Transient inversion: Introduced an example with gradient check,

modifying the code to make the gradient check pass (by making it
consistent with discretize then differentiate strategy).

Added Hessian inversion with associated test case, and verified
that it converges as expected.

Also added code functionality to facilitate consistent gradient
inverse for time steppers beyond implicit Euler.

ref #42

examples/simpleTransient/adjoint.i

@@ -76,6 +76,7 @@
     coord_y = src_values/coordy
     time = src_values/time
     value = src_values/values
+    time_shift_factor = 0.5
   []
 []
 
@@ -97,6 +98,9 @@
   solve_type = NEWTON
   petsc_options_iname = '-pc_type -pc_hypre_type'
   petsc_options_value = 'hypre boomeramg'
+  [TimeIntegrator]
+    type = ImplicitEuler
+  []
 []
 
 [Outputs]

examples/simpleTransient/forward.i

@@ -24,13 +24,13 @@
   []
 []
 
-[ICs]
-  [initial]
-    type = FunctionIC
-    variable = u
-    function = exact
-  []
-[]
+#[ICs]
+#  [initial]
+#    type = FunctionIC
+#    variable = u
+#    function = exact
+#  []
+#[]
 
 [Kernels]
   [dt]
@@ -76,10 +76,12 @@
 
   num_steps = 100
   end_time = 1
-
   solve_type = NEWTON
   petsc_options_iname = '-pc_type -pc_hypre_type'
   petsc_options_value = 'hypre boomeramg'
+  [TimeIntegrator]
+    type = ImplicitEuler
+  []
 []
 
 [Reporters]