A Theoretical Framework for Calibration in Computer Models

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 v Supported by NSF DMS and DOE ASCR programs 1

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

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

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

Notation and Assumptions • 5

Parameter identifiability • 6

Physical and Computer Experiments • 8

Fill distance • 9

Kernel Interpolation and Gaussian Process Models • 10

Kennedy-O’Hagan Method • 11

Technical Assumptions • 12

Simplified Kennedy-O’Hagan Method • 13

Reproducing Kernel Hilbert Space • 14

Limiting Value of Likelihood Calibration • 15

Insight on calibration inconsistency • 16

Comparison between two norms • 17

An Illustrative Example • 18

An Illustrative Example (cont’d) • 19

Estimation Efficiency • 20

Convergence Rate • 23

Conclusions • 24