MLMCPy 0.0.1

Multi-Level Monte Carlo with Python

MLMCPy - Multi-Level Monte Carlo with Python

General

MLMCPy is an open source implementation of the Multi-Level Monte Carlo (MLMC) method in Python. It was developed with ease of use in mind.

Dependencies

MLMCPy is intended for use with Python 2.7.

Required packages:

• numpy
• scipy

Optional packages:

• mpi4py (for using with mpirun)
• pytest (for running unit tests)

Tests

Well over one hundred tests are included to thoroughly test MLMCPy.

Example usage

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

'''
This script demonstrates MLMCPy for simulating a spring-mass system with a 
random spring stiffness to estimate the expected value of the maximum 
displacement using multi-level Monte Carlo. Here, we use Model and RandomInput
objects with functional forms as inputs to MLMCPy. See the
/examples/spring_mass/from_data/ for an example of using precomputed data
in files as inputs.
'''

# 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.

mlmc_simulator = MLMCSimulator(stiffness_distribution, models)

[estimates, sample_sizes, variances] = \
    mlmc_simulator.simulate(epsilon=1e-1,
                            initial_sample_sizes=100,
                            verbose=True)

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Authors

Luke Morrill
Georgia Tech

James Warner
UQ Center of Excellence
NASA Langley Research Center
[e-mail]

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

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Notices

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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