SURE 2023 Kickoff

Transcript

1. SUREly Speedy Simulations
   Sou-Cheng Choi and Fred J. Hickernell
   Illinois Institute of Technology
   Dept Applied Math Ctr Interdisc Scientific Comput
   SAS
   Office of Research
   schoi32@iit.edu
   hickernell@iit.edu sites.google.com/iit.edu/fred-j-hickernell
   Looking forward to working with you this summer
   Slides at speakerdeck.com/fjhickernell/sure-kickoff-2023-may-22-talk and
   www.overleaf.com/9233123434vvgphqsccgfc
   Jupyter notebook with computations and figures here Visit us at qmcpy.org
   SURE Kickoff, revised Monday 22nd May, 2023
   

   View Slide

2. Why Quasi-Monte Carlo Is Faster Problems Next Steps References
   Uncertainty in a Cantilevered Beam1
   u(x) = g(Z, x) = beam deflection
   = solution of a differential equation boundary value problem
   x = position
   Z ∼ U[1, 1.2]3
   defines uncertainty in Young’s modulus = the randomness in the problem
   µ(x) = expected or mean value of the beam deflection
   = E[g(Z, x)] =
   [0,1]3
   g(z, x) dz ≈
   1
   n
   n
   i=1
   g(Zi
   , x)
   sample mean or
   Monte Carlo estimate
   µ(end) = 1037 How to choose Z1
   , Z2
   , . . .?
   1M. Parno and L. Seelinger (2022). Uncertainty propagation of material properties of a cantilevered beam. url:
   https://um-bridge-benchmarks.readthedocs.io/en/docs/forward-benchmarks/muq-beam-propagation.html. 2/10
   

   View Slide

3. Why Quasi-Monte Carlo Is Faster Problems Next Steps References
   Simple vs. Quasi-Monte Carlo for the Cantilevered Beam via QMCPy2
   g(Z, x) = beam deflection from DE solver
   Z ∼ U[1, 1.2]3
   defines uncertainty x = position
   µ(x) = E[g(Z, x)] ≈
   1
   n
   n
   i=1
   g(Zi
   , x)
   Monte Carlo estimate
   µ(end) = 1037
   10−2 10−1 100 101 102
   Tolerance, ε
   sec
   min
   hr
   Time (s)
   Lattice
   (ε−1)
   IID
   (ε−2)
   10−2 10−1 100 101 102
   Tolerance, ε
   102
   103
   104
   105
   106
   n
   Lattice
   (ε−1)
   IID
   (ε−2) 0 10 20 30
   Position
   0
   250
   500
   750
   1000
   Mean Deflection
   Cantilevered Beam
   2S.-C. T. Choi, F. J. H., et al. (2023). QMCPy: A quasi-Monte Carlo Python Library (versions 1–1.4). doi: 10.5281/zenodo.3964489. url:
   https://qmcsoftware.github.io/QMCSoftware/. 3/10
   

   View Slide

4. Why Quasi-Monte Carlo Is Faster Problems Next Steps References
   Simple vs. Quasi-Monte Carlo for the Cantilevered Beam via QMCPy2
   g(Z, x) = beam deflection from DE solver
   µ(x) = E[g(Z, x)] ≈
   1
   n
   n
   i=1
   g(Zi
   , x)
   Monte Carlo estimate
   µ(end) = 1037
   10−2 10−1 100
   Tolerance, ε
   sec
   min
   Time (s)
   Lattice
   (ε−1)
   Lattice Parallel
   (ε−1)
   10−2 10−1 100
   Tolerance, ε
   101
   102
   103
   104
   105
   106
   n
   Lattice
   (ε−1)
   Lattice Parallel
   (ε−1)
   0 10 20 30
   Position
   0
   250
   500
   750
   1000
   Mean Deflection
   Cantilevered Beam
   2S.-C. T. Choi, F. J. H., et al. (2023). QMCPy: A quasi-Monte Carlo Python Library (versions 1–1.4). doi: 10.5281/zenodo.3964489. url:
   https://qmcsoftware.github.io/QMCSoftware/. 3/10
   

   View Slide

5. Why Quasi-Monte Carlo Is Faster Problems Next Steps References
   Simple or Independent and Identically Distributed (IID) Monte Carlo
   0 1
   xi,1
   0
   1
   xi,2
   n = 64
   0 1
   xi,1
   0
   1
   xi,2
   n = 128
   0 1
   xi,1
   0
   1
   xi,2
   n = 256
   0 1
   xi,1
   0
   1
   xi,2
   n = 512
   IID Points
   Gaps and clusters, uneven
   4/10
   

   View Slide

6. Why Quasi-Monte Carlo Is Faster Problems Next Steps References
   Low Discrepancy Points Fill Space More Evenly Than IID Points
   0 1
   xi,1
   0
   1
   xi,2
   n = 64
   0 1
   xi,1
   0
   1
   xi,2
   n = 128
   0 1
   xi,1
   0
   1
   xi,2
   n = 256
   0 1
   xi,1
   0
   1
   xi,2
   n = 512
   Lattice Points
   0 1
   xi,1
   0
   1
   xi,2
   n = 64
   0 1
   xi,1
   0
   1
   xi,2
   n = 128
   0 1
   xi,1
   0
   1
   xi,2
   n = 256
   0 1
   xi,1
   0
   1
   xi,2
   n = 512
   Sobol’ Points
   5/10
   

   View Slide

7. Why Quasi-Monte Carlo Is Faster Problems Next Steps References
   Low Discrepancy Points Look “Good” in All Coordinate Projections
   0 1
   xi,1
   0
   1
   xi,2
   0 1
   xi,2
   0
   1
   xi,3
   0 1
   xi,5
   0
   1
   xi,11
   0 1
   xi,14
   0
   1
   xi,15
   Projections of Lattice Points
   0 1
   xi,1
   0
   1
   xi,2
   0 1
   xi,2
   0
   1
   xi,3
   0 1
   xi,5
   0
   1
   xi,11
   0 1
   xi,14
   0
   1
   xi,15
   Projections of Sobol’ Points
   6/10
   

   View Slide

8. Why Quasi-Monte Carlo Is Faster Problems Next Steps References
   Low Discrepancy Points Can Be Randomized
   0 1
   xi,1
   0
   1
   xi,2
   0 1
   xi,1
   0
   1
   xi,2
   0 1
   xi,1
   0
   1
   xi,2
   0 1
   xi,1
   0
   1
   xi,2
   Randomized Lattice Points
   0 1
   xi,1
   0
   1
   xi,2
   0 1
   xi,1
   0
   1
   xi,2
   0 1
   xi,1
   0
   1
   xi,2
   0 1
   xi,1
   0
   1
   xi,2
   Randomized Sobol’ Points
   7/10
   

   View Slide

9. Why Quasi-Monte Carlo Is Faster Problems Next Steps References
   Research Problems for Your Consideration
   Explore quasi-Monte Carlo aka low discrepancy sampling performance using QMCPy on other
   uncertainty quantification problems in the UM-Bridge suite3
   Link QMCPy with popular finite element solver FEniCSx 4
   Implement quasi-Monte Carlo Gaussian random fields (in Julia)
   github.com/PieterjanRobbe/GaussianRandomFields.jl in QMCPy
   Environment, Sustainability, and Government (ESG) investing using QMC
   ▶ comparing Sharpe ratio or other ratios (e.g., Traynor ratio) in Markowitz asset allocation model
   ▶ incorporating backtest and comparing to benchmark returns e.g., S&P500
   ▶ parallelization — important when choosing from a large pool of candidate assets
   (Big) data compression with QMC and deep learning5
   3A. Davis, M. Parno, A. Reinarz, and L. Seelinger (2022). UQ and Model Bridge (UM-Bridge). url:
   https://um-bridge-benchmarks.readthedocs.io/en/docs/.
   4M. E. Rognes A. Logg K. B. Ølgaard and G. N. Wells (2012). “FFC: the FEniCS Form Compiler”. In: Automated Solution of Differential Equations
   by the Finite Element Method. Ed. by K.-A. Mardal A. Logg and G. N. Wells. Vol. 84. Lecture Notes in Computational Science and Engineering.
   Springer. Chap. 11.
   5J. Dick and M. Feischl (2021). “A quasi-Monte Carlo data compression algorithm for machine learning”. In: J. Complexity 67, p. 101587. issn:
   0885-064X. doi: https://doi.org/10.1016/j.jco.2021.101587. url:
   https://www.sciencedirect.com/science/article/pii/S0885064X2100042X. 8/10
   

   View Slide

10. Why Quasi-Monte Carlo Is Faster Problems Next Steps References
    Dip Your Toes in
    All actions can be undone. These actions only need to be done once.
    1 Go to qmcpy.org , where we introduce and explain our QMCPy library
    a Read the blog Why Add Q to MC?
    b Select the menu item GitHub to go to the GitHub repository for QMCPy
    2 The QMCPy GitHub software, documentation, and demos reside at the QMCPy GitHub repository
    a Create a GitHub account if you do not have one
    b Optional: you may wish to install a GitHub desktop client like this or that
    c Scroll down to the README
    d Click on the Contributing
    3 To contribute to QMCPy you need to set up your environment. While at Contributing
    a Open a terminal and set up your programming environment following the instructions.
    b Install Visual Studio Code or your favorite integrated development environment (IDE)
    c In your environment run pip install jupyterlab
    d Move to the SURE2023 branch of QMCPY by typing git checkout SURE2023
    e Type jupyter-lab to start Jupyter Lab
    f Run the notebook used to make the figures for these slides
    4 Join our Speedy Simulations Slack workspace here
    9/10
    

    View Slide

11. Why Quasi-Monte Carlo Is Faster Problems Next Steps References
    References
    A. Logg K. B. Ølgaard, M. E. Rognes and G. N. Wells (2012). “FFC: the FEniCS Form Compiler”. In:
    Automated Solution of Differential Equations by the Finite Element Method. Ed. by
    K.-A. Mardal A. Logg and G. N. Wells. Vol. 84. Lecture Notes in Computational Science and
    Engineering. Springer. Chap. 11.
    Choi, S.-C. T. et al. (2023). QMCPy: A quasi-Monte Carlo Python Library (versions 1–1.4). doi:
    10.5281/zenodo.3964489. url: https://qmcsoftware.github.io/QMCSoftware/.
    Davis, A. et al. (2022). UQ and Model Bridge (UM-Bridge). url:
    https://um-bridge-benchmarks.readthedocs.io/en/docs/.
    Dick, J. and M. Feischl (2021). “A quasi-Monte Carlo data compression algorithm for machine
    learning”. In: J. Complexity 67, p. 101587. issn: 0885-064X. doi:
    https://doi.org/10.1016/j.jco.2021.101587. url:
    https://www.sciencedirect.com/science/article/pii/S0885064X2100042X.
    Parno, M. and L. Seelinger (2022). Uncertainty propagation of material properties of a cantilevered
    beam. url: https://um-bridge-benchmarks.readthedocs.io/en/docs/forward-
    benchmarks/muq-beam-propagation.html.
    10/10
    

    View Slide