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
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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
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 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
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
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