diff --git a/src/Assembly.jl b/src/Assembly.jl
index e0b3374..a5f10c9 100644
--- a/src/Assembly.jl
+++ b/src/Assembly.jl
@@ -262,34 +262,37 @@ function create_from_nz_locally_assembled(
   a,
   callback::Function = rows -> nothing
 )
-  # Recover some data
   I,J,V = get_allocations(a)
-  test_gids, trial_gids = axes(a)
+  test_ids = partition(axes(a,1))
+  trial_ids = partition(axes(a,2))
 
-  rows = remove_ghost(unpermute(test_gids))
+  rows = map(remove_ghost,map(unpermute,test_ids))
   b = callback(rows)
 
-  # convert I and J to global dof ids
-  to_global_indices!(I,test_gids;ax=:rows)
-  to_global_indices!(J,trial_gids;ax=:cols)
+  map(map_local_to_global!,I,test_ids)
+  map(map_local_to_global!,J,trial_ids)
 
-  # Create the range for cols
-  cols = _setup_prange(trial_gids,J;ax=:cols)
+  # TODO: replace_ghost or union_ghost?
+  # Actually, do we want to change the ghosts at all? We could return the unpermuted trial_ids
+  J_owners = find_owner(trial_ids,J)
+  cols = map(replace_ghost,map(unpermute,trial_ids),J,J_owners)
 
-  # Convert again I,J to local numeration
-  to_local_indices!(I,rows;ax=:rows)
-  to_local_indices!(J,cols;ax=:cols)
+  map(map_global_to_local!,I,rows)
+  map(map_global_to_local!,J,cols)
+
+  assembled = true
+  a_sys = change_axes(a,(PRange(rows),PRange(cols)))
+  values = map(create_from_nz,local_views(a_sys))
+  A = PSparseMatrix(values,rows,cols,assembled)
 
-  A = _setup_matrix(I,J,V,rows,cols)
   return A, b
 end
 
 function create_from_nz_assembled(
-  a, 
+  a,
   callback::Function = rows -> nothing,
-  async_callback::Function = b -> nothing
+  async_callback::Function = b -> empty_async_task
 )
-  # Recover some data
   I,J,V = get_allocations(a)
   test_ids = partition(axes(a,1))
   trial_ids = partition(axes(a,2))
@@ -298,36 +301,44 @@ function create_from_nz_assembled(
   map(map_local_to_global!,I,test_ids)
   map(map_local_to_global!,J,trial_ids)
 
-  # Move (I,J,V) triplets to the owner process of each row I.
+  # Overlapped COO communication and vector assembly
   rows = map(unpermute,test_ids)
-  # rows = replace_ghost(unpermute(test_gids),I,find_owner(test_gids,I))
   t = PartitionedArrays.assemble_coo!(I,J,V,rows)
-
-  # Overlap CSC communication with rhs assembly
   b = callback(rows)
-
-  # Wait the transfer to finish
   wait(t)
 
-  # Create the Prange for the cols
-  J_owners = find_owner(trial_ids,J)
-  cols = map(union_ghost,map(unpermute,trial_ids),J,J_owners)
-
   # Overlap rhs communications with CSC compression
   t2 = async_callback(b)
+  J_owners = find_owner(trial_ids,J)
+  cols = map(replace_ghost,map(unpermute,trial_ids),J,J_owners) # TODO: replace_ghost or union_ghost?
 
-  # Convert again I,J to local numeration
   map(map_global_to_local!,I,rows)
   map(map_global_to_local!,J,cols)
 
+  assembled = true
   a_sys = change_axes(a,(PRange(rows),PRange(cols)))
   values = map(create_from_nz,local_views(a_sys))
-  A = PSparseMatrix(values,rows,cols,true)
+  A = PSparseMatrix(values,rows,cols,assembled)
 
-  # Wait the transfer to finish
-  if !isa(t2,Nothing)
-    wait(t2)
-  end
+  wait(t2)
+  return A, b
+end
+
+function create_from_nz_subassembled(
+  a,
+  callback::Function = rows -> nothing,
+  async_callback::Function = b -> empty_async_task
+)
+  rows = partition(axes(a,1))
+  cols = partition(axes(a,2))
+
+  b = callback(rows)
+  t2 = async_callback(b)
+
+  assembled = false
+  values = map(create_from_nz,local_views(a))
+  A = PSparseMatrix(values,rows,cols,assembled)
 
+  wait(t2)
   return A, b
 end
diff --git a/src/PArraysExtras.jl b/src/PArraysExtras.jl
index be90762..4d87e2b 100644
--- a/src/PArraysExtras.jl
+++ b/src/PArraysExtras.jl
@@ -111,6 +111,10 @@ function SparseArrays.findnz(A::PartitionedArrays.SubSparseMatrix)
   return I[mask], J[mask], V[mask]
 end
 
+# Async tasks
+
+const empty_async_task = PartitionedArrays.FakeTask(x -> nothing)
+
 # Linear algebra
 
 function LinearAlgebra.axpy!(α,x::PVector,y::PVector)