Embed Size (px)
Citation preview
A Statistical Inverse Analysis For Model A Statistical Inverse Analysis For Model CalibrationCalibration
TFSA09, February 5, 2009
Outline:
Introduction and Motivation:
• Why statistical inverse analysis?
Proposed Approach:
Numerical Example
• Bayesian framework
Conclusion and Future Direction
Motivation:Motivation:
Reality
Computational Model
Mathematical Model
Valid
ati
on
Verifica
tio
n
Pre
dic
tio
n
Coding
Assimilation
Qualification
Why Statistical Inverse Analysis?
Input uncertainty
1
Not Always Possible!
Motivation:Motivation:
Reality
Computational Model
Mathematical Model
Valid
ati
on
Verifica
tio
n
Pre
dic
tio
n
Coding
Assimilation
Qualification
Why Statistical Inverse Analysis?
Input uncertainty
1
Motivation: HyShotII Flight ExperimentMotivation: 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 ExperimentMotivation: 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 sensor Temperature sensors
Given noisy measurements of bottom pressure infer the inflow pressure and Mach number and their
uncertainties
Objective:
Supersonic Shock Train: Setup
Problem Setup:
S1 S2 S3 S4 S5 S6 S7 S8
Pressure sensorsBump
Computational Model:
Supersonic Shock Train: Computational Model
• 2D Euler equations• Steady state
Pressure Distribution:
S1 S2 S3 S4 S5 S6 S7 S8
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
