Uncertainties in Modeling & Simulation -...
Transcript of Uncertainties in Modeling & Simulation -...
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Uncertainties in Modeling & Simulation
Prof. Yan WangWoodruff School of Mechanical Engineering
Georgia Institute of TechnologyAtlanta, GA 30332, [email protected]
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Multiscale Systems Engineering Research Group
Learning Objectives
To understand fundamental concepts of Modeling & Simulation (M&S)
To understand the major sources of epistemic uncertainty in M&S
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How to study a systemSystem
Experiment with actual
system
Experiment with a model of actual system
Physical model
Mathematical model
Analytical solution
Simulation
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What is Modeling?
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Why mathematical modeling?
Advantages
Disadvantages
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An example of modeling
Free fall model
DFdvg
dt m= =
D DUF Fdv cg v
dt m m mF cv+ −
= = = −
1 2DD U
FFdvdt m
dvc v cF dtm
− +=
+=
( )0 2DD U
dvv c
dFFdv
dt m mt
cF γ− +++= =
Resistance/ Drag force
White noise
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Mathematical model
Dependent variable=f(Independent variable)
High dimensional
Parametric systems
“Noisy” systems
( )y f x=
( )1 2, , ...y f x x=
( ),y f x u=
( ), ,y f x u γ=
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Complexity of Mathematical Models
Simple
Complex
Linear
Nonlinear
Algebraic Equation/ Closed-form
Differential Equation
Static
Dynamic
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Model Taxonomy
System model
Deterministic Stochastic
Static Dynamic Static Dynamic
Continuous Discrete Continuous Discrete
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•Molecular Dynamics / Force Field
Modeling & Simulation at Multiple Scales
Various methods used to simulate at different length and time scales
nm μm mm m
pico-snsμs
ms
Length Scales
Time Scales
femto-s
s
•Tight Binding
•Kinetic Monte Carlo
•Finite Element Analysis
•Dislocation Dynamics
•Quantum Monte Carlo•Self-Consistent Field (Hartree-Fock)
•Density Functional Theory
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Simulation-based Designmodel refinement model refinement
mathematical model(first-principles,
empirical, multiscale)
mathematical modelmathematical model(first(first--principles, principles,
empirical, empirical, multiscalemultiscale))
computer simulation
computer computer simulationsimulation
validationvalidationvalidation
Simulation-based designSimulationSimulation--based designbased design
scalable algorithms & scalable algorithms & solverssolvers
data/observationsdata/observationsdata/observations
geometry modeling &geometry modeling &discretizationdiscretization schemesschemes
visualizationvisualizationdata mining/sciencedata mining/science
optimizationoptimizationuncertainty quantificationuncertainty quantification
verificationverificationverificationapproximation approximation error controlerror control
numerical modelnumerical modelnumerical model
parameter inversionparameter inversiondata assimilationdata assimilationmodel/data error controlmodel/data error control
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Two types of “uncertainties”Aleatory uncertainty (variability, irreducible uncertainty, random error)
inherently associated with the randomness/fluctuation (e.g. environmental stochasticity, inhomogeneity of materials, fluctuation of measuring instruments)can only be reduced by taking average of multiple measurements.
Epistemic uncertainty (incertitude, reducible uncertainty, systematic error)
imprecision comes from scientific ignorance, inobservability, lack of knowledge, etc.can be reduced by additional empirical effort (such as calibration).
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Random Error
Determines the precision of any measurement
Always present in every physical measurement
Better apparatus
Better procedure
Repeat
Estimate
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Systematic ErrorDetermines the accuracyof any measurement
May be present in every physical measurement
calibration
uniform or controlled conditions (e.g., avoid systematic changes in temperature, light intensity, air currents, etc.)
Identify & eliminate or reduce
chronological data
24
24.08
24.16
24.24
24.32
0 20 40 60 80 100
trial
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Uncertainties in Modeling & SimulationAleatory Uncertainty:
inherent randomness in the system. Also known as:
• stochastic uncertainty
• variability
• irreducible uncertainty
Epistemic Uncertainty:due to lack of perfect knowledge about the system. Also known as:
• Incertitude
• system error
• reducible uncertainty
Lack of data
Input Uncertainties
Conflicting Information
Conflicting Beliefs
Lack of Introspection
Measurement Errors
Lack of information
about dependency
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Approximations in simulation
From mathematical models to numerical models
Taylor series
Functional analysis
From numerical models to computer codesDiscretization (differentiation, integration)
Searching algorithms (solving equations, optimization)
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Mathematical models Numerical modelsApproximation in Taylor Series
Truncation
( ) ( ) ( ) ( ) ( )( ) ( ) ( )
2
0 0 0
2
0 0
'
'
f x f x f x x x O h
f x f x h O h
= + − +
= + +
( ) ( ) ( ) ( ) ( )2 3
0 0 0
1' ''
2f x f x f x h f x h O h= + + +
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Mathematical models Numerical modelsFunctional Analysis
Convert complex functions into simple and computable ones by transformation in vector spaces
Fourier analysis
Wavelet transform
Polynomial chaos expansion
Spectral methods
Mesh-free methods
…
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Mathematical models Numerical modelsFunctional Analysis
Approximate the original by linear combinations of basis functions ’s as
In a vector space (e.g. Hilbert space) with an infinite number of dimensions
( )f x
0( ) ( )
N
i iif x c x
=≈ ∑ ψ
( )ixψ
0( ) ( )
i iif x c x
∞
== ∑ ψ
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Mathematical models Numerical modelsFunctional Analysis
An inner product is defined as a “projection”in the vector space, such as
Typically we choose orthogonal basis functions ’s such that
for orthonormal basis functions
, : ( ) ( ) ( )f g f x g x W x dx∞
−∞= ∫
,f g
( )ixψ
( , , ), ( ) ( ) ( )
0 ( )i j i j
Constant i j i jx x W x dx
i j
∞
−∞
⎧ ∀ =⎪= = ⎨ ≠⎪⎩∫ψ ψ ψ ψ
1 ( ), ( ) ( ) ( )
i jx x W x dx
∞ ⎧
0 ( )i j i j i j−∞
=⎪= = ⎨ ≠⎪⎩∫ψ ψ ψ ψ
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Mathematical models Numerical modelsFunctional Analysis
The coefficients ’s are computed by
The computable function is
with truncation!
ic
,
,i
i
i i
fc =
ψ
ψ ψ
0( ) ( )
N
i iif x c x
=≈ ∑ ψ
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Numerical Model Computer CodeCompute integrals
QuadratureApproximate the integrand function by a polynomial of certain degree
Approximate the integral by the weighted sum of regularly sampled functional values
• e.g. Simpson’s 3/8 rule
( ) ( ) ( ) ( ) ( )3
0
(3)0 1 2 3
33 3
8
x
x
hI f x dx f x f x f x f x⎡ ⎤≈ = + + +⎣ ⎦∫
x
f(x)
f(x0)
x0
f(x1)f(x2)
f(x3)
x1 x2 x3
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Numerical Model Computer CodeCompute integrals
Monte Carlo simulationLet p(u) denote uniform density function over [a, b]
Let Ui denote i th uniform random variable generated by Monte Carlo according to the density p(u)
Then, for “large” N
Variance reduction (importance sampling) to improve efficiency
1
( ) ( )b N
iia
b af x dx f U
N =
−≈ ∑∫
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Numerical Model Computer CodeCompute derivatives
Finite-divided-difference methodsApproximated derivatives come from Taylor series
• e.g. forward-finite-difference
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2 21 1' 'i i i i i i if x f x f x x x O h f x f x h O h+ += + − + = + +
( ) ( ) ( ) ( )1' i ii
f x f xf x O h
h+ −
= +
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Floating-Point Representation
How does computer represent numbers?
Imperfect worldPerfect world
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Ariane 5
Exploded 37 seconds after liftoff
Cargo worth $500 million
WhyComputed horizontal velocity as floating point number
Converted to 16-bit integer
Worked OK for Ariane 4
Overflowed for Ariane 5• Used same software
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Do you trust your computer?
Single precision: f = 1.172603…
Double precision: f = 1.1726039400531…
Extended precision: f = 1.172603940053178…
Correct one is: f = -0.8273960599468213
yxyyyyxxyyxf
25.5)212111(75.333),( 8462226 ++−−−+=
f(x = 77617, y = 33096) = ?
Rump’s function:
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Another Story
On February 25, 1991
A Patriot missile battery assigned to protect a military installation at Dhahran, Saudi Arabia
But … failed to intercept a Scud missile
28 soldiers died
… an error in computer arithmetic
1101.0 ≠×
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IEEE 574 Standard
If E = 2w–1 and T ≠0, then v is NaN regardless of S.If E = 2w–1 and T =0 , then v = (–1)S×∞.If 1≤E ≤2w–2, then v = (–1)S×2E–bias×(1+21–p×T);
normalized numbers have an implicit leading significand bit of 1.If E =0 and T ≠0, v = (–1)S×2emin×(0+21–p ×T);
denormalized numbers have an implicit leading significand bit of 0.If E =0 and T =0 , then v = (–1)S×0 (signed zero)
where bias=2w–1–1 and emin = 2–2w–1=1–bias
TE
w bits t = p – 1 bits
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Distribution of Values
6-bit IEEE-like formatw = 3 exponent bits
t = 2 fraction/mantissa bits
bias = 3
-15 -10 -5 0 5 10 15Denormalized Normalized Infinity
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Distribution of Values(zoom-in view)
6-bit IEEE-like formatw = 3 exponent bits
t = 2 fraction/mantissa bits
bias = 3
-1 -0.5 0 0.5 1Denormalized Normalized Infinity
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Round-Off Errors
Overflow error – “not large enough”Underflow error – “not small enough”Rounding error – “chopping”
http://www.cs.utah.edu/~zachary/isp/applets/FP/FP.html
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Rounding
Round by chopping (round toward zero)Truncate base expansion after (p-1)st digit
Machine epsilon
Round to nearest (round to even)Last digit is even in case of tie
Machine epsilon
pmachine
−= 1βε
pmachine
−= 1
21 βε
machinexxxfl ε≤−)(
EppE ddddD β
ββββ ⎟⎟
⎠
⎞⎜⎜⎝
⎛++++±=× −
−11
221
0
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Cancellation
Subtraction between two p-digit numbers having the same sign and similar magnitudes yields result with fewer than pdigits.
Significant digits of two numbers cancel.
Despite exactness of result, cancellation often implies serious loss of information.
Relative uncertainty in difference is largely due to previous rounding errors.
011)1()1( =−=−−+ εε
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Special Numbers
Expression Result
0.0 / 0.0 NaN
1.0 / 0.0 Infinity
NaN < 1.0 false
NaN == NaN false
0.0 == -0.0 true
NaN + 1.0 NaN
Infinity + Infinity Infinity
NaN == 1.0 false
-1.0 / 0.0 -Infinity
Infinity + 1.0 Infinity
NaN > 1. 0 false
standard range of values permitted by the encoding (from 1.4e-45 to 3.4028235e+38 for float)
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Floating point Hazards
The result is 2.600000000000001
The result is 0.29 28
This expression does NOT equal to this expression when
0.0 - f -f
! (f >= g)
true
f
f is 0
f < g f or g is NaN
f == f f is NaN
f + g – g g is infinity or NaN
double s=0; for (int i=0; i<26; i++) s += 0.1; System.out.println(s);
double d = 29.0 * 0.01; System.out.println(d); System.out.println((int) (d * 100));
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Comparing Floating Point Numbers
Try to avoid floating point comparison directlyTesting if a floating number is greater than or less than zero is even risky.Instead, you should compare the absolute value of the difference of two floating numbers with some pre-chosen epsilon value, and test if they are "close enough“If the scale of the underlying measurements is unknown, the test “abs(a/b - 1) < epsilon” is more robust.Don’t use floating point numbers for exact values
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Multiscale Systems Engineering Research Group
Uncertainties in M&S
Model errors due to approximations in truncation or sampling
Taylor approximation
Functional analysis
Numerical errors due to floating-point representation
Round-off errors
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Multiscale Systems Engineering Research Group
Summary
Modeling is abstraction
M&S always has approximationsinvolved, which are important sources of epistemic uncertainty.
Computer tricks us
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Multiscale Systems Engineering Research Group
Further ReadingsGoldberg, D. (1991) “What every computer scientist should know about floating-point arithmetic,” ACM Computing Surveys, 23(1), 5-48