France 2008 1

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Recent advances in Recent advances in Global Sensitivity Analysis Global Sensitivity Analysis

techniquestechniquesS. Kucherenko

Imperial College London, UKs.kucherenko@imperial.ac.uk

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Introduction of Global Sensitivity Analysis and Introduction of Global Sensitivity Analysis and Sobol’ Sobol’ Sensitivity IndicesSensitivity Indices Why Quasi Monte Carlo methods (Why Quasi Monte Carlo methods (Sobol’ sequence sampling) Sobol’ sequence sampling) are much more efficient than are much more efficient than MonteMonte Carlo (random sampling) ? Carlo (random sampling) ?

Effective dimensions and their link with Sobol’ Effective dimensions and their link with Sobol’ Sensitivity Sensitivity IndicesIndices

Classification of functions based on global sensitivity indicesClassification of functions based on global sensitivity indices

Link between Link between Sobol’ Sobol’ Sensitivity IndicesSensitivity Indices and Derivative based and Derivative based Global Sensitivity MeasuresGlobal Sensitivity Measures

Quasi Randon Sampling - Quasi Randon Sampling - High Dimensional Model High Dimensional Model Representation with polynomial approximationRepresentation with polynomial approximation

Application of parametric GSA for optimal experimental design Application of parametric GSA for optimal experimental design

Outline

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Model Output

......

xi : input factors

Propagation of uncertainty

Input

x1

x2

…

x3

…1 2 n

…

x4

xk

y

)(xp

)(xfY

)(Yp

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Consider a model

x is a vector of input variables

Y is the model output.

1 1

0 1,2,..., 1 21

1

...

0

( ) , ... , ,..., ,

( , ,..., ) 0, , 1i s is k

k

i i ij i j k ki i j i

i i i

Y f x f f x f x x f x x x

f x x dx k k s

klji

ijlji

ij

k

ii SSSS ,...,2,1

1

...1

Variance decomposition:

Sobol’ SI:

Sensitivity Indices (SI)Sensitivity Indices (SI)

2 2 2 2, 1,2,...,i iji i j n

ANOVA decomposition (HDMR):

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Sobol’ Sensitivity Indices (SI)Sobol’ Sensitivity Indices (SI)

Definition:

- partial

variances

- variance

Requires 2n integral evaluations for calculations

Sensitivity indices for subsets of variables:

Introduction of the total variance:

Corresponding global sensitivity indices:

1 1

2 2... ... /

s si i i iS

1 1 1 1

12 2... ...

0

,..., ,...,s si i i i i is i isf x x dx x

1

220

0

f x f dx 1

2... si i

,x y z

1

1

2 2, ,

1 ...s

s

m

y i is i i

222

ztoty

,/ 22 yyS ./ 22 toty

totyS

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0 1toty yS S

How to use Sobol’ Sensitivity Indices?How to use Sobol’ Sensitivity Indices?

ytoty SS

iS totiS

0totiS f x

accounts for all interactions between y and z, x=(y,z).

The important indices in practice are and

does not depend on ;

does only depend on ;

corresponds to the absence of interactions between

and other variables

If then function has additive structure:

Fixing unessential variables

If does not depend on so it can be fixed

complexity reduction, from to variables

ix

1iS f x ixtotii SS ix

n

siS

1

,1 0 i ii

f x f f x

1totzS f x z

0,f x f y z znn n

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Evaluation of Sobol’ Sensitivity Evaluation of Sobol’ Sensitivity IndicesIndices

1

0

1 2

0

1 2 200

1( , ) ( , ') ',

1[ ( , ) ( ', )] ',

2

( , )

y

toty

S f y z f y z dydzdzD

S f y z f y z dydzdzD

D f y z dydz f

Straightforward use of Anova decomposition requires Straightforward use of Anova decomposition requires

22nn integral evaluations – not practical ! integral evaluations – not practical !

There are efficient formulas for evaluationThere are efficient formulas for evaluation

of Sobol’ Sensitivity Indices ( Sobol’ 1990):of Sobol’ Sensitivity Indices ( Sobol’ 1990):

Evaluation is reduced to high-dimensional Evaluation is reduced to high-dimensional

integration. Monte Carlo method is the only integration. Monte Carlo method is the only

way to deal with such problemsway to deal with such problems

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Original vrs Improved formulae for Original vrs Improved formulae for evaluation ofevaluation of

Sobol’ Sensitivity IndicesSobol’ Sensitivity Indices

1 200

1 2 200

1 200

112

0 00

1

0

( , ) ( , ') '

( , )

for small indices 1

( , ) ( , ') '

loss of accuracy

Notice that ( , ) ( ', ')

( , ) ( , ') ( ', ') '

( ,

' '

y

y

y

f y z f y z dydzdz fS

f y z dydz f

S

f y z f y z dydzdz f

f f y z dyd

f y z f y z f y z dydzdz

z f y z d dz

f

y

Sy

1

0 00

much more accurate ( Kucherenko, Mauntz, 2002)

Requires ( +2) model evalution original Sobol' formulas (2 +1)

The same model evaluations ca

) ( , )

n be used for computing second

z f f y z f dydz

N n N n

order indices

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Improved formula for Sobol’ Sensitivity Improved formula for Sobol’ Sensitivity IndicesIndices

Comparison between improved and original formulas for Si

-25

-20

-15

-10

-5

0

11 13 15 17 19 21

log 2

(Err

or)

improved Sobol original Sobol

log2(N)

1

71

6( ) ,

( 1)(2 1)

180, 5.110

nT

i ii

f x ix S Sn n n

n S

Comparison between improved and original formulas for Si

1.00E-07

1.00E-06

1.00E-05

1.00E-04

1.00E-03

1.00E-02

1.00E-01

16 18 20 22 24

log

2(S

1)

improved Sobol original Sobol analytical value

log2(N)

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Comparison deterministic and Monte Carlo Comparison deterministic and Monte Carlo integration methodsintegration methods

n

n/p

2

50d

[ ] ( )

Deterministic integration method of p-order,

k points in each direction: N = k

Error: ( ), N =O(1/ ) .

Estimate: 10 , 2, 50

N =10 the total number of particles in the unive

nH

p

I f f x dx

O k

p n

rse

[ ] is impossible to evalua

"Curse of Dimensiona

te

li "

!

ty

I f

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Monte Carlo integration methods Monte Carlo integration methods

1

22

2 1/ 21/ 2

[ ] [ ( )]

1Monte Carlo : [ ] ( )

{ } is a sequence of random points in

Error: [ ] [ ]

( )Expectation:

Convergence does not d

( )

( )= ( ( ))

epent on dimensiona

N

N ii

ni

N

N

I f E f x

I f f zN

z H

I f I f

fE

N

fE

N

lity

but it is slow

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How to improve MC ?How to improve MC ?

1/ 2

II. Use better ( more uniformly di

( )Slow convergence: =

How to

stributed )

sequences.

improve MC ?

I. Decrease ( ) variance reduction.

Discrepancy is a measure of deviation from uniformity:

( )

N

f

N

f

Q y

n1 2

( )

( )

1/ 2 1/ 2

, ( ) [0, ) [0, ) ... [0, ),

( ) volume of

sup ( ) ,

random sequences: (ln ln ) / ~ 1/

n

n

Q yN

Q y H

N

H Q y y y y

m Q Q

ND m Q

N

D N N N

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Sobol’ Sequences vrs Random numbersSobol’ Sequences vrs Random numbersand regular gridand regular grid

Unlike random numbers, successive Sobol’ points “know" about the position of previously sampled points and fill the gaps between them

Regular Grid/ 64 Points Random Numbers/ 64 Points Sobol Numbers/ 64 Points

Sobol Numbers/ 256 PointsRandom Numbers/ 256 PointsRegular Grid/ 256 Points

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Quasi random sequencesQuasi random sequences

(ln )( ) Low discrepancy sequences (LDS)

Convergence: [ ] [ ] ( )

( )

Assymptotically ~ (1/ ) much higher than

Variation of

(ln

~

)

(1/ )

n

N

QMC N

Q

N

M

n

QM

C

MC

C

ND c d

NI f I f V f D

V f f

O N

NO N

O N

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What is the optimal way to arrange N What is the optimal way to arrange N points in two dimensions?points in two dimensions?

Regular Grid Sobol’ Sequence

Low dimensional projections of low discrepancy sequences are better distributed than higher dimensional projections

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Comparison between Sobol sequencesComparison between Sobol sequences

and random numbersand random numbers

 

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Normally distributed Sobol’ Normally distributed Sobol’ SequencesSequences

Normal probability plots Histograms

Uniformly distributed Sobol’ sequences can be Uniformly distributed Sobol’ sequences can be

transformed to any other distribution with a known transformed to any other distribution with a known

distribution functiondistribution function

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Are QMC efficient for high dimensional Are QMC efficient for high dimensional problems ?problems ?

21

(ln )

Assymptotically ~ (1/ )

but increseas with u

not achievable for prac

ntil exp( )

50, 5 10 tical applications

n

QMC

QMC

QMC

O N

NO N

N N n

n N

“For high-dimensional problems (n > 12),

QMC offers no practical advantage over Monte Carlo”

( Bratley, Fox, and Niederreiter (1992)) ?!

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DiscrepancyDiscrepancyI. Low DimensionsI. Low Dimensions

 

0.0001

0.001

1024 2048 4096 8192

T

N

Discrepancy, n=5

RandomHaltonSobol

1e-06

1e-05

0.0001

0.001

0.01

128 256 512 1024 2048 4096 8192 16384 32768 65536

T

N

Discrepancy, n=20

RandomHaltonSobol

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DiscrepancyDiscrepancyII. High DimensionsII. High Dimensions

 

1e-11

1e-10

1e-09

1e-08

1e-07

1e-06

1e-05

0.0001

0.001

128 256 512 1024 2048 4096 8192 16384 32768

T

N

Discrepancy, n=50

RandomHaltonSobol

1e-20

1e-19

1e-18

1e-17

1e-16

128 256 512 1024 2048 4096 8192 16384

T

N

Discrepancy, n=100

RandomSobol

MC in high-dimensions has smaller discrepancy

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Is MC more efficient for high-dimensional Is MC more efficient for high-dimensional problems than QMC ?problems than QMC ?

Pros: MC in high-dimensions has smaller discrepancy Some studies show degradation of the convergence

rate of QMC methods in high-dimensions to O(1/√N)

Cons:

Huge success of QMC methods in finance: QMC methods were proven to be much more efficient than MC even for problems with thousands of variables

Many tests showed superior performance of QMC methods for high-dimensional integration

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Effective dimensionEffective dimension

0

The effective dimension of ( ) in the superposition sense

is the smallest integer such that

Let u be a car

(1 ),

dinality of a set of variables .

It means that ( ) is almost a sum

1S

S

uu d

f x

d

S

u

f x

1

{1,2,..., }

1,

The function ( ) has effective dimension in the truncation sense if

E

of -dimensional functions.

does not depend on the

xa

order in which

mple:

1

(1 ),T

n

i

S

i TS

u

S

T

ud

n

d

f x

f x

x d d

d

d

S

can be reduced

the input variables are sampled,

- depends on the order by reodering variables T Td d

___________________________________________________________

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For many problems only low order terms in the ANOVA decomposition are important

Consider an approximation error

Theorem 1:

Link between an approximation error and

effective dimension in superposition sense

Approximation errorsApproximation errors

Set of variables can be regarded as not important if If and

Consider an approximation error

Theorem 2:

Link between an approximation error and effective dimension in truncation sense

1 1

1

1 1

1

0 ...1 ...

2

...1 ...

( ) ( ,..., )

1( , ) ( ) ( )

( , ) 1 ( ,..., )

( , )

s s

s

s s

s

d s

i i i is i i

d s

i i i is i i

h x f f x x

f h f x h x dx

f h S x x

f h

20 0

0

0

1( ) ( ) ( , )

( ) 2

( ) 2

totz

z f x f y z dxD

z S

z

E

E

:S dd

nd

0( ) ,f x f y z1totzS

z0z z

( , )f h

:T dd

0( )z

1( ,..., )nx x x

1 1( , ) : ( ,..., ), ( ,..., )d d nx y z y x x z z x ___________________________________________________________________

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Type B. Dominant low order indices

1

1n

ii

SS nd

Classification of functionsClassification of functions

Type B,C. Variables are equally important

T Ti j TS S nd

Type A. Variables are not equally important

T Ty z

y zT

S Sn

n nd

Type C. Dominanthigher order indices

1

1n

ii

SS nd

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Sensitivity indices for type A functionsSensitivity indices for type A functions

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Integration error vs. N. Type AIntegration error vs. N. Type A(a) f(x) = ∑n

j=1(-1)i ij=1 xj, n = 360, (b) f(x) = s

i=1 │4xi-2│/(1+a

i), n = 100

{1,2} {3,4,...360}

{1,2} {3,4,...100}

0.94, 0.1

0.64

T T

T T

S S

S S

1/ 2

2

1

1( )

KkN

k

I IK

(a)

(b)

-20

-15

-10

-5

0

8 11 14 17 20

log2(N)

log

2( ) QMC (-0.94)

MC (-0.52)

-14

-10

-6

-2

8 11 14 17 20

log2(N)

log

2(

)

QMC (-0.69)

MC (-0.49)

~ , 0 1N

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Sensitivity indices for type B functionsSensitivity indices for type B functions Dominant low order indices

1

1n

ii

SS nd

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Integration error vs. N. Type BIntegration error vs. N. Type B Dominant low order indices

360

)11()(1

/1

n

xnxfn

i

ni

360

5.0)(

1

n

n

xnxf

n

i

i

-22

-18

-14

-10

8 11 14 17 20

log2(N)

log

2( ) QMC (-0.66)

MC (-0.50)

-19

-15

-11

-7

8 11 14 17 20

log2(N)

log

2(

)

QMC (-0.66)

MC (-0.53)

(a)

(b)

1

1n

ii

SS nd

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Sensitivity indices for type C functionsSensitivity indices for type C functions Dominant higher order indices

1

1n

ii

SS nd

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The integration error vs. N. Type CThe integration error vs. N. Type C Dominant higher order indices:

1/

1

( ) (1/ 2)

20

nn

ii

f x x

n

-8

-6

-4

-2

0

8 11 14 17 20

log2(N)

log

2(

)

QMC (-0.46)

MC (-0.45)

-7

-5

-3

-1

8 11 14 17 20

log2(N)

log

2)

QMC (-0.44)

MC (-0.44)

(a)

(b)

1

1

4 2( ) , 0

1

4 2

20

ni i

ii i

n

ii

x af x a

a

x

n

1

1n

ii

SS nd

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The Morris method

Model ),..,( 1 kxxyy

Elementary Effect for the ith input factor in a point Xo

),...,(),..,,,..,,(

),...,(00000

,000

00 111211

kkiiik

xxyxxxxxxyxxEEi

x1

x2

(x01, x0

2) (x01+, x0

2)

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x1

x2

x1

x2

x..

xr

r elem. effects EE1i EE2

i … EEri are

computed at X1 , … , Xr and then averaged.

Average of EEi’s (xi)

Standard deviation of the EEi’s σ (xi)

The EEi is still a local measure Solution: take the average of several EE

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A graphical representation of results

DK5 ZJ3

DK3

DJ3

ZK5

ZK4

DJ4

DK2

ZJ6

ZK1

0,00E+00

1,00E-01

2,00E-01

3,00E-01

4,00E-01

5,00E-01

6,00E-01

7,00E-01

8,00E-01

9,00E-01

0,00E+00 5,00E-02 1,00E-01 1,50E-01 2,00E-01 2,50E-01 3,00E-01 3,50E-01 4,00E-01 4,50E-01

mu

sigm

a

Factors can be screened on the (xi), σ (xi) plane

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Implemention of the Morris method

r trajectories of (k+1) sample points are generated, each providing one EE per input

x1

x2

x3

Y1 Y2

Y3

Y4

A trajectory of the EE design

Total cost = r (k + 1)r is in the range 4 -10

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.]1,0[~

0

1

24)(

UX

a

a

aXXg

i

i

i

iiii

)(1

i

k

ii Xgy

*(xi) and STi give similar ranking

Problems: large Δ -> incorrect *(xi)

a=99a=9a=0.9

A comparison with variance-based methods:*(xi) is related to STi

Test: the g-function of Sobol’

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Derivative based Global Sensitivity Measures

Morris measure in the limit Δ → 0

1 1

1

0 0 0 0

0

0 0

*

2

( ,..., ) lim ( ,..., ).

( ,..., ) .

.

k k

k

n

n

i i

ii

i iH

iiH

E x x EE x x

fE x x

x

M E dx

fdx

x

x1

x2

x1

x2

x..

xr

Sample X1 , … , Xr Sobol points, estimate finite differences E1

i ,E2i

… Eri and then averaged.

Average of Ei’s M*(xi)

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The integration error vs. N. Type AThe integration error vs. N. Type A g-function of Sobol’ .

(a)

(b)

1

1 2 3

4 2( ) ,

1

0, 6.52

ni i

i i

n

x af x

a

a a a a

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Comparison of Sobol’ SI and Derivative Comparison of Sobol’ SI and Derivative based Global Sensitivity Measuresbased Global Sensitivity Measures

(a)

(b)

1

1 2 3

4 2( ) ,

1

0, 6.52

ni i

i i

n

x af x

a

a a a a

(c) There is a link between and i tot

iS

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Comparison of Sobol’ SI and Comparison of Sobol’ SI and Derivative based Global Sensitivity Derivative based Global Sensitivity

MeasuresMeasures

iv1. Small values of imply small values of . 2. For highly nonlinear functions ranking based on global SI can be very different from that based on derivative based sensitivity measures

2

2

Assume that

then

i

tot ii

f x L

vS

D

Theorem

totiS

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For many problems only low order terms in the ANOVA decomposition are important.

01

( ) ,n

i i ij i ji i j i

h x f f x f x x

1 1

1,

1 ,n n

i iji i i j

nTi i ij

j j i

S S

S S S

Sobol’ SI:

Quasi Randon Sampling Quasi Randon Sampling HDMRHDMR

is a metamodel (HDMR), Rabitz et al:

1 1

1

1 1

1

0 ...1 ...

2

...1 ...

( ) ( ,..., )

1( , ) ( ) ( )

( , ) 1 ( ,..., )

s s

s

s s

s

d s

i i i is i i

d s

i i i is i i

h x f f x x

f h f x h x dx

f h S x x

( )h x

It is assumed that effective dimension in superposition sense ds=2.

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Polynomial Approximation

Properties:

Orthonormal polynomial

base

First few Legendre polynomials:

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Global Sensitivity Analysis Global Sensitivity Analysis (HDMR)(HDMR)

The number of function evaluations is N(n+2) for original Sobol’ method N for sensitivity indices based on RS-HDMR

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How to define maximum polynomial How to define maximum polynomial order ?order ?

Homma-Saltelli function

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RMSE for Homma-Saltelli functionRMSE for Homma-Saltelli function

Root mean square error:

QMC outperforms MC

RS-HDMR hashigher convergencethan Sobol SI method

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g-function: with 2 important and 8 unimportant variables

Sobol g-functionSobol g-function

QRS-HDMRconverges

faster

Values of Sitot

can be inaccurate.

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Sobol g-functionSobol g-function

Error measure:Error measure:

Function ApproximationFunction Approximation

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QRS-HDMR method requires 10 to 103 times less model evaluations than Sobol SI method !

Computational costsComputational costs

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Optimal experimental design (OED) for Optimal experimental design (OED) for parameter estimationparameter estimation

Find values of experimentally manipulable variables (controls) and the

time sampling strategy for a set of Nexp experiments which provides

maximum information for the subsequent parameter estimation problem

UL uuu

Non-linear programming

problem (NLP) with partial

differential-algebraic

(PDAEs) constraints

subject to:

System dynamics (ODEs, DAEs)

Other algebraic constraints

Upper and lower bounds:

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Case study: fed-batch reactorCase study: fed-batch reactor

11111 ypyur

dt

dym

2212

12 yuup

yr

dt

dy m

2

2

5.0

5.0

y

yrm

Biomass:

Substrate:

Reaction rate:

Parameters to be estimated: p1, p2

0.05 < p1 < 0.98, 0.05 < p2 < 0.98

Control variables: u1, u2

Dilution factor: 0.05 < u1 < 0.5

Feed substrate concentration:

5 < u2 < 50

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OED traditional approachOED traditional approach

2

1

A-optimality

E-optimality

D-optimality

Fisher Information Matrix ( FIM ) based criteria:

A criterion =

D criterion =

E criterion =

Modified-E criterion =

FIMFIM

min

maxmin

FIMminmax

FIMdetmax

1min FIMtrace

N

iii

T

i tp

yWt

p

yFIM

1

Main drawback: based on local SI non-realistic linear and local assumptions

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Parametric GSAParametric GSA

Optimal experimental design: identification of a set of experiments with conditions that

deliver measurement data that are the most sensitive to the unknown parameters

Nonlinear dynamic model: Y ( , , )

- uncertain parameters,

- control variables,

- time

Fi depend on parameters !nd ( , ), ( , )

Solve: max ( ( , ))

OED *for parameter estimation

Ti i

iu

f p u t

p

u

t

S u t S u t

F S u t

u

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Application of ParametricApplication of Parametric GSA for GSA for parameter optimizationparameter optimization

Main advantage: based on global SI allows to consider a range of

values for the parameters to be estimated

objective function:

Application of Global Optimization method

1

, ,N T

i i i ii

GSIM Q u t W Q u t

1,1 1,2 1,

2,1 2,2 2,

,1 ,2 ,

, , ,

, , ,

, , ,

i i p i

i i p ii

s i s i s p i

S u t S u t S u t

S u t S u t S u tQ t

S u t S u t S u t

1 1 1

1 2

1 2

, , ,

, , ,

i i ip

i

s s si i i

p

y y yu t u t u t

p p p

Q t

y y yu t u t u t

p p p

GSIMu

detmax

1

, ,N T

i i i ii

FIM Q u t W Q u t

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Case study: fed-batch reactorCase study: fed-batch reactor

11111 ypyur

dt

dym

2212

12 yuup

yr

dt

dy m

2

2

5.0

5.0

y

yrm

Biomass:

Substrate:

Reaction rate:

Parameters to be estimated: p1, p2

0.05 < p1 < 0.98, 0.05 < p2 < 0.98

Control variables: u1, u2

Dilution factor: 0.05 < u1 < 0.5

Feed substrate concentration:

5 < u2 < 50

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Optimal Experimental DesignOptimal Experimental Design

Problem constraints: Experiment duration: 10 h Number of measurement times: 10 Controls varied every 2 hours

Results:

0 2 4 6 8 100

10

20

30

40

Time (h)

Co

nce

ntr

atio

n (

g/l)

y1y2

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

Time (h)

u1

(h

-1)

0 2 4 6 8 100

10

20

30

40

50

u2

(g

/l)

Optimal input profile for u1 and u2 :

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Setting of the Parameter Estimation Setting of the Parameter Estimation ProblemProblem

Steps to find p:

Take experimental or generated pseudo-experimental points

Maximum likelihood optimization

subject to:

System dynamics (ODEs, DAEs)

Other algebraic constraints

Upper and lower bounds:

UL ppp

Non-linear programming

problem (NLP) with partial

differential-algebraic

(PDAEs) constraints

p: set of parameters to be estimated : model prediction

: measurements variance : experimental measures

pyki

kiy~2ki

NE

i

NV

j

NM

k ijk

ijkijkijkml

i ypypJ

1 1 1

2

212~

2

1exp2

y

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Results of Results of parameter parameter estimationestimation

p1 = 0.37 ± 0.02, p2 = 0.72 ± 0.12

p1 = 0.5 ± 0.05 , p2 = 0.5 ± 0.11

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

p1

Den

sity

0 0.2 0.4 0.6 0.8 10

5

10

15

20

p2

Den

sity

0 0.2 0.4 0.6 0.8 10

5

10

15

p1

Den

sity

0 0.2 0.4 0.6 0.8 10

5

10

15

20

p2

Den

sity

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PublicationsPublications

Hung WY, Kucherenko S., Samsatli N.J. and Shah N., The Proceedings of the 2003 Summer Computer Simulation Conference, Canada. Simulation Series, V 35, N3, pp. 101-106 (2003)Hung W.Y., Kucherenko S., Samsatli N.J. and Shah N (2004). Journal of the Operational Research Society 55, 801-813.Sobol’ I., Kucherenko S. Monte Carlo Methods and Simulation, 11, 1, 1-9 (2005).Sobol’ I., Kucherenko S. Wilmott, 56-61, 1 (2005).Kucherenko S., Shah N. Wilmott, 82-91, 4 (2007). Sobol, I.M., S. Tarantola, D. Gatelli, S.S. Kucherenko, W. Mauntz Reliability Engineering & System Safety, 957-960, 92 (2007 ). Rodriguez-Fernandez M., Kucherenko S., Pantelides C., Shah N. Proc. ESCAPE17, V. Plesu and P.S. Agachi (Editors), p66-71, (2007)Kucherenko S., Mauntz W. Submitted to Journal of Comp. Physics (2007). S. Kucherenko. Fifth International Conference on Sensitivity Analysis of Model Output, Budapest, (2007)S. Kucherenko, M. Rodriguez-Fernandez, C. Pantelides, N. Shah. Submitted to Reliability Engineering Systems Safety (2007)D. Gatelli, S. Kucherenko, M. Ratto, S. Tarantola, Submitted to Reliability Engineering Systems Safety (2007)I.M. Sobol’, S. Kucherenko. Submitted to Journal of Comp. Physics (2008).

Application of Global Sensitivity Analysis to Biological ModelsA.Kiparissides, M.Rodriguez-Fernandez, S. Kucherenko, A. Mantalaris, E.Pistikopoulos Application of Global Sensitivity Analysis to Biological Models, Submitted to ESCAPE18 (2008).

 

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SummarySummary

Quasi MC methods based on Sobol’ sequences outperform MCuasi MC methods based on Sobol’ sequences outperform MC

The error generated by the factors fixing is bounded by the total The error generated by the factors fixing is bounded by the total sensitivitysensitivity index of the fixed factors index of the fixed factors Functions can be classified according to their effective dimensionFunctions can be classified according to their effective dimension

The method of derivative based global sensitivity measures The method of derivative based global sensitivity measures (DGSM) is more efficient than the Morris and the Sobol’ SI (DGSM) is more efficient than the Morris and the Sobol’ SI methods. There is a link between DGSM and Sobol’ SImethods. There is a link between DGSM and Sobol’ SI

Quasi Randon Sampling - Quasi Randon Sampling - High Dimensional Model Representation High Dimensional Model Representation with polynomial approximation can be orders of magnitude more with polynomial approximation can be orders of magnitude more efficient than Sobol’ SI for evaluation of main effectsefficient than Sobol’ SI for evaluation of main effects

Application of global SI to OED results in the reduction of the Application of global SI to OED results in the reduction of the required experimental work and the increased accuracy of required experimental work and the increased accuracy of parameter estimationparameter estimation

 

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Thank you for inviting me !

AcknowledgmentsAcknowledgments

Prof. Sobol’

Imperial College London, UK:

N. Shah, M. Rodríguez Fernández, B. Feil, W. Mauntz,

C. Pantelides

Joint Research Centre, ISPRA, Italy:

S. Tarantola, D. Gatelli, M. Ratto

Financial support:

EPSRC Grant EP/D506743/1