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Uncertainty Quantification, Inverse Problemsand
many applications
Alexander Litvinenko
Center for UncertaintyQuantification
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http://sri-uq.kaust.edu.sa/
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Uncertainty Quantification (UQ)
Parameters are affected by uncertainty (because they are notperfectly known or because they are intrinsically variable).
Goal: develop effective ways to include, treat and reduceuncertainty in a mathematical models.
Model uncertainties by random variables and random fields.
Realisations of random fields (porosity, permeability)Center for UncertaintyQuantification
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Motivation: Minimising the risk w.r.t. the cost
Risk under uncertainties is a high-dimensional functional.Goal: To find an optimal balance between risk and cost.(Detailed risk-cost understanding is useful for better decisionmaking)Difficulty: Many parameters are uncertain
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Overview of uncertainty quantification
ConsiderA(u; q) = f
Uncertain Input:1. Parameter q := q(ω) (assume moments/cdf/pdf/quantiles
of q are given)2. Boundary and initial conditions, right-hand side3. Geometry of the domain
Uncertain solution:1. mean value and variance of u2. exceedance probabilities P(u > u∗)3. probability density functions (pdf) of u.
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Examples: Projects 2009-2014:
I Virtual Materials and Structures(Use simulations to reduce experiments!)
I Fundamentals of Future Civil Aircraft.(New composite materials, work of D. Joergens and M.Krosche, TU Braunschweig)
I UQ in Heat and Moisture Transport in HeterogeneousMedia.(Low-energy green housing, moisture damage processesin historic buildings, thermal expansion and contraction,work of B. Rosic and Anna Kucherova, Prague).
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Example:
Multiscale phenomena and UQ in the concrete
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Multiscale phenomena
Durability of structures is influenced by moisture damageprocesses.
Figure : 3 scales of the concrete (macro, intermediate and micro)Center for UncertaintyQuantification
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Example:
Uncertainty Quantification in the compositematerials
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Example: Composite materials
Laminated composite plates consist of jointly glued layers ofcarbon fibres.Uncertain Quantity of Interest: Young’s modulus, shearmodulus, ...
Different orientation of carbon fibers in different layers.
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Composite material with 15 layers.
Four material fields of a laminate layer.Center for UncertaintyQuantification
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Statistical model
GOAL: describe the stochastic material properties of eachmaterial block
Original image, discrete phases and smooth approximation
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Example
UQ in moisture damage processescan induce large displacements and extensive damage tostructural materialsApplications: historic buildings, green houses, basement...
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Example: Moisture
ground moisture movement through concrete foundations.Taken from Internet
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Example 2: Moisture
Moisture propagation is modeled via system of
1. diffusion equation2. mass conservation3. energy conservation
Porosity and permeability are not exactly known and difficult tomeasure.We model these porosity and permeability by random fieldsκ(x , ω).
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A big example:UQ in numerical aerodynamics
(described by Navier-Stokes + turbulence modeling)
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Example: uncertainties in free stream turbulence
α
v
v
u
u’
α’
v1
2
Random vectors v1(θ) and v2(θ) model free stream turbulence
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Example: UQ
Assume that RVs α and Ma are Gaussian with
mean st. dev.σ
σ/mean
α 2.79 0.1 0.036Ma 0.734 0.005 0.007
Then uncertainties in the solution lift CL and drag CD are
CL 0.853 0.0174 0.02CD 0.0206 0.003 0.146
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500 MC realisations of cp in dependence on αi and Mai
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Example: prob. density and cumuli. distrib. functions
0.75 0.8 0.85 0.9 0.950
5
10
15
20
25Lift: Comparison of densities
0.005 0.01 0.015 0.02 0.025 0.03 0.0350
50
100
150Drag: Comparison of densities
0.75 0.8 0.85 0.9 0.950
0.2
0.4
0.6
0.8
1Lift: Comparison of distributions
0.005 0.01 0.015 0.02 0.025 0.03 0.0350
0.2
0.4
0.6
0.8
1Drag: Comparison of distributions
sgh13
sgh29
MC
Figure : First row: density functions and the second row: distributionfunctions of lift and drag correspondingly.
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Example: 3sigma intervals
Figure : 3σ interval, σ standard deviation, in each point of RAE2822airfoil for the pressure (cp) and friction (cf) coefficients.
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X
Z
0.2 0 0.2 0.4 0.6 0.8 1 1.2
0.4
0.2
0
0.2
0.4
0.6
0.8
pressure
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
X
Z
0 0.2 0.4 0.6 0.8 10.3
0.2
0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7 pressure
0.038
0.034
0.03
0.026
0.022
0.018
0.014
0.01
0.006
0.002
(left) a mean value of the pressure and (right) a variance of thepressure).
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Stochastical Methods overview
1. Monte Carlo Simulations (easy to implement,parallelisable, expensive, dim. indepen.).
2. Stoch. collocation methods with global polynomials (easyto implement, parallelisable, cheaper than MC, dim.depen.).
3. Stoch. collocation methods with local polynomials (easy toimplement, parallelisable, cheaper than MC, dim. depen.)
4. Stochastic Galerkin (difficult to implement, non-trivialparallelisation, the cheapest from all, dim. depen.)
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Karhunen-Loeve Expansion
The Karhunen-Loeve expansion is the series
κ(x , ω) = µk (x) +∞∑
i=1
√λiki(x)ξi(ω), where
ξi(ω) are random variables and ki are basis functions.
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Truncated Polynomial Chaos Expansion
ξ(ω) ≈Z∑
k=0
ak Ψk (θ1, θ2, ..., θM), where Z =(M + p)!
M!p!
- Z is large, EXPENSIVE TO COMPUTE!M = 9, p = 2, Z = 55M = 9, p = 4, Z = 715M = 100, p = 4, Z ≈ 4 · 106.How to store and to handle so many coefficients ?
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Discrete form
Take weak formulation of the diffusion equation, apply KLE andPCE to the test function v(x , ω), solution u(x , ω) and κ(x , ω),obtain
Ku =
m−1∑`=0
∑γ∈JM,p
∆(γ) ⊗ K `
u = p, (1)
where ∆(γ) are some discrete operators which can becomputed analytically, K ` ∈ Rn×n are the stiffness matrices.
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Galerkin stiffness matrix K
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Example: Lorenz 63
Is a system of ODEs. Has chaotic solutions for certainparameter values and initial conditions.
x = σ(ω)(y − x)
y = x(ρ(ω)− z)− yz = xy − β(ω)z
Initial state q0(ω) = (x0(ω), y0(ω), z0(ω)) are uncertain.
(Here could be a system of chemical equation with uncertaincoefficients...)
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Trajectories of x,y and z in time. After each update (newinformation coming) the uncertainty drops. (O. Pajonk)
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Software: Stochastic Galerkin library
1. Type in your terminalgit clone git://github.com/ezander/sglib.git
2. To initialize all variables, run startup.m
You will find:generalised PCE, sparse grids, (Q)MC, stochastic Galerkin,linear solvers, KLE, covariance matrices, statistics, quadratures(multivariate Chebyshev, Laguerre, Lagrange, Hermite ) etc
There are: many examples, many test, rich demos
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Inverse Problems
Stochastic solution of the ellipticequation with uncertain coefficients
(describes the flow of a fluid through aporous medium.)
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Example 5: 1D elliptic PDE with uncertain coeffs
−∇ · (κ(x , ξ)∇u(x , ξ)) = f (x , ξ), x ∈ [0,1]
Measurements are taken at x1 = 0.2, and x2 = 0.8. The meansare y(x1) = 10, y(x2) = 5 and the variances are 0.5 and 1.5correspondingly.
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Example 5: updating of the solution u
0 0.2 0.4 0.6 0.8 1−20
−10
0
10
20
30
0 0.2 0.4 0.6 0.8 1−20
−10
0
10
20
30
Figure : Original and updated solutions, mean value plus/minus 1,2,3standard deviations
See more in sglib by Elmar Zander
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Example 5: Updating of the parameter
0 0.2 0.4 0.6 0.8 1−1
−0.5
0
0.5
1
1.5
2
0 0.2 0.4 0.6 0.8 1−1
−0.5
0
0.5
1
1.5
2
Figure : Original and updated parameter q.
See more in sglib by Elmar Zander
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LECTURE 3
Lecture 3: Kriging accelerated byorders of magnitude: combining
low-rank covarianceapproximations with
FFT-techniques
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Kriging, combining low-rank approx. with FFT-techniques
Let m be number of measurement points, n number ofestimation points. y ∈ Rm vector of measurement values.
Let s ∈ Rn be the (kriging) vector to be estimate with meanµs = 0 and cov. matrix Qss ∈ Rn×n:
s = Qsy Q−1yy y︸ ︷︷ ︸ξ
, (2)
where Qsy ∈ Rn×m cross covariance matrix and Qyy ∈ Rm×m
auto covariance matrix.
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Numerics: an example from geology
Domain: 20m × 20m × 20m, 25,000× 25,000× 25,000 dofs.4,000 measurements randomly distributed within the volume,with increasing data density towards the lower left back cornerof the domain.The covariance model is anisotropic Gaussian with unitvariance and with 32 correlation lengths fitting into the domainin the horizontal directions, and 64 correlation lengths fittinginto the vertical direction.
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Example: Kriging
A series of zooms into the respective lower left back corner with zoomfactors (sampling rates) of 4, 16, 64 for the top right, bottom left andbottom right plots, respectively. Color scale: showing the 95%confidence interval [µ− 2σ, µ+ 2σ].
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Take to home: we offer
Effective approaches and solution techniques for modeling ofuncertainties in reservoirs, new composite materials, chemicalprocesses,..
Spectrum of tasks:1. Simulations in virtual labour. Statistical/stochastic
description of composite materials2. Experimental design (increase information gain from
experiments)3. Robust design (stable with respect to uncertain
parameters)4. Risk assessment and prediction of rare events under
uncertainties
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