2025-11-12 Multigrid

2025-11-12 Multigrid#

Last time#

  • Transformed quadrature

  • Singular integrals

  • Adaptive integration

Today#

  • Multigrid

    • Spectral perspective

    • Factorization perspective

    • Algebraic multigrid

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

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])
        end
    end
    sparse(rows, cols, vals)
end

advdiff_matrix (generic function with 1 method)

Geometric multigrid and the spectrum#

Multigrid uses a hierarchy of scales to produce an operation “\(M^{-1}\)” that can be applied in \(O(n)\) (“linear”) time and such that \(\lVert I - M^{-1} A \rVert \le \rho < 1\) independent of the problem size. We’ll consider the one dimensional Laplacian using the stencil

\[ \Big(\frac{h^2}{2}\Big) \frac{1}{h^2} \begin{bmatrix} -1 & 2 & -1 \end{bmatrix} \]

function laplace1d(n)
    "1D Laplacion with Dirichlet boundary conditions eliminated"
    h = 2 / (n + 1)
    x = LinRange(-1, 1, n+2)[2:end-1]
    A = diagm(0 => ones(n),
        -1 => -.5*ones(n-1),
        +1 => -.5*ones(n-1))
    x, A # Hermitian(A)
end

laplace1d (generic function with 1 method)

x, A = laplace1d(100)
#A[1:4,1:4]
L, X = eigen(A)
@show L[1:2]
@show L[end-1:end]
scatter(L, title="cond = $(L[end]/L[1])", legend=:none)

L[1:2] = [0.00048371770801235576, 0.0019344028664056293]
L[end - 1:end] = [1.9980655971335943, 1.999516282291988]

What do the eigenvectors look like?#

x, A = laplace1d(100)
L, X = eigen(A)
plot(x, X[:,1:3], label=L[1:3]', legend=:bottomleft)

plot(x, X[:, end-2:end], label=L[end-2:end]', legend=:bottomleft)

Condition number grows with grid refinement#

ns = 2 .^ (2:8)
eigs = vcat([eigvals(laplace1d(n)[2])[[1, end]] for n in ns]'...)

scatter(ns, eigs, label=["min eig(A)" "max eig(A)"])
plot!(n -> 1/n^2, label="\$1/n^2\$", xscale=:log10, yscale=:log10)

Fourier analysis perspective#

Consider the basis \(\phi(x, \theta) = e^{i \theta x}\). If we choose the grid \(x \in h \mathbb Z\) with grid size \(h\) then we can resolve frequencies \(\lvert \theta \rvert \le \pi/h\).

function symbol(S, theta)
    if length(S) % 2 != 1
        error("Length of stencil must be odd")
    end
    w = length(S) ÷ 2
    phi = exp.(1im * (-w:w) * theta')
    vec(S * phi) # not! (S * phi)'
end

function plot_symbol(S, deriv=2; plot_ref=true, n_theta=30)
    theta = LinRange(-pi, pi, n_theta)
    sym = symbol(S, theta)
    rsym = real.((-1im)^deriv * sym)
    fig = plot(theta, rsym, marker=:circle, label="stencil")
    if plot_ref
        plot!(fig, th -> th^deriv, label="exact")
    end
    fig
end

plot_symbol (generic function with 2 methods)

plot_symbol([1 -2 1])
#plot!(xlims=(-1, 1))

Analytically computing smallest eigenvalue#

The longest wavelength for a domain size of 2 with Dirichlet boundary conditions is 4. The frequency is \(\theta = 2\pi/\lambda = \pi/2\). The symbol function works on an integer grid. We can transform via \(\theta \mapsto \theta h\).

scatter(ns, eigs, label=["min eig(A)" "max eig(A)"])
plot!(n -> 1/n^2, label="\$1/n^2\$", xscale=:log10, yscale=:log10)
theta_min = pi ./ (ns .+ 1)
symbol_min = -real(symbol([1 -2 1], theta_min))
scatter!(ns, symbol_min / 2, shape=:utriangle)

Damping properties of Richardson/Jacobi relaxation#

Recall that we would like for \(I - w A\) to have a small norm, such that powers (repeat iterations) will cause the error to decay rapidly.

x, A = laplace1d(40)
ws = LinRange(.3, 1.001, 50)
radius = [opnorm(I - w*A) for w in ws]
plot(ws, radius, xlabel="w")

  • The spectrum of \(A\) runs from \(\theta_{\min}^2\) up to 2. If \(w > 1\), then \(\lVert I - w A \rVert > 1\) because the operation amplifies the high frequencies (associated with the eigenvalue of 2).

  • The value of \(w\) that minimizes the norm is slightly less than 1, but the convergence rate is very slow (only barely less than 1).

Symbol of damping#

w = .7
plot_symbol([0 1 0] - w * [-.5 1 -.5], 0; plot_ref=false)

  • Evidently it is very difficult to damp low frequencies.

  • This makes sense because \(A\) and \(I - wA\) move information only one grid point per iteration.

  • It also makes sense because the polynomial needs to be 1 at the origin, and the low frequencies have eigenvalues very close to zero.

Coarse grids: Make low frequencies high#

As in domain decomposition, we will express our “coarse” subspace, consisting of a grid \(x \in 2h\mathbb Z\), in terms of its interpolation to the fine space. Here, we’ll use linear interpolation.

function interpolate(m, stride=2)
    s1 = (stride - 1) / stride
    s2 = (stride - 2) / stride
    P = diagm(0 => ones(m),
        -1 => s1*ones(m-1), +1 => s1*ones(m-1),
        -2 => s2*ones(m-2), +2 => s2*ones(m-2))
    P[:, 1:stride:end]
end
n = 100; x, A = laplace1d(n)
P = interpolate(n, 2)
plot(x, P[:, 4:6], marker=:auto, xlims=(-1, 0))

L, X = eigen(A)
u_h = X[:, 20]
u_2h = .5 * P' * u_h
plot(x, [u_h, P * u_2h])

Galerkin approximation of \(A\) in coarse space#

\[ A_{2h} u_{2h} = P^T A_h P u_{2h} \]
x, A = laplace1d(n)
P = interpolate(n)
@show size(A)
A_2h = P' * A * P
@show size(A_2h)
L_2h = eigvals(A_2h)
plot(L_2h, title="cond $(L_2h[end]/L_2h[1])")

size(A) = (100, 100)
size(A_2h) = (50, 50)

my_spy(A_2h)

Coarse grid correction#

Consider the \(A\)-orthogonal projection onto the range of \(P\),

\[ S_c = P A_{2h}^{-1} P^T A \]

Sc = P * (A_2h \ P' * A)
Ls, Xs = eigen(I - Sc)
scatter(real.(Ls))

  • This spectrum is typical for a projector. If \(u\) is in the range of \(P\), then \(S_c u = u\). Why?

  • For all vectors \(v\) that are \(A\)-orthogonal to the range of \(P\), we know that \(S_c v = 0\). Why?

A two-grid method#

w = .66
T = (I - w*A) * (I - Sc) * (I - w*A)
Lt = eigvals(T)
scatter(Lt)

  • Can analyze these methods in terms of frequency.

  • LFAToolkit

Multigrid as factorization#

We can interpret factorization as a particular multigrid or domain decomposition method.

We can partition an SPD matrix as

\[\begin{split}A = \begin{bmatrix} F & B^T \\ B & D \end{bmatrix}\end{split}\]
and define the preconditioner by the factorization
\[\begin{split} M = \begin{bmatrix} I & \\ B F^{-1} & I \end{bmatrix} \begin{bmatrix} F & \\ & S \end{bmatrix} \begin{bmatrix} I & F^{-1}B^T \\ & I \end{bmatrix} \end{split}\]
where \(S = D - B F^{-1} B^T\). \(M\) has an inverse
\[\begin{split} \begin{bmatrix} I & -F^{-1}B^T \\ & I \end{bmatrix} \begin{bmatrix} F^{-1} & \\ & S^{-1} \end{bmatrix} \begin{bmatrix} I & \\ -B F^{-1} & I \end{bmatrix} . \end{split}\]

Define the interpolation

\[\begin{split} P_f = \begin{bmatrix} I \\ 0 \end{bmatrix}, \quad P_c = \begin{bmatrix} -F^{-1} B^T \\ I \end{bmatrix} \end{split}\]
and rewrite as
\[\begin{split} M^{-1} = [P_f\ P_c] \begin{bmatrix} F^{-1} P_f^T \\ S^{-1} P_c^T \end{bmatrix} = P_f F^{-1} P_f^T + P_c S^{-1} P_c^T.\end{split}\]
The iteration matrix is thus
\[ I - M^{-1} A = I - P_f F^{-1} P_f^T J - P_c S^{-1} P_c^T A .\]

Permute into C-F split#

m = 100
x, A = laplace1d(m)
my_spy(A)

idx = vcat(1:2:m, 2:2:m)
J = A[idx, idx]
my_spy(J)

Coarse basis functions#

F = J[1:end÷2, 1:end÷2]
B = J[end÷2+1:end,1:end÷2]
P = [-F\B'; I]
my_spy(P)

idxinv = zeros(Int64, m)
idxinv[idx] = 1:m
Pp = P[idxinv, :]
plot(x, Pp[:, 5:7], marker=:auto, xlims=(-1, -.5))

From factorization to algebraic multigrid#

  • Factorization as a multigrid (or domain decomposition) method incurs significant cost in multiple dimensions due to lack of sparsity.

    • We can’t choose enough coarse basis functions so that \(F\) is diagonal, thereby making the minimal energy extension \(-F^{-1} B^T\) sparse.

  • Algebraic multigrid

    • Use matrix structure to aggregate or define C-points

    • Create an interpolation rule that balances sparsity with minimal energy

Aggregation#

Form aggregates from “strongly connected” dofs.

agg = 1 .+ (0:m-1) .÷ 3
mc = maximum(agg)
T = zeros(m, mc)
for (i, j) in enumerate(agg)
    T[i,j] = 1
end
plot(x, T[:, 3:5], marker=:auto, xlims=(-1, -.5))

Sc = T * ((T' * A * T) \ T') * A
w = .67; k = 3
E = (I - w*A)^k * (I - Sc) * (I - w*A)^k
scatter(eigvals(E), ylims=(-1, 1))

  • simple and cheap method

  • stronger smoothing (bigger k) doesn’t help much; need more accurate coarse grid

Smoothed aggregation#

\[ P = (I - w A) T \]
w = .67
P = (I - w * A) * T
plot(x, P[:, 3:5], marker=:auto, xlims=(-1,-.5))

Sc = P * ((P' * A * P) \ P') * A
w = .67; k = 4
E = (I - w*A)^k * (I - Sc) * (I - w*A)^k
scatter(eigvals(E), ylims=(-1, 1))

  • Eigenvalues are closer to zero; stronger smoothing (larger k) helps.

  • Smoother can be made stronger using Chebyshev (like varying the damping between iterations in Jacobi)

Multigrid in PETSc#

Geometric multigrid#

  • -pc_type mg

    • needs a grid hierarchy (automatic with DM)

    • PCMGSetLevels()

    • PCMGSetInterpolation()

    • PCMGSetRestriction()

  • -pc_mg_cycle_type [v,w]

  • -mg_levels_ksp_type chebyshev

  • -mg_levels_pc_type jacobi

  • -mg_coarse_pc_type svd

    • -mg_coarse_pc_svd_monitor (report singular values/diagnose singular coarse grids)

Algebraic multigrid#

  • -pc_type gamg

  • -pc_type ml

    • similar to gamg with different defaults

  • -pc_type hypre

    • Classical AMG (based on C-F splitting)

    • Manages its own grid hierarchy

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