POF darts: Geometric Adaptive Sampling for Probability of Failure

Mohamed S. Ebeida

Sandia National Laboratories

SIAM conference on Uncertainty Quantification

March, 21st 2014

“Puff and Darts” game, circa 1902, courtesy FCIT.

Mohamed Ebeida

Scott Mitchell Laura

Swiler

“POF Darts” researchers hard at work, circa 2013.

Automated Iterative Analysisof Computational Models

Automate typical “parameter variation” studies with various advanced methods and a generic interface to your

simulationDAKOTA

optimization, sensitivity analysis,parameter estimation,

uncertainty quantification

Computational Model (simulation)•Black box: any code: mechanics, circuits, high energy physics, biology, chemistry

•Semi-intrusive: Matlab, ModelCenter, Python SIERRA multi-physics, SALINAS, Xyce

response metrics

parameters(design, UC,

state)

• Can support experimental testing: examine many accident conditions with computer models, then physically test a few worst-case conditions.

DAKOTA Analysis: Iterating over Parameters of Computational Models

Matlab ODE Epidemic Model

disease kinetic parameters

epidemic size, duration, severity

Xyce, Spice Circuit Model

resistances, via diameters

voltage drop, peak current

Abaqus, Sierra, CM/ CFD Model

material props, boundary, initial

conditions temperature, stress, flow rate

Cantilever Beam Model

load, modulus

stress, displacement

• 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)

We are interested in Estimating Probability of Failure

u1

u2

00.5

11.5

22.5

33.5

44.5

5

30 36 42 48 54 60 66 72 78 84

Temperature [deg C]

Final Temperature Values

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

u1

u2

Main Challenges

1. Accurate Estimation of K

2. Efficient Disk packing in high dimensions

3. The Gap between the lower and Upper bounds

Estimation of K

So far we tried two methods …

1. Approximating K by the gradient at disk center using central difference add an additional cost of 2d function evaluations per disk

2. 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

Efficient disk packing

We have been working for a while solving this problem

A talk about kd-dart for that purpose is Next!

• 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 Meshes

Ebeida, MitchellIMR 2011

• Delaunay Meshes– Protect boundary with random balls– Efficient and Good Delaunay Meshes from Random Points

Ebeida, Mitchell, Davidson, Patney, Knupp, OwensSIAM GD/SPM 2011 Computer Aided Design

• MPS with varying radii– Adaptive and Hierarchical Point Clouds– Variable Radii Poisson-disk sampling

Mitchell, Rand, Ebeida, BajajCCCG 2012

Main Published Results

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

Filling the Gap

100 200 300

Filling the Gap

400 500 600

Filling the Gap

700 1000 10000

Filling the Gap

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 ..

Filling the Gap … Deploying a surrogate

• Smooth Herbie Results

POF = 0.008472

25 (0.013567) 50 (0.006836) 50 (0.00829)

Filling the Gap … Deploying a surrogate

• Non-Smooth Herbie Results

POF = 0.0149532757

50 (0.02294) 100 (0.013995) 150 (0.014816)

Handling functions with multiple response

• Text_book example (Dakota)POF = 0.2888745, 0.1922799

100 (1st response) 100 (2nd response) 200 (2nd response)

Handling functions with multiple response

• Text_book example (Dakota) POF = 0.2888745, 0.1922799

200 (1st response, 0.28900) 200 (2nd response, 0.192373)

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

Thanks! … Questions?