MultilevelEstimators.jl Documentation

For installation instructions, see Installation.

For an example on how to use this package, see Example.

A full description of the functionality is described in the Manual.

Installation

MultilevelEstimators is not added to the Julia package manager (just yet), but, the package can easily be installed by cloning the git repository. From the Julia REPL, type ] to enter Pkg mode and run

pkg> add https://github.com/PieterjanRobbe/MultilevelEstimators.jl

This will install the main functionality.

For automatic generation of reports and figures you will need the Reporter.jl package

pkg> add https://github.com/PieterjanRobbe/Reporter.jl

Finally, to run the example problems, you can install the Multigrid solvers and the package to solve lognormal diffusion problems:

pkg> add https://github.com/PieterjanRobbe/SimpleMultigrid.jl
[...]

pkg> add https://github.com/PieterjanRobbe/NotSoSimpleMultigrid.jl
[...]

pkg> add https://github.com/PieterjanRobbe/LognormalDiffusionProblems.jl
[...]

Features

This package features

• Implementation of Multilevel and Multi-Index (Quasi-)Monte Carlo methods.

• Native support for parallelization on multicore machines.

• Full control of advanced algorithm options such as variance regression, mean square error splitting, ...

• Automatic generation of reports with print-quality figures using Reporter.jl.

Most recent addition to the package is unbiased Multilevel and Multi-Index Monte Carlo with automatic learning of the discrete distribution of samples across all levels or mult-indices. (See index set type U.)

References

The algorithms implemented in this package are loosely based on the following papers:

• Basic Multilevel Monte Carlo method:

  [1] Giles, M. B. Multilevel Monte Carlo Path Simulation. Operations Research 56.3 (2008): 607-617.

  [2] Cliffe, K. A., Giles, M. B., Scheichl, R., and Teckentrup, A. L. Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients. Computing and Visualization in Science, 14.1, (2011): 3-15.

• Variance regression and continuation:

  [3] Collier, N., Haji-Ali, A. L., Nobile, F., von Schwerin, E., and Tempone, R. A Continuation Multilevel Monte Carlo Algorithm. BIT Numerical Mathematics 55.2 (2015): 399-432.

• Quasi-Monte Carlo and Multilevel Quasi-Monte Carlo methods:

  [4] Giles, M. B., and Waterhouse, B. J. Multilevel Quasi-Monte Carlo Path Simulation. Advanced Financial Modelling, Radon Series on Computational and Applied Mathematics (2009): 165-181.

• Multi-Index (Quasi-)Monte Carlo methods:

  [5] Robbe, P., Nuyens, D., and Vandewalle, S. A Multi-Index Quasi-Monte Carlo Algorithm for Lognormal Diffusion Problems. SIAM Journal on Scientific Computing 39.5 (2017): S851-S872.

• Adaptive Multi-Index methods:

  [6] Robbe, P., Nuyens, D., and Vandewalle, S. A Dimension-Adaptive Multi-Index Monte Carlo Method Applied to a Model of a Heat Exchanger. International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing. Springer Proceedings in Mathematics & Statistics 241 (2018): 429-445.

• Unbiased estimation:

  [7] Robbe, P., Nuyens, D. and Vandewalle, S. Recycling Samples in the Multigrid Multilevel (Quasi-)Monte Carlo Method. SIAM Journal on Scientific Computing, to appear (2019).

  [8] Robbe, P., Nuyens, D. and Vandewalle, S. Enhanced Multi-Index Monte Carlo by means of Multiple Semi-coarsened Multigrid for Anisotropic Diffusion Problems. In preparation (2019).