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

2. Automated Iterative Analysis of Computational Models
Automate typical “parameter variation” studies with various advanced methods and a generic interface to your simulation
DAKOTA optimization, 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
Emphasize parameters in, responses out
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
Cantilever Beam Model
load, modulus
stress, displacement
Abaqus, Sierra, CM/ CFD Model
material props, boundary, initial conditions
temperature, stress, flow rate
Xyce, Spice Circuit Model
resistances, via diameters
voltage drop, peak current
Matlab ODE Epidemic Model
disease kinetic parameters
epidemic size, duration, severity

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 darts Extending 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
𝒓 𝒊 = 𝒇 𝒖𝒊 − 𝒇 𝒇𝒂𝒊𝒍𝒖𝒓𝒆 𝑲
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 u1 u2

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, Owens Eurographics 2012
Voronoi Meshes
Sites interior, close to domain boundary are OK, not the dual of a body-fitted Delaunay Mesh
Uniform Random Voronoi Meshes Ebeida, Mitchell IMR 2011
Delaunay Meshes
Protect boundary with random balls
Efficient and Good Delaunay Meshes from Random Points Ebeida, Mitchell, Davidson, Patney, Knupp, Owens SIAM GD/SPM 2011  Computer Aided Design
MPS with varying radii
Adaptive and Hierarchical Point Clouds
Variable Radii Poisson-disk sampling Mitchell, Rand, Ebeida, Bajaj CCCG 2012

10. Main Published Results
Simulation of Propagating fractures
Mesh Generation for modeling and simulation of carbon sequestration processes Ebeida, Knupp, Leung, Bishop, Martinez SciDAC 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, Owens HPG 2012
Reducing Sample size while respecting sizing function
A simple algorithm that replaces 2 disks with one while maintaining coverage and conflict conditions
Sifted Disks Ebeida, Mahmoud, Awad, Mohammad, Mitchell, Rand, Owens EG 2013
MPS with improved Coverage
Using rc < rf
Improving spatial coverage while preserving blue noise Ebeida, Awad, Ge, Mahmoud, Mitchell, Knupp, Wei SIAM 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 Herbie Results
POF =
25 ( )
50 ( )
50 ( )

17. Filling the Gap … Deploying a surrogate
Non-Smooth Herbie Results
POF =
50 ( )
100 ( )
150 ( )

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

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

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?