Parameter Study

A parameter study is defined here as performing a simulation with varying uncertain parameters and computing quantities of interest for each set of uncertain parameters. This example is a more detailed example of what is presented in Monte Carlo Example, which is also a parameter study.

Problem Statement

To demonstrate how to perform a parameter study the transient heat equation will be used.

(1)var element = document.getElementById("moose-equation-64cd4b5f-2b32-42cc-af63-2b9b203c9d6a");katex.render("\\frac{\\partial T}{\\partial t} - \\nabla\\cdot\\gamma\\nabla T + s = 0,", element, {displayMode:true,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});

where $var element = document.getElementById("moose-equation-f7d0113f-71e8-4280-b708-de7f7a0e1929");katex.render("T", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ is temperature, $var element = document.getElementById("moose-equation-50d692fc-e187-4e85-8042-f5cd3fb7e3aa");katex.render("t", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ is time, $var element = document.getElementById("moose-equation-429660c4-9982-4d28-aa26-509801b38237");katex.render("\\gamma", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ is diffusivity, and $var element = document.getElementById("moose-equation-aaf9600d-21b6-4733-9fab-dbdf220e3d0e");katex.render("s", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ is a heat source term. For this example this problem is shall be solved on a 2D unit square domain. The left side is subjected to a Dirichlet condition $var element = document.getElementById("moose-equation-e5868afb-e3a7-4a8f-a359-3195e362d9b7");katex.render("T=T_0", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ . The right side of the domain is subject to a Neumann condition $var element = document.getElementById("moose-equation-e3accb0c-bbfc-434e-8883-75673b2e42c2");katex.render("\\gamma\\nabla T \\cdot \\hat{n} = q_0", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ , where $var element = document.getElementById("moose-equation-1d6c389b-eaa7-4dee-a1cb-d2dfa72160c1");katex.render("\\hat{n}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ is the outward facing normal vector from the boundary.

The boundary conditions $var element = document.getElementById("moose-equation-adeb357b-9f9d-4d3a-b552-9e2157fd9c84");katex.render("T_0", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ and $var element = document.getElementById("moose-equation-5ac84341-0607-463a-9d4c-de15401e0932");katex.render("q_0", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ as well as the diffusivity ( $var element = document.getElementById("moose-equation-f1980d03-b05f-439c-9109-100f3a73ecde");katex.render("\\gamma", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ ) and the source term $var element = document.getElementById("moose-equation-77635ba0-e3aa-4ac5-b471-392730cced21");katex.render("s", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ are assumed to be known. However, these parameters have some associated uncertainty. The effect of the uncertainty on the average temperature and heat flux on the left boundary after one second of simulation time is desired, defined as $var element = document.getElementById("moose-equation-5e3fa325-661b-465c-ab6e-a940f23147cb");katex.render("T_{avg} = \\bar{T}(t=1)", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ and $var element = document.getElementById("moose-equation-72f81a99-577d-4bc3-a920-2e4d803a5406");katex.render("q_{left} = q(t=1,x=0)", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ .

Heat Equation Problem

The first step in performing a parameter study is to create an input file that solves your problems without uncertain parameters. For the example problem this will be done assuming $var element = document.getElementById("moose-equation-6be27c22-a349-481e-bc71-5b9028983ec0");katex.render("\\gamma=1", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ , $var element = document.getElementById("moose-equation-168ddaeb-cff6-49a3-ab14-cba25d851171");katex.render("s=1", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ , $var element = document.getElementById("moose-equation-78802235-9f5e-462a-a735-fa4aba28c6b1");katex.render("T_0=-10", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ , and $var element = document.getElementById("moose-equation-7cd7548d-2399-41dc-9895-58de04188b7b");katex.render("q_0=-100", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ .

The complete input file for this problem is provided in Listing 1. The only item that is atypical from a MOOSE simulation input file is the existence of the Controls block, which here simply creates a SamplerReceiver object. This block is required for the parameter study, but shall be discussed in Transfers Block section.

Listing 1: Complete input file for example heat equation problem to be used in parameter study.

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 10
  ny = 10
[]

[Variables/T]
[]

[Kernels]
  [time]
    type = ADTimeDerivative
    variable = T
  []
  [diff]
    type = ADMatDiffusion
    variable = T
    diffusivity = diffusivity
  []
  [source]
    type = ADBodyForce
    variable = T
    value = 1
    function = 1
  []
[]

[BCs]
  [left]
    type = ADDirichletBC
    variable = T
    boundary = left
    value = -10
  []
  [right]
    type = ADNeumannBC
    variable = T
    boundary = right
    value = -100
  []
[]

[Materials/constant]
  type = ADGenericConstantMaterial
  prop_names = 'diffusivity'
  prop_values = 1
[]

[Executioner]
  type = Transient
  solve_type = NEWTON
  num_steps = 4
  dt = 0.25
[]

[Postprocessors]
  [T_avg]
    type = AverageNodalVariableValue
    variable = T
  []
  [q_left]
    type = ADSideDiffusiveFluxAverage
    variable = T
    boundary = left
    diffusivity = diffusivity
  []
[]

[Controls/stochastic]
  type = SamplerReceiver
[]

[Outputs]
[]

Executing this file using the StochasticToolsApp can be done as follows


cd moose/modules/stochastic_tools/examples/parameter_study
../../stochastic_tools-opt -i diffusion.i

The simulation will perform four timesteps and should result in $var element = document.getElementById("moose-equation-e3a2fd94-1980-4ac2-bd18-26cc693d0112");katex.render("T_{avg}=-5.12", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ and $var element = document.getElementById("moose-equation-9c01ae01-9b9c-4aab-b360-30f28bb75ba4");katex.render("q_{left}=-7.77", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ .

Master Input

To perform a parameter study an input file that will drive the stochastic simulations is required. This file will be referred to as the master input file and is responsible for defining the uncertain parameters, running the stochastic simulations, and collecting the stochastic results. The following sub-sections will step through each portion of the master file to explain the purpose. The complete analysis is discussed in Stochastic Results.

StochasticTools Block

The StochasticTools block sets up the master file to be a driver for stochastic simulations but itself is not performing any sort of simulation. Setting up a master without a simulation itself is the default behavior for this block, as such it is empty in this example.

[StochasticTools]
[]

Distributions and Samplers Blocks

The Distributions block defines the statistical distribution for each of the uncertain parameters ( $var element = document.getElementById("moose-equation-f81ebc08-c005-4c06-a561-f41cd113c9f2");katex.render("T_0", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ , $var element = document.getElementById("moose-equation-ed6ef9b6-bf11-4d01-baba-43eff1c3bfbd");katex.render("q_0", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ , $var element = document.getElementById("moose-equation-2412dff9-ad68-4344-a88b-86f5e8a1d7ea");katex.render("\\gamma", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ , and $var element = document.getElementById("moose-equation-f30ce208-4fd2-465e-86dc-4b8afa7e0858");katex.render("s", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ ) to be defined. For this example, the diffusivity ( $var element = document.getElementById("moose-equation-a67405f0-4825-4e0a-9ca0-8b8d9aba2104");katex.render("\\gamma", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ ) is defined by a uniform distribution, the flux ( $var element = document.getElementById("moose-equation-47072ac3-0faa-49f2-bff2-a863828dc784");katex.render("q_0", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ ) by a three-parameter Weibull, and $var element = document.getElementById("moose-equation-e2e46ed9-a8f5-47fd-bf55-2748fe38fd3b");katex.render("T_0", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ and $var element = document.getElementById("moose-equation-ec3f20c1-fb03-49bd-a0d8-5181db082267");katex.render("s", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ a normal distribution.

The Samplers block defines how the distributions shall be sampled for the parameter study. In this case a Latin hypercube sampling strategy is employed. A total of 5000 samples are being used.

[Samplers]
  [hypercube]
    type = LatinHypercube
    num_rows = 5000
    distributions = 'gamma q_0 T_0 s'
  []
[]

For this problem the Sampler object is setup to run in "batch-restore" mode, which is a mode of operation for memory efficient creation of sub-applications. Please see Stochastic Tools Batch Mode for more information.

The input distributions for each of these four terms are included in Figure 1–Figure 4 for 5000 samples.

[Distributions]
  [gamma]
    type = Uniform
    lower_bound = 0.5
    upper_bound = 2.5
  []
  [q_0]
    type = Weibull
    location = -110
    scale = 20
    shape = 1
  []
  [T_0]
    type = Normal
    mean = -10
    standard_deviation = 1.5
  []
  [s]
    type = Normal
    mean = 1
    standard_deviation = 0.25
  []
[]

Figure 1: Input distribution of diffusivity ( $var element = document.getElementById("moose-equation-5617ed47-8137-4796-92b3-aa17098bed3a");katex.render("\\gamma", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ ).

Figure 2: Input distribution of flux boundary condition ( $var element = document.getElementById("moose-equation-4b44a471-b368-4048-acd1-2168e4d6804f");katex.render("q_0", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ ).

Figure 3: Input distribution of temperature boundary condition ( $var element = document.getElementById("moose-equation-594a83b5-7157-43f1-bc57-67cae5d72df3");katex.render("T_0", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ ).

Figure 4: Input distribution of source term ( $var element = document.getElementById("moose-equation-1a52d7d1-f571-45e6-b675-d06ae8f1e445");katex.render("s", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ ).

Transfers Block

The Transfers block serves two purposes. First, the "parameters" sub-block instantiates a SamplerParameterTransfer object that transfers each row of data from the Sampler object to a sub-application simulation. Within the sub-application the parameters listed in the in this sub-block replace the values in the sub-application. This substitution occurs using the aforementioned SamplerReciever object that exists in the Controls block of the sub-application input file. The receiver on the sub-application accepts the parameter names and values from the SamplerParameterTransfer object on the master application.

The results sub-blocks, serves the second purpose by transferring the quantities of interest back to the master application. In this cases those quantities are postprocessors on the sub-application that compute $var element = document.getElementById("moose-equation-38c4dd3d-9a60-4056-8683-9952bb900313");katex.render("T_{avg}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ and $var element = document.getElementById("moose-equation-eff71f80-92fa-49fd-a831-2c7f514782bf");katex.render("q_{left}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ . These computed results are transferred from the sub-application to a VectorPostprocessor object (as discussed next).

[Transfers]
  [parameters]
    type = SamplerParameterTransfer
    multi_app = runner
    sampler = hypercube
    parameters = 'Materials/constant/prop_values Kernels/source/value BCs/right/value BCs/left/value'
    to_control = 'stochastic'
  []
  [results]
    type = SamplerPostprocessorTransfer
    multi_app = runner
    sampler = hypercube
    to_vector_postprocessor = results
    from_postprocessor = 'T_avg q_left'
  []
[]

VectorPostprocessor Block

The VectorPostprocessor block, as mentioned above, is used to collect the stochastic results. The "results" sub-block instantiates a StochasticResults object, which is designed for this purpose. The resulting object will contain a vector for each of the quantities of interest: $var element = document.getElementById("moose-equation-7acb7f73-3b7c-43a8-bb52-eb9186934cf2");katex.render("T_{avg}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ and $var element = document.getElementById("moose-equation-6f6c53b7-f980-40d3-9a72-c742ec48f853");katex.render("q_{left}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ .

The "samples" sub-block simply is used as a means to output the sample data, it is used here for presenting the results in the next section.

This Statistics object is designed to compute multiple statistics and confidence intervals for each of the input vectors. In this case it computes the mean for $var element = document.getElementById("moose-equation-aa598ece-4481-4a05-bb50-7d36f4468f57");katex.render("T_{avg}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ and $var element = document.getElementById("moose-equation-03b02baf-6a2c-47c2-896f-5107194ab138");katex.render("q_{left}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ vectors in the "results" object as well as the 5% and 95% confidence level intervals. Please refer to the documentation for the Statistics object for further documentation and capabilities for this object.

[VectorPostprocessors]
  [results]
    type = StochasticResults
  []
  [samples]
    type = SamplerData
    sampler = hypercube
  []
  [stats]
    type = Statistics
    vectorpostprocessors = results
    compute = 'mean'
    ci_method = 'percentile'
    ci_levels = '0.05'
  []
[]

Outputs Block

The Outputs block enables the output of the VectorPostprocessor data using the comma separated files.

[Outputs]
  csv = true
  execute_on = 'FINAL'
[]

Stochastic Results

Quantity of Interest Distributions

The resulting distributions are for the quantities of interest: $var element = document.getElementById("moose-equation-49cd1f93-8ca3-4cb8-aefe-7837c6668ebd");katex.render("T_{avg}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ and $var element = document.getElementById("moose-equation-16492a5b-cf12-417f-9c5b-fa30d9c05aa0");katex.render("q_{left}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ are presented in Figure 5 and Figure 6.

Figure 5: Resulting distribution of quantity of interest: $var element = document.getElementById("moose-equation-a6efbdb5-5892-4fc6-975e-cbda2599cc6e");katex.render("T_{avg}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ .

Figure 6: Resulting distribution of quantity of interest: $var element = document.getElementById("moose-equation-9d92c597-13ac-4d4b-9df2-af0f31a6d060");katex.render("q_{left}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ .

Statistics

Listing 2 includes the computed statistics and confidence level intervals as output by the VectorPostprocessor object for the example heat conduction problem with 5000 samples. The results of which can be written as:

$var element = document.getElementById("moose-equation-bd048a6d-e3f9-4006-8135-965eea84a6fd");katex.render("\\overline{T}_{avg} = -22.69,\\,95\\%\\, CI[-22.92, -22.46]", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$

$var element = document.getElementById("moose-equation-5a493dfb-9bf9-4a37-8b42-4ca65990008b");katex.render("\\overline{q}_{left} = -67.11,\\,95\\%\\, CI[-67.78, -66.44]", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$

Listing 2: Resulting statistics and confidence level intervals for the computed quantities of interest.

results_results:T_avg,results_results:q_left,stat_type
-22.689650670305,-67.109783446031,3
-22.916544124128,-67.782251379634,3.05
-22.461848597339,-66.436882255796,3.95

(modules/stochastic_tools/examples/parameter_study/diffusion.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 10
  ny = 10
[]

[Variables/T]
[]

[Kernels]
  [time]
    type = ADTimeDerivative
    variable = T
  []
  [diff]
    type = ADMatDiffusion
    variable = T
    diffusivity = diffusivity
  []
  [source]
    type = ADBodyForce
    variable = T
    value = 1
    function = 1
  []
[]

[BCs]
  [left]
    type = ADDirichletBC
    variable = T
    boundary = left
    value = -10
  []
  [right]
    type = ADNeumannBC
    variable = T
    boundary = right
    value = -100
  []
[]

[Materials/constant]
  type = ADGenericConstantMaterial
  prop_names = 'diffusivity'
  prop_values = 1
[]

[Executioner]
  type = Transient
  solve_type = NEWTON
  num_steps = 4
  dt = 0.25
[]

[Postprocessors]
  [T_avg]
    type = AverageNodalVariableValue
    variable = T
  []
  [q_left]
    type = ADSideDiffusiveFluxAverage
    variable = T
    boundary = left
    diffusivity = diffusivity
  []
[]

[Controls/stochastic]
  type = SamplerReceiver
[]

[Outputs]
[]

(modules/stochastic_tools/examples/parameter_study/master.i)

[StochasticTools]
[]

[Distributions]
  [gamma]
    type = Uniform
    lower_bound = 0.5
    upper_bound = 2.5
  []
  [q_0]
    type = Weibull
    location = -110
    scale = 20
    shape = 1
  []
  [T_0]
    type = Normal
    mean = -10
    standard_deviation = 1.5
  []
  [s]
    type = Normal
    mean = 1
    standard_deviation = 0.25
  []
[]

[Samplers]
  [hypercube]
    type = LatinHypercube
    num_rows = 5000
    distributions = 'gamma q_0 T_0 s'
  []
[]

[MultiApps]
  [runner]
    type = SamplerFullSolveMultiApp
    sampler = hypercube
    input_files = 'diffusion.i'
    mode = batch-restore
  []
[]

[Transfers]
  [parameters]
    type = SamplerParameterTransfer
    multi_app = runner
    sampler = hypercube
    parameters = 'Materials/constant/prop_values Kernels/source/value BCs/right/value BCs/left/value'
    to_control = 'stochastic'
  []
  [results]
    type = SamplerPostprocessorTransfer
    multi_app = runner
    sampler = hypercube
    to_vector_postprocessor = results
    from_postprocessor = 'T_avg q_left'
  []
[]

[VectorPostprocessors]
  [results]
    type = StochasticResults
  []
  [samples]
    type = SamplerData
    sampler = hypercube
  []
  [stats]
    type = Statistics
    vectorpostprocessors = results
    compute = 'mean'
    ci_method = 'percentile'
    ci_levels = '0.05'
  []
[]

[Outputs]
  csv = true
  execute_on = 'FINAL'
[]

(modules/stochastic_tools/examples/parameter_study/master.i)

[StochasticTools]
[]

[Distributions]
  [gamma]
    type = Uniform
    lower_bound = 0.5
    upper_bound = 2.5
  []
  [q_0]
    type = Weibull
    location = -110
    scale = 20
    shape = 1
  []
  [T_0]
    type = Normal
    mean = -10
    standard_deviation = 1.5
  []
  [s]
    type = Normal
    mean = 1
    standard_deviation = 0.25
  []
[]

[Samplers]
  [hypercube]
    type = LatinHypercube
    num_rows = 5000
    distributions = 'gamma q_0 T_0 s'
  []
[]

[MultiApps]
  [runner]
    type = SamplerFullSolveMultiApp
    sampler = hypercube
    input_files = 'diffusion.i'
    mode = batch-restore
  []
[]

[Transfers]
  [parameters]
    type = SamplerParameterTransfer
    multi_app = runner
    sampler = hypercube
    parameters = 'Materials/constant/prop_values Kernels/source/value BCs/right/value BCs/left/value'
    to_control = 'stochastic'
  []
  [results]
    type = SamplerPostprocessorTransfer
    multi_app = runner
    sampler = hypercube
    to_vector_postprocessor = results
    from_postprocessor = 'T_avg q_left'
  []
[]

[VectorPostprocessors]
  [results]
    type = StochasticResults
  []
  [samples]
    type = SamplerData
    sampler = hypercube
  []
  [stats]
    type = Statistics
    vectorpostprocessors = results
    compute = 'mean'
    ci_method = 'percentile'
    ci_levels = '0.05'
  []
[]

[Outputs]
  csv = true
  execute_on = 'FINAL'
[]

(modules/stochastic_tools/examples/parameter_study/master.i)

[StochasticTools]
[]

[Distributions]
  [gamma]
    type = Uniform
    lower_bound = 0.5
    upper_bound = 2.5
  []
  [q_0]
    type = Weibull
    location = -110
    scale = 20
    shape = 1
  []
  [T_0]
    type = Normal
    mean = -10
    standard_deviation = 1.5
  []
  [s]
    type = Normal
    mean = 1
    standard_deviation = 0.25
  []
[]

[Samplers]
  [hypercube]
    type = LatinHypercube
    num_rows = 5000
    distributions = 'gamma q_0 T_0 s'
  []
[]

[MultiApps]
  [runner]
    type = SamplerFullSolveMultiApp
    sampler = hypercube
    input_files = 'diffusion.i'
    mode = batch-restore
  []
[]

[Transfers]
  [parameters]
    type = SamplerParameterTransfer
    multi_app = runner
    sampler = hypercube
    parameters = 'Materials/constant/prop_values Kernels/source/value BCs/right/value BCs/left/value'
    to_control = 'stochastic'
  []
  [results]
    type = SamplerPostprocessorTransfer
    multi_app = runner
    sampler = hypercube
    to_vector_postprocessor = results
    from_postprocessor = 'T_avg q_left'
  []
[]

[VectorPostprocessors]
  [results]
    type = StochasticResults
  []
  [samples]
    type = SamplerData
    sampler = hypercube
  []
  [stats]
    type = Statistics
    vectorpostprocessors = results
    compute = 'mean'
    ci_method = 'percentile'
    ci_levels = '0.05'
  []
[]

[Outputs]
  csv = true
  execute_on = 'FINAL'
[]

(modules/stochastic_tools/examples/parameter_study/master.i)

[StochasticTools]
[]

[Distributions]
  [gamma]
    type = Uniform
    lower_bound = 0.5
    upper_bound = 2.5
  []
  [q_0]
    type = Weibull
    location = -110
    scale = 20
    shape = 1
  []
  [T_0]
    type = Normal
    mean = -10
    standard_deviation = 1.5
  []
  [s]
    type = Normal
    mean = 1
    standard_deviation = 0.25
  []
[]

[Samplers]
  [hypercube]
    type = LatinHypercube
    num_rows = 5000
    distributions = 'gamma q_0 T_0 s'
  []
[]

[MultiApps]
  [runner]
    type = SamplerFullSolveMultiApp
    sampler = hypercube
    input_files = 'diffusion.i'
    mode = batch-restore
  []
[]

[Transfers]
  [parameters]
    type = SamplerParameterTransfer
    multi_app = runner
    sampler = hypercube
    parameters = 'Materials/constant/prop_values Kernels/source/value BCs/right/value BCs/left/value'
    to_control = 'stochastic'
  []
  [results]
    type = SamplerPostprocessorTransfer
    multi_app = runner
    sampler = hypercube
    to_vector_postprocessor = results
    from_postprocessor = 'T_avg q_left'
  []
[]

[VectorPostprocessors]
  [results]
    type = StochasticResults
  []
  [samples]
    type = SamplerData
    sampler = hypercube
  []
  [stats]
    type = Statistics
    vectorpostprocessors = results
    compute = 'mean'
    ci_method = 'percentile'
    ci_levels = '0.05'
  []
[]

[Outputs]
  csv = true
  execute_on = 'FINAL'
[]

(modules/stochastic_tools/examples/parameter_study/master.i)

[StochasticTools]
[]

[Distributions]
  [gamma]
    type = Uniform
    lower_bound = 0.5
    upper_bound = 2.5
  []
  [q_0]
    type = Weibull
    location = -110
    scale = 20
    shape = 1
  []
  [T_0]
    type = Normal
    mean = -10
    standard_deviation = 1.5
  []
  [s]
    type = Normal
    mean = 1
    standard_deviation = 0.25
  []
[]

[Samplers]
  [hypercube]
    type = LatinHypercube
    num_rows = 5000
    distributions = 'gamma q_0 T_0 s'
  []
[]

[MultiApps]
  [runner]
    type = SamplerFullSolveMultiApp
    sampler = hypercube
    input_files = 'diffusion.i'
    mode = batch-restore
  []
[]

[Transfers]
  [parameters]
    type = SamplerParameterTransfer
    multi_app = runner
    sampler = hypercube
    parameters = 'Materials/constant/prop_values Kernels/source/value BCs/right/value BCs/left/value'
    to_control = 'stochastic'
  []
  [results]
    type = SamplerPostprocessorTransfer
    multi_app = runner
    sampler = hypercube
    to_vector_postprocessor = results
    from_postprocessor = 'T_avg q_left'
  []
[]

[VectorPostprocessors]
  [results]
    type = StochasticResults
  []
  [samples]
    type = SamplerData
    sampler = hypercube
  []
  [stats]
    type = Statistics
    vectorpostprocessors = results
    compute = 'mean'
    ci_method = 'percentile'
    ci_levels = '0.05'
  []
[]

[Outputs]
  csv = true
  execute_on = 'FINAL'
[]

(modules/stochastic_tools/examples/parameter_study/master.i)

[StochasticTools]
[]

[Distributions]
  [gamma]
    type = Uniform
    lower_bound = 0.5
    upper_bound = 2.5
  []
  [q_0]
    type = Weibull
    location = -110
    scale = 20
    shape = 1
  []
  [T_0]
    type = Normal
    mean = -10
    standard_deviation = 1.5
  []
  [s]
    type = Normal
    mean = 1
    standard_deviation = 0.25
  []
[]

[Samplers]
  [hypercube]
    type = LatinHypercube
    num_rows = 5000
    distributions = 'gamma q_0 T_0 s'
  []
[]

[MultiApps]
  [runner]
    type = SamplerFullSolveMultiApp
    sampler = hypercube
    input_files = 'diffusion.i'
    mode = batch-restore
  []
[]

[Transfers]
  [parameters]
    type = SamplerParameterTransfer
    multi_app = runner
    sampler = hypercube
    parameters = 'Materials/constant/prop_values Kernels/source/value BCs/right/value BCs/left/value'
    to_control = 'stochastic'
  []
  [results]
    type = SamplerPostprocessorTransfer
    multi_app = runner
    sampler = hypercube
    to_vector_postprocessor = results
    from_postprocessor = 'T_avg q_left'
  []
[]

[VectorPostprocessors]
  [results]
    type = StochasticResults
  []
  [samples]
    type = SamplerData
    sampler = hypercube
  []
  [stats]
    type = Statistics
    vectorpostprocessors = results
    compute = 'mean'
    ci_method = 'percentile'
    ci_levels = '0.05'
  []
[]

[Outputs]
  csv = true
  execute_on = 'FINAL'
[]

Problem Statement
Heat Equation Problem
Master Input
Stochastic Results

Install MOOSE

New Users

Examples and Tutorials

Application Usage

Physics and Syntax

Application Development

Framework Development

MOOSEDocs

Infrastructure

Questions

Information and Tools

INL Applications and Remote Access

Parameter Study

Problem Statement

Heat Equation Problem

Master Input

StochasticTools Block

Distributions and Samplers Blocks

Transfers Block

VectorPostprocessor Block

Outputs Block

Stochastic Results

Quantity of Interest Distributions

Statistics