Extended the jacobian to functionals involving Skeleton terms

NEWS.md

@@ -7,8 +7,9 @@ and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0
 ## Unreleased
 
 ### Added
-- Added functionality to take gradient of functional involving integration (`DomainContribution`) over Skeleton faces, with respect to the degrees-of-freedom `FEFunction`.  The interface remains the same - `gradient(f,uh)`. Since PR [#797](https://github.com/gridap/Gridap.jl/pull/797)
+- Added functionality to take gradient of functional involving integration (`DomainContribution`) over Skeleton faces, with respect to the degrees-of-freedom of `FEFunction`.  The interface remains the same - `gradient(f,uh)`. Since PR [#797](https://github.com/gridap/Gridap.jl/pull/797)
 - Extended the `MultiField` functional gradient (with respect to degrees-of-freedom of `MultiFieldFEFunction`) to functionals involving Skeleton integration. The interface remains the same `gradient(f,xh)`. Since PR [#799](https://github.com/gridap/Gridap.jl/pull/799)
+- Added functionality to take jacobian of functional involving integration (`DomainContribution`) over Skeleton faces (obtained from testing bilinear form with the whole set of test `fe_basis`), with respect to the degrees-of-freedom of `FEFunction`.  The interface remains the same - `jacobian(f,uh)`. Since PR [#803](https://github.com/gridap/Gridap.jl/pull/803)
 
 ### Fixed
 - Fixed the behavior of `gradient` for functionals involving operations of `CellFields` inside `mean` and `jump` terms of Skeleton Integration terms. Since PR [#800](https://github.com/gridap/Gridap.jl/pull/800)

src/FESpaces/FEAutodiff.jl

@@ -165,9 +165,7 @@ end
   Gridap, where we can use or import required functionalities from other
   modules without any circular dependency
 =#
-function autodiff_array_gradient(
-  a, i_to_x, j_to_i::SkeletonPair)
-
+function autodiff_array_gradient(a, i_to_x, j_to_i::SkeletonPair)
   i_to_xdual = lazy_map(DualizeMap(ForwardDiff.gradient),i_to_x)
 
   # dual output of both sides at once
@@ -191,12 +189,56 @@ function autodiff_array_gradient(
   lazy_map(BlockMap(2,[1,2]),j_to_result_plus,j_to_result_minus)
 end
 
-function autodiff_array_jacobian(
-  a,i_to_x,j_to_i::SkeletonPair)
-  @notimplemented
+function autodiff_array_jacobian(a,i_to_x,j_to_i::SkeletonPair)
+  i_to_xdual = lazy_map(DualizeMap(ForwardDiff.jacobian),i_to_x)
+  j_to_ydual_plus, j_to_ydual_minus = a(i_to_xdual)
+
+  # Bilinear form when tested with test basis functions dv results in
+  # DomainContribution containing vector of 2-block `VectorBlock{Vector}`
+  # each block coming from plus side and minus side of dv.
+  # So we densify each of the `VectorBlock`` into plain vectors and construct
+  # back the jacobian contributions into a MatrixBlock
+
+  densify = DensifyInnerMostBlockLevelMap()
+  j_to_ydual_plus_dense  = lazy_map(densify, j_to_ydual_plus)
+  j_to_ydual_minus_dense = lazy_map(densify, j_to_ydual_minus)
+
+  # Work for plus side
+  j_to_x_plus = lazy_map(Reindex(i_to_x),j_to_i.plus)
+  j_to_cfg_plus = lazy_map(ConfigMap(ForwardDiff.jacobian),j_to_x_plus)
+  j_to_result_plus_dense = lazy_map(AutoDiffMap(ForwardDiff.jacobian),
+                                    j_to_ydual_plus_dense,j_to_x_plus,j_to_cfg_plus)
+
+  # Work for minus side
+  j_to_x_minus = lazy_map(Reindex(i_to_x),j_to_i.minus)
+  j_to_cfg_minus = lazy_map(ConfigMap(ForwardDiff.jacobian),j_to_x_minus)
+  j_to_result_minus_dense = lazy_map(AutoDiffMap(ForwardDiff.jacobian),
+                                     j_to_ydual_minus_dense,j_to_x_minus,j_to_cfg_minus)
+
+  # j_to_result_plus_dense/j_to_result_minus_dense can be (and must be)
+  # laid out into 2x2 block matrices
+  num_dofs_x_cell = lazy_map(length,i_to_x)
+  num_dofs_x_face_and_cell_plus  = lazy_map(Reindex(num_dofs_x_cell),j_to_i.plus)
+  num_dofs_x_face_and_cell_minus = lazy_map(Reindex(num_dofs_x_cell),j_to_i.minus)
+
+  J_11 = lazy_map((x,b)->view(x, 1:b,:),
+                  j_to_result_plus_dense,num_dofs_x_face_and_cell_plus)
+
+  J_21 = lazy_map((x,b)->view(x, b+1:size(x,1),:),
+                  j_to_result_plus_dense,num_dofs_x_face_and_cell_minus)
+
+  J_12 = lazy_map((x,b)->view(x, 1:b,:),
+                  j_to_result_minus_dense,num_dofs_x_face_and_cell_plus)
+
+  J_22 = lazy_map((x,b)->view(x, b+1:size(x,1),:),
+                  j_to_result_minus_dense,num_dofs_x_face_and_cell_minus)
+
+  # Assembly on SkeletonTriangulation expects an array of facets where each
+  # entry is a 2x2-block MatrixBlock with the blocks of the Jacobian matrix
+  bm_jacobian = BlockMap((2,2),[(1,1),(2,1),(1,2),(2,2)])
+  lazy_map(bm_jacobian, J_11, J_21, J_12, J_22)
 end
 
-function autodiff_array_hessian(
-  a,i_to_x,i_to_j::SkeletonPair)
+function autodiff_array_hessian(a,i_to_x,i_to_j::SkeletonPair)
   @notimplemented
 end

test/FESpacesTests/FEAutodiffTests.jl

@@ -11,6 +11,7 @@ using Gridap.TensorValues
 using Gridap.CellData
 using Gridap.ReferenceFEs
 using ForwardDiff
+using SparseArrays
 
 domain = (0,1,0,1)
 partition = (2,2)
@@ -94,9 +95,9 @@ cell_j_auto = get_array(jacobian(u->res(u,dv),uh))
 
 test_array(cell_j_auto,cell_j,≈)
 
-# comparing AD of integration over Skeleton faces with ForwardDiff results
+## comparing AD of integration over Skeleton faces with ForwardDiff results ##
 
-model = CartesianDiscreteModel((0.,1.,0.,1.),(3,3))
+model = CartesianDiscreteModel((0.,1.,0.,1.),(2,2))
 Ω = Triangulation(model)
 Γ = BoundaryTriangulation(model)
 Λ = SkeletonTriangulation(model)
@@ -108,7 +109,7 @@ dΛ = Measure(Λ,2)
 n_Γ = get_normal_vector(Γ)
 n_Λ = get_normal_vector(Λ)
 
-reffe = ReferenceFE(lagrangian,Float64,2)
+reffe = ReferenceFE(lagrangian,Float64,1)
 V = TestFESpace(model,reffe,conformity=:L2)
 
 u(x) = sin(norm(x))
@@ -118,7 +119,7 @@ uh = FEFunction(U,rand(num_free_dofs(U)))
 
 f_Λ(uh) = ∫(mean(uh))*dΛ
 a_Λ(u) = ∫( - jump(u*n_Λ)⊙mean(∇(u))
-            - mean(∇(u))⊙jump(u*n_Λ)
+            - mean(Δ(u))
             + jump(u*n_Λ)⊙jump(u*n_Λ) )dΛ
 
 # functionals having mean and jump of products of FEFunctions/CellFields
@@ -138,6 +139,8 @@ f_Λ_(θ) = f_uh_free_dofs(f_Λ,uh,θ)
 a_Λ_(θ) = f_uh_free_dofs(a_Λ,uh,θ)
 θ = get_free_dof_values(uh)
 
+# testing gradients of functionals involving Skeleton integration terms
+
 gridapgradf = assemble_vector(gradient(f_Λ,uh),U)
 fdgradf = ForwardDiff.gradient(f_Λ_,θ)
 test_array(gridapgradf,fdgradf,≈)
@@ -162,4 +165,75 @@ gridapgradj = assemble_vector(gradient(j_Λ,uh),U)
 fdgradj = ForwardDiff.gradient(j_Λ_,θ)
 test_array(gridapgradj,fdgradj,≈)
 
+# testing jacobians of functionals involving Skeleton inetgration terms
+uh = FEFunction(U,rand(num_free_dofs(U)))
+θ = get_free_dof_values(uh)
+dv = get_fe_basis(V)
+du = get_trial_fe_basis(U)
+
+a(uh,vh) = ∫(mean(uh)*mean(vh))*dΛ
+# b(uh,vh) = ∫(-jump(uh*n_Λ)⊙mean(∇(vh)) +
+#               jump(vh*n_Λ)⊙mean(∇(uh)) +
+#               mean(Δ(uh))*mean(Δ(vh))  +
+#               jump(uh*n_Λ)⊙jump(vh*n_Λ))*dΛ
+# g(uh,vh) = ∫(mean(uh*vh))*dΛ
+h(uh,vh) = ∫(jump(uh*vh*n_Λ)⊙jump(vh*vh*n_Λ))*dΛ
+# j(uh,vh) = ∫(mean(∇(vh)*uh)⊙jump((∇(vh)⊙∇(uh))*n_Λ))*dΛ
+
+# We use the strongly-typed lowel-level interface of SparseMatrixCSC{T,Int} here
+# as ForwardDiff doesn't work directly through `assemble_vector``, this is due
+# to the typing of SparseMatrixCSC{T,Int} and Vector{T} inside high-level
+# `assemble_vector` API (via the `SparseMatrixAssembler`) using the type T of
+# the eltype of FESpace U, which remains Float even though Dual numbers are
+# passed into θ.
+# So as to mitigate this problem, low-level interface of assemble_vector is
+# being used with SparseMatrixCSC{T,Int} and Vector{T} constructed by hand with
+# types of θ which can be dual numbers when using ForwardDiff or even #
+# ReverseDiff types, when using ReverseDiff! This is just for testing purposes
+function _change_input(f,θ,uh)
+  dir = similar(uh.dirichlet_values,eltype(θ))
+  U = uh.fe_space
+  uh = FEFunction(U,θ,dir)
+  T = eltype(θ)
+  matrix_type = SparseMatrixCSC{T,Int}
+  vector_type = Vector{T}
+  assem = SparseMatrixAssembler(matrix_type,vector_type,U,U)
+  assemble_vector(f(uh),assem,U)
+end
+
+function _assemble_jacobian(f,uh)
+  assemble_matrix(jacobian(f,uh),U,V)
+end
+
+f(uh) = a(uh,dv)
+jac_gridap_a = _assemble_jacobian(f,uh)
+collect(get_array(jacobian(f,uh))) # just to check the working
+f_(θ) = _change_input(f,θ,uh)
+jac_forwdiff_a = ForwardDiff.jacobian(f_,θ)
+test_array(jac_gridap_a,jac_forwdiff_a,≈)
+
+# f(uh) = b(uh,dv)
+# jac_gridap_b = _assemble_jacobian(f,uh)
+# f_(θ) = _change_input(f,θ,uh)
+# jac_forwdiff_b = ForwardDiff.jacobian(f_,θ)
+# test_array(jac_gridap_b,jac_forwdiff_b,≈)
+
+# f(uh) = g(uh,dv)
+# jac_gridap_g = _assemble_jacobian(f,uh)
+# f_(θ) = _change_input(f,θ,uh)
+# jac_forwdiff_g = ForwardDiff.jacobian(f_,θ)
+# test_array(jac_gridap_g,jac_forwdiff_g,≈)
+
+f(uh) = h(uh,dv)
+jac_gridap_h = _assemble_jacobian(f,uh)
+f_(θ) = _change_input(f,θ,uh)
+jac_forwdiff_h = ForwardDiff.jacobian(f_,θ)
+test_array(jac_gridap_h,jac_forwdiff_h,≈)
+
+# f(uh) = j(uh,dv)
+# jac_gridap_j = _assemble_jacobian(f,uh)
+# f_(θ) = _change_input(f,θ,uh)
+# jac_forwdiff_j = ForwardDiff.jacobian(f_,θ)
+# test_array(jac_gridap_j,jac_forwdiff_j,≈)
+
 end # module