Quantitative Methods for DecisionQuantitative...

Quantitative Methods for Decision-Quantitative Methods for DecisionMaking Under Uncertainty

Sankaran MahadevanVanderbilt University, Nashville, TN

Email: [email protected]

Vanderbilt University reliability-studies.vanderbilt.edu

Email: [email protected]: www.reliability-studies.vanderbilt.edu

Reliability and Risk Engineering, Analysis, and Management

NSF-IGERT Graduate Program at VanderbiltNSF-IGERT Graduate Program at Vanderbilt

Educational and Research ThemesEducational and Research ThemesMultidisciplinary integrationLarge, complex systemsModeling and simulation Model

Devices, Components

Large Systems

Economic, legal, regulatory, and social perspectives

ModelIntegration

UncertaintyCross-cutting methodologies Uncertainty & Risk

Methods

g gUncertainty quantification, propagation Risk quantificationDecision-making under uncertainty

Participants• 42 graduate students (34 Ph.D., 8 M.S.) • 30 professors Engineering, Math, Economics, Business, Psychology,


Medicine• Summer researchers (undergraduates, high school teachers)


Multiple Resources at VanderbiltpStructural/Mechanical

Systems Reliability(S M h d )CRESP -- Multi-university

Nuclear Waste RiskAssessment Consortium

(D. Kosson)

(S. Mahadevan)Institute for Space and

Defense Electronics(R. Schrimpf)

Institute for SoftwareIntegrated Systems

Multidisciplinary Doctoral ACCRE High Performance Integrated Systems(G. Karsai)Program in Reliability/Risk

Engineering & Management

ACCRE High PerformanceComputing Network

Vanderbilt Center forEnvironmental Management

Systems (VCEMS)(M Abkowitz)

Business systemsRisk Assessment and

Management(B Cooil)

VECTOR TransportationResearch Center


(M. Abkowitz)(B. Cooil) Research Center(M. Abkowitz)


Industry & Government SupportINDUSTRY

BoeingGOVERNMENT

U. S. DOD (AFRL, Army, Navy, AFOSR)

y pp

Kellogg, Brown & RootFedexGeneral Motors

( , y, y, )U. S. DOE (EM)U. S. DOT (FHWA) NASA (LaRC MSFC GRC ARC JPL)

ChryslerGeneral ElectricPratt and Whitney

NASA (LaRC, MSFC, GRC, ARC, JPL)DOE Labs (SNL, LANL, INL, SRNL)Nuclear Regulatory Commission

Pratt and WhitneyBell HelicopterMedtronicUnion Pacific

Federal Aviation Administration

Nature of supportS i t hiUnion Pacific

XeroxSummer internshipsCollaborative research projectsAdvisory committee membershipSeminar speakersLaboratories

S th t R h I tit t


pStudent recruitment• Southwest Research Institute

• Transportation Technology Center

Risk Analysis Issues

• System modeling• Physics-based behavior models finite elements, bond graphs• Surrogate models (GP, PC, RBF, NN)• Fault trees, event trees, petri-nets• Bayes networks

• Risk analysisRisk analysis• Multi-level -- Material component subsystem system • Risk variation over space and time• Multi-physics, multi-scale problems

• Data Uncertainty• Sparse data, interval data, measurement uncertainty• Expert opinion• Expert opinion• Heterogeneous information

• Model Uncertainty


y• Model form, model parameters• Errors some deterministic, some stochastic


Materials durability, fatigue, fracture

y g g y g

Systems health diagnosis and prognosisDecision-making under uncertaintyModel uncertainty, calibration, validationy

Information Uncertainty

FEM and Dynamic

Usage monitoring

Model update

Physical Variability

Uncertainty

Uncertainty Modeling

Analysis

Probabilistic Fatigue

Prognosis

Bayesian Updating

Decision making /Risk management

Variability

Model Uncertainty

Damage Mechanism

Analysis

Prognosis

Inspection/Testing

Reliability update


Analysis g p

Rotorcraft Damage Tolerance (FAA)

Rotorcraft mast• Two-diameter hollow cylinder

Elli ti l f k i fill t

Analyses• Model calibration

C S• Elliptical surface crack in fillet region

• Sub modeling technique Accuracy in stress intensity factor

– Calibrate EIFS, model parameters– Estimate model errors in different stages

of modeling

• Model validationAccuracy in stress intensity factor • Model validation

• Prediction uncertainty quantification

• Global sensitivity analysis

• Load monitoring and updating


Sources of Uncertainty

• Physical variability• Loading• Material Properties

• Data uncertainty• Sparseness of data used to quantify material propertiesSparseness of data used to quantify material properties• Output measurement uncertainty (final crack size, detection probability)

• Model uncertainty/errors• Analysis assumptions LEFM planar crack• Analysis assumptions LEFM, planar crack• Finite element discretization errors• Combination of multiple crack modes• Approximation due to surrogate modelApproximation due to surrogate model• Crack growth law model form• Model parameters crack growth model

initial flaw size


Dynamic Bayes Network

Cycle i Cycle i+1

ai

y

ai+1

yεexp

ΔK ΔKθ aN

C, m, n C, m, n

εcg

ΔKth, σf

εcg

ΔKth, σf

A

Vanderbilt University reliability-studies.vanderbilt.eduSlide 8 of 22

Model Validation Metrics

Null Hypothesis H0: y ∈ f(x)

Alternative Hypothesis: H : y ∉ f(x)

Model response xObserved values y Alternative Hypothesis: H1: y ∉ f(x)y

Classical hypothesis testing

H0: E(y) = E(x) Var(y) = Var(x)

H1: E(y) ≠ E(x) Var(y) ≠ Var(x)t testchi-square test

Bayesian hypothesis testing

B F t

( )( )

( )( )

( )( )⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

j

i

j

i

j

i

MPMP

MPMP

MPMP

nobservationobservatio

nobservationobservatioFrom Bayes

theorem “Confidence”


Bayes Factor B

Confidence = B/B+1

Crack size prediction UQ

Std. Dev.

Bayes Factor Confidence


Sensitivity Analysis

• Local Only one uncertainty is considered and allothers are ‘frozen’ at the mean valuesothers are frozen at the mean values

• Global Analyze sensitivity of output over the entiredomain of inputs rather than at mean valuesp

• First order effects (S) & Total effects (ST)

))|(( XYEV ))|(())|(( XYEVXYVE)(

))|(( ~

YVXYEV

S iiXiXi =

)())|((

1)(

))|(( ~~ ~~

YVXYEV

YVXYVE

S iXXiXXT

iiii

i−==

Initial flaw size Crack model error Crack model parameter C


Cementitious barrier PA

Multi-physics analysis

Diffusion of IonsInputs

- Porosity Damage Chemical Reactions

- Porosity- Composition- Modulus- …..

Progression

Calibration of chemical reaction model parameters (equilibrium

t t )Damage Accumulation

constants)

Durability prediction


Extrapolation from Validation to Application

Bayes Network Use for

Cm

• Calibration

• Validation

• Extrapolationab e

g Extrapolation

c

dg

f

MCMC techniques Gibbs sampling Extrapolation scenarios

• Nominal values to Extreme valuesNominal values to Extreme values

• Test conditions to Use conditions

• Validation variables to Decision variables


• Components to System

UQ in system-level prediction

FoamSystem

level

Level 0 Level 1 Level 2

Material characterization

Component level Sub-system level

Joints

characterization level

Joints

Hardware data and photos courtesy of Sandia National Laboratories IncreasesSystem complexity

IncreaseSources of uncertainty


DecreasesAmount of real data

Bayes Network Implementation

f1Yf

1XFoam Joints

f1Y j

1Y

1

f1ε j

1εf1X j

1X

fθf2ε

jθj

2ε

fX jX

f2Y

f2X

j2Y

j2X

Y = Experimental dataX = FEM prediction1 - Level 12 - Level 2

J = JointsF = Foamθ = Calibration parameters X


2 - Level 2S - System

θ = Calibration parametersε = Error terms

sX

Likelihood Approach to Data Uncertainty

• Likelihood function⎞⎛⎟

⎞⎜⎛ mn bi P – distribution parameters

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛∝ ∏∏ ∫

==

m

iiX

n

i aX PxfdxPxfPL

i

i 11

)|( )|()(P – distribution parameters m – point data sizen – interval data size

interval data sparse point data

• Maximum Likelihood Estimate Maximize L(P)

• To account for uncertainty in P ∫

=)(

)()(PL

PLPf

• Two approaches– Family of distributions for X (for every sample of P probability

distribution for X)Si l di t ib ti f X– Single distribution of X

• Can use non-parametric distributions

∫= dPPfPxfxf )()|()(


• Can use non-parametric distributions

Risk Management: System Health Monitoring

• System integration• Integrate reliability/risk methods with SHM• Integrate diagnosis with prognosis

• Rapid diagnosis and prognosis• Derive damage signatures• Derive damage signatures• Qualitative isolation, then damage quantification

• Uncertainty Quantification• Quantify variability, uncertainty, errors • Estimate Confidence in diagnosis/prognosis


• Estimate Confidence in diagnosis/prognosis

Decision-Making Under Uncertainty

Optimization

A1 A2

MDA

A1 A2

Risk Analysis

• Various stages in life cycle design, operations, maintenance• Multiple objectives, MCDA, decision trees, utility-based formulations• Multi-disciplinary systems

Optimization for reliability and robustness• Optimization for reliability and robustness• Include both aleatory and epistemic uncertainties

• Dynamic, network systems y y• Critical facility protection – design of safeguards/detectors• Transportation networks, supply networks, emergency response systems

• System of systems


• Multiple system linkages• Homeland security, military, commercial applications

Fire Satellite System

Target latitude,Target longitudeTarget size[Altit d ]

Analyses

Multi-disciplinary uncertainty propagation

Design optimi ation for reliabilit rob stness

Orbit

[Altitude]

Orbit period, eclipse period

Design optimization for reliability, robustness

Orbit Power

P_ACS

Orbit Period,Satellite velocity,Max slewing angle

_

I_min, I_max

Attitude


[P_tot], [T_tot], [A_sat], [I_min], [I_max]

D i i

System of Systems Decision-Making Under Uncertainty

System Complexity

Decisionmaking

• Risk-informed• Design ComplexityOptimization

• Non-linear• Stochastic• Static/Dynamic

• Monolithic• Family of Systems• System of Systems

• Design• Operations • Utility theory

SOS UncertaintyModeling • Aleatory• Epistemic

• Coefficient based• System dynamics• Agent based

S t d l

• Analysis• Management

• Information bonded

• Fully rational • Bounded rational

• Surrogate model

Risk

Types of

Human in the loop

• Energy bonded• Hybrid


ypSystems

Pandemic Influenza Risk ManagementCIPDSS (LANL)


Conclusion

System risk assessmentContinuing opportunities for methods development

System risk assessmentRisk variation with time and spaceDynamic, multi‐physics, systems of systemsComputational effortComputational effort

Decision‐making under uncertaintyDesign, operations, maintenance, risk managementData collection Model de elopmentData collection, Model developmentEmbedding flexibility

Include data uncertaintyl d lSparse, noisy, qualitative, missing data, intervals, expert opinion

Multi‐scale fusion of heterogeneous information

Include model uncertainty


yValidation, calibration, error estimation, extrapolation

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