Lecture 1 A Prime on Uncertainty Quantification Americo Barbosa da Cunha Junior Universidade do Estado do Rio de Janeiro (UERJ) NUMERICO – Nucleus of Modeling and Experimentation with Computers numerico.ime.uerj.br
[email protected] www.americocunha.org
UNESP Ilha Solteira 22 - 25 de agosto de 2017 Ilha Solteira - SP, Brasil c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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Predictive Science
c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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Four Paradigms for Science
1
Experimental Science thousand years ago empirical observation/description of natural phenomena
2
Theoretical Science last few hundred years generalizations via models/mathematical equations
3
Computational Science last few decades exploration of complex phenomena via computers
4
Data-driven Science today based on big data sets from several sources information extracted via state-of-art statistical methods T. Hey and S. Tansley and K. Tolle (Editors), The Fourth Paradigm: Data-Intensive Scientific Discovery, Microsoft Research, 2009.
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Modeling and Quantification of Uncertainties in Physical Systems
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ally based predictions of physical reality to make informed decisions. Scientifically based predictions are simply foreocesses based on the methods of science: scientific theories (assertions about the underlying reality that brings about a s (knowledge received through the senses or the use of instruments).* d observation—the fundamental pillars of science—can be cast as mathematical models: mathematical constructs that t knowledge of the system in a usable form. Mathematical models are thus abstractions of physical reality. Fundamental en used successfully for millennia. However, mathematical models generally involve parameters that must be “tuned” nts the particular system or phenomenon about which predictions are to be made. These are the moduli, coefficients, nd initial data, etc., that another within a class cted to characterize the rtunately, these model known with great precierial to material, specicase, or they may not generally involve large ved only with sufficient same time, experimenare often fraught with to imperfections in the impossibility of acquirto the problem at hand. of the greatest triumphs ogy, makes possible the of enormous complexn of computer modeling ispensable pillar of sciFigure 1. Imperfect computational modeling: Imperfections in the mathematical models, . Mathematical models incomplete observational data, observations delivered by imperfect instruments, and corrupeate the computational tion of the model itself in the discretization needed for computation all lead to imperfect paths enable to solution via to knowledge. Reproduced from J.T. Oden, “A Brief View of V & V & UQ,” a presentation to the Board on Mathematical Sciences and Their Applications, National Research Council, introduces more errors. T. Oden, Computer October 2009. Predictions with Quantified Uncertainty, Part I. SIAM News, v. 43, 2010. nd uncertainties infect * Picture from1). this reference ased predictions (see Figure imperfections? is hereJrthat old ideas—from philosopher Karland Popper (1902–1994) theologian and mathematician c A. It
Cunha (UERJ) Modeling Quantification ofand Uncertainties in Physical Systems
Computational Predictive Science
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Computational Prediction: the engineer’s perspective
designed system manufacturing process (variabilities)
mathematical modeling (model uncertainty)
real system real input
(real parameters)
real response
model input + model parameters
computational model
model response
(data uncertainty) (uncertain system)
C. Soize, A comprehensive overview of a non-parametric probabilistic approach of model uncertainties for predictive models in structural dynamics. Journal of Sound and Vibration, 288: 623–652, 2005. c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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Model vs Reality Reality is too complex to be understood in every detail Reality is partially understood through models Models are idealizations of reality
Model 6= Reality ”All models are wrong but some are useful” George E. P. Box
Models must capture main features of reality
Model = Caricature of Reality
c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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Model vs Reality Reality is too complex to be understood in every detail Reality is partially understood through models Models are idealizations of reality
Model 6= Reality ”All models are wrong but some are useful” George E. P. Box
Models must capture main features of reality
Model = Caricature of Reality
model c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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Model vs Reality Reality is too complex to be understood in every detail Reality is partially understood through models Models are idealizations of reality
Model 6= Reality ”All models are wrong but some are useful” George E. P. Box
Models must capture main features of reality
Model = Caricature of Reality
model c A. Cunha Jr (UERJ)
reality Modeling and Quantification of Uncertainties in Physical Systems
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For every reality several models are possible
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Modeling and Quantification of Uncertainties in Physical Systems
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For every reality several models are possible Albert Einstein
* Pictures from Google Images. c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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For every reality several models are possible Albert Einstein
* Pictures from Google Images. c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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For every reality several models are possible Albert Einstein
Prof. Samuel da Silva (UNESP Ilha Solteira)
* Pictures from Google Images. c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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For every reality several models are possible Albert Einstein
Prof. Samuel da Silva (UNESP Ilha Solteira)
* Pictures from Google Images. c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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For every reality several models are possible Albert Einstein
Prof. Samuel da Silva (UNESP Ilha Solteira)
Models with different levels of fidelity can be constructed * Pictures from Google Images. c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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Construction of a predictive model It is usually a three-step task: 1st step: physical modeling postulate hypotheses about system 2nd step: mathematical modeling translate hypotheses into equations 3rd step: computational modeling discretization of model equations implementation into a computer code Hypotheses may be translate into equations via: physical laws phenomenological relationships ad-hoc considerations data-driven information c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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Model example 1: vehicle dynamics Real system
* Pictures from (i) Peugeot website; (ii) Google Images; (iii) C.Soize. c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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Model example 1: vehicle dynamics Simple model Real system
m z¨(t)+c z(t)+k ˙ z(t) = F sin(ω t)
* Pictures from (i) Peugeot website; (ii) Google Images; (iii) C.Soize. c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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Model example 1: vehicle dynamics 16
A - Nominal and stochastic computational elastoac
Complex model
Real system
Nominal model - Structure: 978,733 DOF of displacement (left fi - Acoustic cavity: 8,139 DOF of pressure (right Reduced nominal model - Structure : 1722 elastic modes - Acoustic cavity: 57 acoustic modes T Stochastic reduced model constructed with the nonparametric of both system parameters uncertainties and model uncertain T cavity and vib - Uncertainties: structure, acoustic
ρ¨ u + c u˙ = ∇ · σ σ = σ
2 = ∇u + ∇u σ = C : Uncertainties and Stochastic C. SOIZE, Universit´ e Paris-Est + b.c./i.c.
* Pictures from (i) Peugeot website; (ii) Google Images; (iii) C.Soize. c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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Model example 2: epidemiological forecast βhIv
Sh
αh
Eh
γ
Ih
Rh
Human Population
δ
Sv
(βvIh)1N
Ev
αv
δ
Iv
δ
δ
Vector Population
dSh dt dEh dt dIh dt
= −βh Sh Iv
= βh Sh Iv − αh Eh
dSv
dEv dt
= αh Eh − γ Ih
dIv dt
dRh dt
= γ Ih
+ initial conditions
= δ − βv Sv
dt
dC dt
= βv Sv
Ih N
Ih N
− δ Sv
− (δ + αv ) Ev
= αv Ev − δ Iv
= αh Eh
E. Dantas, M. Tosin and A. Cunha Jr, Calibration of a SEIR–SEI epidemic model to describe Zika virus outbreak in Brazil, 2017 (under review). c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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Uncertainty Quantification
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Uncertainty Quantification (UQ) What is Uncertainty Quantification? Uncertainty quantification (UQ) is multidisciplinar area involving engineering, mathematics, and computer science. It deals with quantitative characterization and reduction of uncertainties in applications. Why Uncertainty Quantification? Decision Making Some kind of certification is essential for high-risk decisions Model Validation Verify model limitations is necessary for accurate predictions Robust Design/Optimization Devices with low sensitivity to variations are often required c A. Cunha Jr (UERJ)
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In a simplistic way UQ aims to:
3
simulation
system response
2 1 0 −1 −2 −3 0
1
2
3
4
5
6
input
c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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In a simplistic way UQ aims to:
3
simulation experiment
system response
2 1 0 −1 −2 −3 0
1
2
3
4
5
6
input
c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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In a simplistic way UQ aims to:
3
simulation experiment
system response
2 1 0 −1 −2 −3 0
1
2
3
4
5
6
input
c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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In a simplistic way UQ aims to:
3
simulation experiment confidence band
system response
2 1 0 −1 −2 −3 0
1
2
3
4
5
6
input
c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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In a simplistic way UQ aims to: (i) add error bars to simulations 3
simulation experiment confidence band
system response
2 1 0 −1 −2 −3 0
1
2
3
4
5
6
input
c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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In a simplistic way UQ aims to: (i) add error bars to simulations 3
simulation experiment confidence band
system response
2 1 0 −1 −2 −3 0
1
2
3
4
5
6
input
experimental
simulation
OPTIMAL
c A. Cunha Jr (UERJ)
experimental
simulation
VIOLATION
Modeling and Quantification of Uncertainties in Physical Systems
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In a simplistic way UQ aims to: (i) add error bars to simulations 3
simulation experiment confidence band
system response
2 1 0 −1 −2 −3 0
1
2
3
4
5
6
input
(ii) define a precise notion of validated model experimental
simulation
OPTIMAL
c A. Cunha Jr (UERJ)
experimental
simulation
VIOLATION
Modeling and Quantification of Uncertainties in Physical Systems
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UQ Vocabulary
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Modeling and Quantification of Uncertainties in Physical Systems
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Errors vs Uncertainties errors and uncertainties in UQ 6= errors and uncertainties in metrology Uncertainties: associated to variabilities intrinsic to the system of interest and potential lack of knowledge about the physics (aleatory or epistemic) Errors: associated to the translation of a mathematical model into a computational model/code (discretization, round-off, bugs)
G. Iaccarino Quantification of Uncertainty in Flow Simulations Using Probabilistic Methods, VKI Lecture Series, Stanford University, 2008 c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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Errors vs Uncertainties errors and uncertainties in UQ 6= errors and uncertainties in metrology Uncertainties: associated to variabilities intrinsic to the system of interest and potential lack of knowledge about the physics (aleatory or epistemic) Errors: associated to the translation of a mathematical model into a computational model/code (discretization, round-off, bugs) Uncertainties: physical nature
G. Iaccarino Quantification of Uncertainty in Flow Simulations Using Probabilistic Methods, VKI Lecture Series, Stanford University, 2008 c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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Errors vs Uncertainties errors and uncertainties in UQ 6= errors and uncertainties in metrology Uncertainties: associated to variabilities intrinsic to the system of interest and potential lack of knowledge about the physics (aleatory or epistemic) Errors: associated to the translation of a mathematical model into a computational model/code (discretization, round-off, bugs) Uncertainties: physical nature Errors: mathematical nature G. Iaccarino Quantification of Uncertainty in Flow Simulations Using Probabilistic Methods, VKI Lecture Series, Stanford University, 2008 c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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Aleatory Uncertainties (data uncertainties)
Characteristics: induced by variabilities in real system arises naturally from observations impossible to be eliminated/reduced (irreducible) Examples: geometric dimensions material properties measurement noise etc G. Iaccarino Quantification of Uncertainty in Flow Simulations Using Probabilistic Methods, VKI Lecture Series, Stanford University, 2008 c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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Epistemic Uncertainties (model uncertainties) Characteristics: induced by lack of knowledge of physics arises from modeling hypotheses can be reduced/eliminated (reducible) Examples: geometric form boundary conditions constitutive equations turbulence models etc G. Iaccarino Quantification of Uncertainty in Flow Simulations Using Probabilistic Methods, VKI Lecture Series, Stanford University, 2008 c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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Verification and Validation (V&V)
Verification Are we solving the equation right?
Validation Are we solving the right equation?
G. Iaccarino Quantification of Uncertainty in Flow Simulations Using Probabilistic Methods, VKI Lecture Series, Stanford University, 2008 c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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Verification and Validation (V&V)
Verification Are we solving the equation right? It is an exercise in mathematics. Validation Are we solving the right equation?
G. Iaccarino Quantification of Uncertainty in Flow Simulations Using Probabilistic Methods, VKI Lecture Series, Stanford University, 2008 c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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Verification and Validation (V&V)
Verification Are we solving the equation right? It is an exercise in mathematics. Validation Are we solving the right equation? It is an exercise in physics.
G. Iaccarino Quantification of Uncertainty in Flow Simulations Using Probabilistic Methods, VKI Lecture Series, Stanford University, 2008 c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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An example in V&V
y
y0
c A. Cunha Jr (UERJ)
g
Modeling and Quantification of Uncertainties in Physical Systems
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An example in V&V
y
Mathematical model:
y0
g
m y¨ (t) = −m g y˙ (0) = v0 y (0) = y0
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Modeling and Quantification of Uncertainties in Physical Systems
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Verification of the equation solution Mathematical model m y¨ (t) = −m g + initial conditions
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Modeling and Quantification of Uncertainties in Physical Systems
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Verification of the equation solution Mathematical model m y¨ (t) = −m g + initial conditions Numerical (Runge-Kutta) yn+1 = yn +
hn (k1 +2 k2 +2 k3 +k4 ) 6
tn+1 = tn +hn
c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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Verification of the equation solution Mathematical model m y¨ (t) = −m g
solution verification 2500
+ initial conditions
yn+1 = yn +
hn (k1 +2 k2 +2 k3 +k4 ) 6
tn+1 = tn +hn
heigth (m)
2000
Numerical (Runge-Kutta)
1500 1000 500 simulation 0 0
5
10
15
20
time
c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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Verification of the equation solution Mathematical model m y¨ (t) = −m g
solution verification 2500
+ initial conditions
yn+1 = yn +
hn (k1 +2 k2 +2 k3 +k4 ) 6
tn+1 = tn +hn
heigth (m)
2000
Numerical (Runge-Kutta)
1500 1000 500 simulation 0 0
Reference (analytical)
5
10
15
20
time
1 y (t) = − g t 2 + v0 t + y0 2
c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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Verification of the equation solution Mathematical model m y¨ (t) = −m g
solution verification 2500
+ initial conditions
2000
yn+1
hn = yn + (k1 +2 k2 +2 k3 +k4 ) 6
heigth (m)
Numerical (Runge-Kutta)
1500 1000 500
tn+1 = tn +hn
simulation analytical 0
Reference (analytical)
0
5
10
15
20
time
1 y (t) = − g t 2 + v0 t + y0 2
c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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Verification of the equation solution Mathematical model m y¨ (t) = −m g
solution verification
−12
1
+ initial conditions
yn+1
hn = yn + (k1 +2 k2 +2 k3 +k4 ) 6
0.8
absolute error
Numerical (Runge-Kutta)
x 10
0.6 0.4 0.2
tn+1 = tn +hn
0
Reference (analytical)
0
5
10
15
20
time
1 y (t) = − g t 2 + v0 t + y0 2
c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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Verification of the equation solution Mathematical model m y¨ (t) = −m g
solution verification
−12
1
+ initial conditions
yn+1
hn = yn + (k1 +2 k2 +2 k3 +k4 ) 6
0.8
absolute error
Numerical (Runge-Kutta)
x 10
0.6 0.4 0.2
tn+1 = tn +hn
0
Reference (analytical)
0
5
10
15
20
time
1 y (t) = − g t 2 + v0 t + y0 2 The model equation is well solved c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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Validation of the model model validation 2500
Toy model: m y¨ (t) = −m g
heigth (m)
2000 1500 1000
y˙ (0) = v0
500
y (0) = y0
0
toy model
c A. Cunha Jr (UERJ)
0
5
10
15
20
time
Modeling and Quantification of Uncertainties in Physical Systems
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Validation of the model model validation 2500
Toy model: m y¨ (t) = −m g
heigth (m)
2000 1500 1000
y˙ (0) = v0
500
y (0) = y0
0
toy model experiment 0
5
10
15
20
time
c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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Validation of the model model validation 2500
Toy model: m y¨ (t) = −m g
heigth (m)
2000 1500 1000
y˙ (0) = v0
500
y (0) = y0
0
toy model experiment 0
5
10
15
20
time
The mathematical model is not representative
c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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Validation of the model model validation 2500
Improved model: 2 1 m y¨ (t) = −m g + ρ A CD y˙ (t) 2
heigth (m)
2000 1500 1000
y˙ (0) = v0
500
y (0) = y0
0
toy model experiment 0
5
10
15
20
time
c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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Validation of the model model validation 2500
Improved model: 2 1 m y¨ (t) = −m g + ρ A CD y˙ (t) 2
heigth (m)
2000 1500 1000
y˙ (0) = v0
500
y (0) = y0
0
improved model toy model experiment 0
5
10
15
20
time
c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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Validation of the model model validation 2500
Improved model: 2 1 m y¨ (t) = −m g + ρ A CD y˙ (t) 2
heigth (m)
2000 1500 1000
y˙ (0) = v0
500
y (0) = y0
0
improved model toy model experiment 0
5
10
15
20
time
An improved model enhance the predictions
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Modeling and Quantification of Uncertainties in Physical Systems
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Calibration of the model
Forward Problem
model parameters p
model observable φ(p)
computational model
p∗
φ1, φ2, · · · , φM
fitting parameters
field observations
Inverse Problem
* Left pictures from Introduction to stormwater modeling, Minnesota Stormwater Manual c A. Cunha Jr (UERJ)
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Calibration vs Validation
* Picture from NHI course on Travel Demand Forecasting c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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Sensitivity Analysis 6= UQ Sensitivity Analysis: based on derivatives ∂ui /∂ξj measure response sensitivity to changes on a certain input characterization of input variabilities is not required (deterministic method) valid only for a fixed (nominal) set of parameters (local analysis) large sensibility =⇒ 6 large uncertainties
Uncertainty Quantification: based on propagation of uncertainties identify overall output uncertainty characterization of input variabilities is required (stochastic method) G. Iaccarino Quantification of Uncertainty in Flow Simulations Using Probabilistic Methods, VKI Lecture Series, Stanford University, 2008 c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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Computers & Uncertainties
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Modeling and Quantification of Uncertainties in Physical Systems
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Computation under uncertainty
ξ
computational model
input
u = M(ξ) output
1
Data Representation: characterize inputs uncertainties
2
Uncertainty Propagation: quantify output uncertainties
3
Certification: establish acceptable levels of uncertainty
G. Iaccarino Quantification of Uncertainty in Flow Simulations Using Probabilistic Methods, VKI Lecture Series, Stanford University, 2008 c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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Computation under uncertainty
data representation
ξ
computational model
input
u = M(ξ) output
1
Data Representation: characterize inputs uncertainties
2
Uncertainty Propagation: quantify output uncertainties
3
Certification: establish acceptable levels of uncertainty
G. Iaccarino Quantification of Uncertainty in Flow Simulations Using Probabilistic Methods, VKI Lecture Series, Stanford University, 2008 c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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Computation under uncertainty
uncertainty propagation
ξ
computational model
input
u = M(ξ) output
1
Data Representation: characterize inputs uncertainties
2
Uncertainty Propagation: quantify output uncertainties
3
Certification: establish acceptable levels of uncertainty
G. Iaccarino Quantification of Uncertainty in Flow Simulations Using Probabilistic Methods, VKI Lecture Series, Stanford University, 2008 c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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Computation under uncertainty
certification
ξ
computational model
input
u = M(ξ) output
1
Data Representation: characterize inputs uncertainties
2
Uncertainty Propagation: quantify output uncertainties
3
Certification: establish acceptable levels of uncertainty
G. Iaccarino Quantification of Uncertainty in Flow Simulations Using Probabilistic Methods, VKI Lecture Series, Stanford University, 2008 c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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Computation under uncertainty
data representation
uncertainty propagation
ξ
computational model
input
certification
u = M(ξ) output
1
Data Representation: characterize inputs uncertainties
2
Uncertainty Propagation: quantify output uncertainties
3
Certification: establish acceptable levels of uncertainty
G. Iaccarino Quantification of Uncertainty in Flow Simulations Using Probabilistic Methods, VKI Lecture Series, Stanford University, 2008 c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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Data Representation How to model/characterize uncertainties? Probabilistic approaches: Parametric probabilistic approach Nonparametric probabilistic approach
Nonprobabilistic approaches: Interval analysis Evidency theory Fuzzy logic
Uncertainty characterization must be based on available information experimental data theoretical arguments etc G. Iaccarino Quantification of Uncertainty in Flow Simulations Using Probabilistic Methods, VKI Lecture Series, Stanford University, 2008 c A. Cunha Jr (UERJ)
Modeling and Quantification of Uncertainties in Physical Systems
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Uncertainty Propagation Given: computational model and input uncertainty Find: output uncertainty Uncertainty Propagation
computational model
input uncertainty
output uncertainty
The most complex and computationally intensive step Propagation technique depends on representation approach G. Iaccarino Quantification of Uncertainty in Flow Simulations Using Probabilistic Methods, VKI Lecture Series, Stanford University, 2008 c A. Cunha Jr (UERJ)
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Certification
Specify levels of reliability for predictions: confidence interval/band probability of events of interest boxplot etc
G. Iaccarino Quantification of Uncertainty in Flow Simulations Using Probabilistic Methods, VKI Lecture Series, Stanford University, 2008 * Pictures from Public Domain and PennState course STAT414/415 c A. Cunha Jr (UERJ)
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Key points about UQ
UQ is essential for applied tasks such as decision making, model validation, and robust design UQ is a new discipline with much theory to be developed Until today there is no consensus on UQ basic vocabulary Certification for numerical simulations is a worldwide trend
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References A. Cunha Jr, Modeling and quantification of physical systems uncertainties in a probabilistic framework, In: S. Ekwaro-Osire; A. C. Gon¸calves; F. M. Alemayehu (Org.), Probabilistic Prognostics and Health Management of Energy Systems, Springer International Publishing, p. 127-156, 2017. http://dx.doi.org/10.1007/978-3-319-55852-3_8 R. C. Smith, Uncertainty Quantification: Theory, Implementation, and Applications, SIAM, 2013. T. J. Sulivan, Introduction to Uncertainty Quantification, Springer International Publishing, 2015. C. Soize, Uncertainty Quantification: An Accelerated Course with Advanced Applications in Computational Engineering, Springer International Publishing, 2017. C. Soize Stochastic Models of Uncertainties in Computational Mechanics, Amer Society of Civil Engineers, 2012. R. Ghanem, D. Higdon and H. Owhadi (Editors) Handbook of Uncertainty Quantification, Springer International Publishing, 2017.
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