AdjointSolutionUserObject

Reads a variable from a mesh in one simulation to another specifically for loading forward solution in adjoint simulation during inverse optimization.

Overview

This user-object is meant to be used for transient inverse optimization where the forward solution is needed to compute the adjoint and subsequent gradient (i.e. nonlinear problems and material inversion). The object acts similarly to SolutionUserObject and can be used by objects such as SolutionAux and SolutionFunction. The difference stems from the "reverse_time_end" parameter where the solution loaded is reversed in time. This is due to the reverse time-stepping required to evaluate the adjoint solution. Due to specificity of this object's application, only exodus files can be loaded.

Example Input File Syntax

Here is a material inversion example where the forward and adjoint models are defined as:

[MultiApps<<<{"href": "../../syntax/MultiApps/index.html"}>>>]
  [forward]
    type = FullSolveMultiApp<<<{"description": "Performs a complete simulation during each execution.", "href": "../multiapps/FullSolveMultiApp.html"}>>>
    input_files<<<{"description": "The input file for each App.  If this parameter only contains one input file it will be used for all of the Apps.  When using 'positions_from_file' it is also admissable to provide one input_file per file."}>>> = forward.i
    cli_args<<<{"description": "Additional command line arguments to pass to the sub apps. If one set is provided the arguments are applied to all, otherwise there must be a set for each sub app."}>>> = 'Outputs/csv=false;Outputs/console=false'
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = FORWARD
  []
  [adjoint]
    type = FullSolveMultiApp<<<{"description": "Performs a complete simulation during each execution.", "href": "../multiapps/FullSolveMultiApp.html"}>>>
    input_files<<<{"description": "The input file for each App.  If this parameter only contains one input file it will be used for all of the Apps.  When using 'positions_from_file' it is also admissable to provide one input_file per file."}>>> = gradient.i
    cli_args<<<{"description": "Additional command line arguments to pass to the sub apps. If one set is provided the arguments are applied to all, otherwise there must be a set for each sub app."}>>> = 'Outputs/console=false;UserObjects/load_u/mesh=optimize_grad_out_forward0.e'
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = ADJOINT
  []
[]

The forward app outputs using Exodus, which results in the file optimize_grad_out_forward0.e being generated:

[Outputs<<<{"href": "../../syntax/Outputs/index.html"}>>>]
  csv<<<{"description": "Output the scalar variable and postprocessors to a *.csv file using the default CSV output."}>>> = true
  exodus<<<{"description": "Output the results using the default settings for Exodus output."}>>> = true
[]

The adjoint app reads in this outputted file and a SolutionAux sets the u auxiliary variable to it's values, while reversing the time:

[UserObjects<<<{"href": "../../syntax/UserObjects/index.html"}>>>]
  [load_u]
    type = AdjointSolutionUserObject<<<{"description": "Reads a variable from a mesh in one simulation to another specifically for loading forward solution in adjoint simulation during inverse optimization.", "href": "AdjointSolutionUserObject.html"}>>>
    mesh<<<{"description": "The name of the mesh file (must be xda/xdr or exodusII file)."}>>> = optimize_grad_out_forward0.e
    system_variables<<<{"description": "The name of the nodal and elemental variables from the file you want to use for values"}>>> = 'u'
    reverse_time_end<<<{"description": "End time used for reversing the time integration when evaluating function derivative."}>>> = 1
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = 'timestep_begin'
  []
[]

[AuxKernels<<<{"href": "../../syntax/AuxKernels/index.html"}>>>]
  [u_aux]
    type = SolutionAux<<<{"description": "Creates fields by using information from a SolutionUserObject.", "href": "../auxkernels/SolutionAux.html"}>>>
    variable<<<{"description": "The name of the variable that this object applies to"}>>> = u
    solution<<<{"description": "The name of the SolutionUserObject"}>>> = load_u
    direct<<<{"description": "If true the meshes must be the same and then the values are simply copied over."}>>> = true
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = 'timestep_begin'
  []
[]

Input Parameters

meshThe name of the mesh file (must be xda/xdr or exodusII file).
C++ Type:MeshFileName
Controllable:No
Description:The name of the mesh file (must be xda/xdr or exodusII file).
reverse_time_endEnd time used for reversing the time integration when evaluating function derivative.
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:End time used for reversing the time integration when evaluating function derivative.

Required Parameters

nodal_variable_orderFIRSTSpecifies the order of the nodal solution data.
Default:FIRST
C++ Type:MooseEnum
Options:FIRST, SECOND
Controllable:No
Description:Specifies the order of the nodal solution data.
rotation1_vector0 0 1Vector about which to rotate points of the simulation.
Default:0 0 1
C++ Type:libMesh::VectorValue<double>
Unit:(no unit assumed)
Controllable:No
Description:Vector about which to rotate points of the simulation.
system_variablesThe name of the nodal and elemental variables from the file you want to use for values
C++ Type:std::vector<std::string>
Controllable:No
Description:The name of the nodal and elemental variables from the file you want to use for values

Optional Parameters

allow_duplicate_execution_on_initialFalseIn the case where this UserObject is depended upon by an initial condition, allow it to be executed twice during the initial setup (once before the IC and again after mesh adaptivity (if applicable).
Default:False
C++ Type:bool
Controllable:No
Description:In the case where this UserObject is depended upon by an initial condition, allow it to be executed twice during the initial setup (once before the IC and again after mesh adaptivity (if applicable).
execute_onTIMESTEP_ENDThe list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html.
Default:TIMESTEP_END
C++ Type:ExecFlagEnum
Options:XFEM_MARK, FORWARD, ADJOINT, HOMOGENEOUS_FORWARD, ADJOINT_TIMESTEP_BEGIN, ADJOINT_TIMESTEP_END, NONE, INITIAL, LINEAR, LINEAR_CONVERGENCE, NONLINEAR, NONLINEAR_CONVERGENCE, POSTCHECK, TIMESTEP_END, TIMESTEP_BEGIN, MULTIAPP_FIXED_POINT_END, MULTIAPP_FIXED_POINT_BEGIN, MULTIAPP_FIXED_POINT_CONVERGENCE, FINAL, CUSTOM
Controllable:No
Description:The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html.
execution_order_group0Execution order groups are executed in increasing order (e.g., the lowest number is executed first). Note that negative group numbers may be used to execute groups before the default (0) group. Please refer to the user object documentation for ordering of user object execution within a group.
Default:0
C++ Type:int
Controllable:No
Description:Execution order groups are executed in increasing order (e.g., the lowest number is executed first). Note that negative group numbers may be used to execute groups before the default (0) group. Please refer to the user object documentation for ordering of user object execution within a group.
force_postauxFalseForces the UserObject to be executed in POSTAUX
Default:False
C++ Type:bool
Controllable:No
Description:Forces the UserObject to be executed in POSTAUX
force_preauxFalseForces the UserObject to be executed in PREAUX
Default:False
C++ Type:bool
Controllable:No
Description:Forces the UserObject to be executed in PREAUX
force_preicFalseForces the UserObject to be executed in PREIC during initial setup
Default:False
C++ Type:bool
Controllable:No
Description:Forces the UserObject to be executed in PREIC during initial setup

Execution Scheduling Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector<std::string>
Controllable:No
Description:Adds user-defined labels for accessing object parameters via control logic.
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Controllable:Yes
Description:Set the enabled status of the MooseObject.
use_displaced_meshFalseWhether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.
Default:False
C++ Type:bool
Controllable:No
Description:Whether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.

Advanced Parameters

prop_getter_suffixAn optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:An optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.
use_interpolated_stateFalseFor the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.
Default:False
C++ Type:bool
Controllable:No
Description:For the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.

Material Property Retrieval Parameters

rotation0_angle0Anticlockwise rotation angle (in degrees) to use for rotation about rotation0_vector.
Default:0
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Anticlockwise rotation angle (in degrees) to use for rotation about rotation0_vector.
rotation0_vector0 0 1Vector about which to rotate points of the simulation.
Default:0 0 1
C++ Type:libMesh::VectorValue<double>
Unit:(no unit assumed)
Controllable:No
Description:Vector about which to rotate points of the simulation.
rotation1_angle0Anticlockwise rotation angle (in degrees) to use for rotation about rotation1_vector.
Default:0
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Anticlockwise rotation angle (in degrees) to use for rotation about rotation1_vector.
scale1 1 1 Scale factor for points in the simulation
Default:1 1 1
C++ Type:std::vector<double>
Unit:(no unit assumed)
Controllable:No
Description:Scale factor for points in the simulation
scale_multiplier1 1 1 Scale multiplying factor for points in the simulation
Default:1 1 1
C++ Type:std::vector<double>
Unit:(no unit assumed)
Controllable:No
Description:Scale multiplying factor for points in the simulation
transformation_ordertranslation scaleThe order to perform the operations in. Define R0 to be the rotation matrix encoded by rotation0_vector and rotation0_angle. Similarly for R1. Denote the scale by s, the scale_multiplier by m, and the translation by t. Then, given a point x in the simulation, if transformation_order = 'rotation0 scale_multiplier translation scale rotation1' then form p = R1*(R0*x*m - t)/s. Then the values provided by the SolutionUserObjectBase at point x in the simulation are the variable values at point p in the mesh.
Default:translation scale
C++ Type:MultiMooseEnum
Options:rotation0, translation, scale, rotation1, scale_multiplier
Controllable:No
Description:The order to perform the operations in. Define R0 to be the rotation matrix encoded by rotation0_vector and rotation0_angle. Similarly for R1. Denote the scale by s, the scale_multiplier by m, and the translation by t. Then, given a point x in the simulation, if transformation_order = 'rotation0 scale_multiplier translation scale rotation1' then form p = R1*(R0*x*m - t)/s. Then the values provided by the SolutionUserObjectBase at point x in the simulation are the variable values at point p in the mesh.
translation0 0 0 Translation factors for x,y,z coordinates of the simulation
Default:0 0 0
C++ Type:std::vector<double>
Unit:(no unit assumed)
Controllable:No
Description:Translation factors for x,y,z coordinates of the simulation

Coordinate System Transformation Parameters

Input Files

(modules/optimization/examples/materialTransient/gradient.i)
(modules/optimization/test/tests/userobjects/adjoint_solution/adjoint.i)

(modules/optimization/examples/materialTransient/optimize_grad.i)

[Optimization]
[]

[OptimizationReporter]
  type = GeneralOptimization
  objective_name = objective_value
  parameter_names = 'D'
  num_values = '4'
  initial_condition = '0.01 0.01 0.01 0.01'
  upper_bounds = '1e2'
  lower_bounds = '1e-3'
[]

[Reporters]
  [main]
    type = OptimizationData

    measurement_file = forward_out_data_0011.csv
    file_xcoord = measurement_xcoord
    file_ycoord = measurement_ycoord
    file_zcoord = measurement_zcoord
    file_time = measurement_time
    file_value = simulation_values
  []
[]

[MultiApps]
  [forward]
    type = FullSolveMultiApp
    input_files = forward.i
    cli_args = 'Outputs/csv=false;Outputs/console=false'
    execute_on = FORWARD
  []
  [adjoint]
    type = FullSolveMultiApp
    input_files = gradient.i
    cli_args = 'Outputs/console=false;UserObjects/load_u/mesh=optimize_grad_out_forward0.e'
    execute_on = ADJOINT
  []
[]

[Transfers]
  [to_forward]
    type = MultiAppReporterTransfer
    to_multi_app = forward
    from_reporters = 'main/measurement_xcoord
                      main/measurement_ycoord
                      main/measurement_zcoord
                      main/measurement_time
                      main/measurement_values
                      OptimizationReporter/D'
    to_reporters = 'data/measurement_xcoord
                    data/measurement_ycoord
                    data/measurement_zcoord
                    data/measurement_time
                    data/measurement_values
                    diffc_rep/D_vals'
  []
  [from_forward]
    type = MultiAppReporterTransfer
    from_multi_app = forward
    from_reporters = 'data/misfit_values data/objective_value'
    to_reporters = 'main/misfit_values OptimizationReporter/objective_value'
  []
  [to_adjoint]
    type = MultiAppReporterTransfer
    to_multi_app = adjoint
    from_reporters = 'main/measurement_xcoord
                      main/measurement_ycoord
                      main/measurement_zcoord
                      main/measurement_time
                      main/misfit_values
                      OptimizationReporter/D'
    to_reporters = 'data/measurement_xcoord
                    data/measurement_ycoord
                    data/measurement_zcoord
                    data/measurement_time
                    data/misfit_values
                    diffc_rep/D_vals'
  []
  [from_adjoint]
    type = MultiAppReporterTransfer
    from_multi_app = adjoint
    from_reporters = 'adjoint/inner_product'
    to_reporters = 'OptimizationReporter/grad_D'
  []
[]

[Executioner]
  type = Optimize
  tao_solver = taobqnls
  petsc_options_iname = '-tao_gatol'
  petsc_options_value = '1e-4'
[]

(modules/optimization/examples/materialTransient/forward.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 2
    xmax = 1
    ymax = 1
    nx = 10
    ny = 10
  []
[]

[Variables/u]
  initial_condition = 0
[]

[Kernels]
  [dt]
    type = TimeDerivative
    variable = u
  []
  [diff]
    type = MatDiffusion
    variable = u
    diffusivity = D
  []
  [src]
    type = BodyForce
    variable = u
    value = 1
  []
[]

[BCs]
  [dirichlet]
    type = DirichletBC
    variable = u
    boundary = 'right top'
    value = 0
  []
[]

[Materials]
  [diffc]
    type = GenericFunctionMaterial
    prop_names = 'D'
    prop_values = 'diffc_fun'
    output_properties = 'D'
    outputs = 'exodus'
  []
[]

[Functions]
  [diffc_fun]
    type = NearestReporterCoordinatesFunction
    value_name = 'diffc_rep/D_vals'
    x_coord_name = 'diffc_rep/D_x_coord'
    y_coord_name = 'diffc_rep/D_y_coord'
  []
[]

[Reporters]
  [diffc_rep]
    type = ConstantReporter
    real_vector_names = 'D_x_coord D_y_coord D_vals'
    real_vector_values = '0.25 0.75 0.25 0.75;
                          0.25 0.25 0.75 0.75;
                          1  0.2   0.2   0.05' # Reference solution
    outputs = none
  []
  [data]
    type = OptimizationData
    variable = u
    measurement_points = '0.25 0.25 0  0.25 0.75 0  0.75 0.25 0  0.75 0.75 0
                          0.25 0.25 0  0.25 0.75 0  0.75 0.25 0  0.75 0.75 0
                          0.25 0.25 0  0.25 0.75 0  0.75 0.25 0  0.75 0.75 0
                          0.25 0.25 0  0.25 0.75 0  0.75 0.25 0  0.75 0.75 0
                          0.25 0.25 0  0.25 0.75 0  0.75 0.25 0  0.75 0.75 0
                          0.25 0.25 0  0.25 0.75 0  0.75 0.25 0  0.75 0.75 0
                          0.25 0.25 0  0.25 0.75 0  0.75 0.25 0  0.75 0.75 0
                          0.25 0.25 0  0.25 0.75 0  0.75 0.25 0  0.75 0.75 0
                          0.25 0.25 0  0.25 0.75 0  0.75 0.25 0  0.75 0.75 0
                          0.25 0.25 0  0.25 0.75 0  0.75 0.25 0  0.75 0.75 0'
    measurement_times = '0.1 0.1 0.1 0.1
                         0.2 0.2 0.2 0.2
                         0.3 0.3 0.3 0.3
                         0.4 0.4 0.4 0.4
                         0.5 0.5 0.5 0.5
                         0.6 0.6 0.6 0.6
                         0.7 0.7 0.7 0.7
                         0.8 0.8 0.8 0.8
                         0.9 0.9 0.9 0.9
                         1.0 1.0 1.0 1.0'
    measurement_values = '0 0 0 0
                          0 0 0 0
                          0 0 0 0
                          0 0 0 0
                          0 0 0 0
                          0 0 0 0
                          0 0 0 0
                          0 0 0 0
                          0 0 0 0
                          0 0 0 0'
    objective_name = objective_value
    outputs = none
  []
[]

[Postprocessors]
  [D1]
    type = PointValue
    variable = D
    point = '0.25 0.25 0'
  []
  [D2]
    type = PointValue
    variable = D
    point = '0.75 0.25 0'
  []
  [D3]
    type = PointValue
    variable = D
    point = '0.25 0.75 0'
  []
  [D4]
    type = PointValue
    variable = D
    point = '0.75 0.75 0'
  []
[]

[Executioner]
  type = Transient

  solve_type = NEWTON
  nl_rel_tol = 1e-8
  nl_abs_tol = 1e-12

  dt = 0.1
  num_steps = 10
[]

[Outputs]
  csv = true
  exodus = true
[]

(modules/optimization/examples/materialTransient/gradient.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 2
    xmax = 1
    ymax = 1
    nx = 10
    ny = 10
  []
[]

[Variables/u_adjoint]
  initial_condition = 0
[]

[Kernels]
  [dt]
    type = TimeDerivative
    variable = u_adjoint
  []
  [diff]
    type = MatDiffusion
    variable = u_adjoint
    diffusivity = D
  []
[]

[DiracKernels]
  [misfit]
    type = ReporterTimePointSource
    variable = u_adjoint
    value_name = data/misfit_values
    x_coord_name = data/measurement_xcoord
    y_coord_name = data/measurement_ycoord
    z_coord_name = data/measurement_zcoord
    time_name = data/measurement_time
    reverse_time_end = 1
  []
[]

[BCs]
  [dirichlet]
    type = DirichletBC
    variable = u_adjoint
    boundary = 'right top'
    value = 0
  []
[]

[Materials]
  [diffc]
    type = GenericFunctionMaterial
    prop_names = 'D'
    prop_values = 'diffc_fun'
  []
[]

[Functions]
  [diffc_fun]
    type = NearestReporterCoordinatesFunction
    value_name = 'diffc_rep/D_vals'
    x_coord_name = 'diffc_rep/D_x_coord'
    y_coord_name = 'diffc_rep/D_y_coord'
  []
[]

[Reporters]
  [diffc_rep]
    type = ConstantReporter
    real_vector_names = 'D_x_coord D_y_coord D_vals'
    real_vector_values = '0.25 0.75 0.25 0.75;
                          0.25 0.25 0.75 0.75;
                          0.1  10   10   0.1' # Reference solution
    outputs = none
  []
  [data]
    type = OptimizationData
  []
[]

[AuxVariables/u]
[]

[UserObjects]
  [load_u]
    type = AdjointSolutionUserObject
    mesh = forward_out.e
    system_variables = 'u'
    reverse_time_end = 1
    execute_on = 'timestep_begin'
  []
[]

[AuxKernels]
  [u_aux]
    type = SolutionAux
    variable = u
    solution = load_u
    direct = true
    execute_on = 'timestep_begin'
  []
[]

[VectorPostprocessors]
  [adjoint]
    type = ElementOptimizationDiffusionCoefFunctionInnerProduct
    variable = u_adjoint
    forward_variable = u
    function = diffc_fun
    reverse_time_end = 1
  []
[]

[Executioner]
  type = Transient

  solve_type = NEWTON
  nl_rel_tol = 1e-8
  nl_abs_tol = 1e-12

  dt = 0.1
  num_steps = 10
[]

(modules/optimization/examples/materialTransient/gradient.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 2
    xmax = 1
    ymax = 1
    nx = 10
    ny = 10
  []
[]

[Variables/u_adjoint]
  initial_condition = 0
[]

[Kernels]
  [dt]
    type = TimeDerivative
    variable = u_adjoint
  []
  [diff]
    type = MatDiffusion
    variable = u_adjoint
    diffusivity = D
  []
[]

[DiracKernels]
  [misfit]
    type = ReporterTimePointSource
    variable = u_adjoint
    value_name = data/misfit_values
    x_coord_name = data/measurement_xcoord
    y_coord_name = data/measurement_ycoord
    z_coord_name = data/measurement_zcoord
    time_name = data/measurement_time
    reverse_time_end = 1
  []
[]

[BCs]
  [dirichlet]
    type = DirichletBC
    variable = u_adjoint
    boundary = 'right top'
    value = 0
  []
[]

[Materials]
  [diffc]
    type = GenericFunctionMaterial
    prop_names = 'D'
    prop_values = 'diffc_fun'
  []
[]

[Functions]
  [diffc_fun]
    type = NearestReporterCoordinatesFunction
    value_name = 'diffc_rep/D_vals'
    x_coord_name = 'diffc_rep/D_x_coord'
    y_coord_name = 'diffc_rep/D_y_coord'
  []
[]

[Reporters]
  [diffc_rep]
    type = ConstantReporter
    real_vector_names = 'D_x_coord D_y_coord D_vals'
    real_vector_values = '0.25 0.75 0.25 0.75;
                          0.25 0.25 0.75 0.75;
                          0.1  10   10   0.1' # Reference solution
    outputs = none
  []
  [data]
    type = OptimizationData
  []
[]

[AuxVariables/u]
[]

[UserObjects]
  [load_u]
    type = AdjointSolutionUserObject
    mesh = forward_out.e
    system_variables = 'u'
    reverse_time_end = 1
    execute_on = 'timestep_begin'
  []
[]

[AuxKernels]
  [u_aux]
    type = SolutionAux
    variable = u
    solution = load_u
    direct = true
    execute_on = 'timestep_begin'
  []
[]

[VectorPostprocessors]
  [adjoint]
    type = ElementOptimizationDiffusionCoefFunctionInnerProduct
    variable = u_adjoint
    forward_variable = u
    function = diffc_fun
    reverse_time_end = 1
  []
[]

[Executioner]
  type = Transient

  solve_type = NEWTON
  nl_rel_tol = 1e-8
  nl_abs_tol = 1e-12

  dt = 0.1
  num_steps = 10
[]

(modules/optimization/test/tests/userobjects/adjoint_solution/adjoint.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 10
    ny = 10
  []
[]

[MultiApps]
  [forward]
    type = FullSolveMultiApp
    input_files = forward.i
    execute_on = INITIAL
  []
[]

[AuxVariables/u_reverse]
[]

[UserObjects]
  [u_reverse_solution]
    type = AdjointSolutionUserObject
    mesh = forward_out.e
    system_variables = 'u'
    reverse_time_end = 10
  []
  [terminate]
    type = Terminator
    expression = 'u_reverse_test > 1e-12'
    error_level = ERROR
  []
[]

[AuxKernels]
  [u_reverse_aux]
    type = SolutionAux
    variable = u_reverse
    solution = u_reverse_solution
  []
[]

[Functions]
  [u_reverse_fun]
    type = ParsedFunction
    expression = '(x + y) * (11 - t)'
  []
[]

[Postprocessors]
  [u_reverse_test]
    type = ElementL2Error
    variable = u_reverse
    function = u_reverse_fun
  []
[]

[Problem]
  solve = false
[]

[Executioner]
  type = Transient
  dt = 1
  end_time = 10
[]

Overview
Example Input File Syntax
Input Parameters
Input Files