Transcript Probabilistic Modeling: multiscale and validation.

Probabilistic Modeling, Multiscale and Validation
Roger Ghanem
University of Southern California
Los Angeles, California
PCE Workshop, USC, August 21st 2008.
Outline
 Introduction and Objectives
 Representation of Information
 Model Validation
 Efficiency Issues
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.
Two meaningful questions
Nothing new here.
What is new is:
sensor technology
computing technology
Can/must adapt our “packaging” of
information and knowledge accordingly.
Theoretical basis:
Cameron-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).
Representation of Uncertainty
The random quantities are resolved as surfaces in
a normalized space:
Independent random variables
Multidimensional Orthogonal
Polynomials
These could be, for example:
•
Parameters in a PDE
•
Boundaries in a PDE (e.g. Geometry)
•
Field Variable in a PDE
Dimension of vector
complexity of
reflects
Representation of Uncertainty
Uncertainty in model
parameters
Uncertainty due to small
experimental database or
anything else.
Dimension of vector
complexity of
reflects
Characterization of Uncertainty
 Galerkin Projections
 Maximum Likelihood
 Maximum Entropy
 Bayes Theorem
 Ensemble Kalman Filter
Uncertainty Propagation: Stochastic Projection
Example Application: W76 Foam study
System has 10320 HEX elements.
Stochastic block has 2832 elements.
Foam domain.
1. Modeled as non-stationary random field.
Built-up structure with
shell, foam and devices.
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.
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
Foam study
Statistics of maximum acceleration
Histogram of average of maximum acceleration
Foam study
Statistics of maximum acceleration
Plots of density functions of the maximum acceleration
Effect of missing information
CDF of calibrated stochastic parameters (3 out of 9 shown)
Estimate
%95 probability box
Remarks:
 Confidence intervals are due to finite sample size.
Validation Challenge Problem
Criterion for certifying a design (we would like to assess it without fullscale experiments:
Treated as a random variables:
Sample Mean of
Sample Variance of
0.0835
0.000830
Remark: Based on only 25 samples.
Efficiency Issues: Basis Enrichment
Solution with 3rd
order chaos
Solution with 3rd
order chaos and
enrichment
Exact solution
NANO-RESONATOR WITH RANDOM
GEOMETRY
NANO-RESONATOR WITH RANDOM
GEOMETRY
semiconductor
conductor
OBJETCIVE:
1. Determine requirements on manufacturing tolerance.
2. Determine relationship between manufacturing tolerance
and performance reliability.
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.
TREATMENT OF RANDOM GEOMETRY
Ref: Tartakovska & Xiu, 2006.
GOVERNING EQUATIONS
Elastic BVP for Semiconductor:
Interior Electrostatic BVP :
Exterior Electrostatic BVP :
A couple of realizations of solution (deformed
shape and charge distribution)
More Significant Probabilistic Results
PDF of Vertical
Displacement at tip.
PDF of Maximum Principal
Stress at a point.
Typical Challenge
Comparison of Monte Carlo, Quadrature and Exact
Evaluations of the Element Integrations
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).
Implementation: Sundance
Non-intrusive implementation
Implementation: Dakota
Example Application (non-intrusive)
Example Application (non-intrusive)
joints
Example Application
Example Application
Conclusions
 Personal Experience:
Every time I have come close to concluding on
PCE, new horizons have unfolded in
 Applications
 Models
 Algorithms
 Software