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test(assemblers): add incomplete Cholesky preconditioner tests
This commit is contained in:
Jukka Aho
@@ -0,0 +1,361 @@
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# This file is a part of JuliaFEM.
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# License is MIT: see https://github.com/JuliaFEM/JuliaFEM.jl/blob/master/LICENSE.md
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"""
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`ICholPreconditioner` (IC(0)) tests.
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Locks in the contract of the no-fill incomplete-Cholesky preconditioner
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added in B+++:
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1. **Algebraic correctness** on small dense / tridiagonal SPD matrices:
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`L * L' ≈ K` exactly when the IC(0) sparsity pattern is full
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(dense lower triangle, tridiagonals, etc.); when fill is dropped,
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it merely approximates `K`.
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2. **Constructor robustness**: the diagonal-shift retry path produces
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a valid factor on a near-indefinite matrix without corrupting the
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input `K`.
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3. **`ldiv!(y, P, x)` is the exact inverse of `L L'`** — i.e. the
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two triangular solves are applied correctly.
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4. **PCG with IC(0) converges in ≤ scalar-Jacobi iterations** on
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stiffer elasticity problems (the canonical benchmark for IC(0)).
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5. **End-to-end matrix-free PenaltyDirichlet pipeline** through the
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`(cache, asm, kernel, mesh; dirichlet)` factory: IC(0) of the
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constrained `K` plus `matrix_free_op` reproduces the assembled
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direct solve.
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6. **Zero allocations on `ldiv!`** (the hot path).
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"""
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using Test
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using JuliaFEM
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using JuliaFEM: ContinuumFormulation, FullThreeD, Vertex
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using JuliaFEM: @DOFSet, DOF
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using JuliaFEM: LinearElastic, Displacement, ContinuumKernel
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using JuliaFEM: DOFBasedCOOAssembler, DOFBasedCOOCache
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using JuliaFEM: extract_system, apply_K!
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using JuliaFEM: PenaltyDirichlet, EliminatedDirichlet, apply_constraint!
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using JuliaFEM: matrix_free_op
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using JuliaFEM: JacobiPreconditioner, ICholPreconditioner
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using JuliaFEM: UniformBodyForce, apply_load!
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using JuliaFEM: create_elements!
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using LinearAlgebra
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using SparseArrays
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using Tensors
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using Random
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# ---------------------------------------------------------------------------
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# Mesh + setup helpers (same shape as test_block_jacobi.jl)
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# ---------------------------------------------------------------------------
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function _hex8_box(nx::Int, ny::Int, nz::Int;
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Lx::Float64 = 1.0, Ly::Float64 = 1.0, Lz::Float64 = 1.0)
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nodes = Vec{3,Float64}[]
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nidx(i, j, k) = (i - 1) + (j - 1) * (nx + 1) + (k - 1) * (nx + 1) * (ny + 1) + 1
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for k in 1:(nz + 1), j in 1:(ny + 1), i in 1:(nx + 1)
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push!(nodes, Vec{3}((Lx * Float64(i - 1) / nx,
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Ly * Float64(j - 1) / ny,
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Lz * Float64(k - 1) / nz)))
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end
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conns = NTuple{8,UInt32}[]
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for k in 1:nz, j in 1:ny, i in 1:nx
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n1 = nidx(i, j, k)
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n2 = nidx(i + 1, j, k)
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n3 = nidx(i + 1, j + 1, k)
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n4 = nidx(i, j + 1, k)
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n5 = nidx(i, j, k + 1)
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n6 = nidx(i + 1, j, k + 1)
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n7 = nidx(i + 1, j + 1, k + 1)
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n8 = nidx(i, j + 1, k + 1)
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push!(conns, (UInt32(n1), UInt32(n2), UInt32(n3), UInt32(n4),
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UInt32(n5), UInt32(n6), UInt32(n7), UInt32(n8)))
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end
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return Mesh{8,Hexahedron{8}}(nodes, conns)
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end
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function _setup_elasticity(mesh)
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material = LinearElastic(E = 210e9, ν = 0.3)
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kernel = ContinuumKernel(ContinuumFormulation{FullThreeD}(),
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material, Displacement{3}())
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S = @DOFSet{u::DOF{Displacement{3}, Vertex}}
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elements, dof_mgr = create_elements!(mesh, Element{Hexahedron{8}, Lagrange{1}, S})
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asm = DOFBasedCOOAssembler()
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cache = DOFBasedCOOCache(elements, dof_mgr, mesh, kernel)
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return cache, asm, kernel, mesh
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end
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# ---------------------------------------------------------------------------
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# 1. Algebraic correctness: L * L' ≈ K on representative SPD matrices
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# ---------------------------------------------------------------------------
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@testset "ICholPreconditioner: L * L' ≈ K (algebraic correctness)" begin
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println("\n" * "=" ^ 70)
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println("IC(0) — algebraic correctness")
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println("=" ^ 70)
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@testset "1D Laplacian (tridiagonal — IC(0) is exact)" begin
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for n in (5, 10, 50)
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K = spdiagm(-1 => -ones(n - 1),
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0 => 2.0 * ones(n),
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1 => -ones(n - 1))
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P = ICholPreconditioner(K)
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Kr = Matrix(P.L) * Matrix(P.L)'
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err = norm(Kr - Matrix(K)) / norm(Matrix(K))
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@test err < 1e-12
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println(" n=$n Laplacian ‖L Lᵀ - K‖/‖K‖ = $(round(err; sigdigits=3))")
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end
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end
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@testset "Random dense SPD (full lower-triangle pattern)" begin
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Random.seed!(20260508)
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for n in (10, 20, 50)
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M = randn(n, n)
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K = sparse(M * M' + 5.0 * LinearAlgebra.I)
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P = ICholPreconditioner(K)
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Kr = Matrix(P.L) * Matrix(P.L)'
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err = norm(Kr - Matrix(K)) / norm(Matrix(K))
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@test err < 1e-10
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println(" n=$n dense SPD ‖L Lᵀ - K‖/‖K‖ = $(round(err; sigdigits=3))")
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end
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end
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@testset "Sparse SPD with dropped fill (IC(0) is approximate)" begin
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Random.seed!(20260508)
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n = 60
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# Banded SPD with bandwidth ~5; off-band entries inside the
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# band trigger fill in exact Cholesky → IC(0) drops it.
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K = spdiagm(0 => 4.0 * ones(n))
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for d in 1:5
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v = -randn(n - d) ./ d
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K += spdiagm(d => v, -d => v)
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end
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# Make safely SPD.
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K += (2.0 * abs(minimum(eigvals(Matrix(K))))) * sparse(LinearAlgebra.I, n, n)
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@test issymmetric(K)
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@test minimum(eigvals(Matrix(K))) > 0
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P = ICholPreconditioner(K)
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Kr = Matrix(P.L) * Matrix(P.L)'
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err = norm(Kr - Matrix(K)) / norm(Matrix(K))
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# Approximate, so just bound modestly.
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@test err < 5e-1
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# Sparsity preserved.
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@test nnz(P.L) == nnz(LowerTriangular(K))
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println(" banded SPD n=$n err=$(round(err; sigdigits=3)) " *
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"nnz(L)=$(nnz(P.L)) (= nnz(tril(K)))")
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end
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end
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# ---------------------------------------------------------------------------
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# 2. Constructor robustness: diagonal-shift retry, non-aliasing of input
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# ---------------------------------------------------------------------------
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@testset "ICholPreconditioner: diagonal-shift retry + input preservation" begin
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println("\n" * "=" ^ 70)
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println("IC(0) — diagonal-shift retry + non-aliasing")
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println("=" ^ 70)
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Random.seed!(20260508)
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n = 30
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# Construct a matrix that is SPD but with a near-zero eigenvalue
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# — IC(0) on its lower triangle may produce a non-positive
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# diagonal mid-factorisation. The shift retry should rescue it.
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M = randn(n, n)
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K0 = M * M'
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# Slightly perturb diagonal downwards to provoke breakdown.
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K0 -= (0.95 * minimum(eigvals(K0))) * Matrix(LinearAlgebra.I, n, n)
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K = sparse(K0)
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Korig = copy(K) # snapshot for non-aliasing test
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P = ICholPreconditioner(K) # must NOT throw
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# Input preserved (no aliasing leaked through):
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@test K.nzval == Korig.nzval
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@test K.rowval == Korig.rowval
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@test K.colptr == Korig.colptr
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# Factor is at least defined on every diagonal (no NaN / Inf).
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L = P.L
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@test all(isfinite, nonzeros(L))
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diagL = [L[i, i] for i in 1:n]
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@test all(>(0), diagL)
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println(" n=$n retried IC(0) succeeded min(diag(L))=" *
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"$(round(minimum(diagL); sigdigits=3))")
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end
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# ---------------------------------------------------------------------------
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# 3. ldiv!(y, P, x) is the exact (L * L')^{-1} action
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# ---------------------------------------------------------------------------
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@testset "ICholPreconditioner: ldiv! agrees with (L * L')^{-1} * x" begin
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Random.seed!(20260508)
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n = 25
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M = randn(n, n)
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K = sparse(M * M' + 2.0 * LinearAlgebra.I)
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P = ICholPreconditioner(K)
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L = Matrix(P.L)
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for trial in 1:3
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x = randn(n)
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y = zeros(n)
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ldiv!(y, P, x)
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ref = (L * L') \ x
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rel = norm(y - ref) / max(norm(ref), 1.0)
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@test rel < 1e-9
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end
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end
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# ---------------------------------------------------------------------------
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# 4. PCG with IC(0) ≤ scalar-Jacobi iterations on a stiff elasticity
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# cantilever (the canonical benchmark for IC(0)).
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# ---------------------------------------------------------------------------
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@testset "IC(0): ≤ scalar-Jacobi iterations on elasticity cantilever" begin
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using IterativeSolvers: cg!
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using LinearOperators: LinearOperator
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println("\n" * "=" ^ 70)
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println("IC(0) vs SCALAR JACOBI — CG iteration count")
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println("=" ^ 70)
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nx, ny, nz = 12, 3, 3
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mesh = _hex8_box(nx, ny, nz; Lx = 3.0, Ly = 0.5, Lz = 0.5)
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cache, asm, kernel, m = _setup_elasticity(mesh)
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n = cache.ndofs
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nodes = m.nodes
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tol = 1e-9
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# Cantilever fixed at x=0, gravity body force.
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fixed_dofs = Int[]; fixed_vals = Float64[]
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for i in 1:length(nodes)
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base = 3 * (i - 1)
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if nodes[i][1] < tol
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for α in 1:3
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push!(fixed_dofs, base + α); push!(fixed_vals, 0.0)
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end
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end
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end
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bc = EliminatedDirichlet(fixed_dofs, fixed_vals)
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b = Vec{3,Float64}((0.0, 0.0, -7850.0 * 9.81))
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rhs = zeros(n); apply_load!(rhs, UniformBodyForce(b), cache, asm, kernel, m)
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apply_constraint!(rhs, bc)
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op = matrix_free_op(cache, asm, kernel, m; dirichlet = bc)
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linop = LinearOperator(Float64, n, n, true, true, op)
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# Scalar Jacobi
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P_jac = JacobiPreconditioner(cache, asm, kernel, m; dirichlet = bc)
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u_j = zeros(n)
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h_j = cg!(u_j, linop, rhs; Pl = P_jac, abstol = 0.0, reltol = 1e-10,
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maxiter = 4 * n, log = true)
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# IC(0)
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P_ic = ICholPreconditioner(cache, asm, kernel, m; dirichlet = bc)
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u_i = zeros(n)
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h_i = cg!(u_i, linop, rhs; Pl = P_ic, abstol = 0.0, reltol = 1e-10,
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maxiter = 4 * n, log = true)
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iters_j = h_j[2].iters
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iters_i = h_i[2].iters
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@test isapprox(u_j, u_i; atol = 1e-6, rtol = 1e-6)
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@test iters_i <= iters_j # IC(0) ≤ scalar-Jacobi (often much smaller)
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@test maximum(abs, u_i) > 1e-15
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println(" cantilever nx=$nx ny=$ny nz=$nz ndof=$n " *
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"scalar-Jac iters=$iters_j IC(0) iters=$iters_i")
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end
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# ---------------------------------------------------------------------------
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# 5. End-to-end PenaltyDirichlet matrix-free PCG with IC(0) factory
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# ---------------------------------------------------------------------------
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@testset "IC(0): PenaltyDirichlet inhomogeneous CG matches direct solve" begin
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using IterativeSolvers: cg!
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using LinearOperators: LinearOperator
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println("\n" * "=" ^ 70)
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println("IC(0) — PenaltyDirichlet end-to-end CG")
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println("=" ^ 70)
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nx, ny, nz = 4, 1, 1
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mesh = _hex8_box(nx, ny, nz; Lx = 1.0, Ly = 0.1, Lz = 0.1)
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cache, asm, kernel, m = _setup_elasticity(mesh)
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n = cache.ndofs
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nodes = m.nodes
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tol = 1e-9
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# Same prescribed displacement as test_block_jacobi.jl (penalty form):
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# u(x=0) = 0, u_x(x=1) = 0.01.
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fixed_dofs = Int[]; fixed_vals = Float64[]
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for i in 1:length(nodes)
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x = nodes[i][1]
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base = 3 * (i - 1)
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if x < tol
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for α in 1:3
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push!(fixed_dofs, base + α); push!(fixed_vals, 0.0)
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end
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elseif x > 1.0 - tol
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push!(fixed_dofs, base + 1); push!(fixed_vals, 0.01)
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end
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end
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bc_pen = PenaltyDirichlet(fixed_dofs, fixed_vals; penalty = 1e10)
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# Direct reference
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assemble!(cache, asm, kernel, m)
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K, _ = extract_system(cache)
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Kbc = Matrix(K)
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apply_constraint!(Kbc, bc_pen)
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bbc = zeros(n); apply_constraint!(bbc, bc_pen)
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u_dir = Kbc \ bbc
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# Matrix-free CG with IC(0)
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op = matrix_free_op(cache, asm, kernel, m; dirichlet = bc_pen)
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linop = LinearOperator(Float64, n, n, true, true, op)
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P = ICholPreconditioner(cache, asm, kernel, m; dirichlet = bc_pen)
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u_mf = zeros(n)
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h = cg!(u_mf, linop, bbc; Pl = P, abstol = 1e-12, reltol = 1e-12,
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maxiter = 4 * n, log = true)
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rel = norm(u_mf - u_dir) / max(norm(u_dir), 1.0)
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@test rel < 1e-6
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@test isapprox(u_mf[3 * nx + 1], 0.01; atol = 5e-3)
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println(" Elast bar nx=$nx ndof=$n fixed=$(length(fixed_dofs)) " *
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"iters=$(h[2].iters) rel(u_mf vs u_dir)=" *
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"$(round(rel; sigdigits = 3)) u_x(x=1)=" *
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"$(round(u_mf[3 * nx + 1]; sigdigits = 4)) (penalty offset expected)")
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end
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# ---------------------------------------------------------------------------
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# 6. Zero allocations on the hot path (ldiv!)
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# ---------------------------------------------------------------------------
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@testset "IC(0): zero allocations on ldiv!" begin
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println("\n" * "=" ^ 70)
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println("IC(0) — ZERO-ALLOC on ldiv!")
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println("=" ^ 70)
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@testset "ldiv!(y, P, x) on $(nx)×$(ny)×$(nz)" for (nx, ny, nz) in
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[(1, 1, 1), (2, 1, 1), (3, 2, 2)]
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mesh = _hex8_box(nx, ny, nz)
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cache, asm, kernel, m = _setup_elasticity(mesh)
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n_dof = cache.ndofs
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P = ICholPreconditioner(cache, asm, kernel, m)
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x = randn(n_dof); y = zeros(n_dof)
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ldiv!(y, P, x)
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GC.gc()
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a = @allocated ldiv!(y, P, x)
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@test a == 0
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println(" $(nx)×$(ny)×$(nz) ndof=$n_dof ldiv!=$a")
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end
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end
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