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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