Monte Carlo Example

Possibly the simplest example of a stochastic analysis is to perform Monte Carlo analysis of a given simulation. That is, solve a single problem and vary parameters within this simulation with a random set of perturbed parameters.

Problem Statement

The first step is to define the simulation to perform, in this case the simulation considered is a 1D transient diffusion equation with Dirichlet boundary conditions on each end of the domain: find uu such that

∂u∂t+∂2u∂x2=0x∈[0,1],\frac{\partial u}{\partial t} + \frac{\partial^2 u}{\partial x^2} = 0 \quad x \in [0,1],

where u(0)=U(0,0.5)u(0) = U(0, 0.5), u(1)=U(1,2)u(1) = U(1,2), and U(a,b)U(a,b) defines a continuous uniform distribution with aa and bb defining the lower and upper limits of the distribution, respectively.

Sub-Application Input

The problem defined above, with respect to the MultiApps system, is a sub-application. The complete input file for the problem is provided in Listing 1. The only item required to enable the stochastic analysis is the Controls block, which contains a SamplerReceiver object, the use of which will be explained in the following section.

Listing 1: Complete input file for executing the transient diffusion problem.

[Mesh]
  type = GeneratedMesh
  dim = 1
  nx = 10
[]

[Variables]
  [u]
  []
[]

[Kernels]
  [diff]
    type = Diffusion
    variable = u
  []
  [time]
    type = TimeDerivative
    variable = u
  []
[]

[BCs]
  [left]
    type = DirichletBC
    variable = u
    boundary = left
    value = 0
  []
  [right]
    type = DirichletBC
    variable = u
    boundary = right
    value = 1
  []
[]

[Executioner]
  type = Transient
  num_steps = 5
  dt = 0.01
  solve_type = PJFNK
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
[]

[Controls]
  [stochastic]
    type = SamplerReceiver
  []
[]

[Postprocessors]
  [left_bc]
    type = PointValue
    point = '0 0 0'
    variable = u
  []
  [right_bc]
    type = PointValue
    point = '1 0 0'
    variable = u
  []
[]

[Outputs]
  csv = true
[]
(contrib/moose/modules/stochastic_tools/test/tests/transfers/monte_carlo/sub.i)

Master Input

The master application, with respect to the MultiApps system, is the driver of the stochastic simulations, by itself it does not perform a solve. The complete input file for the master application is shown in Listing 2, but the import sections will be detailed individually.

First, Distributions for each of the two stochastic boundary conditions are defined.

[Distributions]
  [uniform_left]
    type = Uniform
    lower_bound = 0
    upper_bound = 0.5
  []
  [uniform_right]
    type = Uniform
    lower_bound = 1
    upper_bound = 2
  []
[]
(contrib/moose/modules/stochastic_tools/test/tests/transfers/monte_carlo/monte_carlo.i)

Second, a MonteCarloSampler is defined that utilizes the two distributions and creates the Monte Carlo samples.

[Samplers]
  [sample]
    type = MonteCarlo
    num_rows = 10
    distributions = 'uniform_left uniform_right'
    execute_on = INITIAL # create random numbers on initial and use them for each timestep
  []
[]
(contrib/moose/modules/stochastic_tools/test/tests/transfers/monte_carlo/monte_carlo.i)

Notice, that this sampler only executes on "initial", which means that the random numbers are created once during the initial setup of the problem and not changed again during the simulation.

Next, a SamplerTransientMultiApp object is created. This object creates and runs a sub-application for each sample provided by the sampler object.

[MultiApps]
  [sub]
    type = SamplerTransientMultiApp
    input_files = sub.i
    sampler = sample
  []
[]
(contrib/moose/modules/stochastic_tools/test/tests/transfers/monte_carlo/monte_carlo.i)

Finally, the SamplerParameterTransfer is utilized to communicate the sampler data to the sub-application. The 'parameters' input lists the parameters on the sub-applications to perturb.

[Transfers]
  [sub]
    type = SamplerParameterTransfer
    to_multi_app = sub
    sampler = sample
    parameters = 'BCs/left/value BCs/right/value'
    execute_on = INITIAL
    check_multiapp_execute_on = false
  []
[]
(contrib/moose/modules/stochastic_tools/test/tests/transfers/monte_carlo/monte_carlo.i)

Listing 2: Complete input file for master application for executing Monte Carlo stochastic simulations.

[StochasticTools]
  auto_create_executioner = false
[]

[Distributions]
  [uniform_left]
    type = Uniform
    lower_bound = 0
    upper_bound = 0.5
  []
  [uniform_right]
    type = Uniform
    lower_bound = 1
    upper_bound = 2
  []
[]

[Samplers]
  [sample]
    type = MonteCarlo
    num_rows = 10
    distributions = 'uniform_left uniform_right'
    execute_on = INITIAL # create random numbers on initial and use them for each timestep
  []
[]

[MultiApps]
  [sub]
    type = SamplerTransientMultiApp
    input_files = sub.i
    sampler = sample
  []
[]

[Transfers]
  [sub]
    type = SamplerParameterTransfer
    to_multi_app = sub
    sampler = sample
    parameters = 'BCs/left/value BCs/right/value'
    execute_on = INITIAL
    check_multiapp_execute_on = false
  []
[]

[Executioner]
  type = Transient
  num_steps = 5
  dt = 0.01
[]

[Outputs]
  execute_on = 'INITIAL TIMESTEP_END'
[]
(contrib/moose/modules/stochastic_tools/test/tests/transfers/monte_carlo/monte_carlo.i)