An Efficient Computational Framework for Uncertainty Quantification in Multiscale Systems

Materials Process Design and Control LaboratoryCornell University

An efficient computational framework for uncertainty quantification in

multiscale systems

1

Materials Process Design and Control LaboratorySibley School of Mechanical and Aerospace Engineering

169 Frank H. T. Rhodes HallCornell University

Ithaca, NY 14853-3801

Email: [email protected]: http://mpdc.mae.cornell.edu/

Xiang MaPresentation for Thesis Defense (B-Exam)

Date: Aug 18, 2010


Acknowledgement

2

SPECIAL COMMITTEE: Prof. Nicholas Zabaras, M & A.E., Cornell UniversityProf. Subrata Mukherjee, T & A.M., Cornell UniversityProf. Phaedon-Stelios Koutsourelakis, CEE., Cornell UniversityProf. Gennady Samorodnitsky, OR & IE., Cornell University

FUNDING SOURCES: Air Force of Scientific Research (AFOSR)National Science Foundation (NSF)Sibley School of Mechanical & Aerospace Engineering

Materials Process Design and Control Laboratory (MPDC)


Outline of the presentation

3

Motivation: Coupling multiscaling and uncertainty analysis

Mathematical representation of uncertainty

Problem definition: Stochastic multiscale flow in porous media

Adaptive Sparse Grid Collocation (ASGC) Method for the solution of SPDEs

ASGC coupled with Adaptive High Dimensional Model Representation technique for high-dimensional SPDEs

Mixed Finite Element Heterogeneous Multiscale Method

Numerical Examples


Need for uncertainty analysis

4

All physical systems have inherent associated randomness

Why uncertainty modeling?Assess product and process reliability

Estimate confidence level in model predictions

Identify relative sources of randomness

Provide robust design solutions

SOURCES OF UNCERTAINTIES

• Multiscale nature inherently statistical

• Uncertainties in process conditions

• Material heterogeneity

• Model formulation – approximations,

assumptions


Why uncertainty and multiscaling

5

Uncertainties introduced across various length scales have a non-trivial interactions.

MacroMesoMicro

Use micro averaged models for resolving physical scales

Imprecise boundary conditions

Initial perturbations

Physical properties, structure follow a statistical description



6







Numerical Examples



7

Sample space Sigma-algebra Probability measure

Math: Probability space (Ω, F, P)

SSample space of elementary events

Real line

MAP

Collection of all possible outcomes

Each outcome is mapped to a corresponding real value

Stochastic variable can be regarded as a function taking value in the stochastic space.

: ( , , )X x t Ω

A stochastic process is a random field with variations across space and time.


Spectral stochastic representation

8

: ( , , )X x t ΩA stochastic process = spatially, temporally varying random function

CHOOSE APPROPRIATE BASIS FOR THE

PROBABILITY SPACE

HYPERGEOMETRIC ASKEY POLYNOMIALS

PIECEWISE POLYNOMIALS (FE TYPE)

SPECTRAL DECOMPOSITION

COLLOCATION, MC (DELTA FUNCTIONS)

GENERALIZED POLYNOMIAL CHAOS EXPANSION

SUPPORT-SPACE REPRESENTATION

KARHUNEN-LOÈVE EXPANSION

SPARSE GRID, CUBATURE, LH


Stochastic collocation based framework

9

Function value at any point is simply

Stochastic function in 2 dimensions

Need to represent this function

Sample the function at a finite set of points

Use polynomials to get a approximate representation

: Nf →( )

1{ }i MM i==Θ Y

( ) ( )

1( ) ( ) ( )

Mi i

ii

f f a=

= ∑Y Y YI

( )f YISpatial domain is approximated using a FEM discretization.

Stochastic domain is approximated using multidimensional interpolating functions


Collocation based framework for UQ

10

Pre A-exam developments:1) Stochastic natural convection using generalized polynomial chaos expansion.

2) Preliminary studies on the effect of stochastic discontinuities.

3) A framework for solving stochastic partial differential equations based on adaptive sparse grid collocation method (ASGC).

Post A-exam developments:1) An efficient framework for Bayesian inference approach based on ASGC.

2) High Dimensional Stochastic Model Representation (HDMR) technique for the solution of high-dimensional SPDEs.

3) Analysis of stochastic multiscale systems: Uncertainty across length- scales

4) Data-driven approaches to construct stochastic input models from limited information: explore the non-linearity structure of data using Kernel Principal Component Analysis (KPCA).



11







Numerical Examples


Flow through heterogeneous media

12

Necessitates stochastic multiscaling

Ground water remediation, nuclear contamination and enhanced oil recovery

Availability of enough input data and comparative results

Pose and solve questions that provide some insights

Ground water remediation and contamination control

Oil recovery

http://cgr.ebs.ogi.edu/merl/ScreenGrabs/Ch1_Pollutants.png

http://cgr.ebs.ogi.edu/merl/ScreenGrabs/Ch1_Pollutants.png

http://www.snf-oil.com/images/shema_Enhanced_Oil_Recovery.jpg


Problem definition

13

with the boundary condition

Length scale of the system,

Length scale of permeability variation,

But exact permeability unknown. Some statistics or limited data

Permeability is a realization from corresponding probability space ( , )K ω⋅ ∈Ω

Basic equations for flow and transport in a domain

Ll L<<

Multiscale paradigm


Karhunen-Loéve expansion

14

1. Representation of random process

- Karhunen-Loeve, Polynomial Chaos expansions

2. Infinite dimensions to finite dimensions

- depends on the covariance

Karhunen-Loèvè expansion

Based on the spectral decomposition of the covariance kernel of the stochastic process

Random process Mean

Set of random variables to

be found

Eigenpairs of covariance

kernel

• Need to know covariance

• Converges uniformly to any second order process

Set the number of stochastic dimensions, N

Dependence of variables

Pose the (N+d) dimensional problem( )1( , ) , , , NK x K x Y Yω = …

( ) ( )1

, N

i ii

Yρ ρ=

= ∀ ∈ Γ∏Y Y1

NN

ii=

Γ ≡ Γ ∈∏Stochastic space: PDF:


Stochastic variational formulation

15

Define the deterministic functional spaces:

and duality product Define the stochastic functional spaces:

Tensor product to perform the functional spaces:

Stochastic variational form: Find


Stochastic collocation method

16

The weak form is equivalent to: for a.e. the following deterministic equations hold:

As before, stochastic variable can be regarded as a function taking value in the stochastic space.

This nature is utilized by the stochastic collocation method to construct the interpolant of the stochastic function. Therefore, we only need the solution to thedeterministic problem at the collocation points.

Random variable ξ

SSample space of elementary events

Real line

MAP

Collection of all possible outcomes

Each outcome is mapped to a corresponding real value



17







Numerical Examples


Sparse grid collocation

18

1D sampling

2D sampling

In higher dimensions, not all the points are equally important.

Some points contribute less to the accuracy of the solution (e.g. points where the function is very smooth, regions where the function is linear, etc.). Discard the points that contribute less: SPARSE GRID COLLOCATION1

Tensor product

1 .S. Smolyak, Quadrature and interpolation formulas for tensor product of certain classes of functions, Soviet Math. Dokl, 4 (1963) 240-243.

Materials Process Design and Control LaboratoryCornell University 19

Choice of collocation points and nodal basis functions

In the context of incorporating adaptivity, we use the Newton-Cotes grid with equidistant support nodes and the linear hat function as the univariate nodal basis.

ijY12i i

jY −− 12i ijY −+

1

In this manner, one ensures a local support and that discontinuities in the stochastic space can be resolved. The piecewise linear basis functions is defined as


Conventional sparse grid collocation (CSGC)

20

LET OUR BASIC 1D INTERPOLATION SCHEME BE SUMMARIZED AS

IN MULTIPLE DIMENSIONS, THIS CAN BE WRITTEN AS

Denote the one dimensional interpolation formula as

In higher dimensions, a simple case is the tensor product formula

1i i iU U −Δ = −

Here, we define the hierarchical surplus as:

0 0U =

Using the 1D formula, the sparse interpolant , where is the depth of sparse grid interpolation and is the number of stochastic dimensions, is given by the Smolyak algorithm as

( )00,q q≥ ∈ N,q NA q


Nodal basis versus hierarchical basis

2121

Nodal basis Hierarchical basis


Adaptive sparse grid collocation

22

11y

21w

22w

21y 2

2y

31y 3

2y

31w 3

2w

41y 4

2y

Anisotropic sampling for interpolating functions with steep gradients and other localized phenomena

1D tree-like data structure:

2D:

Define a threshold value. If magnitude of the hierarchical surplus is greater than this threshold, refine around this point.

Add 2N neighbor points. Scales linearly instead of O(2N) of other stochastic adaptive methods


Definition of the error indicator

23

The mean of the random solution can be evaluated as follows:

Denoting we rewrite the mean as

We now define the error indicator as follows:

In addition to the surpluses, this error indicator incorporates information from the basis functions. This forces the error to decrease to a sufficiently small value for a large interpolation level. This error indicator guarantees that the refinement would stop at a certain interpolation level.


Adaptive sparse grid collocation: Algorithm

24


Adaptive sparse grid collocation

2525

Ability to detect and reconstruct steep gradients3 2 2 3

1( , )|10 | 10

f x yx y− −=

− − + 2 2 1

1( , )| 0.3 | 10

f x yx y −=

− − +


Application to Rayleigh-Bénard instability

26

Cold wall θ = -0.5

hot wall θ(ω)

Insu

late

d

Insu

late

d

Filled with air (Pr = 0.7), Ra = 2500, which is larger than the critical Rayleigh number, so that convection can be initialized by varying the hot wall temperature. Thus, the hot wall temperature is assumed to be a random variable.

Investigation: Stochastic simulation to find the critical wall temperature.

0.4 0.3h Yθ = + [0,1]Y ∼

( , , ) 0t ω∇ ⋅ =u x2( , , ) ( , , ) ( , , ) ( , , ) Pr ( , , ) Pr Ra ( , , )t t t p t t t

tω ω ω ω ω θ ω∂

+ ⋅∇ = −∇ + ∇ +∂

u x u x u x x u x x

2( , , ) ( , , ) ( , , ) ( , , )t t t tt

θ ω ω θ ω θ ω∂+ ⋅∇ = ∇

∂x u x x x


Rayleigh-Bénard instability: stochastic formulation

27

0.541

θ

Nonlinear: convection

0.60.50.40.30.20.10

-0.1-0.2-0.3-0.4

Max = 0.667891

Linear: conduction

0.350.250.150.05

-0.05-0.15-0.25-0.35-0.45

Max = 0.436984


Rayleigh-Bénard instability: Mean

28

1.61.20.80.40

-0.4-0.8-1.2-1.6

Max = 1.73709

1.61.20.80.40

-0.4-0.8-1.2-1.6

Max = 1.73123

0.50.40.30.20.10

-0.1-0.2-0.3-0.4

Max = 0.55

0.50.40.30.20.10

-0.1-0.2-0.3-0.4

Max = 0.549918

1.61.20.80.40

-0.4-0.8-1.2-1.6

Max = 1.73715

1.61.20.80.40

-0.4-0.8-1.2-1.6

Max = 1.73128

ASGC (top) MC (bottom)

u v θ


Rayleigh-Bénard instability: Variance

29

0.0120.0110.010.0090.0080.0070.0060.0050.0040.0030.0020.001

Max = 0.0129218

32.62.21.81.410.60.2

Max = 3.30617

32.62.21.81.410.60.2

Max = 3.28535

32.62.21.81.410.60.2

Max = 3.27903

32.62.21.81.410.60.2

Max = 3.29984

0.0120.0110.010.0090.0080.0070.0060.0050.0040.0030.0020.001

Max = 0.0128783

ASGC (top) MC (bottom)

u v θ



30






Numerical Examples


Motivation of HDMR

31

Conventional and adaptive collocation methods are not suitable for high-dimensional problems due to their weakly dependence on the dimensionality (logarithmic) in the error estimate.

Stochastic elliptic: 25, 1/16cN L= =

Although ASGC can alleviate this problem to some extent, its performance depends on the regularity of the problem and the method is only effective when some random dimensions are more important than others.

These modeling issues for high-dimensional stochastic problems motivate the use of the High Dimensional Model Representation (HDMR) technique.


High dimensional model representation (HDMR)

32

( ) ( ) ( ) ( )

( ) ( )

1 2 1 2 1 2 3 1 2 31 2 1 2 3

1 1

1

01

12 1

, , ,

, , , ,s s

s

N N N

i i i i i i i i i i i ii i i i i i

N

i i i i N Ni i

f f f Y f Y Y f Y Y Y

f Y Y f Y Y

= < < <

< <

= + + + +

+ + +

∑ ∑ ∑

∑

Y

… …

In this expansion:

denotes the zeroth-order effect which is a constant.

The component function gives the effect of the variable acting independently of the other input variables.

The component function describes the interactive effects of the variables and . Higher-order terms reflect the cooperative effects ofincreasing numbers of variables acting together to impact upon .

The last term gives any residual dependence of all the variables locked together in a cooperative way to influence the output .

( )i if YiY

( )1 2 1 2,i i i if Y Y

1iY

2iY

( )12 1, ,N Nf Y Y… …

0f

( )f Y

( )f Y

Several low-dimensional problems


CUT-HDMR

33

This equation is often written in a more compact notation:

for a given set where denotes the set of coordinate indices and . Here, denotes the - dimensional vector containing those components of whose indices belong to the set , where is the cardinality of the corresponding set , i.e. .

For example, if , then and implies

In CUT-HDMR, a reference point is first chosen

where the notation means that the components of other than those indices that belong to the set equal to those of the reference point.

Mean of the random input vector is chosen as the reference point.


CUT-HDMR coupled with ASGC

34

Therefore, the -dimensional stochastic problem is transformed to several lower-order -dimensional problems which can easily solved by ASGC:

N

where are the hierarchical surpluses for different sub-problems indexed by and is only a function of the coordinates belonging to

Then the mean of the HDMR expansion is simply

In other words, instead of solving the N - dimensional problem directly using ASGC, which is impractical for extremely high dimensional problems, we only need to solve several one- or two- dimensional problems, which can be solved efficiently via ASGC.


Adaptive HDMR1

35

1 .X. Ma, N. Zabaras, An adaptive high-dimensional stochastic model representation technique for the solution of stochastic partial differential equations, JCP, 229 (2010) 3884-3915.

Identify the important dimensions through the first-order expansion

Then we define the important dimensions as those whose weights are larger than a predefined error threshold . Only higher-order terms which consist of only these important dimensions are considered.

We only construct the higher-order terms which is of importance:

Stop the construction if relative error is small than a threshold


Adaptive HDMR algorithm

36


Stochastic Porous Media Flow

37

injection well

production well

1 1×

. in in

fK p

∇ == − ∇u

uDD

20.25, 2.0, 500 :L Nσ= = =

0 0.2 0.4 0.6 0.8 14

4.5

5

5.5

6

6.5

7

7.5x 10

-3

x

Stan

dard

Dev

iatio

n

MC: 1,000,000HDMR: ε = 10-6, θ1 = 2 × 10 -4

σ2 = 2.0

-0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.060

10

20

30

40

50

60

70

v

PDF

MCθ1 = 10-2

θ1 = 10-3

θ1 = 2 × 10-4

σ2 = 2.0

ε = 10-6

Standard deviation of velocity at y = 0.5 PDF of velocity at point ( 0, 0.5) Convergence of the normalized errors of the standard deviation

30 30×Grid:

Highest stochastic dimension problem reported based on non-MC method

[ 1,1]iY −∼


Stochastic Natural Convection

38

0.0750.070.0650.060.0550.050.0450.040.0350.030.0250.020.0150.010.005

3.532.521.510.50

-0.5-1-1.5-2-2.5-3-3.5

3.532.521.510.50

-0.5-1-1.5-2-2.5-3-3.5

1.31.21.110.90.80.70.60.50.40.30.20.1

0.450.40.350.30.250.20.150.10.050

-0.05-0.1-0.15-0.2-0.25-0.3-0.35-0.4-0.45

1.110.90.80.70.60.50.40.30.20.1

Mean

Std

u v θ

10, [ 1,1]iN Y= −∼

Pr 1.0, 1000Ra= =

hot w

all θ

(ω)

Cold w

all θ = -0.5

Insulated

Insulated

Grid: 30 30×

0.1, 0.05L σ= = 0.1, 1.0L σ= = 0.1, 2.0L σ= =



39







Numerical Examples


Spatial finite element discretization

40

Permeability is defined as a cell-wise constant on a fine scale grid

A coarse-scale grid is also defined, where we are seeking the coarse-scale solution. This grid is assumed to be conforming to the fine-scale grid where the permeability is defined.

element face

element face


Mixed finite element method

41

The mixed finite element method for the elliptic problem on the coarse grid :

where the lowest-order Raviart-Thomas space for velocity

and piecewise constant approximation for pressureE

Find

E

( )1 2,L Lx x ( )1 2,R Lx x

( )1 2,R Rx x( )1 2,L Rx x

3cψ1

cψ

2cψ

4cψ

is the value of the coarse-scale flux at the middle point of the side


Multiscale bilinear form

42

All the fine-scale information of the permeability is incorporated in the bilinear form, i.e. .

Define the global matrix for the bilinear form: , where

2x2: Numerical quadrature

where

1ξ

× ×

× ×2ξ

4ξ 3ξ

It is clear that the realization of the permeability field at the quadrature point is not able to capture the full information at the subgrid scale in the coarse element.

and are the quadrature points and weights.


Modified multiscale bilinear form

43

Therefore, we need to modify the bilinear form at the quadrature points following the framework of heterogeneous multiscale method1:

where is the solution to the following local subgrid problem in the sampling domain

with appropriate boundary condition. can be considered as the subgridpressure.

1 . W. E, B. Engquist, The heterogeneous multi-scale methods, Comm. Math. Sci 1 (2002) 87-132.

http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.cms/1118150402


Choice of the sampling domain

44

We would like to take the sampling domain the same as the coarse element, in order to capture all the fine-scale information of the permeability within the current coarse element.

In other words, the subgrid problem is solved on the same coarse element for each quadrature point. Since the governing equations and the solution domain are the same for all the subgrid problem, the only difference between them is the boundary conditions applied on the coarse element, which is in general plays a significant role in the accuracy of the multiscale method.

kξk

E Eδ =

• •

• •

• •

• •

• •

• •

•

•

•

•


Choice of boundary condition

45

Due the theory of HMM, the Neumann boundary condition is

E1,2 ( )ckψ ξ 1,2 ( )c

kψ ξ

constant

In another view, the boundary condition is equivalent to apply the total flux applied along the coarse element boundary , which is

Hence we modify the boundary condition to

E

where denotes the value of the ith coarse-scale RT0 basis function at the kth quadrature point.

ie jeiKjK

ixΔ jxΔ

Transmissibility in the x-direction: 1

2 | |a

jiv a

i j

xxTK K

ν−

⎛ ⎞ΔΔ= +⎜ ⎟⎜ ⎟

⎝ ⎠

Transmissibility is a measure of the ability of the interface to transport the flow across the boundary of the element.


Choice of boundary condition

46

The modified boundary condition is

E1ν

2ν

3ν

4ν

4

11 11

/ | |i ii

Q T T ν=∑

4

11 21

/ | |i ii

Q T T ν=∑4

11 31

/ | |i ii

Q T T ν=∑

4

11 41

/ | |i ii

Q T T ν=∑

as example11Q

5ν

6ν

7ν

8ν

8

11 55

/ | |i ii

Q T T ν=∑

8

11 65

/ | |i ii

Q T T ν=∑

8

11 75

/ | |i ii

Q T T ν=∑

8

11 85

/ | |i ii

Q T T ν=∑

The sum of the flux applied on the fine-scale element is equal to the total flux applied on the same coarse element boundary

We just redistribute the total flux on the coarse-scale element boundary according to the ability to transport the flow at the interface of each fine-scale element . This is clearly a better choice for boundary condition since it determines the flow conditions across the inter-block boundaries.


Subgrid problem in the coarse element

47

Therefore, our subgrid problem in one coarse-element E is defined as follows: For each quadrature point we seek the solution to the following subgrid problem for each coarse-scale RT0 basis function

subject to the following Neumann boundary condition

E E

0ik ⋅ =u n

0ik ⋅ =u n


The coarse-scale problem

48

We will define the modified bilinear form as: for any

The MxHMM version of the Darcy’s equation on the coarse scale reads: Fine the coarse scale such that

with the boundary condition

The only difference is the modified bilinear form, which needs the solution of the local subgrid problem.

It is through these subgrid problems and the mixed formulation that the effect of the heterogeneity on coarse-scale solutions can be correctly captured.

where is defined through the subgrid problem.


Reconstructing the fine-scale velocity

49

The idea is to solve Darcy’s equations within each coarse element using Neumann boundary condition given by the coarse-scale flux :

In order to make the problem well-posed, pressure is fixed in one fine-scale element with the coarse-scale pressure.

Again, using mixed finite element, the velocity field is conservative on both fine and coarse scales.

E E

cp p=


Solution of the transport equation

50

The weak formulation is find such that

Let be the time step and denote by the approximation of the water saturation in element , then the discrete equation is

where

Degree of freedom in RT0 basis, can directly obtained from the solution of multiscale problem.


Quality assessment

51

To assess the quality of our multiscale approach, we will use the so called water cut curve , which defines the fraction of water in the produced fluid, i.e.

where is the flow rate of produced oil at the outlet boundary and is the flow rate of the produced water.

The water cut curve can be calculated as

where refers to the outflow boundary condition.

The dimensionless time is measure in pore volume injected (PVI):

where is the total pore volume of the system, which is equal to the area of the domain here and is the total flow rate.


Solution Methodology

52

Generate the permeability sample given the collocation

point, set coarse discretization

Compute the stiffness matrix for each coarse element

Compute the stochastic coarse-scale fluxes

Solve the subgrid problems for each basis function at

quadrature points

POSTPROCESSING: Compute the statistics

of the solution

Solve stochastic multiscale problem

with HDMR

Generate collocation point

Reconstruct the fine-scale velocity

Solve the subgridproblems with coarse-

scale flux

Solve the transport problem

Return function value at collocation point



53







Numerical Examples


Example 1: Realistic permeability

54

In order to verify the accuracy of the deterministic multiscale code, in this example, we consider a deterministic permeability field from the top layer of the SPE10 comparative project.

100p =

We assume the flow is from bottom to top with the boundary conditions:

The logarithm of the fine-scale permeability is defined on 60x220 grid as shown on the right.

For saturation equation, we assume zero initial conditions and the following boundary condition:

0p =


55

Contour of x-velocity for various coarse grids

( )a Fine Scale ( ) 30 110b × ( )15 55c × ( )10 44d × ( ) 6 22e ×


Contour of y-velocity for various coarse grids

56

( )a Fine Scale ( ) 30 110b × ( )15 55c × ( )10 44d × ( ) 6 22e ×


Saturation contour for various coarse grids at 0.4 PVI

57

( )a Fine Scale ( ) 30 110b × ( )15 55c × ( )10 44d × ( ) 6 22e ×


Saturation Movies

58

( )a Fine Scale ( )15 55b ×


Water cut curves for various coarse grids

59

PVI

F(t)

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Reference30 x 11015 x 5510 x 446 x 22


Relative errors for various coarse grids

60

For velocity:

For saturation:

For water cut:

The reference solution is taken from the solution of the fine-scale problem using mixed finite element method directly.


Example 2: stochastic simulation

61

1p = 0p =

0⋅ =u n

0⋅ =u n

Fine-scale grid: 64 64×

Coarse-scale grid: 8 8×

where is a zero mean Gaussian random field with covariance function. is the correlation length and is the standard deviation

Y

[ ]20,1Ω =

Here are assumed i.i.d uniform random variables on [-1,1]. For comparison, the reference solution is taken from 106 MC samples, where each direct problem is solved using the fine-scale solver.

The stochastic problem is solved using HDMR, where the solution of each deterministic problem at the collocation point is from the multiscale solver. We fix and investigate the effect of the anisotropy of the random field.


Example 2: Isotropic random field

62

In this problem, we take Due to the slow decay of the eigenvalue, the KLE is truncated after 100 terms. Therefore, the stochastic dimension is 100.

The problem is solved with HDMR where each sub-problem is solved through ASGC. We take First, we show the results at 0.2 PVI.

It is interesting to note that the mean is nearly the same as the homogeneous solution with the mean permeability. This is called “heterogeneity-induced dispersion”, where the heterogeneity smoothes the water saturation profile in the ensemble sense, although individual realization shows heterogeneity.


Isotropic field: std of saturation at 0.2 PVI

63

The result indicates that higher water saturation variance are concentrated near displacement fronts, which are areas of steep saturation gradients.


Isotropic filed: Convergence of HDMR

64

The normalized error is defined the same as before.

Number of important dimensionsiN − Total number of expansion component functionscN −

From the table, it is seen that the results are indeed quite accurate despite the fact that 64-fold upscaling is used to solve the deterministic problem and adaptive methods are used to solve the stochastic problem.


Isotropic filed: Interpolation properties

65

We randomly generate one input vector and reconstruct the saturation from HDMR. At the same time, we run a deterministic problem with the fine-scale model and the same realization of the random input vector.

The two results are exactly the same.


Isotropic field: PDF and CDF

66

PDF and CDF are plotted at the point (0.2, 0), where the highest standard deviation of saturation occurs.

The results are obtained from the saturation realizations through HDMR.


Example 2: Anisotropic random field

67

In this problem, we take Due to the increasing of correlation length in x dimension, the KLE is truncated after 50 terms. Therefore, the stochastic dimension is 50. We first solve this problem again at time 0.2 PVI using HDMR with ASGC. We take the relevant parameters as

We get the same results as in the isotropic case.


Anisotropic field: std of saturation at 0.2 PVI

68

It is interesting to note that the shape of contour is nearly the same as that of isotropic random field. Only the value of standard deviation are different.

The introduction of anisotropy has the effect of increasing the output uncertainty.

Materials Process Design and Control LaboratoryCornell University 69

Anisotropic field: Convergence of HDMR

According to our previous numerical results, larger uncertainty requires more expansion terms. Here, indeed more expansion terms and collocation points are needed compared with that of isotropic case.

In addition, the highest HDMR expansion order is 3. There are 3 third-order component functions, which indicating the existence of higher-order cooperative effects among the inputs.


Anisotropic field: Results at 0.2 PVI

70


Conclusions

71

An efficient computational framework developed for analysis of complex multiscale systems.

The key aspects of these developments is to utilize the adaptive HDMR coupled with ASGC for solving stochastic PDEs involved.

The most important rational that has made this technique important is its non-intrusive character, where only repetitive function calls are required at a much less number of sampling points than that of MC method.

We have developed a black box stochastic toolkit that can seamlessly link with any deterministic simulator to facilitate stochastic analysis.


Uncertainty Quantification Across Length Scales

72

Uncertain heat flux on the nozzle flap

Limited information about the microstructure

of the material

Uncertainty in initial and/or boundary (operating) conditions at the macro scale +

Topological uncertainty at lower scales

How do mechanical properties and damage evolve?

Will the device fail? With what probability?

- Simple one way coupling: Upscale the property statistics to the macro scale and use to compute statistics

- Two way coupling: For tightly coupled problems, solve micro-macro stochastic PDEs simultaneously. Need information theoretic models to propagate uncertainty across scales


Continuum-Discrete Stochastic Coupling

73

Averaged properties

Microstructure

Atomistic model

Continuum model

Exact microstructural features unknown. Experiments only provide limited statistical information

- Where does fracture occur? Depends on the local microstructure => need a stochastic framework for analysis.

- Stochastic homogenization can only be applied hierarchically/adaptively (e.g. near or far from crack tips)

- Assumes significance in the robust design of materials that withstand failure

Is it possible to couple discrete simulators (e.g. MD, MC) with a continuum stochastic (higher scale) simulator?

- Since lower scales posses higher information content, estimate statistics and utilize it to run macro simulator

- Issues with computational efficiency (model reduction)


Publications

74

Xiang Ma and N. Zabaras, “A stabilized stochastic finite element second-order projection method for modeling natural convection in random porous media", J. Computational Physics, Vol. 227, pp. 8448-8471, 2008.

Xiang Ma and N. Zabaras, “An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations", J. Computational Physics, Vol. 208, pp. 3084-3113, 2009.

Xiang Ma and N. Zabaras, “An efficient Bayesian inference approach to inverse problems based on an adaptive sparse grid collocation method", Inverse Problems, Vol. 25, 035013 (27pp) , 2009. (selected as the highlights of 2009 in Inverse Problems)

Xiang Ma and N. Zabaras, “An adaptive high-dimensional stochastic model representation technique for the solution of stochastic partial differential equations", J. Computational Physics, Vol. 229, pp. 3884 -3915, 2010.

Xiang Ma and N. Zabaras, “A stochastic mixed finite element heterogeneous multiscale method for flow in porous media", J. Computational Physics, submitted.

Xiang Ma and N. Zabaras, “Kernel principal component analysis for stochastic input model generation", J. Computational Physics, submitted.

An Efficient Computational Framework for Uncertainty Quantification in Multiscale Systems

Documents

uncertainty quantification

stochastic multiscale

highdimensional spdes

probability space

statistical uncertainties

solution of spdesasgc

presentation3 motivation

presentation6 motivation