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

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