Q_{n+1}^* Q_{n+1} H_n y = y^* H_n^* H_n y .$

GMRES#

$A Q_n = Q_{n+1} H_n$

GMRES (Generalized Minimum Residual) minimizes

$\lVert A x - b \rVert$
over the subspace $$Q_n$$. I.e., $$x = Q_n y$$ for some $$y$$. By the Arnoldi recurrence, this is equivalent to
$\lVert Q_{n+1} H_n y - b \lVert$
which can be solved by minimizing
$\lVert H_n y - Q_{n+1}^* b \rVert .$
Since $$q_0 = b/\lVert b \lVert$$, the least squares problem is to minimize
$\Big\lVert H_n y - \lVert b \rVert e_0 \Big\rVert .$
The solution of this least squares problem is achieved by incrementally updating a $$QR$$ factorization of $$H_n$$.

Notes#

• The solution $$x_n$$ constructed by GMRES at iteration $$n$$ is not explicitly available. If a solution is needed, it must be constructed by solving the $$(n+1)\times n$$ least squares problem and forming the solution as a linear combination of the $$n$$ vectors $$Q_n$$. The leading cost is $$2mn$$ where $$m \gg n$$.

• The residual vector $$r_n = A x_n - b$$ is not explicitly available in GMRES. To compute it, first build the solution $$x_n$$ as above.

• GMRES minimizes the 2-norm of the residual $$\lVert r_n \rVert$$ which is equivalent to the $$A^* A$$ norm of the error $$\lVert x_n - x_* \rVert_{A^* A}$$.

More notes on GMRES#

• GMRES needs to store the full $$Q_n$$, which is unaffordable for large $$n$$ (many iterations). The standard solution is to choose a “restart” $$k$$ and to discard $$Q_n$$ and start over with an initial guess $$x_k$$ after each $$k$$ iterations. This algorithm is called GMRES(k). PETSc’s default solver is GMRES(30) and the restart can be controlled using the run-time option -ksp_gmres_restart.

• Most implementations of GMRES use classical Gram-Schmidt because it is much faster in parallel (one reduction per iteration instead of $$n$$ reductions per iteration). The PETSc option -ksp_gmres_modifiedgramschmidt can be used when you suspect that classical Gram-Schmidt may be causing instability.

• There is a very similar (and older) algorithm called GCR that maintains $$x_n$$ and $$r_n$$. This is useful, for example, if a convergence tolerance needs to inspect individual entries. GCR requires $$2n$$ vectors instead of $$n$$ vectors, and can tolerate a nonlinear preconditioner. FGMRES is a newer algorithm with similar properties to GCR.

Experiments in PETSc#

• PETSc = Portable Extensible Toolkit for Scientific computing

• ./configure, make

• Depends on BLAS and LAPACK (default system or package manager)

• or ./configure --download-f2cblaslapack --download-blis

• Depends on MPI for parallelism (package manager)

• or ./configure --download-mpich (or --download-openmpi)

• docker run -it --rm jedbrown/petsc

• Lots of examples (mostly C and Fortran, some Python)

• Experimental bindings in Rust, Julia

• We’ll use src/snes/tutorials/ex15.c

• “p-Bratu”: combines p-Laplacian with Bratu nonlinearity

$-\nabla\cdot\big(\kappa(\nabla u) \nabla u\big) - \lambda e^u = f$

• Newton solver with Krylov on each iteration

Simple preconditioners#

Jacobi -pc_type jacobi#

$P_{\text{Jacobi}}^{-1} = D^{-1}$

where $$D$$ is the diagonal of $$A$$.

Gauss-Seidel -pc_type sor#

$P_{GS}^{-1} = (L+D)^{-1}$

where $$L$$ is the (strictly) lower triangular part of $$A$$. The upper triangular part may be used instead, or a symmetric form

$P_{SGS}^{-1} = (L+U)^{-1} A \Big( I - (L+D)^{-1} \Big) .$

Over-relaxation#

-pc_sor_omega 1. is default (Gauss-Seidel)

Run p-Laplacian example#

$cd src/snes/tutorials$ make ex15
$./ex15 -da_refine 2 -dm_view$ ./ex15 -ksp_monitor -pc_type jacobi
\$ ./ex15 -snes_view


Experiments#

• How does iteration count vary under grid refinement?

• How sensitive is it to parameters

• p-Laplacian -p $$> 1$$ and -epsilon $$> 0$$

• Bratu -lambda $$< 6.8$$

• How sensitive to -ksp_gmres_restart?

• -ksp_monitor_true_residual

• -ksp_view_eigenvalues

Symmetric problems#

Lanczos iteration: like GMRES for symmetric problems#

If $$A$$ is symmetric, then $$H = Q^T A Q$$ is also symmetric. Since $$H$$ is Hessenberg, this means $$H$$ is tridiagonal. Instead of storing $$Q_n$$, it is sufficient to store only the last two columns since the iteration satisfies a 3-term recurrence. The analog of GMRES for the symmetric case is called MINRES and is also useful for solving symmetric indefinite problems.

Instead of minimizing the $$A^T A$$ norm of the error, the Conjugate Gradient method minimizes the $$A$$ norm of the error. For $$A$$ to induce a norm, it must be symmetric positive definite. Jeremy Shewchuck’s guide to CG is an excellent resource.