Added example for the amg preconditioner

Author: fverdugo

docs/Manifest.toml

@@ -2,7 +2,7 @@
 
 julia_version = "1.10.0"
 manifest_format = "2.0"
-project_hash = "e37729e95710f39fa1cd840707ef95d2e3bbf22c"
+project_hash = "6cb04e0444d1618368044337280d73d6270dffbd"
 
 [[deps.ANSIColoredPrinters]]
 git-tree-sha1 = "574baf8110975760d391c710b6341da1afa48d8c"
@@ -313,7 +313,13 @@ version = "2.8.1"
 deps = ["CircularArrays", "Distances", "FillArrays", "IterativeSolvers", "LinearAlgebra", "MPI", "Printf", "Random", "SparseArrays", "SparseMatricesCSR"]
 path = ".."
 uuid = "5a9dfac6-5c52-46f7-8278-5e2210713be9"
-version = "0.3.4"
+version = "0.4.3"
+
+[[deps.PartitionedSolvers]]
+deps = ["IterativeSolvers", "LinearAlgebra", "PartitionedArrays", "Random", "SparseArrays"]
+path = "../extensions/PartitionedSolvers"
+uuid = "11b65f7f-80ac-401b-9ef2-3db765482d62"
+version = "0.1.0"
 
 [[deps.Pkg]]
 deps = ["Artifacts", "Dates", "Downloads", "FileWatching", "LibGit2", "Libdl", "Logging", "Markdown", "Printf", "REPL", "Random", "SHA", "Serialization", "TOML", "Tar", "UUIDs", "p7zip_jll"]

docs/Project.toml

@@ -3,3 +3,4 @@ Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4"
 IterativeSolvers = "42fd0dbc-a981-5370-80f2-aaf504508153"
 Literate = "98b081ad-f1c9-55d3-8b20-4c87d4299306"
 PartitionedArrays = "5a9dfac6-5c52-46f7-8278-5e2210713be9"
+PartitionedSolvers = "11b65f7f-80ac-401b-9ef2-3db765482d62"

docs/examples.jl

@@ -208,7 +208,40 @@ psparse!(A,V,cacheA) |> wait
 r = A*x - b
 norm(r)
 
+# # Distributed algebraic multigrid (AMG)
+#
+# So far, we have use a conjugate gradient method to solve the linear system. However, this is approach does not scale well
+# to larger problems and one needs to consider a preconditioner. A distributed algebraic multigrid (AMG) preconditioner is available in `PartitonedSolvers`, an extension
+# package of `PartitionedArrays` that provides parallel solvers for systems built with matrices and vectors from `PartitionedArrays`.
+#
+# First, let us solve a larger problem without a preconditioner. To this end we use function `laplace_matrix` that builds a Laplace matrix of
+# arbitrary size.
+
+using PartitionedArrays: laplace_matrix
+
+nodes_per_dir = (40,40,40)
+parts_per_dir = (2,2,1)
+nparts = prod(parts_per_dir)
+parts = LinearIndices((nparts,))
+A = laplace_matrix(nodes_per_dir,parts_per_dir,parts)
+x_exact = pones(partition(axes(A,2)))
+b = A*x_exact
+
+# Now, we have a matrix and a rhs vector. Let us solve the system:
+
+x = similar(b,axes(A,2))
+x .= 0
+_, history = IterativeSolvers.cg!(x,A,b;log=true)
+history
+
+# Now solve the system while using an AMG preconditioner.
+
+using PartitionedSolvers: amg, preconditioner
 
+x .= 0
+Pl = preconditioner(amg(),x,A,b)
+_, history = IterativeSolvers.cg!(x,A,b;Pl,log=true)
+history