diff --git a/src/ad_api.jl b/src/ad_api.jl
index c9867b25..a142f93a 100644
--- a/src/ad_api.jl
+++ b/src/ad_api.jl
@@ -82,13 +82,18 @@ end
 
 """
     get_residual_nnzj(b::ADModelBackend, nvar, nequ)
+    get_residual_nnzj(nls::AbstractNLSModel, nvar, nequ)
 
-Return `get_nln_nnzj(b.jacobian_residual_backend, nvar, nequ)`.
+Return the number of nonzeros elements in the residual Jacobians.
 """
 function get_residual_nnzj(b::ADModelBackend, nvar, nequ)
   get_nln_nnzj(b.jacobian_residual_backend, nvar, nequ)
 end
 
+function get_residual_nnzj(nls::AbstractNLSModel, nvar, nequ)
+  nls.nls_meta.nnzj
+end
+
 function get_residual_nnzj(
   b::ADModelBackend{GB, HvB, JvB, JtvB, JB, HB, GHJ, HvBLS, JvBLS, JtvBLS, JBLS, HBLS},
   nvar,
@@ -118,6 +123,27 @@ function get_nln_nnzh(nlp::AbstractNLPModel, nvar)
   nlp.meta.nnzh
 end
 
+"""
+    get_residual_nnzh(b::ADModelBackend, nvar)
+    get_residual_nnzh(nls::AbstractNLSModel, nvar)
+
+Return the number of nonzeros elements in the residual Hessians.
+"""
+function get_residual_nnzh(b::ADModelBackend, nvar)
+  get_nln_nnzh(b.hessian_residual_backend, nvar)
+end
+
+function get_residual_nnzh(nls::AbstractNLSModel, nvar)
+  nls.nls_meta.nnzh
+end
+
+function get_residual_nnzh(
+  b::ADModelBackend{GB, HvB, JvB, JtvB, JB, HB, GHJ, HvBLS, JvBLS, JtvBLS, JBLS, HBLS},
+  nvar) where {GB, HvB, JvB, JtvB, JB, HB, GHJ, HvBLS, JvBLS, JtvBLS, JBLS, HBLS <: AbstractNLPModel}
+  nls = b.hessian_residual_backend
+  nls.nls_meta.nnzh
+end
+
 throw_error(b) =
   throw(ArgumentError("The AD backend $b is not loaded. Please load the corresponding AD package."))
 gradient(b::ADBackend, ::Any, ::Any) = throw_error(b)
@@ -289,6 +315,43 @@ function NLPModels.hess_coord!(
   return vals
 end
 
+function NLPModels.hess_structure_residual!(
+  b::ADBackend,
+  nls::AbstractADNLSModel,
+  rows::AbstractVector{<:Integer},
+  cols::AbstractVector{<:Integer},
+)
+  n = nls.meta.nvar
+  pos = 0
+  for j = 1:n
+    for i = j:n
+      pos += 1
+      rows[pos] = i
+      cols[pos] = j
+    end
+  end
+  return rows, cols
+end
+
+function NLPModels.hess_coord_residual!(
+  b::ADBackend,
+  nls::AbstractADNLSModel,
+  x::AbstractVector,
+  v::AbstractVector,
+  vals::AbstractVector,
+)
+  F = get_F(nls, b)
+  Hx = hessian(b, x -> dot(F(x), v), x)
+  k = 1
+  for j = 1:(nls.meta.nvar)
+    for i = j:(nls.meta.nvar)
+      vals[k] = Hx[i, j]
+      k += 1
+    end
+  end
+  return vals
+end
+
 function NLPModels.hprod!(
   b::ADBackend,
   nlp::ADModel,
diff --git a/src/nls.jl b/src/nls.jl
index c7c81e7b..0edcfef2 100644
--- a/src/nls.jl
+++ b/src/nls.jl
@@ -165,8 +165,9 @@ function ADNLSModel!(
 
   meta = NLPModelMeta{T, S}(nvar, x0 = x0, nnzh = nnzh, name = name, minimize = minimize)
   nls_nnzj = get_residual_nnzj(adbackend, nvar, nequ)
+  nls_nnzh = get_residual_nnzh(adbackend, nvar)
   nls_meta =
-    NLSMeta{T, S}(nequ, nvar, nnzj = nls_nnzj, nnzh = div(nvar * (nvar + 1), 2), lin = linequ)
+    NLSMeta{T, S}(nequ, nvar, nnzj = nls_nnzj, nnzh = nls_nnzh, lin = linequ)
   return ADNLSModel(meta, nls_meta, NLSCounters(), adbackend, F!, (cx, x) -> cx)
 end
 
@@ -210,8 +211,9 @@ function ADNLSModel!(
     minimize = minimize,
   )
   nls_nnzj = get_residual_nnzj(adbackend, nvar, nequ)
+  nls_nnzh = get_residual_nnzh(adbackend, nvar)
   nls_meta =
-    NLSMeta{T, S}(nequ, nvar, nnzj = nls_nnzj, nnzh = div(nvar * (nvar + 1), 2), lin = linequ)
+    NLSMeta{T, S}(nequ, nvar, nnzj = nls_nnzj, nnzh = nls_nnzh, lin = linequ)
   return ADNLSModel(meta, nls_meta, NLSCounters(), adbackend, F!, (cx, x) -> cx)
 end
 
@@ -272,8 +274,9 @@ function ADNLSModel!(
     minimize = minimize,
   )
   nls_nnzj = get_residual_nnzj(adbackend, nvar, nequ)
+  nls_nnzh = get_residual_nnzh(adbackend, nvar)
   nls_meta =
-    NLSMeta{T, S}(nequ, nvar, nnzj = nls_nnzj, nnzh = div(nvar * (nvar + 1), 2), lin = linequ)
+    NLSMeta{T, S}(nequ, nvar, nnzj = nls_nnzj, nnzh = nls_nnzh, lin = linequ)
   return ADNLSModel(meta, nls_meta, NLSCounters(), adbackend, F!, c!)
 end
 
@@ -422,8 +425,9 @@ function ADNLSModel!(
     minimize = minimize,
   )
   nls_nnzj = get_residual_nnzj(adbackend, nvar, nequ)
+  nls_nnzh = get_residual_nnzh(adbackend, nvar)
   nls_meta =
-    NLSMeta{T, S}(nequ, nvar, nnzj = nls_nnzj, nnzh = div(nvar * (nvar + 1), 2), lin = linequ)
+    NLSMeta{T, S}(nequ, nvar, nnzj = nls_nnzj, nnzh = nls_nnzh, lin = linequ)
   return ADNLSModel(meta, nls_meta, NLSCounters(), adbackend, F!, clinrows, clincols, clinvals, c!)
 end
 
@@ -639,8 +643,9 @@ function ADNLSModel!(
     minimize = minimize,
   )
   nls_nnzj = get_residual_nnzj(adbackend, nvar, nequ)
+  nls_nnzh = get_residual_nnzh(adbackend, nvar)
   nls_meta =
-    NLSMeta{T, S}(nequ, nvar, nnzj = nls_nnzj, nnzh = div(nvar * (nvar + 1), 2), lin = linequ)
+    NLSMeta{T, S}(nequ, nvar, nnzj = nls_nnzj, nnzh = nls_nnzh, lin = linequ)
   return ADNLSModel(meta, nls_meta, NLSCounters(), adbackend, F!, c!)
 end
 
@@ -744,8 +749,9 @@ function ADNLSModel!(
     minimize = minimize,
   )
   nls_nnzj = get_residual_nnzj(adbackend, nvar, nequ)
+  nls_nnzh = get_residual_nnzh(adbackend, nvar)
   nls_meta =
-    NLSMeta{T, S}(nequ, nvar, nnzj = nls_nnzj, nnzh = div(nvar * (nvar + 1), 2), lin = linequ)
+    NLSMeta{T, S}(nequ, nvar, nnzj = nls_nnzj, nnzh = nls_nnzh, lin = linequ)
   return ADNLSModel(meta, nls_meta, NLSCounters(), adbackend, F!, clinrows, clincols, clinvals, c!)
 end
 
@@ -864,11 +870,7 @@ function NLPModels.hess_structure_residual!(
   cols::AbstractVector{<:Integer},
 )
   @lencheck nls.nls_meta.nnzh rows cols
-  n = nls.meta.nvar
-  I = ((i, j) for i = 1:n, j = 1:n if i ≥ j)
-  rows .= getindex.(I, 1)
-  cols .= getindex.(I, 2)
-  return rows, cols
+  return hess_structure_residual!(nls.adbackend.hessian_residual_backend, nls, rows, cols)
 end
 
 function NLPModels.hess_coord_residual!(
@@ -881,23 +883,7 @@ function NLPModels.hess_coord_residual!(
   @lencheck nls.nls_meta.nequ v
   @lencheck nls.nls_meta.nnzh vals
   increment!(nls, :neval_hess_residual)
-  F = get_F(nls, nls.adbackend.hessian_residual_backend)
-  Hx = hessian(nls.adbackend.hessian_residual_backend, x -> dot(F(x), v), x)
-  k = 1
-  for j = 1:(nls.meta.nvar)
-    for i = j:(nls.meta.nvar)
-      vals[k] = Hx[i, j]
-      k += 1
-    end
-  end
-  return vals
-end
-
-function NLPModels.jth_hess_residual(nls::ADNLSModel, x::AbstractVector, i::Int)
-  @lencheck nls.meta.nvar x
-  increment!(nls, :neval_jhess_residual)
-  F = get_F(nls, nls.adbackend.hessian_residual_backend)
-  return Symmetric(hessian(nls.adbackend.hessian_residual_backend, x -> F(x)[i], x), :L)
+  return hess_coord_residual!(nls.adbackend.hessian_residual_backend, nls, x, v, vals)
 end
 
 function NLPModels.hprod_residual!(
diff --git a/src/predefined_backend.jl b/src/predefined_backend.jl
index 6e2d0ffd..740fde7e 100644
--- a/src/predefined_backend.jl
+++ b/src/predefined_backend.jl
@@ -3,14 +3,14 @@ default_backend = Dict(
   :hprod_backend => ForwardDiffADHvprod,
   :jprod_backend => ForwardDiffADJprod,
   :jtprod_backend => ForwardDiffADJtprod,
-  :jacobian_backend => SparseADJacobian, # ForwardDiffADJacobian
-  :hessian_backend => SparseADHessian, # ForwardDiffADHessian
+  :jacobian_backend => SparseADJacobian,
+  :hessian_backend => SparseADHessian,
   :ghjvprod_backend => ForwardDiffADGHjvprod,
   :hprod_residual_backend => ForwardDiffADHvprod,
   :jprod_residual_backend => ForwardDiffADJprod,
   :jtprod_residual_backend => ForwardDiffADJtprod,
-  :jacobian_residual_backend => SparseADJacobian, # ForwardDiffADJacobian,
-  :hessian_residual_backend => ForwardDiffADHessian,
+  :jacobian_residual_backend => SparseADJacobian,
+  :hessian_residual_backend => SparseADHessian,
 )
 
 optimized = Dict(
@@ -18,14 +18,14 @@ optimized = Dict(
   :hprod_backend => ReverseDiffADHvprod,
   :jprod_backend => ForwardDiffADJprod,
   :jtprod_backend => ReverseDiffADJtprod,
-  :jacobian_backend => SparseADJacobian, # ForwardDiffADJacobian
-  :hessian_backend => SparseReverseADHessian, # ForwardDiffADHessian,
+  :jacobian_backend => SparseADJacobian,
+  :hessian_backend => SparseReverseADHessian,
   :ghjvprod_backend => ForwardDiffADGHjvprod,
   :hprod_residual_backend => ReverseDiffADHvprod,
   :jprod_residual_backend => ForwardDiffADJprod,
   :jtprod_residual_backend => ReverseDiffADJtprod,
-  :jacobian_residual_backend => SparseADJacobian, # ForwardDiffADJacobian
-  :hessian_residual_backend => ForwardDiffADHessian,
+  :jacobian_residual_backend => SparseADJacobian,
+  :hessian_residual_backend => SparseReverseADHessian,
 )
 
 generic = Dict(
diff --git a/src/sparse_hessian.jl b/src/sparse_hessian.jl
index 4140762b..8d899666 100644
--- a/src/sparse_hessian.jl
+++ b/src/sparse_hessian.jl
@@ -180,8 +180,16 @@ function NLPModels.hess_structure!(
   return rows, cols
 end
 
+function NLPModels.hess_structure_residual!(
+  b::Union{SparseADHessian, SparseReverseADHessian},
+  nls::AbstractADNLSModel,
+  rows::AbstractVector{<:Integer},
+  cols::AbstractVector{<:Integer},
+)
+  return hess_structure!(b, nls, rows, cols)
+end
+
 function sparse_hess_coord!(
-  ℓ::Function,
   b::SparseADHessian{Tag, GT, S, T},
   x::AbstractVector,
   obj_weight,
@@ -219,7 +227,6 @@ function sparse_hess_coord!(
 end
 
 function sparse_hess_coord!(
-  ℓ::Function,
   b::SparseReverseADHessian{T, S, Tagf, F, Tagψ, P},
   x::AbstractVector,
   obj_weight,
@@ -265,8 +272,7 @@ function NLPModels.hess_coord!(
   obj_weight::Real,
   vals::AbstractVector,
 )
-  ℓ = get_lag(nlp, b, obj_weight, y)
-  sparse_hess_coord!(ℓ, b, x, obj_weight, y, vals)
+  sparse_hess_coord!(b, x, obj_weight, y, vals)
 end
 
 function NLPModels.hess_coord!(
@@ -277,8 +283,7 @@ function NLPModels.hess_coord!(
   vals::AbstractVector,
 )
   b.y .= 0
-  ℓ = get_lag(nlp, b, obj_weight, b.y)
-  sparse_hess_coord!(ℓ, b, x, obj_weight, b.y, vals)
+  sparse_hess_coord!(b, x, obj_weight, b.y, vals)
 end
 
 function NLPModels.hess_coord!(
@@ -286,13 +291,23 @@ function NLPModels.hess_coord!(
   nlp::ADModel,
   x::AbstractVector,
   j::Integer,
-  vals::AbstractVector{T},
-) where {T}
+  vals::AbstractVector,
+)
   for (w, k) in enumerate(nlp.meta.nln)
     b.y[w] = k == j ? 1 : 0
   end
-  obj_weight = zero(T)
-  ℓ = get_lag(nlp, b, obj_weight, b.y)
-  sparse_hess_coord!(ℓ, b, x, obj_weight, b.y, vals)
+  obj_weight = zero(eltype(x))
+  sparse_hess_coord!(b, x, obj_weight, b.y, vals)
   return vals
 end
+
+function NLPModels.hess_coord_residual!(
+  b::Union{SparseADHessian, SparseReverseADHessian},
+  nls::AbstractADNLSModel,
+  x::AbstractVector,
+  v::AbstractVector,
+  vals::AbstractVector,
+)
+  obj_weight = zero(eltype(x))
+  sparse_hess_coord!(b, x, obj_weight, v, vals)
+end
diff --git a/test/runtests.jl b/test/runtests.jl
index a69dc634..b98044b0 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -29,6 +29,7 @@ end
 
 @testset "Basic Hessian derivative test" begin
   include("sparse_hessian.jl")
+  include("sparse_hessian_nls.jl")
 end
 
 for problem in NLPModelsTest.nlp_problems ∪ ["GENROSE"]
diff --git a/test/sparse_hessian.jl b/test/sparse_hessian.jl
index 7c5c0529..15327e97 100644
--- a/test/sparse_hessian.jl
+++ b/test/sparse_hessian.jl
@@ -1,5 +1,7 @@
 list_sparse_hess_backend =
-  ((ADNLPModels.SparseADHessian, Dict()), (ADNLPModels.ForwardDiffADHessian, Dict()))
+  ((ADNLPModels.SparseADHessian, Dict()),
+   (ADNLPModels.SparseReverseADHessian, Dict()),
+   (ADNLPModels.ForwardDiffADHessian, Dict()))
 
 dt = (Float32, Float64)
 
diff --git a/test/sparse_hessian_nls.jl b/test/sparse_hessian_nls.jl
new file mode 100644
index 00000000..4cc056ee
--- /dev/null
+++ b/test/sparse_hessian_nls.jl
@@ -0,0 +1,45 @@
+list_sparse_hess_backend = (
+  (ADNLPModels.SparseADHessian, Dict()),
+  (ADNLPModels.SparseReverseADHessian, Dict()),
+  (ADNLPModels.ForwardDiffADHessian, Dict()),
+)
+
+dt = (Float32, Float64)
+
+@testset "Basic Hessian of residual derivative with backend=$(backend) and T=$(T)" for T in dt,
+  (backend, kw) in list_sparse_hess_backend
+
+  F!(Fx, x) = begin
+    Fx[1] = x[1] - 1
+    Fx[2] = 10 * (x[2] - x[1]^2)
+    Fx[3] = x[2] + 1
+    Fx
+  end
+  x0 = T[-1.2; 1.0]
+  nvar = 2
+  nequ = 3
+  nls = ADNLPModels.ADNLSModel!(F!, x0, 3, hessian_residual_backend = backend; kw...)
+
+  x = rand(T, nvar)
+  v = rand(T, nequ)
+  rows, cols = zeros(Int, nls.nls_meta.nnzh), zeros(Int, nls.nls_meta.nnzh)
+  vals = zeros(T, nls.nls_meta.nnzh)
+  hess_structure_residual!(nls, rows, cols)
+  hess_coord_residual!(nls, x, v, vals)
+  @test eltype(vals) == T
+  H = Symmetric(sparse(rows, cols, vals, nvar, nvar), :L)
+  @test H == [-20 * v[2] 0; 0 0]
+
+  # Test also the implementation of the backends
+  b = nls.adbackend.hessian_residual_backend
+  @test nls.nls_meta.nnzh == ADNLPModels.get_nln_nnzh(b, nvar)
+  ADNLPModels.hess_structure_residual!(b, nls, rows, cols)
+  ADNLPModels.hess_coord_residual!(b, nls, x, v, vals)
+  @test eltype(vals) == T
+  H = Symmetric(sparse(rows, cols, vals, nvar, nvar), :L)
+  @test H == [-20 * v[2] 0; 0 0]
+
+  nls = ADNLPModels.ADNLSModel!(F!, x0, 3, matrix_free = true; kw...)
+  @test nls.adbackend.hessian_backend isa ADNLPModels.EmptyADbackend
+  @test nls.adbackend.hessian_residual_backend isa ADNLPModels.EmptyADbackend
+end
diff --git a/test/sparse_jacobian.jl b/test/sparse_jacobian.jl
index 20b4d05e..0a522eb1 100644
--- a/test/sparse_jacobian.jl
+++ b/test/sparse_jacobian.jl
@@ -1,8 +1,10 @@
 list_sparse_jac_backend = (
-  (ADNLPModels.SparseADJacobian, Dict()), # default
+  (ADNLPModels.SparseADJacobian, Dict()),
   (ADNLPModels.ForwardDiffADJacobian, Dict()),
 )
+
 dt = (Float32, Float64)
+
 @testset "Basic Jacobian derivative with backend=$(backend) and T=$(T)" for T in dt,
   (backend, kw) in list_sparse_jac_backend
 
diff --git a/test/sparse_jacobian_nls.jl b/test/sparse_jacobian_nls.jl
index 23df0dc3..0144c50f 100644
--- a/test/sparse_jacobian_nls.jl
+++ b/test/sparse_jacobian_nls.jl
@@ -1,8 +1,10 @@
 list_sparse_jac_backend = (
-  (ADNLPModels.SparseADJacobian, Dict()), # default
+  (ADNLPModels.SparseADJacobian, Dict()),
   (ADNLPModels.ForwardDiffADJacobian, Dict()),
 )
+
 dt = (Float32, Float64)
+
 @testset "Basic Jacobian of residual derivative with backend=$(backend) and T=$(T)" for T in dt,
   (backend, kw) in list_sparse_jac_backend