Make orthogonalization method chooseable for gmres (#293)

Author: jkrch

docs/src/linear_systems/gmres.md

@@ -13,9 +13,9 @@ gmres!
 
 The implementation pre-allocates a matrix $V$ of size `n` by `restart` whose columns form an orthonormal basis for the Krylov subspace. This allows BLAS2 operations when updating the solution vector $x$. The Hessenberg matrix is also pre-allocated.
 
-Modified Gram-Schmidt is used to orthogonalize the columns of $V$.
+By default, modified Gram-Schmidt is used to orthogonalize the columns of $V$, since it is numerically more stable than classical Gram-Schmidt. Modified Gram-Schmidt is however inherently sequential, and if stability is not a concern, classical Gram-Schmidt can be used, which is implemented using BLAS2 operations. As a compromise the "DGKS criterion" can be used, which conditionally applies classical Gram-Schmidt repeatedly to stabilize it, and is typically one to two times slower than classical Gram-Schmidt. 
 
 The computation of the residual norm is implemented in a non-standard way, namely keeping track of a vector $\gamma$ in the null-space of $H_k^*$, which is the adjoint of the $(k + 1) \times k$ Hessenberg matrix $H_k$ at the $k$th iteration. Only when $x$ needs to be updated is the Hessenberg matrix mutated with Givens rotations.
 
 !!! tip
-    GMRES can be used as an [iterator](@ref Iterators). This makes it possible to access the Hessenberg matrix and Krylov basis vectors during the iterations.
+    GMRES can be used as an [iterator](@ref Iterators). This makes it possible to access the Hessenberg matrix and Krylov basis vectors during the iterations.

src/gmres.jl

@@ -28,7 +28,7 @@ Residual(order, T::Type) = Residual{T, real(T)}(
     one(real(T))
 )
 
-mutable struct GMRESIterable{preclT, precrT, solT, rhsT, vecT, arnoldiT <: ArnoldiDecomp, residualT <: Residual, resT <: Real}
+mutable struct GMRESIterable{preclT, precrT, solT, rhsT, vecT, arnoldiT <: ArnoldiDecomp, residualT <: Residual, resT <: Real, orthmethT}
     Pl::preclT
     Pr::precrT
     x::solT
@@ -44,6 +44,8 @@ mutable struct GMRESIterable{preclT, precrT, solT, rhsT, vecT, arnoldiT <: Arnol
     maxiter::Int
     tol::resT
     β::resT
+
+    orth_meth::orthmethT
 end
 
 converged(g::GMRESIterable) = g.residual.current ≤ g.tol
@@ -66,7 +68,8 @@ function iterate(g::GMRESIterable, iteration::Int=start(g))
     g.arnoldi.H[g.k + 1, g.k] = orthogonalize_and_normalize!(
         view(g.arnoldi.V, :, 1 : g.k),
         view(g.arnoldi.V, :, g.k + 1),
-        view(g.arnoldi.H, 1 : g.k, g.k)
+        view(g.arnoldi.H, 1 : g.k, g.k),
+        g.orth_meth
     )
 
     # Implicitly computes the residual
@@ -109,7 +112,8 @@ function gmres_iterable!(x, A, b;
                          reltol::Real = sqrt(eps(real(eltype(b)))),
                          restart::Int = min(20, size(A, 2)),
                          maxiter::Int = size(A, 2),
-                         initially_zero::Bool = false)
+                         initially_zero::Bool = false,
+                         orth_meth::OrthogonalizationMethod = ModifiedGramSchmidt())
     T = eltype(x)
 
     # Approximate solution
@@ -126,7 +130,8 @@ function gmres_iterable!(x, A, b;
 
     GMRESIterable(Pl, Pr, x, b, Ax,
         arnoldi, residual,
-        mv_products, restart, 1, maxiter, tolerance, residual.current
+        mv_products, restart, 1, maxiter, tolerance, residual.current,
+        orth_meth
     )
 end
 
@@ -163,6 +168,7 @@ Solves the problem ``Ax = b`` with restarted GMRES.
 - `Pr`: right preconditioner;
 - `log::Bool`: keep track of the residual norm in each iteration;
 - `verbose::Bool`: print convergence information during the iterations.
+- `orth_meth::OrthogonalizationMethod = ModifiedGramSchmidt()`: orthogonalization method (ModifiedGramSchmidt(), ClassicalGramSchmidt(), DGKS())
 
 # Return values
 
@@ -184,15 +190,17 @@ function gmres!(x, A, b;
                 maxiter::Int = size(A, 2),
                 log::Bool = false,
                 initially_zero::Bool = false,
-                verbose::Bool = false)
+                verbose::Bool = false,
+                orth_meth::OrthogonalizationMethod = ModifiedGramSchmidt())
     history = ConvergenceHistory(partial = !log, restart = restart)
     history[:abstol] = abstol
     history[:reltol] = reltol
     log && reserve!(history, :resnorm, maxiter)
 
     iterable = gmres_iterable!(x, A, b; Pl = Pl, Pr = Pr,
                                abstol = abstol, reltol = reltol, maxiter = maxiter,
-                               restart = restart, initially_zero = initially_zero)
+                               restart = restart, initially_zero = initially_zero,
+                               orth_meth = orth_meth)
 
     verbose && @printf("=== gmres ===\n%4s\t%4s\t%7s\n","rest","iter","resnorm")
 

src/orthogonalize.jl

@@ -7,9 +7,10 @@ struct ClassicalGramSchmidt <: OrthogonalizationMethod end
 struct ModifiedGramSchmidt <: OrthogonalizationMethod end
 
 # Default to MGS, good enough for solving linear systems.
-@inline orthogonalize_and_normalize!(V::StridedMatrix{T}, w::StridedVector{T}, h::StridedVector{T}) where {T} = orthogonalize_and_normalize!(V, w, h, ModifiedGramSchmidt)
+@inline orthogonalize_and_normalize!(V::StridedMatrix{T}, w::StridedVector{T}, h::StridedVector{T}) where {T} = 
+	orthogonalize_and_normalize!(V, w, h, ModifiedGramSchmidt())
 
-function orthogonalize_and_normalize!(V::StridedMatrix{T}, w::StridedVector{T}, h::StridedVector{T}, ::Type{DGKS}) where {T}
+function orthogonalize_and_normalize!(V::StridedMatrix{T}, w::StridedVector{T}, h::StridedVector{T}, ::DGKS) where {T}
     # Orthogonalize using BLAS-2 ops
     mul!(h, adjoint(V), w)
     mul!(w, V, h, -one(T), one(T))
@@ -37,7 +38,7 @@ function orthogonalize_and_normalize!(V::StridedMatrix{T}, w::StridedVector{T},
     nrm
 end
 
-function orthogonalize_and_normalize!(V::StridedMatrix{T}, w::StridedVector{T}, h::StridedVector{T}, ::Type{ClassicalGramSchmidt}) where {T}
+function orthogonalize_and_normalize!(V::StridedMatrix{T}, w::StridedVector{T}, h::StridedVector{T}, ::ClassicalGramSchmidt) where {T}
     # Orthogonalize using BLAS-2 ops
     mul!(h, adjoint(V), w)
     mul!(w, V, h, -one(T), one(T))
@@ -49,7 +50,7 @@ function orthogonalize_and_normalize!(V::StridedMatrix{T}, w::StridedVector{T},
     nrm
 end
 
-function orthogonalize_and_normalize!(V::StridedVector{Vector{T}}, w::StridedVector{T}, h::StridedVector{T}, ::Type{ModifiedGramSchmidt}) where {T}
+function orthogonalize_and_normalize!(V::StridedVector{Vector{T}}, w::StridedVector{T}, h::StridedVector{T}, ::ModifiedGramSchmidt) where {T}
     # Orthogonalize using BLAS-1 ops
     for i = 1 : length(V)
         h[i] = dot(V[i], w)
@@ -63,7 +64,7 @@ function orthogonalize_and_normalize!(V::StridedVector{Vector{T}}, w::StridedVec
     nrm
 end
 
-function orthogonalize_and_normalize!(V::StridedMatrix{T}, w::StridedVector{T}, h::StridedVector{T}, ::Type{ModifiedGramSchmidt}) where {T}
+function orthogonalize_and_normalize!(V::StridedMatrix{T}, w::StridedVector{T}, h::StridedVector{T}, ::ModifiedGramSchmidt) where {T}
     # Orthogonalize using BLAS-1 ops and column views.
     for i = 1 : size(V, 2)
         column = view(V, :, i)

test/orthogonalize.jl

@@ -34,7 +34,7 @@ m = 3
     end
 
     # Assuming V is a matrix
-    @testset "Using $method" for method = (DGKS, ClassicalGramSchmidt, ModifiedGramSchmidt)
+    @testset "Using $method" for method = (DGKS(), ClassicalGramSchmidt(), ModifiedGramSchmidt())
 
         # Projection size
         h = zeros(T, m)
@@ -55,7 +55,7 @@ m = 3
 
         # Orthogonalize w in-place
         w = copy(w_original)
-        nrm = orthogonalize_and_normalize!(V_vec, w, h, ModifiedGramSchmidt)
+        nrm = orthogonalize_and_normalize!(V_vec, w, h, ModifiedGramSchmidt())
 
         is_orthonormalized(w, h, nrm)
     end