Multi-Level Monte Carlo with Python

MLMCPy - Multi-Level Monte Carlo with Python

General

MLMCPy is an open source Python implementation of the Multi-Level Monte Carlo (MLMC) method for uncertainty propagation. Once a user defines their computational model and specifies the uncertainty in the model input parameters, MLMCPy can be used to estimate the expected value of a quantity of interest to within a specified precision. Support is available to perform the required model evaluations in parallel (if mpi4py is installed) and extensions of the MLMC method are provided to calculate more advanced statistics (e.g., covariance, CDFs).

Dependencies

MLMCPy is intended for use with Python 2.7 and relies on the following packages:

• numpy
• scipy
• mpi4py (optional for running in parallel)
• pytest (optional for running unit tests)

Example Usage

'''
Simple example of propagating uncertainty through a spring-mass model using MLMC.
Estimates the expected value of the maximum displacement of the system when the spring
stiffness is a random variable. See the /examples/spring_mass/from_model/ for more details.
'''

import numpy as np
import sys

from MLMCPy.input import RandomInput
from MLMCPy.mlmc import MLMCSimulator

# Add path for example SpringMassModel to sys path.
sys.path.append('./examples/spring_mass/from_model/spring_mass')
import SpringMassModel

# Step 1 - Define random variable for spring stiffness:
# Need to provide a sampleable function to create RandomInput instance in MLMCPy
def beta_distribution(shift, scale, alpha, beta, size):
return shift + scale*np.random.beta(alpha, beta, size)

stiffness_distribution = RandomInput(distribution_function=beta_distribution,
shift=1.0, scale=2.5, alpha=3., beta=2.)

# Step 2 - Initialize spring-mass models. Here using three levels with MLMC.
# defined by different time steps
model_level1 = SpringMassModel(mass=1.5, time_step=1.0)
model_level2 = SpringMassModel(mass=1.5, time_step=0.1)
model_level3 = SpringMassModel(mass=1.5, time_step=0.01)
models = [model_level1, model_level2, model_level3]

# Step 3 - Initialize MLMC & predict max displacement to specified error (0.1).
mlmc_simulator = MLMCSimulator(stiffness_distribution, models)
[estimates, sample_sizes, variances] = mlmc_simulator.simulate(epsilon=1e-1)


Getting Started

The best way to get started with MLMCPy is to take a look at the scripts in the examples/ directory. A simple example of propagating uncertainty through a spring mass system can be found in the examples/spring_mass/from_model directory. There is a second example that demonstrates the case where a user has access to input-output data from multiple levels of models (rather than a model they can directly evaluate) in the examples/spring_mass/from_data/ directory. For more information, see the source code documentation in docs/MLMCPy_documentation.pdf (a work in progress).

Tests

The tests can be performed by running "py.test" from the tests/ directory to ensure a proper installation.

Developers

UQ Center of Excellence
NASA Langley Research Center
Hampton, Virginia

This software was funded by and developed under the High Performance Computing Incubator (HPCI) at NASA Langley Research Center.

Contributors: James Warner (james.e.warner@nasa.gov), Luke Morrill, Juan Barrientos

Copyright 2018 United States Government as represented by the Administrator of the National Aeronautics and Space Administration. No copyright is claimed in the United States under Title 17, U.S. Code. All Other Rights Reserved.

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