1. A Theoretical Framework for Calibration in Computer Models: Parametrization, Estimation and Convergence Properties Rui Tuo and C. F. Jeff Wu CAS and Oak Ridge Nat Labs Georgia Institute of Technology 1  Supported by NSF DMS and DOE ASCR programs

2. Calibration Parameters Consider a computer experiment problem with both computer output and physical response. –Physical experiment has control variables. –Computer/simulation code is deterministic. –Computer input involves control variables and calibration parameters. Calibration parameters represent inherent attributes of the physical system, not observed or controllable in physical experiment, e.g., material porosity or permeability in comp. material design. 2

3. Calibration Problems In many cases, the true value of the calibration parameters cannot be measured physically. Kennedy and O’Hagan (2001) described the calibration problems as: –“Calibration is the activity of adjusting the unknown (calibration) parameters until the outputs of the (computer) model fit the observed data.” 3

4. A Spot Welding Example Consider a spot welding example from Bayarri et al. (2007). Two sheets of metal are compressed by water- cooled copper electrodes under an applied load. Control variables –Applied load –Direct current of magnitude Calibration parameter –Contact resistance at the faying surface 4

5. Notation and Assumptions 5

6. Parameter identifiability 6

7. 7

8. Physical and Computer Experiments 8

9. Fill distance 9

10. Kernel Interpolation and Gaussian Process Models 10

11. Kennedy-O’Hagan Method 11

12. Technical Assumptions 12

13. Simplified Kennedy-O’Hagan Method 13

14. Reproducing Kernel Hilbert Space 14

15. Limiting Value of Likelihood Calibration 15

16. Insight on calibration inconsistency 16

17. Comparison between two norms 17

18. An Illustrative Example 18

19. An Illustrative Example (cont’d) 19

20. Estimation Efficiency 20

21. 21

22. 22

23. Convergence Rate 23

24. Conclusions 24