2022-10-10 Iterative Solvers#

Last time#

  • Request RC accounts

  • Sparse direct solvers

    • impact of ordering on formulation

    • cost scaling


  • Classical iterative methods

  • Concept of preconditioning

  • Krylov methods (focus on GMRES)

  • PETSc experiments

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)

Request an RC account#


  • This gives ssh login access. We’ll use Alpine, which is a modern CPU and GPU cluster.

  • This will be good for benchmarking and larger runs. We’ll also use GPUs later in class.


  • We’ll start using PETSc this week.

  • You can build PETSc on your laptop. You’ll need C development tools.

    • Linux: use package manager (apt install, dnf install, …)

    • OSX: install xcode, many people like homebrew as a package manager

    • Windows: WSL then follow Linux instructions (or MSYS2 or Cygwin)

    • Any: install Docker, then use our image


How expensive how fast?#

Suppose we have a second order accurate method in 3D.

n = 2. .^ (2:13)
N = n.^3
error = (50 ./ n) .^ 2
seconds = 1e-10 * N.^2
hours = seconds / 3600
cloud_dollars = 3 * hours
kW_hours = 0.2 * hours
barrel_of_oil = kW_hours / 1700
kg_CO2 = kW_hours * 0.709
cost = cloud_dollars
plot(cost, error, xlabel="cost", ylabel="error", xscale=:log10, yscale=:log10)

Outlook on sparse direct solvers#

  • Sparse direct works well for 2D and almost-2D problems to medium large sizes

    • High order FD methods make sparse direct cry

    • High order finite element are okay, but not high-continuity splines

    • Almost-2D includes a lot of industrial solid mechanics applications

      • The body of a car, the frame of an airplane

  • Sparse direct is rarely usable in “fully 3D” problems

    • “thick” structures

      • soil mechanics, hydrology, building foundations, bones, tires

    • fluid mechanics

      • aerodynamics, heating/cooling systems, atmosphere/ocean

  • Setup cost (factorization) is much more expensive than solve

    • Amortize cost in time-dependent problems

      • Rosenbrock methods: factorization reused across stages

      • “lag” Jacobian in Newton (results in “modified Newton”)

      • “lag” preconditioner with matrix-free iterative methods (Sundials, PETSc)

    • Factorization pays off if you have many right hand sides

Why iterative solvers over direct solvers?#

  • Less reliable, more leaky abstraction

  • More sensitive to problem formulation

  • Slower for small problems

  • Several different strategies, each with tuning knobs

  • Accuracy tolerances needed

\(O(N)\) solvers available for many important problems#

  • High-order discretization can be okay

Gradient descent#

Suppose \(A\) is a symmetric positive definite matrix and consider the scalar functional

\[f(u) = \frac 1 2 u^T A u - b^T u . \]

Then the gradient is

\[\nabla_u f = A u - b .\]
x = LinRange(-4, 4, 40)
A = [2 -1; -1 8]
b = [1, 1] 
f(u) = .5 * u' * A * u - b' * u
contour(x, x, (u1, u2) -> f([u1, u2]), aspect_ratio=:equal)

Aside: Derivative of a dot product#

Let \(f(\boldsymbol x) = \boldsymbol y^T \boldsymbol x = \sum_i y_i x_i\) and compute the derivative

\[ \frac{\partial f}{\partial \boldsymbol x} = \begin{bmatrix} y_0 & y_1 & \dotsb \end{bmatrix} = \boldsymbol y^T . \]

Note that \(\boldsymbol y^T \boldsymbol x = \boldsymbol x^T \boldsymbol y\) and we have the product rule,

\[ \frac{\partial \lVert \boldsymbol x \rVert^2}{\partial \boldsymbol x} = \frac{\partial \boldsymbol x^T \boldsymbol x}{\partial \boldsymbol x} = 2 \boldsymbol x^T . \]


\[ \frac{\partial \lVert \boldsymbol x - \boldsymbol y \rVert^2}{\partial \boldsymbol x} = \frac{\partial (\boldsymbol x - \boldsymbol y)^T (\boldsymbol x - \boldsymbol y)}{\partial \boldsymbol x} = 2 (\boldsymbol x - \boldsymbol y)^T .\]

Aside: Variational notation#

It’s convenient to express derivatives in terms of how they act on an infinitessimal perturbation. So we might write

\[ \delta f = \frac{\partial f}{\partial x} \delta x .\]

(It’s common to use \(\delta x\) or \(dx\) for these infinitesimals.) This makes inner products look like a normal product rule

\[ \delta(\mathbf x^T \mathbf y) = (\delta \mathbf x)^T \mathbf y + \mathbf x^T (\delta \mathbf y). \]

A powerful example of variational notation is differentiating a matrix inverse

\[ 0 = \delta I = \delta(A^{-1} A) = (\delta A^{-1}) A + A^{-1} (\delta A) \]

and thus

\[ \delta A^{-1} = - A^{-1} (\delta A) A^{-1} \]

Try gradient descent#

\[ u_{k+1} = u_k - \omega \nabla_u f \]
function grad_descent(loss, grad, u0; omega=1e-3, tol=1e-5)
    """Minimize loss(c) via gradient descent with initial guess u0
    using learning rate gamma.  Declares convergence when gradient
    is less than tol or after 500 steps.
    u = copy(u0)
    uhist = [copy(u)]
    lhist = [loss(u)]
    for it in 1:500
        g = grad(u)
        u -= omega * g
        push!(uhist, copy(u))
        push!(lhist, loss(u))
        if norm(g) < tol
    (u, hcat(uhist...), lhist)
grad_descent (generic function with 1 method)

Visualize gradient descent#

A = [1 1; 1 30]
loss(u) = .5 * u' * A * u
grad(u) = A * u
u, uhist, lhist = grad_descent(loss, grad, [.9, .9],
plot(lhist, yscale=:log10)
plot(uhist[1, :], uhist[2, :], marker=:circle)
x = LinRange(-1, 1, 30)
contour!(x, x, (x,y) -> loss([x, y]))

Richardson iteration#

The simplest iterative method is Richardson’s method, which solves \(A u = b\) by the iteration

\[ u_{k+1} = u_k + \omega (b - A u_k) \]
where \(\omega > 0\) is a damping parameter and \(u_0\) is an initial guess (possibly the zero vector).

  • Algebraically equivalent to gradient descent when \(A\) is SPD

  • Non-symmetric matrices are harder to visualize

Richardson convergence#

\[ u_{k+1} = u_k + \omega (b - A u_k) \]

If \(b = A u_*\), this iteration is equivalent to

(38)#\[\begin{align} u_{k+1} - u_* &= (u_k - u_*) - \omega A (u_k - u_*) \\ &= (I - \omega A) (u_k - u_*) . \end{align}\]

It is convenient for convergence analysis to identify the “error” \(e_k = u_k - u_*\), in which this becomes

\[ e_{k+1} = (I - \omega A) e_k \]
\[ e_k = (I - \omega A)^k e_0 \]
in terms of the initial error. Evidently powers of the iteration matrix \(I - \omega A\) tell the whole story.

Suppose that the eigendecomposition

\[ X \Lambda X^{-1} = I - \omega A \]
exists. Then
\[ (I - \omega A)^k = (X \Lambda X^{-1})^k = X \Lambda^k X^{-1} \]
and the convergence (or divergence) rate depends only on the largest magnitude eigenvalue. This analysis isn’t great for two reasons:

  1. Not all matrices are diagonalizable.

  2. The matrix \(X\) may be very ill-conditioned.

Aside: Schur decomposition#

We can repair these weaknesses by using the Schur decomposition

\[ Q R Q^h = I - \omega A \]
where \(R\) is right-triangular and \(Q\) is unitary (i.e., orthogonal if real-valued; \(Q^h\) is the Hermitian conjugate of \(Q\)). The Schur decomposition always exists and \(Q\) has a condition number of 1.

  • Where are the eigenvalues in \(R\)?

Evidently we must find \(\omega\) to minimize the maximum eigenvalue of \(I - \omega A\). We can do this if \(A\) is well conditioned, but not in general.


Question: What is the condition number of the Laplacian on 100 evenly spaced points?#

  • How does it scale under grid refinement?

A = advdiff_matrix(20)
omega = .002
ev = eigvals(Matrix(I - omega * A))
scatter(real.(ev), imag.(ev), xlim=(-2, 2), ylim=(-1, 1))

Monic polynomials small on the spectrum#

Equivalently to finding \(\omega\) such that \(\lVert I - \omega A \rVert\) is minimized, we may seek a monic polynomial \(p(z) = 1 - \omega z\) that minimizes

\[ \max_{\lambda \in \sigma(A)} \lvert p(\lambda) \rvert . \]

This concept can be extended to higher degree polynomials, which is essentially what Krylov methods do (discovering the polynomial adaptively, weighted by the right hand side).

ev = eigvals(Matrix(A))
scatter(real.(ev), zero.(ev))
plot!(x -> (1 - omega * x)^5)


Preconditioning is the act of creating an “affordable” operation “\(P^{-1}\)” such that \(P^{-1} A\) (or \(A P^{-1}\)) is is well-conditoned or otherwise has a “nice” spectrum. We then solve the system

\[ P^{-1} A x = P^{-1} b \quad \text{or}\quad A P^{-1} \underbrace{(P x)}_y = b \]

in which case the convergence rate depends on the spectrum of the iteration matrix

\[ I - \omega P^{-1} A . \]

  • The preconditioner must be applied on each iteration.

  • It is not merely about finding a good initial guess.

There are two complementary techniques necessary for efficient iterative methods:

  • “accelerators” or Krylov methods, which use orthogonality to adaptively converge faster than Richardson

  • preconditioners that improve the spectrum of the preconditioned operator

Although there is ongoing research in Krylov methods and they are immensely useful, I would say preconditioning is 90% of the game for practical applications, particularly as a research area.

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.