analyse de sensibilité déterministe pour la simulation numérique du
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Analyse de sensibilite deterministe
pour la simulation numerique
du transfert de contaminants
Estelle Marchand
financial support from ANDRA
additional support from GDR MOMAS
12 decembre 2007
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Outline
1 Introduction
2 Statistical Analyses
3 Deterministic uncertainty and sensitivity analysis
First order approximation
Singular Value Decomposition
SVD and statistical analysis
4 Applications
Flow problem
Transport problem
5 Implementation aspects
Generic tools
C++ implementation of PIO
A generic tool for deterministic sensitivity analysis
6 Conclusions
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Outline
1 Introduction
2 Statistical Analyses
3 Deterministic uncertainty and sensitivity analysis
First order approximation
Singular Value Decomposition
SVD and statistical analysis
4 Applications
Flow problem
Transport problem
5 Implementation aspects
Generic tools
C++ implementation of PIO
A generic tool for deterministic sensitivity analysis
6 Conclusions
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Context
Long term management of radionuclear waste
long life waste : separation / transmutation → CEA
reversible storage in deep geological layers → ANDRA
storage in subsurface
Storage in deep geological layers
waste packages
elaborated barrier
geological barrier
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Motivation
Modeling flow and transport around a nuclear waste storage site
subsurface properties,
contaminants properties:
K ΦRi Di csati
F : numerical
solverssafety indicators
(doses, water flow,
molar flow, concentrations)
Uncertainties about input data:
measurement techniques, spatial variability→
uncertainties about
safety indicators
Question: how to evaluate the influence of the uncertainties about the
input parameters on the safety indicators?
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Sensitivity analysis
Evaluation of the uncertainties about the safety indicators due to the
uncertainties about the input parameters.
Hierarchical classification of the input parameters according to their
influence on the safety indicators.
Two approaches Probabilistic Deterministic
(Monte Carlo)
Type of analysis global local
Implementation simple more complex
Computation time very important smaller
The two approaches are complementary.
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Outline
1 Introduction
2 Statistical Analyses
3 Deterministic uncertainty and sensitivity analysis
First order approximation
Singular Value Decomposition
SVD and statistical analysis
4 Applications
Flow problem
Transport problem
5 Implementation aspects
Generic tools
C++ implementation of PIO
A generic tool for deterministic sensitivity analysis
6 Conclusions
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Quick overview of Monte Carlo methods — 1
Sampling
Generate a sample of vectors of input parameters (LHS...)
Apply F to each vector
Uncertainty analysis
Analyse the sample of output variables
scatterplots
estimation of means, of standard deviations
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Quick overview of Monte Carlo methods — 2
Sensitivity analysis
Analyse correlations between input and output variables to
determine which inputs influence the outputs:
computation of statistical indicators
difficulties solution
Deal with strong use of
nonlinearities of F rank transformation
Separate influences of F partial correlations,
and of input correlations standard regressions
F must be monotoneous
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Outline
1 Introduction
2 Statistical Analyses
3 Deterministic uncertainty and sensitivity analysis
First order approximation
Singular Value Decomposition
SVD and statistical analysis
4 Applications
Flow problem
Transport problem
5 Implementation aspects
Generic tools
C++ implementation of PIO
A generic tool for deterministic sensitivity analysis
6 Conclusions
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Uncertainty analysis of the linearized problem
F : p 7→ Φvector of input parameters vector of output variables
Local study: analysis of the linearized problem dΦ = F ′ (p)dp
Computation of
uncertainties about output variables of the linearized problem:
(
|dp|, F ′)
→ upper bound for |dΦi | =∑
j
∣
∣
∣F ′
ij
∣
∣
∣
∣
∣dpj∣
∣
statistical indicators for the linearized problem
computed using formulas without sampling from F ′ and
correlations and standard deviations of the input parameters
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Singular Value Decomposition
Local study: analysis of the linearized problem dΦ = F ′ (p)dp
Definition
SVD of F ′ (p): F ′ (p) = USVT where
S: diagonal matrix
s1 ≥ s2 ≥ ... ≥ 0: singular values =
√
eigenvalues of F ′(p)TF ′(p)
U and V : orthogonal matrices
uk , vk columns of U, V : singular vectors
F ′ (p)vk = skukor vk ∈ ker F
′ (p)
or uk ∈ (im F ′ (p))⊥
→ Hierarchical classification of inputs according to influence on outputs
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SVD and statistical analysis
Proper orthogonal decompositions: Bibliographic study
Simplify the simulation of a random vector x by decomposingx =
∑
skukwith u orthogonal basis and sk random variables
variance concentrated in the first terms of the decomposition
by eigenvalue decomposition of the covariance matrix of x
approximation using a sample of x : SVD of the sample
Extension to sensitivity analysis?
y = F (x), x random vector, F deterministic model
We assume components of x are independent and E(xxT ) = I.
We compute SVD of F = Σy ,xΣ−1x ,x
We obtain changes of variables in input and output spaces s. t.
new input components are still independent
each new output is correlated with at most one new input
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Computation of the derivatives of F
Approximate differentiation
Divided differences (DD) are either expensive or inaccurate
→ to be used for validation only
Exact differentiation
Adjoint or reverse mode / direct mode
Automatic differentiation (AD) by operator overloading with AdolC
Manual differentiation (MD): implementation of analytical formulas
Performance comparison
AD with AdolC is competitive with MD concerning running time
Memory management is difficult with AD
AD is more convenient than DD to validate MD.
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Outline
1 Introduction
2 Statistical Analyses
3 Deterministic uncertainty and sensitivity analysis
First order approximation
Singular Value Decomposition
SVD and statistical analysis
4 Applications
Flow problem
Transport problem
5 Implementation aspects
Generic tools
C++ implementation of PIO
A generic tool for deterministic sensitivity analysis
6 Conclusions
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Choice of the model F
Flow model: stationary Darcy’s law
div~u = q Ω~u = −K~∇p Ωp = p ΓD
~u · ~ν = gN ΓN
Fi associates water flow Φi through outlet channel Sito input parameters logK:
Fi : logK→ Φi =
∫
Si
~u · ~ν .
K is piecewise constant.
Computed with a mixed hybrid finite element formulation.
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Hydraulic test case
Display of a stream line 2D vertical cut
Simplified 3D hydrological model
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Spectrum of possible input parameters
Global probabilistic study
Truncated lognormal laws of bounds min, max
Variable Kv (m.s−1) Kh/Kv K4/K3 K5 (m.s−1) K6/K5
min,max 10−14, 10−12 1, 102 10, 103 10−9, 10−7 10, 103
K3/Kv ; uniform law of bounds min = 10, max = 1000
Choice of points for local deterministic studies with 6 parameters
Most probable set of parameters (1 study)
One parameter is set to min or max (2 × 6 studies)
All parameters are set to min or max (26 studies)
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Sensitivity results for most probable set of parameters
F ′ (logK ) vk = skuk
Singular valuessk
1 2 3 410
−5
10−4
10−3
10−2
10−1
100
singular value index
sin
gu
lar
va
lue
input space singular vectorsvk
h v 3 4 5 6 7 8 9 10 11 12_13−1
−0.5
0
0.5
1
parameter value index (permeability)
1th input sv
2th input sv
3th input sv
4th input sv
5th input sv
6th input sv
7th input sv
8th input sv
9th input sv
10th input sv
11th input sv
12th input sv
output space singular vectorsuk
1 2 3 4−1
−0.5
0
0.5
1
mesure index (velocity flow)
1th output sv
2th output sv
3th output sv
4th output sv
Hierarchical classification of local influences:
log(K6) on Φ4
log(K5) on Φ3
log(Kv ) on Φ1 + Φ2
log(K3) on Φ1 − Φ2 (almost zero)
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Sensitivity results for a less probable set of parameters
F ′ (logK ) vk = skuk
Chosen parameters: maxK6 and max (logK3 − logKv )Singular valuessk
1 2 3 410
−6
10−5
10−4
10−3
10−2
10−1
100
singular value index
sin
gu
lar
va
lue
input space singular vectorsvk
h v 3 4 5 6 7 8 9 10 11 12_13−1
−0.5
0
0.5
1
parameter value index (permeability)
1th input sv
2th input sv
3th input sv
4th input sv
5th input sv
6th input sv
7th input sv
8th input sv
9th input sv
10th input sv
11th input sv
12th input sv
output space singular vectorsuk
1 2 3 4−1
−0.5
0
0.5
1
mesure index (velocity flow)
1th output sv
2th output sv
3th output sv
4th output sv
Modifications of the local influences:
log(K6) + log(K8) on Φ4
v3 and v4 slightly modified
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Sensitivity results for a less probable set of parameters
F ′ (logK ) vk = skuk
Chosen parameters: max (logK5 − logKv )Singular valuessk
1 2 3 410
−4
10−3
10−2
10−1
100
singular value index
sin
gu
lar
va
lue
input space singular vectorsvk
h v 3 4 5 6 7 8 9 10 11 12_13−1
−0.5
0
0.5
1
parameter value index (permeability)
1th input sv
2th input sv
3th input sv
4th input sv
5th input sv
6th input sv
7th input sv
8th input sv
9th input sv
10th input sv
11th input sv
12th input sv
output space singular vectorsuk
1 2 3 4−1
−0.5
0
0.5
1
mesure index (velocity flow)
1th output sv
2th output sv
3th output sv
4th output sv
Local influences are reordered:
ln(K6) on Φ4
ln(Kv ) on Φ1 + Φ2
ln(K5) on Φ3
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Results of probabilistic study: indicator of correlation
between the water flows and the input parameters (∗)
(∗) Laurent Loth and Guillaume Pepin, ANDRA
Φ1 Φ3 Φ4
logKh 0.10 0.03 0.01
logKv 1.00 0.05 0.01
logK3 0.08 0.00 0.03
logK4 0.02 0.01 0.02
logK5 0.12 0.93 0.14
logK6 0.08 0.47 1.00
indicator = mean of the coefficients of Spearman, PRCC, SRRC
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Choice of the model F
Transport model
ω∂c
∂t+ div
(
c~u− D~∇c)
+ λc = q Ω
c = cd ΓD(
c~u− D~∇c)
· ~ν = gF ΓF~∇c · ~ν = gN ΓN
F ni associates contaminant flow Ψni through outlet channel Si at tnto weighted (log of) K, ω, D and q:
F ni : input parameters→ Ψni =
∫
Si
(
c~u− D~∇c)n
· ~ν .
~u from flow equation
Parameters are piecewise constant.Computed with a splitting method
explicit for convection (for precision), with finite volumes
implicit for diffusion (for stability), with mixed finite elements
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Flow+transport test case
Simplified model
Two dimensional computation over a vertical non planar cut
Computation over zones 1 to 6
11 uncertain parameters
q (source term)
ω over 1, 2, 3, 4 and 5, 6
D over 1, 2, 3, 4 and 5, 6
kh, kv , k3, k4, k5, k6
Computation for one set of parameters
Output = fluxes(
Ψtn1 ,Ψtn2 ,Ψtn3 ,Ψtn4
)
, at a given date tn
Output = fluxes(
Ψt1i ,Ψt2i , ...)
, through a given outlet channel SiE. Marchand (ANDRA & INRIA) Deterministic sensitivity analysis 12-12-07 24 / 43
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Sensitivity results for most probable set of parameters,
tn = 106 years
F ′ (logK ) vk = skuk
Singular valuessk
1 2 3 410
−14
10−12
10−10
10−8
10−6
10−4
10−2
100
singular value index
sin
gu
lar
va
lue
input space singular vectorsvk
q om_134 om_56 D_134 D_56 kh kv k3 k4 k5 k6−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
parameter value index
output space singular vectorsuk
S1 S2 S3 S4
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
mesure index (velocity flow)
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Sensitivity results for most probable set of parameters,
output = Ψt12 , ..., Ψ
t52
F ′ (logK ) vk = skuk
Singular valuessk
1 2 3 4 510
−15
10−10
10−5
100
singular value index
sin
gu
lar
va
lue
input space singular vectorsvk
q om_134 om_56 D_134 D_56 kh kv k3 k4 k5 k6−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
parameter value index
output space singular vectorsuk
t1 t2 t3 t4 t5−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
mesure index (velocity flow)
Diffusion parameters then convection parameters.
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Probabilistic results (ANDRA)
Most influent parameters: Φ1,2,3,4, D1,2,3,4, kv
For S1 et S2
weak influence of param zones 5,6
time decreasing influence of diffusion parameters
For S3
less weak influence of param zones 5,6
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Conclusion about comparisons
Flow
Easy to choose a non linear scaling
Log parameterization is consistant with probabilistic data
Transport
Scaling more difficult
Stronger coupling between parameters → consistance with
probabilistic data difficult
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Outline
1 Introduction
2 Statistical Analyses
3 Deterministic uncertainty and sensitivity analysis
First order approximation
Singular Value Decomposition
SVD and statistical analysis
4 Applications
Flow problem
Transport problem
5 Implementation aspects
Generic tools
C++ implementation of PIO
A generic tool for deterministic sensitivity analysis
6 Conclusions
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Motivation
Possible structure for a program
Generic: algorithmic part, difficult to set up
ex. domain decomposition algorithm
Specific: computational part, heavy, already existing in libraries
ex. solve flow equation
Ideas
Avoid reimplementing the generic part for each specific application
Safe and expressive functional language used for the generic part
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Paradigm for the implementation of generic tools – 1
Paradigm
Heavy computations → external programs (computational servers)
Master program in Caml
Communications between external and master programs → PIO
To deal with heterogeneity of codes
Polyglot Input Output, agnostic communication library
Simple, robust protocoleAscii and binary modes
Polyglot: implemented in Caml, C++, C, Fortran
in project: Matlab, Java
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Paradigm for the implementation of generic tools – 2
Heavy comp. C++
Worker (C++)
PIO C++
PIO Caml
Driver (Caml)
Generic algorithm Caml
= librariesor modules
= process
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Protocole of communication PIO
Channels = standard i/o channels or file streams
A communication can be
a Task communication: a function name and its arguments,
a Result communication: a sequence of typed values,an Error communication: an exception value.
When a Task communication is received
Try to apply the function given by its nameIf success: Result communication sent back
Otherwise: Error communication sent back
→ a communication is always sent back
Values: one string, one integer, one float, one integer couple, one
integer triple, one float couple, or one float triple, or vectors of
such, or float matrices.
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Caml: a strongly typed functional language
Distributed by INRIA since 1985 (http://caml.inria.fr/)
⇒ Unison, MLdonkey, Coq, FFTW, cdtools, . . .
High-level language based on clear theoretical foundation and
formal semantics
Strong typing: static and automatic
No automatic type conversion ⇒ safety
Segmentation fault or bus error never happens
A program that compiles is close to be correct
Functional language ⇒ close to the mathematics
# let compose f g = function x -> g (f x);;
compose : (α → β) → (β → γ) → α → γ
Polymorphism, modules, objects ⇒ complex structures/algorithms
Implementing with Caml requires a different way of thinking
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C++ implementation of PIO
Based on preexisting Caml implementation
Difficulty: late knowledge of concrete data types during
communication
Implementation of a general UNION type using standard C++
library
Implementation of the UNION type
From Caml and C union
class Elem<...> to simulate free pointers ie“Something (of fixed type) or nothing”
One vector of length 0 (“empty”) or of length 1 (“full”)
Identifier, safe constructors, safe assignment
A UNION class
Several Elem<...> attributes: exactly one is “full”
Identifier, safe constructors, safe assignement
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General types for PIO
Information over communication channels
Most general type: “Communication”
UNION composed of: Task, Result, Error
General type of data: “Typed value”
for the arguments of a Task or the components of a Result
UNION composed of: Matrix type[1,2], Vector type,
Lexem
Matrix type[1,2], Vector type, Lexem
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Using a generic tool is easy
The user writes (several) independent external programs in any
programing language
Only his main function must follow a particular specification
(infinite loop receiving orders and returning results using PIO).
Examples are provided in each language
It is not necessary to know Caml language to use the generic tool
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Generic tool for deterministic sensitivity analysis
Steps for an analysis
Initialize: read mesh, compute values ... Returns the size of the
Jacobian matrix
Compute rows or columns of F ′
Assemble the Jacobian matrix
Parameterize
Compute SVD
We can use several generic tools together (for example computation of
a column of F ′ using domain decomposition)
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Outline
1 Introduction
2 Statistical Analyses
3 Deterministic uncertainty and sensitivity analysis
First order approximation
Singular Value Decomposition
SVD and statistical analysis
4 Applications
Flow problem
Transport problem
5 Implementation aspects
Generic tools
C++ implementation of PIO
A generic tool for deterministic sensitivity analysis
6 Conclusions
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Conclusions
Local deterministic sensitivity analysis for a flow and transportproblem
Exact computation of the Jacobian matrix,
use AdolC for validation
SVD of Jacobian matrix provides locally a hierarchy for influences
of the inputs on the outputs
Cheap → several local analyses
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Conclusions
Comparison with global probabilistic results for the flow problem
For this example: weak variability of the results of the
deterministic study ↔ weak nonlinearity of the model in the
spectrum of possible input parameters
For this example probabilistic and deterministic studies provide
very similar results
A strong variability of the deterministic results should lead to be
wary of the results of the probabilistic study
Difficulties for the transport problem
Scaling
Consistant parameterization
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Conclusions
Software development
Development of a generic tool for sensitivity analysis
PIO C++
Perspective
Use it with parallel domain decomposition, with
Neumann-Neumann preconditioning and balancing
Go on connecting probabilitic and deterministic analyses with
numerical applications
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THANK YOU FOR YOUR ATTENTION
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