Uncertainty quantification of structures with unknown probabilistic dependency Robert L. Mullen

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Uncertainty quantification of structures with
unknown probabilistic dependencyRobert L. Mullen
Seminar NIST April 3th 2015
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Rafi MuhannaSchool of Civil and Environmental
Engineering Georgia Institute of Technology
Atlanta, GA 30332, USA
Dr.M.V.Rama Rao Department of Civil
Engineering Vasavi College of Engineering,
Hyderabad 500 031 INDIA
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Imprimers
Ivo Babushka Vladik Kreinovich Ray Moore A.
Neumier Scott Ferson
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Motivation A toy problem.
A truss structure (extendable to general
FEA) Loading given in terms of random variables
(RV) (extendable to parameters given by RV) p
f (P1, P2, P3) vs p f1(P1), pf2(P2),
pf3(P3) What can one compute about structural
response when dependency between p1, p2, p3 is
unknown?
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Common Engineering Solution Assume
independence between P1, P2, P3 use Monte Carlo
simulation and forget about it.
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Common Engineering Solution Assume
independence between P1, P2, P3 and forget about
it.
Assuming independence can result in significant
underestimation of risk (Ferson et al. 2004).
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• Outline
• Introduction
• Representing dependency
• Parametric models (and associated
  statistical measures)
• Copulas
• Representation of Ignorance
• Any dependency bounds
• Numerical methods for P-box calculations
• Results
• Conclusion

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Multi-dimensional PDF
Two dimensional
N dimensional
Equation images from Wikipedia http//en.wikipedia
.org/wiki/Multivariate_normal_distribution
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Realizations of Gaussian RV
Calculated using Matlab program copulaa1 Robert
L Mullen 2012
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Sum of two standard Gaussian RV
Calculated using Matlab program copula1 Robert L
Mullen 2012
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Other distributions can be constructed by
transforming marginal distribution
Theorem 2.1. (from Devroye , Springer 1986) Let
F be a continuous distribution function on R with
inverse F-1 defined by F -1 ( u ) inf
aF(a)u, OltU lt1 . If U is a uniform O,1
random variable, then F-1(U) has distribution
function F. Also, If X has distribution function
F , then F ( X ) is uniformly distributed on
0,1.
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Other distributions can be constructed by
transforming marginal distribution
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Correlated uniform distributions
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Sum of two uniform RV
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Examples of dependendtuncorrelated variables
Picture from Scott Ferson, Roger B. Nelsen,
Janos Hajagos, Daniel J. Berleant, Jianzhong
Zhang, W. Troy Tucker, Lev R. Ginzburg and
William L. Oberkampf, SAND2004-3072.
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Copulas
Copulas define dependency between random variable
with any marginal distribution. The marginal
distributions are transformed to uniform
distribution between 0,1, the Copula is then
the CDF of the transformed (ie. Uniform)
variables.
Sklar, A. (1959). Fonctions de répartition à n
dimensions et leurs marges. Publications de
LInstitut de Statistiques de l'Université de
Paris 8 229-231.
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FréchetHoeffding copula bounds
Figure by Matteo Zandi,Graph of the
Fréchet-Hoeffding copula limits 22 November
2010 Wikimedia commons
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FréchetHoeffding copula bounds
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FréchetHoeffding Copulas allow for bounding
operation on RV with given marginal distributions
with no assumption on dependency
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• Outline
• Introduction
• Representing dependency
• Representation of Ignorance
• Entropy and Information
• Intervals and P-box Representation
• Any dependency bounds
• Numerical methods for P-box calculations
• Results
• Conclusion

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Ignorance and Information
 
Claude E. Shannon Warren Weaver 1949, 1971
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Entropy
Maximum Entropy results in Equal probabilities
for no information Principle of insufficient
reason However, this results in no difference
between being ignorant of the outcome behavior
and a system that has well known behavior of a
uniform distributed outcome.
Laplace 1812. See Stigler, Stephen M. (1986)
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Entropy II
Uniqueness of the entropy expression only holds
for a finite set of events. Other expressions
have the same four properties of entropy in the
countable (less studies) and the non-countable
cases. (see Jos Uffink 1997 for example).
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Maximum entropy III

• Depends on the choice of scale
• A solution in terms of stiffness is
• incompatible with one based on flexibility
• even though the information is equivalent
• Warner North interprets Ed Jaynes as saying that
  two states of information that are judged to be
  equivalent should lead to the same probability
  assignments . Maxent doesnt (Ferson 2014)

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Representation of Ignorance

• Entropy Probability, Subjective probability
• Fuzzy sets, fuzzy-probabilistic methods (Möller,
  B., Graf, W. and Beer, M 2000)
• Collection of model results
• Bounds, Confidence intervals, Intervals

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Interval Approach

• Only range of information (tolerance) is
  available
• Represents an uncertain quantity by giving a
  range of possible values
• How to define bounds on the possible ranges of
  uncertainty?
• experimental data, measurements, expert knowledge

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Interval Representation

• Interval number represents a range of possible
  values within a closed set

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Interval vectors and matrices

• Interval Vectors and Matrices
• An interval matrix is such matrix that contains
  all real matrices whose elements are obtained
  from all possible values between the lower and
  upper bounds of its interval components

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Interval Representation - Vectors
y
3
1
x
2
4
Interval
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Interval Operations

• Let x a, b and y c, d be two interval
  numbers
• 1. Addition
• x y a, b c, d a c, b d
• 2. Subtraction
• x ? y a, b ? c, d a ??d, b ??c
• 3. Multiplication
• xy min(ac,ad,bc,bd), max(ac,ad,bc,bd)
• 4. Division
• 1 / x 1/b, 1/a

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Properties of Interval Arithmetic

• Let x, y and z be interval numbers
• 1. Commutative Law
• x y y x
• xy yx
• 2. Associative Law
• x (y z) (x y) z
• x(yz) (xy)z
• 3. Distributive Law does not always hold, but
• x(y z) ??xy xz

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Probability box (Pbox)

• A set of allowable CDF defined by upper and
  lower bounding functions.
• Ferson S, Kreinovich V, Ginzburg L, Myers DS,
  Sentz K, Constructing probability boxes and
  Dempster-Shafer structures, Tech. Rep.
  SAND2002-4015, Sandia National Laboratories,
  2003.
• Williamson R. Probabilistic arithmetic. PhD
  thesis, University of Queensland, Austrialia,
  1989.

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Pbox representation for Imprecise Probability
Figure from Scott Ferson and Vladik
Kreinovich-REC 2006
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Cumulative probability
0
1.0
2.0
3.0
0.0
X
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• Outline
• Introduction
• Representing dependency
• Representation of Ignorance
• Entropy and Information
• Intervals and P-box Representation
• Any dependency bounds
• Numerical methods for P-box calculations
• Results
• Conclusion

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An even simpler Toy problem posed by
Kolmogorov (Frank, Nelsen, Schweizer, Probab. Th.
Rel. Fields 74, 199-211 (1987). Specifically,
what are the optimal bounds for the probability
distribution of the sum of two random variables X
and Y whose individual distribution Fx and Fy are
given?
Makarov, G.D. Estimates for the distribution
function of a sum of two random variables when
the marginal distributions are fixed. Theor.
Probab. Appl. 26, 803-806 (1981)
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Independent
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Correlated
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Family of dependency that constructs bounds
c2alpha-c1 if (c2lt0) c2c21 end Xc1c2
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• Outline
• Introduction
• Representing dependency
• Representation of Ignorance
• Any dependency bounds
• Numerical methods for P-box calculations
• Results
• Conclusion

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Techniques for operating on Pboxes

• Monte Carlo Integration (Zhang 2001)
• Discrete approximation of Probability bounds
  (Williamson and Downs 1990)
• Interval extension to Polynomial Chaos (Redhorse
  and Benjamin 2004)

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P-box Monte Carlo method

• Perform Interval calculation of realization
  (using Interval Finite Element)
• Construct interval representation of random input
  variables from Pbox
• Collect Interval Response
• Sort Lower and upper bounds and construct Pbox
  for Response variable.
• Zhang, H., Mullen, R. L. and Muhanna, R. L.
  Interval Monte Carlo methods for structural
  reliability, Structural Safety, 2010

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Generation of random intervals from a
probability-box
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Finite Element Interval Monte Carlo for Pbox
analysis

• In steady-state analysis-variational formulation

• Invoking the y of ?, that is ?? 0

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However

• Solution to linear system of interval equations
• Is NP (Vladik Kreinovich)
• Compute approximate bounds using fix point
  theorem.

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Required Improvements to Linear Pbox Finite
Element Methods to solve nonlinear problems

• Sharp solutions to systems of equations
• Improve sharpness of Secondary quantities
  (stress/strain).
• Prevent accumulation of errors in iterative
  correctors

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Error in secondary quantities
Conventional Finite Element
Secondary quantities such as stress/strain
calculated from displacement have shown
significant overestimation of interval bounds
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Use constraints to augment original Variational
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• Use secant stiffness methods to prevent iterative
  accumulation of interval overestimation (Rao et.
  al. 2013)

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Discrete P-box Random Set

• Construct Data type for a discrete Pbox
• Define overloaded operators for discrete Pbox
  variable
• Apply conventional deterministic (scalar)
  algorithms using Pbox variables
• Result is in the form of a discrete Pbox

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Interval based discrete Pbox

• Proved interval bounds for a given range of the
  CDF
• Uniform discretization for algorithmic simplicity
• Preserve the guaranteed enclosure philosophy from
  Interval Arithmetic
• Extensions to higher order discretization and
  non-uniform discretization are possible.
• Extreme case of Any Dependency or Independent

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P-Box defined by bounding lognormal distribution
with mean of 2.47, 11.08 and standard deviation
of 2.76, 12.38.
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Best fit vs. Guaranteed enclosure
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Polynomial Chaos Methods

• Expand upper and lower CDF as a random variable
  using the same orthogonal polynomial basis.
• Replace polynomial coefficients by a single
  interval number

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Polynomial Chaos Methods

• Current Research Issues
• Galerkin projection and truncated series are
  estimates and not bounds.
• Possible for bounds to cross.

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• Outline
• Introduction
• Representing dependency
• Representation of Ignorance
• Any dependency bounds
• Numerical methods for P-box calculations
• Results
• Conclusion

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A truss structure
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Property Value
cross-sectional areas for elements A1 to A6 10.32 cm2
cross-sectional areas for elements A7 to A15 6.45 cm2
Elastic modulus 200 GPa
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Confidence structures (c-boxes) are imprecise
generalizations of confidence distributions. Fers
on, S., O'Rawe, J., and Balch, M. (2014)
Statistics of random loadings acting on the truss Statistics of random loadings acting on the truss Statistics of random loadings acting on the truss
90 confidence interval 99confidence interval
Mean ln P1 4.4465, 4.5199 4.4258, 4.5407
Mean ln P2 5.5452, 5.6186 5.5244, 5.6393
Mean ln P3 4.4465, 4.5199 4.4258, 4.5407
ln Standard dev. P1, P2, P3 0.09975 0.09975
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Example 2
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Nonlinear Parameters

•  

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Cubic ModelVertical Deflection at Center
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Ramberg-Osgood modelStrain Element 2
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• Outline
• Introduction
• Representing dependency
• Representation of Ignorance
• Any dependency bounds
• Numerical methods for P-box calculations
• Results
• Conclusion

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Outline Introduction Representing dependency
Parametric models (and associated
statistical measures) Copulas Any
dependency bounds Intervals and P-box
Representation Results Conclusion
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Cubic ModelStrain Element 2
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Ramberg-Osgood modelVertical Deflection at Center
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Examples Continuum Load Uncertainty

• Square plate with opening

44.22 kPa
0.0 kPa
-10.65 kPa
0.063 kPa
Contour values of maximum ?yy (kPa)
Contour values of minimum ?yy (kPa)
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Efficiency study
Nv Sens. Present
246 1.06 0.72
648 64.05 10.17
1210 965.86 59.7
1692 4100 156.3
2254 14450 358.8
2576 32402 528.45
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Summary

• Behavior of Linear and Nonlinear of structures
  with unknown dependency among random parameters
  and loading can be bounded using concept of Pbox
  structures

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Rene Magritte, Clairvoyance, 1936
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