Probabilistic Modeling, Multiscale and Validation

1
Probabilistic Modeling, Multiscale and Validation

• Roger Ghanem
• University of Southern California
• Los Angeles, California

PCE Workshop, USC, August 21st 2008.
2
Outline

• Introduction and Objectives
• Representation of Information
• Model Validation
• Efficiency Issues

3
Introduction

• Objectives
• Determine certifiable confidence in model-based
  predictions
• Certifiable amenable to analysis
• Accept the possibility that certain statements,
  given available resources, cannot be certified.
• Compute actions to increase confidence in model
  predictions change the information available to
  the prediction.
• More experimental/field data, more detailed
  physics, more resolution for numerics
• Stochastic models package information in a manner
  suitable for analysis
• Adapt this packaging to the needs of our
  decision-maker
• Craft a mathematical model that is parameterized
  with respect to the relevant uncertainties.

4
Two meaningful questions
Nothing new here.
What is new is sensor technology
computing technology
Can/must adapt our packaging of information and
knowledge accordingly.
5
Theoretical basisCameron-Martin Theorem
The polynomial chaos decomposition of any
square-integrable functional of the Brownian
motion converges in mean-square as N goes to
infinity.
For a finite-dimensional representation, the
coefficients are functions of the missing
dimensions they are random variables (Slud,1972).
6
Representation of Uncertainty

• The random quantities are resolved as surfaces in
• a normalized space
• These could be, for example
• Parameters in a PDE
• Boundaries in a PDE (e.g. Geometry)
• Field Variable in a PDE

Independent random variables
Multidimensional Orthogonal Polynomials
Dimension of vector reflects complexity of
7
Representation of Uncertainty
Uncertainty due to small experimental database or
anything else.
Uncertainty in model parameters
Dimension of vector reflects complexity of
8
Characterization of Uncertainty

• Galerkin Projections
• Maximum Likelihood
• Maximum Entropy
• Bayes Theorem
• Ensemble Kalman Filter

9
Characterization of UncertaintyMaximum Entropy
Estimation / Spatio-Temporal Processes
Temperature is measured as function of time along
cables. Temperature fluctuations affect sound
speed in the ocean.
10
Characterization of Uncertainty Maximum Entropy
Estimation / Spatio-Temporal Processes
Temperature time histories, , at
various depths.
11
Characterization of UncertaintyMaximum Entropy
Estimation with Histogram Constraints

• Reduced order model of
• KL expansion
  

Spearman Rank Correlation Coefficient is also
matched
A typical plot of marginal pdf for a
Karhunen-Loeve variable.
12
Uncertainty Propagation Stochastic Projection
13
Example Application W76 Foam study
System has 10320 HEX elements. Stochastic block
has 2832 elements.
Foam domain.
1. Modeled as non-stationary random field. 2.
Accounting for random and structured
variations 3. Limited observations are assumed
selected 30 locations on the foam. Limited
statistical observations Correlation estimator
from small sample size interval bounds on
correlation matrix.
Built-up structure with shell, foam and devices.
14
Example Application W76 Foam study

• Polynomial Chaos representation of epistemic
  information
• Constrained polynomial chaos construction
• Radial Basis function consistent spatial
  interpolation
• Cubature integration in high-dimensions

15
Foam studyStatistics of maximum acceleration
Histogram of average of maximum acceleration
16
Foam studyStatistics of maximum acceleration
Plots of density functions of the maximum
acceleration
17
Effect of missing information
18
CDF of calibrated stochastic parameters (3 out of
9 shown)
Estimate 95 probability box
Remarks

• Confidence intervals are due to finite sample
  size.

19
Validation Challenge Problem
Criterion for certifying a design (we would like
to assess it without full-scale experiments
Treated as a random variables
Sample Mean of Sample Variance of
0.0835 0.000830
Remark Based on only 25 samples.
20
Efficiency Issues Basis Enrichment
Solution with 3rd order chaos
Solution with 3rd order chaos and enrichment
Exact solution
21
NANO-RESONATOR WITH RANDOM GEOMETRY
22
NANO-RESONATOR WITH RANDOM GEOMETRY
semiconductor
conductor

• OBJETCIVE
• Determine requirements on manufacturing
  tolerance.
• Determine relationship between manufacturing
  tolerance and performance reliability.

23
APPROACH

• Define the problem on some underlying
  deterministic geometry.
• Define a random mapping from the deterministic
  geometry to the random geometry.
• Approximate this mapping using a polynomial
  chaos decomposition.
• Solve the governing equations using coupled
  FEM-BEM.
• Compare various implementations.

24
TREATMENT OF RANDOM GEOMETRY
Ref Tartakovska Xiu, 2006.
25
GOVERNING EQUATIONS
Elastic BVP for Semiconductor
Interior Electrostatic BVP
Exterior Electrostatic BVP
26
A couple of realizations of solution (deformed
shape and charge distribution)
27
More Significant Probabilistic Results
PDF of Vertical Displacement at tip.
PDF of Maximum Principal Stress at a point.
28
Typical Challenge
29
Comparison of Monte Carlo, Quadrature and Exact
Evaluations of the Element Integrations
30
Using Components of Existing Analysis Software
Only one deterministic solve required. Minimal
change to existing codes. Need iterative
solutions with multiple right hand
sides. Integrated into ABAQUS (not commercially).
31
Implementation Sundance
32
Non-intrusive implementation
33
Implementation Dakota
34
Example Application (non-intrusive)
35
Example Application (non-intrusive)
joints
36
Example Application
37
Example Application
38
Conclusions

• Personal Experience
• Every time I have come close to concluding on
  PCE, new horizons have unfolded in
• Applications
• Models
• Algorithms
• Software