UQ PROTOCOLS WITH LEGACY DATA - HOUMAN OWHADI M. ORTIZ, M. MCKERNS, C. SCOVEL A. LASHGARI, B. LI, L. LUCAS, T. SULLIVAN, U. TOPCU, F. THEIL

UQ protocols with legacy data
        Houman Owhadi
     M. Ortiz, M. McKerns, C. Scovel
        A. Lashgari, B. Li, L. Lucas,
      T. Sullivan, U. Topcu, F. Theil

     PSAAP Review Caltech. October 2010.

The current UQ team

Michael Ortiz        Houman Owhadi          Clint Scovel

 Bo Li           Mike McKerns    Tim Sullivan     Florian Theil

The UQ challenge in the certification context

We want to certify that

Problem

 and

G and P at Caltech
                           Unknown exact response function
 Uncertainties= known inputs                             Performance measure(s)
                                   G
Projectile velocity
                                                             Perforation
                                                                area
  Plate thickness

 Plate Obliquity

Year 1 information on P and G

  and

But we know

How do we know that?

Expert opinion

Use empirical mean + margin hit

DATA on demand protocol

Year 1 approach: Plug the information into
McDiarmid’s concentration of measure inequality

Sufficient condition for

DATA on the demand protocol
  First: use the model F to bound the diameter DG
                      Modeled response function   Performance measure(s)
 Known inputs
                               F
Projectile velocity
                                                       Perforation area
Plate thickness

Plate Obliquity

Use an optimization algorithm to compute

  DF −G is a measure of model-system mismatch

Year 2: Generalization to Unknown unknowns and
uncontrollable variables

The computation of the validation diameter requires
separate experiments with identical velocities and UU

Projectile Velocities can be measured but not controlled

UU may be correlated to other input random variables
          Developed stochastic optimization
                algorithms to bound

Year 2: UQ without integral testing

Use hierarchical structures to bound diameters

Year 2: Sharper bounds via domain decomposition
           Gather diameter+mean information in
           sub-domains and plug into McDiarmid

            Automated partition rule

Theorem
If F is continuous then in the limit as the number of iterations
goes to infinity, the upper bound obtained by the domain
decomposition algorithm converges to the p.o.f.

Year 3: Legacy data protocol
   and

But we know

Year 3: Legacy data protocol

Linear Program for McDiarmid sub-diameters
Theorem

Year 3: Optimal bounds on uncertainties
We know

              ⇔

Plug the info into an optimization problem instead of McD

McDiarmid inequality

Reduction of optimization variables

Reduction of optimization variables

Theorem

Explicit Solutions
Theorem   N =2                                (m = 0)

Theorem N = 3, has an explicit solution too

Theorem   Other cases

An important observation
Theorem     N =2                        (m = 0)

Corollary

   Contrary to the sensitivity analysis
   paradigm, input uncertainties
   do not necessarily propagate to output
   uncertainties!

Optimal certification bounds with Legacy data

   and

But we know

Reduction to a finite dimensional optimization problem

Theorem

   This observation, together with McShane's extension theorem,
   leads to a finite-dimensional reduced optimization problem that has
   the same extreme values as the infinite-dimensional problem

What about other types of information?

       Optimal Uncertainty Quantification (2010).
H. Owhadi, C. Scovel, T. Sullivan, M.McKerns and M. Ortiz.
                   arXiv:1009.0679v1

Reduction of optimization variables

Selection of optimal experiments

Experiments

Ex:

Min Overlap= Best experiment

Review team recommendations
  Solve simple UQ problem, obtain solution by traditional
    methods (e.g., Monte Carlo) compare results

 Done that in
                       Optimal Uncertainty Quantification (2010).
      H. Owhadi, C. Scovel, T. Sullivan, M.McKerns and M. Ortiz. arXiv:1009.0679v1

Answered that too (by predicting the outcomes of possible experiments)

Upper-bounds on the probability of non perforation
              with a surrogate model for hypervelocity impact

 One should be careful with such comparisons in presence of asymmetric information

The real question is how to construct a selective information set A.

Review team recommendations
 • Emphasize the precise and limited definition of UU in the
   UQ approach and emphasize that the concept of UU is a
   means to an end rather than the end itself

Done that in
                      Optimal Uncertainty Quantification (2010).
     H. Owhadi, C. Scovel, T. Sullivan, M.McKerns and M. Ortiz. arXiv:1009.0679v1

Review team recommendations

Review team recommendations
• Three scenarios for which predictions under untested
  conditions can be made:
   – Change of geometry with target and projectile materials
     remaining the same and projectile velocity within the range of
     experimental capability
   – Same geometry, target and projectile materials, but with
     projectile velocity marginally outside the experimental capability
   – Same geometry, but different materials; for example simply
     replacing the tantalum of the target and projectile with another
     body-centered cubic metal not yet studied by the Caltech team

          The method has been developed
     (prediction of the outcomes of experiments)
               We just need a well defined information set
    (some information/constraints on G and P in the uncharted domain)
                       in order to get useful bounds

On the origins of the information set

                Experimental data

Physical laws                   Expert Judgment

                 Information/
                 constraints
                 on G and P

Review team recommendations
 • Use established definitions and terminology from the
   larger V&V community when reporting results to the
   outside world

     Okay but the UQ problem has never been well posed
so that terminology is ambiguous and not universally accepted

          UQ is currently at the stage that probability
  theory was before its rigorous formalization by Kolmogorov

                     Optimal Uncertainty Quantification (2010).
    H. Owhadi, C. Scovel, T. Sullivan, M.McKerns and M. Ortiz. arXiv:1009.0679v1

Suggests that the development of a well posed UQ framework
 may lead to non trivial and worthwhile questions and results

Review team recommendations
  • Diversify your portfolio to encompass methodologies for
    prediction of outcome, in addition to certification

Done

Review team recommendations
  • Be specific as to when a computation represents a
    prediction as opposed to part of the V&V effort

Done

          Without priors there is no maximum likelihood and
             predictions are intervals not specific values
           They are obtained by solving finite-dimensional
                (min and max) optimization problems

                        Optimal Uncertainty Quantification (2010).
       H. Owhadi, C. Scovel, T. Sullivan, M.McKerns and M. Ortiz. arXiv:1009.0679v1

Review team recommendations
• Apply same quality V&V practices to smaller length scale
  codes that are applied to the continuum codes

        See talks by M. Ortiz and M. Aivazis