1. A Statistical Inverse Analysis For Model Calibration TFSA09, February 5, 2009

2. Outline: Introduction and Motivation: Why statistical inverse analysis? Proposed Approach: Numerical Example Bayesian framework Conclusion and Future Direction

3. Motivation: Reality Computational Model Mathematical Model Validation Verification Prediction Coding Assimilation Qualification Why Statistical Inverse Analysis? Input uncertainty 1 Not Always Possible!

4. Motivation: Reality Computational Model Mathematical Model Validation Verification Prediction Coding Assimilation Qualification Why Statistical Inverse Analysis? Input uncertainty 1

5. Motivation: HyShotII Flight Experiment UQ Challenges: No direct measurements of: Flight Mach number Angle of attack Vehicle altitude Objective: Validation of computational tools against flight measurements Model uncertainties Photo: Chris Stacey, The University of Queensland

6. Motivation: HyShotII Flight Experiment Inverse Analysis Objective: Given noisy measurements of pressure and temperature infer: Flight Mach number Angle of attack Vehicle altitude and their uncertainties. Combustor pressure sensors Intake pressure sensors Nose pressure sensorTemperature sensors

7. Given noisy measurements of bottom pressure infer the inflow pressure and Mach number and their uncertainties Objective: Supersonic Shock Train: Setup Problem Setup: S1S1 S2S2 S3S3 S4S4 S5S5 S6S6 S7S7 S8S8 Pressure sensors Bump

8. Computational Model: Supersonic Shock Train: Computational Model 2D Euler equations Steady state Pressure Distribution: S1S1 S2S2 S3S3 S4S4 S5S5 S6S6 S7S7 S8S8

9. Supersonic Shock Train: Bayesian Inverse Analysis Measurement Uncertainties Model predictionObservation Prior distribution to parameters Bayes’ Formula Bayesian estimate Posterior distribution of parameters

10. Numerical Results: Posterior Distribution Exact Estimate Sensor 1:

11. Exact Estimate Sensors 1,2: Numerical Results: Posterior Distribution

12. Exact Estimate Sensors 1,2,3: Numerical Results: Posterior Distribution

13. Exact Estimate Sensors 1,…,4: Numerical Results: Posterior Distribution

14. Exact Estimate Sensors 1,…,5: Numerical Results: Posterior Distribution

15. Exact Estimate Sensors 1,…,6: Numerical Results: Posterior Distribution

16. Exact Estimate Sensors 1,…,7: Numerical Results: Posterior Distribution

17. Exact Estimate Sensors 1,…,8: Numerical Results: Posterior Distribution

18. Conclusion and Future Directions: We presented a statistical inverse analysis: Infer inflow conditions and their uncertainties based on noisy response measurements Use the existing deterministic solvers HyShotII flight conditions based on the available flight data More challenging applications Combustor pressure sensors Intake pressure sensors Nose pressure sensor Temperature sensors