2022-10-12 Krylov and preconditioning#

Last time#

  • Classical iterative methods

  • Concept of preconditioning


  • Krylov methods (focus on GMRES)

  • PETSc experiments

  • Simple preconditioners

using Plots
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))

function advdiff_matrix(n; kappa=1, wind=[0, 0])
    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
    stencil = stencil_advect + stencil_diffuse
    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)]
            mask = in_domain.(neighbors)
            append!(rows, idx.(repeat([(i,j)], 5))[mask])
            append!(cols, idx.(neighbors)[mask])
            append!(vals, stencil[mask])
    sparse(rows, cols, vals)
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

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


\[ A Q_n = Q_{n+1} H_n \]

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 \(H_n\).


  • 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

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


-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


  • 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.

Conjugate Gradients: changing the norm#

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.