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POF darts: Geometric Adaptive Sampling for Probability of Failure

Scott Mitchell. Mohamed Ebeida. Laura Swiler. “Puff and Darts” game, circa 1902, courtesy FCIT. POF darts: Geometric Adaptive Sampling for Probability of Failure. “POF Darts” researchers hard at work, circa 2013. Mohamed S. Ebeida Sandia National Laboratories

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POF darts: Geometric Adaptive Sampling for Probability of Failure

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  1. Scott Mitchell Mohamed Ebeida Laura Swiler “Puff and Darts” game, circa 1902, courtesy FCIT. POF darts: Geometric Adaptive Sampling for Probability of Failure “POF Darts” researchers hard at work, circa 2013. Mohamed S. Ebeida Sandia National Laboratories SIAM conference on Uncertainty Quantification March, 21st 2014

  2. Automated Iterative Analysisof Computational Models Automate typical “parameter variation” studies with various advanced methods and a generic interface to your simulation DAKOTAoptimization, sensitivity analysis,parameter estimation,uncertainty quantification response metrics parameters(design, UC, state) • Computational Model (simulation) • Black box: any code: mechanics, circuits, high energy physics, biology, chemistry • Semi-intrusive: Matlab, ModelCenter, Python SIERRA multi-physics, SALINAS, Xyce • Can support experimental testing: examine many accident conditions with computer models, then physically test a few worst-case conditions.

  3. DAKOTA Analysis: Iterating over Parameters of Computational Models disease kinetic parameters epidemic size, duration, severity resistances, via diameters voltage drop, peak current Matlab ODE Epidemic Model Xyce, Spice Circuit Model material props, boundary, initial conditions temperature, stress, flow rate stress, displacement load, modulus Abaqus, Sierra, CM/ CFD Model Cantilever BeamModel

  4. We are interested in Estimating Probability of Failure • Device subject to heating (experiment or computational simulation) • Uncertainty in composition/ environment (thermal conductivity, density, boundary), parameterized by u1, …, uN • Response temperature f(u)=T(u1, …, uN)calculated by heat transfer code • Given distributions of u1,…,uN, • UQ methods calculate : • Probability(T≥Tcritical) u2 u1

  5. POF dartsExtending Lipschitzian Optimization to UQ • Let f(u) be Lipschitz continuous • One may sample a point ui, evaluate f(ui) and construct a sphere centered around this point with a radius • This disk would lie entirely in failure or non-failure region • Next sample should be picked outside all prior disks • Finally, volume of failure (red) disks gives a lower bound on POF while volume of non-failure (green) disks gives an upper bound u2 u1

  6. Main Challenges • Accurate Estimation of K • Efficient Disk packing in high dimensions • The Gap between the lower and Upper bounds

  7. Estimation of K So far we tried two methods … • Approximating K by the gradient at disk center using central difference  add an additional cost of 2d function evaluations per disk • Using prior samples to approximate K … less function evaluations, works as good as 1 if not better  Either way if the remaining white space is relatively small and still have budget we increase K and shrink all disks to create more room for new samples

  8. Efficient disk packing We have been working for a while solving this problem  A talk about kd-dart for that purpose is Next!

  9. Main Published Results • First E(n log n) algorithm with provably correct output • Efficient Maximal Poisson-Disk Sampling,Ebeida, Patney, Mitchell, Davidson, Knupp, Owens, SIGGRAPH 2011 • Simpler, less memory, provably correct, faster in practice but no run-time proof • A Simple Algorithm for Maximal Poison-Disk Sampling in High Dimensions,Ebeida, Mitchell, Patney, Davidson, OwensEurographics 2012 • Voronoi Meshes • Sites interior, close to domain boundary are OK, not the dual of a body-fitted Delaunay Mesh • Uniform Random Voronoi MeshesEbeida, MitchellIMR 2011 • Delaunay Meshes • Protect boundary with random balls • Efficient and Good Delaunay Meshes from Random PointsEbeida, Mitchell, Davidson, Patney, Knupp, OwensSIAM GD/SPM 2011  Computer Aided Design • MPS with varying radii • Adaptive and Hierarchical Point Clouds • Variable Radii Poisson-disk samplingMitchell, Rand, Ebeida, BajajCCCG 2012

  10. Main Published Results • Simulation of Propagating fractures • Mesh Generation for modeling and simulation of carbon sequestration processesEbeida, Knupp, Leung, Bishop, MartinezSciDAC 2011 • Hyperplanes for integration, MPS and UQ • K-d darts,Ebeida, Patney, Mitchell, Dalbey, Davidson, Owens, TOG 2014 • Rendering using line darts • High quality parallel depth of field using line samples,Tzeng, Patney, Davidson, Ebeida, Mitchell, OwensHPG 2012 • Reducing Sample size while respecting sizing function • A simple algorithm that replaces 2 disks with one while maintaining coverage and conflict conditions • Sifted DisksEbeida, Mahmoud, Awad, Mohammad, Mitchell, Rand, OwensEG 2013 • MPS with improved Coverage • Using rc < rf • Improving spatial coverage while preserving blue noiseEbeida, Awad, Ge, Mahmoud, Mitchell, Knupp, WeiSIAM GD/SPM 2013  Computer Aided Design

  11. Filling the Gap 100 200 300

  12. Filling the Gap 400 500 600

  13. Filling the Gap 700 1000 10000

  14. Filling the Gap

  15. Filling the Gap … Deploying a surrogate • After we finish the disk packing step, instead of solving a union volume problem (which is challenging by itself. We build a surrogate and evaluate POF directly from it. • Initial results: this new approach reduces the count of function evaluation significantly even with Noisy functions • Very recent … still testing ..

  16. Filling the Gap … Deploying a surrogate • Smooth HerbieResults POF = 0.008472 25 (0.013567) 50 (0.00829) 50 (0.006836)

  17. Filling the Gap … Deploying a surrogate • Non-Smooth HerbieResults POF = 0.0149532757 100 (0.013995) 150 (0.014816) 50 (0.02294)

  18. Handling functions with multiple response • Text_book example (Dakota) POF = 0.2888745, 0.1922799 100 (1st response) 100 (2nd response) 200 (2nd response)

  19. Handling functions with multiple response • Text_book example (Dakota) POF = 0.2888745, 0.1922799 200 (2nd response, 0.192373) 200 (1st response, 0.28900)

  20. Summary and Future work • We developed POF-darts as an extension of Lipschitzian optimization to UQ problems • We are currently investigating more interaction between POF-darts disk packing and Various surrogates within Dakota • We have introduced new sampling techniques based on computational geometry to generate well spaced point sets without suffering from the Curse-Of-Dimensionality • Very few steps in what seems to be a new fruitful path for various applications

  21. Thanks! … Questions?

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