InversePowerMethod

Inverse power method for eigenvalue problems.

Overview

Eigenvalue executioners such as this one intend on solving the eigenvalue problem described by:

where and are linear or nonlinear operators represented by kernels. To differentiate the kernels from the kernels, we must derive all kernels from EigenKernel. Currently we are only interested in the absolute minimum eigenvalue and the corresponding eigenvector of the system. We are also not seeking the solutions of a general nonlinear eigenvalue problem, where the operators have nonlinear dependency on the eigenvalue.

The inverse power method algorithm

  1. Initialization

  2. Update x and k

  3. Check the convergence

    and

    When either of them is not true, return Step 2, otherwise exit.

We notice immediately that remains constant during the iteration, so if we make equal to 1, the algorithm can be simplified a little:

  1. Initialization

  2. Update x and k

  3. Check the convergence

    and

    When either of them is not true, return Step 2, otherwise exit.

Also in this simplified algorithm, the solution is automatically normalized making . We can do postprocessing to normalize the solution so that , where can be any norm and is a scalar constant.

If the minimum eigenvalue and the second smallest eigenvalue are close, i.e. the dominance ratio is about equal to one, the inverse power iteration converges very slowly. In such a case, we can apply accelerations, such as Chebyshev acceleration, based on the on-the-fly estimation of the dominance ratio.

The inverse power method is appealing because we can apply matrix-free schemes on evaluating . We can use PJFNK for inverting and we do not have to exactly assemble matrix for the preconditioning purpose.

Input Parameters

  • bx_normTo evaluate |Bx| for the eigenvalue

    C++ Type:PostprocessorName

    Controllable:No

    Description:To evaluate |Bx| for the eigenvalue

Required Parameters

  • Chebyshev_acceleration_onTrueIf Chebyshev acceleration is turned on

    Default:True

    C++ Type:bool

    Controllable:No

    Description:If Chebyshev acceleration is turned on

  • eig_check_tol1e-06Eigenvalue convergence tolerance

    Default:1e-06

    C++ Type:double

    Controllable:No

    Description:Eigenvalue convergence tolerance

  • k01Initial guess of the eigenvalue

    Default:1

    C++ Type:double

    Controllable:No

    Description:Initial guess of the eigenvalue

  • max_power_iterations300The maximum number of power iterations

    Default:300

    C++ Type:unsigned int

    Controllable:No

    Description:The maximum number of power iterations

  • min_power_iterations1Minimum number of power iterations

    Default:1

    C++ Type:unsigned int

    Controllable:No

    Description:Minimum number of power iterations

  • sol_check_tol1.79769e+308Convergence tolerance on |x-x_previous| when provided

    Default:1.79769e+308

    C++ Type:double

    Controllable:No

    Description:Convergence tolerance on |x-x_previous| when provided

  • splittingTop-level splitting defining a hierarchical decomposition into subsystems to help the solver.

    C++ Type:std::vector<std::string>

    Controllable:No

    Description:Top-level splitting defining a hierarchical decomposition into subsystems to help the solver.

  • verboseFalseSet to true to print additional information

    Default:False

    C++ Type:bool

    Controllable:No

    Description:Set to true to print additional information

  • xdiffTo evaluate |x-x_previous| for power iterations

    C++ Type:PostprocessorName

    Controllable:No

    Description:To evaluate |x-x_previous| for power iterations

Optional Parameters

  • accept_on_max_fixed_point_iterationFalseTrue to treat reaching the maximum number of fixed point iterations as converged.

    Default:False

    C++ Type:bool

    Controllable:No

    Description:True to treat reaching the maximum number of fixed point iterations as converged.

  • auto_advanceFalseWhether to automatically advance sub-applications regardless of whether their solve converges, for transient executioners only.

    Default:False

    C++ Type:bool

    Controllable:No

    Description:Whether to automatically advance sub-applications regardless of whether their solve converges, for transient executioners only.

  • custom_abs_tol1e-50The absolute nonlinear residual to shoot for during fixed point iterations. This check is performed based on postprocessor defined by the custom_pp residual.

    Default:1e-50

    C++ Type:double

    Controllable:No

    Description:The absolute nonlinear residual to shoot for during fixed point iterations. This check is performed based on postprocessor defined by the custom_pp residual.

  • custom_ppPostprocessor for custom fixed point convergence check.

    C++ Type:PostprocessorName

    Controllable:No

    Description:Postprocessor for custom fixed point convergence check.

  • custom_rel_tol1e-08The relative nonlinear residual drop to shoot for during fixed point iterations. This check is performed based on the postprocessor defined by custom_pp residual.

    Default:1e-08

    C++ Type:double

    Controllable:No

    Description:The relative nonlinear residual drop to shoot for during fixed point iterations. This check is performed based on the postprocessor defined by custom_pp residual.

  • direct_pp_valueFalseTrue to use direct postprocessor value (scaled by value on first iteration). False (default) to use difference in postprocessor value between fixed point iterations.

    Default:False

    C++ Type:bool

    Controllable:No

    Description:True to use direct postprocessor value (scaled by value on first iteration). False (default) to use difference in postprocessor value between fixed point iterations.

  • disable_fixed_point_residual_norm_checkFalseDisable the residual norm evaluation thus the three parameters fixed_point_rel_tol, fixed_point_abs_tol and fixed_point_force_norms.

    Default:False

    C++ Type:bool

    Controllable:No

    Description:Disable the residual norm evaluation thus the three parameters fixed_point_rel_tol, fixed_point_abs_tol and fixed_point_force_norms.

  • fixed_point_abs_tol1e-50The absolute nonlinear residual to shoot for during fixed point iterations. This check is performed based on the main app's nonlinear residual.

    Default:1e-50

    C++ Type:double

    Controllable:No

    Description:The absolute nonlinear residual to shoot for during fixed point iterations. This check is performed based on the main app's nonlinear residual.

  • fixed_point_algorithmpicardThe fixed point algorithm to converge the sequence of problems.

    Default:picard

    C++ Type:MooseEnum

    Options:picard, secant, steffensen

    Controllable:No

    Description:The fixed point algorithm to converge the sequence of problems.

  • fixed_point_force_normsFalseForce the evaluation of both the TIMESTEP_BEGIN and TIMESTEP_END norms regardless of the existence of active MultiApps with those execute_on flags, default: false.

    Default:False

    C++ Type:bool

    Controllable:No

    Description:Force the evaluation of both the TIMESTEP_BEGIN and TIMESTEP_END norms regardless of the existence of active MultiApps with those execute_on flags, default: false.

  • fixed_point_max_its1Specifies the maximum number of fixed point iterations.

    Default:1

    C++ Type:unsigned int

    Controllable:No

    Description:Specifies the maximum number of fixed point iterations.

  • fixed_point_min_its1Specifies the minimum number of fixed point iterations.

    Default:1

    C++ Type:unsigned int

    Controllable:No

    Description:Specifies the minimum number of fixed point iterations.

  • fixed_point_rel_tol1e-08The relative nonlinear residual drop to shoot for during fixed point iterations. This check is performed based on the main app's nonlinear residual.

    Default:1e-08

    C++ Type:double

    Controllable:No

    Description:The relative nonlinear residual drop to shoot for during fixed point iterations. This check is performed based on the main app's nonlinear residual.

  • relaxation_factor1Fraction of newly computed value to keep.Set between 0 and 2.

    Default:1

    C++ Type:double

    Controllable:No

    Description:Fraction of newly computed value to keep.Set between 0 and 2.

  • transformed_postprocessorsList of main app postprocessors to transform during fixed point iterations

    C++ Type:std::vector<PostprocessorName>

    Controllable:No

    Description:List of main app postprocessors to transform during fixed point iterations

  • transformed_variablesList of main app variables to transform during fixed point iterations

    C++ Type:std::vector<std::string>

    Controllable:No

    Description:List of main app variables to transform during fixed point iterations

Fixed Point Iterations Parameters

  • auto_initializationTrueTrue to ask the solver to set initial

    Default:True

    C++ Type:bool

    Controllable:No

    Description:True to ask the solver to set initial

  • 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:No

    Description:Set the enabled status of the MooseObject.

  • outputsVector of output names where you would like to restrict the output of variables(s) associated with this object

    C++ Type:std::vector<OutputName>

    Controllable:No

    Description:Vector of output names where you would like to restrict the output of variables(s) associated with this object

  • skip_exception_checkFalseSpecifies whether or not to skip exception check

    Default:False

    C++ Type:bool

    Controllable:No

    Description:Specifies whether or not to skip exception check

  • time0System time

    Default:0

    C++ Type:double

    Controllable:No

    Description:System time

Advanced Parameters

  • automatic_scalingFalseWhether to use automatic scaling for the variables.

    Default:False

    C++ Type:bool

    Controllable:No

    Description:Whether to use automatic scaling for the variables.

  • compute_scaling_onceTrueWhether the scaling factors should only be computed once at the beginning of the simulation through an extra Jacobian evaluation. If this is set to false, then the scaling factors will be computed during an extra Jacobian evaluation at the beginning of every time step.

    Default:True

    C++ Type:bool

    Controllable:No

    Description:Whether the scaling factors should only be computed once at the beginning of the simulation through an extra Jacobian evaluation. If this is set to false, then the scaling factors will be computed during an extra Jacobian evaluation at the beginning of every time step.

  • ignore_variables_for_autoscalingList of variables that do not participate in autoscaling.

    C++ Type:std::vector<std::string>

    Controllable:No

    Description:List of variables that do not participate in autoscaling.

  • off_diagonals_in_auto_scalingFalseWhether to consider off-diagonals when determining automatic scaling factors.

    Default:False

    C++ Type:bool

    Controllable:No

    Description:Whether to consider off-diagonals when determining automatic scaling factors.

  • resid_vs_jac_scaling_param0A parameter that indicates the weighting of the residual vs the Jacobian in determining variable scaling parameters. A value of 1 indicates pure residual-based scaling. A value of 0 indicates pure Jacobian-based scaling

    Default:0

    C++ Type:double

    Controllable:No

    Description:A parameter that indicates the weighting of the residual vs the Jacobian in determining variable scaling parameters. A value of 1 indicates pure residual-based scaling. A value of 0 indicates pure Jacobian-based scaling

  • scaling_group_variablesName of variables that are grouped together for determining scale factors. (Multiple groups can be provided, separated by semicolon)

    C++ Type:std::vector<std::vector<std::string>>

    Controllable:No

    Description:Name of variables that are grouped together for determining scale factors. (Multiple groups can be provided, separated by semicolon)

Solver Variable Scaling Parameters

  • compute_initial_residual_before_preset_bcsFalseUse the residual norm computed *before* preset BCs are imposed in relative convergence check

    Default:False

    C++ Type:bool

    Controllable:No

    Description:Use the residual norm computed *before* preset BCs are imposed in relative convergence check

  • n_max_nonlinear_pingpong100The maximum number of times the non linear residual can ping pong before requesting halting the current evaluation and requesting timestep cut

    Default:100

    C++ Type:unsigned int

    Controllable:No

    Description:The maximum number of times the non linear residual can ping pong before requesting halting the current evaluation and requesting timestep cut

  • nl_abs_div_tol1e+50Nonlinear Absolute Divergence Tolerance. A negative value disables this check.

    Default:1e+50

    C++ Type:double

    Controllable:No

    Description:Nonlinear Absolute Divergence Tolerance. A negative value disables this check.

  • nl_abs_step_tol0Nonlinear Absolute step Tolerance

    Default:0

    C++ Type:double

    Controllable:No

    Description:Nonlinear Absolute step Tolerance

  • nl_abs_tol1e-50Nonlinear Absolute Tolerance

    Default:1e-50

    C++ Type:double

    Controllable:No

    Description:Nonlinear Absolute Tolerance

  • nl_div_tol1e+10Nonlinear Relative Divergence Tolerance. A negative value disables this check.

    Default:1e+10

    C++ Type:double

    Controllable:No

    Description:Nonlinear Relative Divergence Tolerance. A negative value disables this check.

  • nl_forced_its0The Number of Forced Nonlinear Iterations

    Default:0

    C++ Type:unsigned int

    Controllable:No

    Description:The Number of Forced Nonlinear Iterations

  • nl_max_funcs10000Max Nonlinear solver function evaluations

    Default:10000

    C++ Type:unsigned int

    Controllable:No

    Description:Max Nonlinear solver function evaluations

  • nl_max_its50Max Nonlinear Iterations

    Default:50

    C++ Type:unsigned int

    Controllable:No

    Description:Max Nonlinear Iterations

  • nl_rel_step_tol0Nonlinear Relative step Tolerance

    Default:0

    C++ Type:double

    Controllable:No

    Description:Nonlinear Relative step Tolerance

  • nl_rel_tol1e-08Nonlinear Relative Tolerance

    Default:1e-08

    C++ Type:double

    Controllable:No

    Description:Nonlinear Relative Tolerance

  • num_grids1The number of grids to use for a grid sequencing algorithm. This includes the final grid, so num_grids = 1 indicates just one solve in a time-step

    Default:1

    C++ Type:unsigned int

    Controllable:No

    Description:The number of grids to use for a grid sequencing algorithm. This includes the final grid, so num_grids = 1 indicates just one solve in a time-step

  • residual_and_jacobian_togetherFalseWhether to compute the residual and Jacobian together.

    Default:False

    C++ Type:bool

    Controllable:No

    Description:Whether to compute the residual and Jacobian together.

  • snesmf_reuse_baseTrueSpecifies whether or not to reuse the base vector for matrix-free calculation

    Default:True

    C++ Type:bool

    Controllable:No

    Description:Specifies whether or not to reuse the base vector for matrix-free calculation

  • solve_typePJFNK: Preconditioned Jacobian-Free Newton Krylov JFNK: Jacobian-Free Newton Krylov NEWTON: Full Newton Solve FD: Use finite differences to compute Jacobian LINEAR: Solving a linear problem

    C++ Type:MooseEnum

    Options:PJFNK, JFNK, NEWTON, FD, LINEAR

    Controllable:No

    Description:PJFNK: Preconditioned Jacobian-Free Newton Krylov JFNK: Jacobian-Free Newton Krylov NEWTON: Full Newton Solve FD: Use finite differences to compute Jacobian LINEAR: Solving a linear problem

Nonlinear Solver Parameters

  • contact_line_search_allowed_lambda_cuts2The number of times lambda is allowed to be cut in half in the contact line search. We recommend this number be roughly bounded by 0 <= allowed_lambda_cuts <= 3

    Default:2

    C++ Type:unsigned int

    Controllable:No

    Description:The number of times lambda is allowed to be cut in half in the contact line search. We recommend this number be roughly bounded by 0 <= allowed_lambda_cuts <= 3

  • contact_line_search_ltolThe linear relative tolerance to be used while the contact state is changing between non-linear iterations. We recommend that this tolerance be looser than the standard linear tolerance

    C++ Type:double

    Controllable:No

    Description:The linear relative tolerance to be used while the contact state is changing between non-linear iterations. We recommend that this tolerance be looser than the standard linear tolerance

  • line_searchdefaultSpecifies the line search type (Note: none = basic)

    Default:default

    C++ Type:MooseEnum

    Options:basic, bt, contact, cp, default, l2, none, project, shell

    Controllable:No

    Description:Specifies the line search type (Note: none = basic)

  • line_search_packagepetscThe solver package to use to conduct the line-search

    Default:petsc

    C++ Type:MooseEnum

    Options:petsc, moose

    Controllable:No

    Description:The solver package to use to conduct the line-search

Solver Line Search Parameters

  • l_abs_tol1e-50Linear Absolute Tolerance

    Default:1e-50

    C++ Type:double

    Controllable:No

    Description:Linear Absolute Tolerance

  • l_max_its10000Max Linear Iterations

    Default:10000

    C++ Type:unsigned int

    Controllable:No

    Description:Max Linear Iterations

  • l_tol0.01Linear Tolerance

    Default:0.01

    C++ Type:double

    Controllable:No

    Description:Linear Tolerance

  • reuse_preconditionerFalseIf true reuse the previously calculated preconditioner for the linearized system across multiple solves spanning nonlinear iterations and time steps. The preconditioner resets as controlled by reuse_preconditioner_max_linear_its

    Default:False

    C++ Type:bool

    Controllable:No

    Description:If true reuse the previously calculated preconditioner for the linearized system across multiple solves spanning nonlinear iterations and time steps. The preconditioner resets as controlled by reuse_preconditioner_max_linear_its

  • reuse_preconditioner_max_linear_its25Reuse the previously calculated preconditioner for the linear system until the number of linear iterations exceeds this number

    Default:25

    C++ Type:unsigned int

    Controllable:No

    Description:Reuse the previously calculated preconditioner for the linear system until the number of linear iterations exceeds this number

Linear Solver Parameters

  • max_xfem_update4294967295Maximum number of times to update XFEM crack topology in a step due to evolving cracks

    Default:4294967295

    C++ Type:unsigned int

    Controllable:No

    Description:Maximum number of times to update XFEM crack topology in a step due to evolving cracks

  • update_xfem_at_timestep_beginFalseShould XFEM update the mesh at the beginning of the timestep

    Default:False

    C++ Type:bool

    Controllable:No

    Description:Should XFEM update the mesh at the beginning of the timestep

Xfem Fixed Point Iterations Parameters

  • mffd_typewpSpecifies the finite differencing type for Jacobian-free solve types. Note that the default is wp (for Walker and Pernice).

    Default:wp

    C++ Type:MooseEnum

    Options:wp, ds

    Controllable:No

    Description:Specifies the finite differencing type for Jacobian-free solve types. Note that the default is wp (for Walker and Pernice).

  • petsc_optionsSingleton PETSc options

    C++ Type:MultiMooseEnum

    Options:-dm_moose_print_embedding, -dm_view, -ksp_converged_reason, -ksp_gmres_modifiedgramschmidt, -ksp_monitor, -ksp_monitor_snes_lg-snes_ksp_ew, -ksp_snes_ew, -snes_converged_reason, -snes_ksp, -snes_ksp_ew, -snes_linesearch_monitor, -snes_mf, -snes_mf_operator, -snes_monitor, -snes_test_display, -snes_view

    Controllable:No

    Description:Singleton PETSc options

  • petsc_options_inameNames of PETSc name/value pairs

    C++ Type:MultiMooseEnum

    Options:-ksp_atol, -ksp_gmres_restart, -ksp_max_it, -ksp_pc_side, -ksp_rtol, -ksp_type, -mat_fd_coloring_err, -mat_fd_type, -mat_mffd_type, -pc_asm_overlap, -pc_factor_levels, -pc_factor_mat_ordering_type, -pc_hypre_boomeramg_grid_sweeps_all, -pc_hypre_boomeramg_max_iter, -pc_hypre_boomeramg_strong_threshold, -pc_hypre_type, -pc_type, -snes_atol, -snes_linesearch_type, -snes_ls, -snes_max_it, -snes_rtol, -snes_divergence_tolerance, -snes_type, -sub_ksp_type, -sub_pc_type

    Controllable:No

    Description:Names of PETSc name/value pairs

  • petsc_options_valueValues of PETSc name/value pairs (must correspond with "petsc_options_iname"

    C++ Type:std::vector<std::string>

    Controllable:No

    Description:Values of PETSc name/value pairs (must correspond with "petsc_options_iname"

Petsc Parameters

  • normal_factorNormalize x to make |x| equal to this factor

    C++ Type:double

    Controllable:No

    Description:Normalize x to make |x| equal to this factor

  • normalizationTo evaluate |x| for normalization

    C++ Type:PostprocessorName

    Controllable:No

    Description:To evaluate |x| for normalization

  • output_before_normalizationTrueTrue to output a step before normalization

    Default:True

    C++ Type:bool

    Controllable:No

    Description:True to output a step before normalization

Normalization Parameters

    Restart Parameters