BiCGStab(ℓ) (#134)

* WIP on BiCGStabl(l)

* Remove unused residual vector

* Export Identity

* Add the BiCGStab(l) test

* Use the convex combination idea

* Switch between convex update and standard behaviour

* Fix for complex numbers & fix ordinary bicgstab(l)

* More testing & benching

* Translate to iterators & remove convex combination for now

* Compatibility with 0.5, and recover the old API with ConvergenceHistry stuff

benchmark/benchmark-linear-systems.jl

@@ -5,6 +5,8 @@ import Base.A_ldiv_B!, Base.\
 using BenchmarkTools
 using IterativeSolvers
 
+include("../test/advection_diffusion.jl")
+
 # A DiagonalMatrix that doesn't check whether it is singular in the \ op.
 immutable DiagonalPreconditioner{T}
     diag::Vector{T}
@@ -64,4 +66,15 @@ function gmres(; n = 100_000, tol = 1e-5, restart::Int = 15, maxiter::Int = 210)
     impr, old
 end
 
+function bicgstabl()
+    A, b = advection_dominated()
+
+    b1 = @benchmark IterativeSolvers.bicgstabl($A, $b, 2, max_mv_products = 1000, convex_combination = false) setup = (srand(1))
+    b2 = @benchmark IterativeSolvers.bicgstabl($A, $b, 2, max_mv_products = 1000, convex_combination = true) setup = (srand(1))
+    b3 = @benchmark IterativeSolvers.bicgstabl($A, $b, 4, max_mv_products = 1000, convex_combination = false) setup = (srand(1))
+    b4 = @benchmark IterativeSolvers.bicgstabl($A, $b, 4, max_mv_products = 1000, convex_combination = true) setup = (srand(1))
+
+    b1, b2, b3, b4
+end
+
 end

src/IterativeSolvers.jl

@@ -21,6 +21,7 @@ include("hessenberg.jl")
 #Linear solvers
 include("stationary.jl")
 include("cg.jl")
+include("bicgstabl.jl")
 include("gmres.jl")
 include("chebyshev.jl")
 include("idrs.jl")

src/bicgstabl.jl

@@ -0,0 +1,254 @@
+export bicgstabl, bicgstabl!, bicgstabl_iterator, bicgstabl_iterator!, BiCGStabIterable
+
+import Base: start, next, done
+
+type BiCGStabIterable{precT, matT, vecT <: AbstractVector, smallMatT <: AbstractMatrix, realT <: Real, scalarT <: Number}
+    A::matT
+    b::vecT
+    l::Int
+
+    x::vecT
+    r_shadow::vecT
+    rs::smallMatT
+    us::smallMatT
+
+    max_mv_products::Int
+    mv_products::Int
+    reltol::realT
+    residual::realT
+
+    Pl::precT
+
+    γ::vecT
+    ω::scalarT
+    σ::scalarT
+    M::smallMatT
+end
+
+bicgstabl_iterator(A, b, l; kwargs...) = bicgstabl_iterator!(zerox(A, b), A, b, l; initial_zero = true, kwargs...)
+
+function bicgstabl_iterator!(x, A, b, l::Int = 2;
+    Pl = Identity(),
+    max_mv_products = min(30, size(A, 1)),
+    initial_zero = false,
+    tol = sqrt(eps(real(eltype(b))))
+)
+    T = eltype(b)
+    n = size(A, 1)
+    mv_products = 0
+
+    # Large vectors.
+    r_shadow = rand(T, n)
+    rs = Matrix{T}(n, l + 1)
+    us = zeros(T, n, l + 1)
+
+    residual = view(rs, :, 1)
+
+    # Compute the initial residual rs[:, 1] = b - A * x
+    # Avoid computing A * 0.
+    if initial_zero
+        copy!(residual, b)
+    else
+        A_mul_B!(residual, A, x)
+        @blas! residual -= one(T) * b
+        @blas! residual *= -one(T)
+        mv_products += 1
+    end
+
+    # Apply the left preconditioner
+    A_ldiv_B!(Pl, residual)
+
+    γ = zeros(T, l)
+    ω = σ = one(T)
+
+    nrm = norm(residual)
+
+    # For the least-squares problem
+    M = zeros(T, l + 1, l + 1)
+
+    # Stopping condition based on relative tolerance.
+    reltol = nrm * tol
+
+    BiCGStabIterable(A, b, l, x, r_shadow, rs, us,
+        max_mv_products, mv_products, reltol, nrm,
+        Pl,
+        γ, ω, σ, M
+    )
+end
+
+@inline converged(it::BiCGStabIterable) = it.residual ≤ it.reltol
+@inline start(::BiCGStabIterable) = 0
+@inline done(it::BiCGStabIterable, iteration::Int) = it.mv_products ≥ it.max_mv_products || converged(it)
+
+function next(it::BiCGStabIterable, iteration::Int)
+    T = eltype(it.b)
+    L = 2 : it.l + 1
+
+    it.σ = -it.ω * it.σ
+
+    ## BiCG part
+    for j = 1 : it.l
+        ρ = dot(it.r_shadow, view(it.rs, :, j))
+        β = ρ / it.σ
+
+        # us[:, 1 : j] .= rs[:, 1 : j] - β * us[:, 1 : j]
+        for i = 1 : j
+            @blas! view(it.us, :, i) *= -β
+            @blas! view(it.us, :, i) += one(T) * view(it.rs, :, i)
+        end
+
+        # us[:, j + 1] = Pl \ (A * us[:, j])
+        next_u = view(it.us, :, j + 1)
+        A_mul_B!(next_u, it.A, view(it.us, :, j))
+        A_ldiv_B!(it.Pl, next_u)
+
+        it.σ = dot(it.r_shadow, next_u)
+        α = ρ / it.σ
+
+        # rs[:, 1 : j] .= rs[:, 1 : j] - α * us[:, 2 : j + 1]
+        for i = 1 : j
+            @blas! view(it.rs, :, i) -= α * view(it.us, :, i + 1)
+        end
+
+        # rs[:, j + 1] = Pl \ (A * rs[:, j])
+        next_r = view(it.rs, :, j + 1)
+        A_mul_B!(next_r, it.A , view(it.rs, :, j))
+        A_ldiv_B!(it.Pl, next_r)
+
+        # x = x + α * us[:, 1]
+        @blas! it.x += α * view(it.us, :, 1)
+    end
+
+    # Bookkeeping
+    it.mv_products += 2 * it.l
+
+    ## MR part
+
+    # M = rs' * rs
+    Ac_mul_B!(it.M, it.rs, it.rs)
+
+    # γ = M[L, L] \ M[L, 1] 
+    F = lufact!(view(it.M, L, L))
+    A_ldiv_B!(it.γ, F, view(it.M, L, 1))
+
+    # This could even be BLAS 3 when combined.
+    BLAS.gemv!('N', -one(T), view(it.us, :, L), it.γ, one(T), view(it.us, :, 1))
+    BLAS.gemv!('N', one(T), view(it.rs, :, 1 : it.l), it.γ, one(T), it.x)
+    BLAS.gemv!('N', -one(T), view(it.rs, :, L), it.γ, one(T), view(it.rs, :, 1))
+
+    it.ω = it.γ[it.l]
+    it.residual = norm(view(it.rs, :, 1))
+
+    it.residual, iteration + 1
+end
+
+# Classical API
+
+bicgstabl(A, b, l = 2; kwargs...) = bicgstabl!(zerox(A, b), A, b, l; initial_zero = true, kwargs...)
+
+function bicgstabl!(x, A, b, l = 2;
+    tol = sqrt(eps(real(eltype(b)))),
+    max_mv_products::Int = min(20, size(A, 1)),
+    log::Bool = false,
+    verbose::Bool = false,
+    Pl = Identity(),
+    kwargs...
+)
+    history = ConvergenceHistory(partial = !log)
+    history[:tol] = tol
+
+    # This doesn't yet make sense: the number of iters is smaller.
+    log && reserve!(history, :resnorm, max_mv_products)
+
+    # Actually perform CG
+    iterable = bicgstabl_iterator!(x, A, b, l; Pl = Pl, tol = tol, max_mv_products = max_mv_products, kwargs...)
+
+    if log
+        history.mvps = iterable.mv_products
+    end
+
+    for (iteration, item) = enumerate(iterable)
+        if log
+            nextiter!(history)
+            history.mvps = iterable.mv_products
+            push!(history, :resnorm, iterable.residual)
+        end
+        verbose && @printf("%3d\t%1.2e\n", iteration, iterable.residual)
+    end
+
+    verbose && println()
+    log && setconv(history, converged(iterable))
+    log && shrink!(history)
+
+    log ? (iterable.x, history) : iterable.x
+end
+
+#################
+# Documentation #
+#################
+
+let
+doc_call = "bicgstab(A, b, l)"
+doc!_call = "bicgstab!(x, A, b, l)"
+
+doc_msg = "Solve A*x = b with the BiCGStab(l)"
+doc!_msg = "Overwrite `x`.\n\n" * doc_msg
+
+doc_arg = ""
+doc!_arg = """`x`: initial guess, overwrite final estimation."""
+
+doc_version = (doc_call, doc_msg, doc_arg)
+doc!_version = (doc!_call, doc!_msg, doc!_arg)
+
+docstring = String[]
+
+#Build docs
+for (call, msg, arg) in (doc_version, doc!_version) #Start
+    push!(docstring, 
+"""
+$call
+
+$msg
+
+# Arguments
+
+$arg
+
+`A`: linear operator.
+
+`b`: right hand side (vector).
+
+`l::Int = 2`: Number of GMRES steps.
+
+## Keywords
+
+`Pl = Identity()`: left preconditioner of the method.
+
+`tol::Real = sqrt(eps(real(eltype(b))))`: tolerance for stopping condition 
+`|r_k| / |r_0| ≤ tol`. Note that:
+
+1. The actual residual is never computed during the iterations; only an 
+approximate residual is used.
+2. If a preconditioner is given, the stopping condition is based on the 
+*preconditioned residual*.
+
+`max_mv_products::Int = min(30, size(A, 1))`: maximum number of matrix
+vector products. For BiCGStab this is a less dubious criterion than maximum
+number of iterations.
+
+# Output
+
+`x`: approximated solution.
+`residual`: last approximate residual norm
+
+# References
+[1] Sleijpen, Gerard LG, and Diederik R. Fokkema. "BiCGstab(l) for 
+linear equations involving unsymmetric matrices with complex spectrum." 
+Electronic Transactions on Numerical Analysis 1.11 (1993): 2000.
+"""
+    )
+end
+
+@doc docstring[1] -> bicgstabl
+@doc docstring[2] -> bicgstabl!
+end

test/advection_diffusion.jl

@@ -0,0 +1,46 @@
+f(x, y, z) = exp.(x .* y .* z) .* sin.(π .* x) .* sin.(π .* y) .* sin.(π .* z)
+
+function advection_dominated(;N = 50, β = 1000.0)
+    # Problem: Δu + βuₓ = f
+    # u = 0 on the boundaries
+    # f(x, y, z) = exp(xyz) sin(πx) sin(πy) sin(πz)
+    # 2nd order central differences (shows serious wiggles)
+
+    # Total number of unknowns
+    n = N^3
+
+    # Mesh width
+    h = 1.0 / (N + 1)
+
+    # Interior points only
+    xs = linspace(0, 1, N + 2)[2 : N + 1]
+
+    # The Laplacian
+    Δ = laplace_matrix(Float64, N, 3) ./ -h^2
+
+    # And the dx bit.
+    ∂x_1d = spdiagm((fill(-β / 2h, N - 1), fill(β / 2h, N - 1)), (-1, 1))
+    ∂x = kron(speye(N^2), ∂x_1d)
+
+    # Final matrix and rhs.
+    A = Δ + ∂x
+    b = reshape([f(x, y, z) for x ∈ xs, y ∈ xs, z ∈ xs], n)
+
+    A, b
+end
+
+function laplace_matrix{T}(::Type{T}, n, dims)
+    D = second_order_central_diff(T, n)
+    A = copy(D)
+
+    for idx = 2 : dims
+        A = kron(A, speye(n)) + kron(speye(size(A, 1)), D)
+    end
+
+    A
+end
+
+second_order_central_diff{T}(::Type{T}, dim) = convert(
+    SparseMatrixCSC{T, Int}, 
+    SymTridiagonal(fill(2 * one(T), dim), fill(-one(T), dim - 1))
+)

test/bicgstabl.jl

@@ -0,0 +1,34 @@
+using IterativeSolvers
+using FactCheck
+using LinearMaps
+
+srand(1234321)
+
+include("advection_diffusion.jl")
+
+facts("bicgstab(l)") do
+
+for T in (Float32, Float64, Complex64, Complex128)
+    context("Matrix{$T}") do
+
+        n = 20
+        A = rand(T, n, n) + 15 * eye(T, n)
+        x = ones(T, n)
+        b = A * x
+
+        for l = (2, 4)
+            context("BiCGStab($l)") do
+
+            # Solve without preconditioner
+            x1, his1 = bicgstabl(A, b, l, max_mv_products = 100, log = true)
+            @fact norm(A * x1 - b) / norm(b) --> less_than(√eps(real(one(T))))
+
+            # Do an exact LU decomp of a nearby matrix
+            F = lufact(A + rand(T, n, n))
+            x2, his2 = bicgstabl(A, b, Pl = F, l, max_mv_products = 100, log = true)
+            @fact norm(A * x2 - b) / norm(b) --> less_than(√eps(real(one(T))))
+            end
+        end
+    end
+end
+end

test/runtests.jl

@@ -13,6 +13,9 @@ include("stationary.jl")
 #Conjugate gradients
 include("cg.jl")
 
+#BiCGStab(l)
+include("bicgstabl.jl")
+
 #GMRES
 include("gmres.jl")