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

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Presentation transcript:

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

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

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

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

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

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

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

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

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

Numerical Results: Posterior Distribution Exact Estimate Sensor 1:

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

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

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

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

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

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

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

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