aurelio-raffa / MultiFidelityABC

MultiFidelityABC

Purpose

This project was created for the 2020 edition of the Bayesian Statistics Course for Mathematical Engineering (Politecnico di Milano).

The goal was to implement and test an Adaptive Multi-Fidelity algorithm for Monte Carlo Markov Chain (MCMC) simulations on Bayesian Inverse Problems, as proposed by Liang Yan and Tao Zhou in the Journal of Computational Physics (volume 381, 15 March 2019, pages 110-128, also available here).

Environment Setup

We use Anaconda to manage our execution environments. Once you have Anaconda installed,

• if your operative system is OSX (64-bit), please open a terminal folder at the /requirements/ directory of this repository and run

  conda create --name myenv --file requirements_osx-64.txt

unfortunately as of today it is not possible to install the fenics package on Windows OS, so please in this case install Windows Subsystem for Linux (WSL) and then refer to the procedure found on the website of the FEniCS Project.

Examples

We have implemented two test cases:

• /source/examples/poisson_multifidelity.py (inspired by this FEniCS tutorial), where we were able to obtain the following results

  true model (MH):
      wall time: .................................. 335.03s
      (avg.) time per iteration: .................. 0.0067s
      effective sample size (min.): ............... 2928 / 50000
      true model eval.: ........................... 50001
  ────────────────────────────────────────────────────────────
  PCE surr. (adap. MH):
      wall time: .................................. 85.62s
      (avg.) time per iteration: .................. 0.0017s
      effective sample size (min.): ............... 2953 / 50000
      true model eval.: ........................... 126
      fitting time: ............................... 0.1264s
      true mod. eval. during fitting: ............. 9
  ────────────────────────────────────────────────────────────
  GPR surr. (adap. MH):
      wall time: .................................. 29.94s
      (avg.) time per iteration: .................. 0.0006s
      effective sample size (min.): ............... 3042 / 50000
      true model eval.: ........................... 113
      fitting time: ............................... 0.1576s
      true mod. eval. during fitting: ............. 20
  

  namely a speedup of a factor 3.9 for the surrogate proposed in the paper versus a speedup factor of more than 11 with our own gaussian process surrogate;
• /source/examples/hyperelastic_multifidelity.py (inspired by the example in this paper), where we obtained

  true model (MH):
      wall time: .................................. 21818.11s
      (avg.) time per iteration: .................. 1.4545s
      effective sample size (min.): ............... 590 / 15000
      true model eval.: ........................... 60004
  ────────────────────────────────────────────────────────────
  PCE surr. (adap. MH):
      wall time: .................................. 117.08s
      (avg.) time per iteration: .................. 0.0078s
      effective sample size (min.): ............... 608 / 15000
      true model eval.: ........................... 240
      fitting time: ............................... 20.0727s
      true mod. eval. during fitting: ............. 64
  ────────────────────────────────────────────────────────────
  GPR surr. (adap. MH):
      wall time: .................................. 98.89s
      (avg.) time per iteration: .................. 0.0066s
      effective sample size (min.): ............... 440 / 15000
      true model eval.: ........................... 240
      fitting time: ............................... 61.6650s
      true mod. eval. during fitting: ............. 200
  

  i.e. a speedup of roughly 186 times for the surrogate presented by the authors versus a speedup of 220 times for our gaussian process surrogate;

the first example is extensively documented in the relative Jupyter notebook (you can find it here);
for the second example we opted for a more traditional script, since its runtime is around 6 hours on a laptop and it leaves little space for interactivity (if you're feeling bored while running it and your device allows for interactive plotting, check out this script enabling real-time traceplots!).

References

• L. Yan and T. Zhou. Adaptive multi-fidelity polynomial chaos approach to bayesian inference in inverse problems. Journal of Computational Physics, Volume 381, 15 March 2019, Pages 110-128, 2018.
• H. P. Langtangen and A. Logg. Solving PDEs in Python - The FEniCS Tutorial I. Springer, 2016.
• M. Hadigola and A. Doostan. Least squares polynomial chaos expansion: A review of sampling strategies. Computer Methods in Applied Mechanics and Engineering, Volume 332, 2017.
• P. Hauseux, J. S. Hale, S. Cotin, S. P.A. Bordas Quantifying the uncertainty in a hyperelastic soft tissue model with stochastic parameters. Applied Mathematical Modelling, Volume 62, Pages 86-102, 2018.