efficient method of solution of large scale engineering problems with interval parameters based on...
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Efficient Method of Solution of Large Scale
Engineering Problems with Interval Parameters Based
on Sensitivity Analysis
Andrzej PownukSilesian University
of Technology, Poland
2/79
Slightly compressible flow- 2D case
tp
B
cVqy
yp
B
kA
yx
xp
BkA
x oc
bsc
yycxxc
)(1 o
o
ppc
BB
),,(),( * txptxpp
),,(),( * txq
ntxp
q
.),(),( 00 xxptxp
3/79
Measurements
R )(:
)( 11
)( 22
)( NN
…
4/79
Example:inexact ruler, …
Accuracy of measurements
exacti xXmaxxi
)(
We can calculate this number in controllable environments (in laboratory).
This error is not connected with probability.
accuracy
5/79
Inexact measurements
)](),([)(ˆ)(
)()( ,)()(
- accuracy of measurements
6/79
Set-valued random variable
R )(ˆ:ˆ
)(ˆ 11
)(ˆ NN
)(ˆ 22
…
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Characteristics of discrete random variable
Mean value
iii
iii NNNE
)(1 ,)(1)(ˆ1)ˆ(
Variance
i
EN
Var i
2
)ˆ()(ˆ1)ˆ(
8/79
Usually we don’t know probability density function (PDF)
Probabilistic methods require assumptions about the probability density function.
1)(2)()(,)()(
tXVartXEXVartXExP
t x
dxet 2
2
21)(
This formula is true only for Gauss PDF.
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Confidence interval})(:{1
P
)}(:{2
})(:{
PP
22
)(f
1
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Interval estimation of probability
RX :
x1,021,1
1 ,0)(
x
xxf X
)(XyY 3xy
y8,0
81,3
11 ,0
)('))(()(3 2
y y
y
yhyhfyf XY
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1 2 x5.1)( XE
1
08333.0))(()(2
1
2 dyxxxfXVar X
]78868.1 ,21132.1[])(,)([ XVarxXVarx
57735.0)()(
)(
dxxfXVarx
XVarxX 59313.0)(
)(
)(
dyyfYVary
YVaryY
Probability for X: Probability for Y:
)(xf X
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,77739.1)( xyy 72261.5)( xyy
dxxfdyyfx
xX
y
yY
)(57735.0)(
x
y
x x
y
y
1
2
2
1
2
2
)(xyy
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x
y
x x
y
y
2P
1P 3P
1P
2P
3P
)( xyy
Updating results using latest information
3
2
1
2
12
P
P
P
Olddata
Newdata
18/79
Properties of confidence intervals
1)Definition of confidence intervals is not based on the probability density function.
2) Confidence intervals can be defined using set-valued random variables
(uncertain measurements).
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Input membership function
x
3
3x
3x
2
2x
2x
1
1x
1x
0
0x
0x
)(xF
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000ˆ)ˆ(ˆ0 yxyx
111ˆ)ˆ(ˆ1 yxyx
222ˆ)ˆ(ˆ2 yxyx
333ˆ)ˆ(ˆ3 yxyx
)(yF
)(xyy ),(xF )]ˆ(),ˆ([)ˆ(ˆ xyxyxy
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Output membership function
0y
0y
0y
1y
1y
1
2y
2y
2
3y
3y
3)(yF
22/79
Interval solutions of the slightly compressible flow equation
,...}ˆ,ˆ:,...),(inf{ ii pp
,...}ˆ,ˆ:,...),(sup{ ii pp
],[ˆ iiii pppp
Similar treatment for saturation.
23/79
Example
Injection well
Production well
24/79
Interval solution (time step 1)p_upper(t) - p_lower(t)
25/79
“Single-region problems”
xx 1 xx 2 xx 3
]2,1[x,321 xxxy
xxxx 321
xxxxxy 2321
2,
41]}2,1[:{ 2 xxx
x
y
1 2-1
1
2
xxy 2
41
26/79
“Multi-region problems”
1x 2x 3x
5,4]}2,1[,,:{ 321321 xxxxxx
Solution of single-region
problem
Solution of multi-region
problem
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More constraints – less uncertainty
,321 xxxy
321 xxx constraints:
Result with constraints(single-region)
Results without constraints(multi-region)
2,
41 5,4
].2,1[ix
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Multi-region case
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Data filealpha_c 5.614583 /* volume conversion factor */beta_c 1.127 /* transmissibility conversion factor */
/* size of the block */
dx 100dy 100h 100
/* time steps */time_step 15number_of_timesteps 10
reservoir_size 20 20
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Interval solution (time step 5)
31/79
Comparison
Single region - Multi-region
[0,55] [psi] [0, 390] [psi]
32/79
Exact solution of equationswith interval parameters
2
1
2
1
2221
1211bb
xx
aaaa
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),( xkpp
**21
2 RLxx
sc ppkAxBqp
*Rpp *
Lpp x
],[
],,[
xxx kkk
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Monotone functions
1x 2x
)( 1xf
)( 2xf
0)(
dxxd f
)(}ˆ:)(sup{ xfxxxfy
)(}ˆ:)(inf{ xfxxxfy
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Extreme value of monotone functions
),...,,( 21 nxxxfy
nn xxxxxx ˆ,...,ˆ,ˆ 2211
nxxx ˆ...ˆˆˆ 21 x
)}ˆ(:)(min{ xxx Verticesyy
)}ˆ(:)(max{ xxx Verticesyy
n2 - calculations of y(x)
36/79
Sensitivity analysis
If 0)( 0
xxf
, then )(),( xyyxyy
If 0)( 0
xxf
, then )(),( xyyxyy
),(xfy ].,[ xxx
]3,1[,2 xxy,2)( x
dxxdy
,422)2(
dxdy ,1)( xyy 9)( xyy
]9,1[ˆ y
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Complexity of the algorithm, which is based on sensitivity analysis
),(xfy .xx
,1xf
,
2xf
nxf
… - n derivatives
),,...,,( 21 nxxxfy .,...,, 21
nxxxfy
We have to calculate the value of n+3 functions.
,......, ixf
00 ,..., ni xxf 1
n
,,1 ,..., nni xxfy
2
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Vector-valued functions nxxxyy ,...,, 2111
nxxxyy ,...,, 2122
nmm xxxyy ,...,, 21
…
In this case we have to repeat previous algorithm m times.We have to calculate the value of m*(n+2) functions.
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Implicit function
)()( xQyxA
)()()( 1 xQxAxy
yxAxQyxAkkk xxx
)()()(
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Sensitivity matrix
n
mmm
n
n
xy
xy
xy
xy
xy
xy
xy
xy
xy
...
............
...
...
21
2
2
2
1
2
1
2
1
1
1
xy
x 2y
2xy
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Sign vector matrix
mn
mm
n
n
SSS
SSSSSS
sign
...............
...
...
21
222
21
112
11
xy 2S
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Independent sign vectors
,ji SS .)1( ji SS
jijiji S *****
** )1(,, SSSSSS
Number of independent sign vectors:
],1[ m
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Complexity of the whole algorithm.
2*p – solutions (p times upper and lower bound).
],1[ mp
.21,12121 mnnpn
)()( xQyxA 1 - solution
n - derivatives .ix
yyxAxQyxA
kkk xxx
)()()(
)(xy
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All sensitivity vector can be calculated
in one system of equationsyxAxQyxA
kkk xxx
)()()(
yAQRHSkk
k xx
],...,[)( 1 nkx
RHSRHSyxA
Complexity of the algorithm: .22,12222 mp
kkx
RHSyxA )(
45/79
Sensitivity analysis method give us the extreme combination of the parameters We know which combination of upper
bound or lower bound generate the exact solution.
We can use these values in the design process.
min,min,1 ,..., nni xxfy max,max,1 ,..., n
ni xxfy
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Example
,
111111111111
1111
4
3
2
1
4
3
2
1
QQQQ
yyyy
],2,1[ix
.
4321
321
43214321
4
3
2
1
xxxxxxx
xxxxxxxx
yyyy
,
2223
3222444
4321
4
4321
4321
4
3
2
1
xxxxx
xxxxxxxx
QQQQ
47/79
Sensitivity matrix
1111011111111111
4
4
3
44
3
3
34
2
3
24
1
3
1
2
4
1
42
3
1
32
2
1
22
1
1
1
xy
xy
xy
xy
xy
xy
xy
xy
xy
xy
xy
xy
xy
xy
xy
xy
xy
x 1y
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Sign vectors
4
4
3
44
3
3
34
2
3
24
1
3
1
2
4
1
42
3
1
32
2
1
22
1
1
1
xy
xy
xy
xy
xy
xy
xy
xy
xy
xy
xy
xy
xy
xy
xy
xy
signsignxyS
4
3
2
1
1111111111111111
1111011111111111
SSSS
S sign
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Independent sign vectors
1111
1111
2*
1**
S
SS
1111111111111111
4
3
2
1
SSSS
50/79
Lower bound- first sign vector
1111
)(
4
3
2
1
1*
xxxx
Sx
2344
))((
))((
))((
))((
))(())(())((
1*4
1*3
1*2
1*1
1*
11*
1*
Sx
Sx
Sx
Sx
SxQSxASxy
y
y
y
y
))(())(())(( 1*
1*
1* SxQSxySxA
]1,1,1,1[1* S]2,1[ix
51/79
Upper bound- first sign vector
2222
)(
4
3
2
1
1*
xxxx
Sx
4688
))((
))((
))((
))((
))(())(())((
1*4
1*3
1*2
1*1
1*
11*
1*
Sx
Sx
Sx
Sx
SxQSxASxy
y
y
y
y
]1,1,1,1[1* S
]2,1[ix
52/79
Lower bound – second sign vector
]1,1,1,1[2* S
2111
)(
4
3
2
1
2*
xxxx
Sx
1555
))((
))((
))((
))((
))(())(())((
2*4
2*3
2*2
2*1
2*
12*
2*
Sx
Sx
Sx
Sx
SxQSxASxy
y
y
y
y
]2,1[ix
53/79
Upper bound – second sign vector
1222
)(
4
3
2
1
2*
xxxx
Sx
5677
))((
))((
))((
))((
))(())(())((
2*4
2*3
2*2
2*1
2*
12*
2*
Sx
Sx
Sx
Sx
SxQSxASxy
y
y
y
y
]1,1,1,1[2* S
]2,1[ix
54/79
Interval solution
1344
))}(()),(()),(()),((min{ 2*
2*
1*
1* SxySxySxySxyy
5688
))}(()),(()),(()),((max{ 2*
2*
1*
1* SxySxySxySxyy
]5 ,1[]6 ,3[]8 ,4[]8 ,4[
y
55/79
Sensitivity in time-dependent problems
),(),( 1 hpQphpA ttt
11
),(),(),(
tt
k
t
kk
tt
hhhphpAhpQphpA
56/79
Sensitivity
57/79
Calculation of total rate and total oil production
PN
wi
wsciT tqtq
1)()(
NTS
iiiTP ttqN
1)(
wf
w
e
cwfscsc pp
srr
B
khppqq
21ln
20
58/79
Interval total rate
PN
wwi
wsciT ptqtq
1),()(
PN
wwi
wsciT ptqtq
1),()(
59/79
Interval total oil production
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Exact value of total rate
PN
wi
wsciT tqtq
1)()(
PP N
w k
iwsc
N
wi
wsc
kiT
k hp
ptq
tqh
tqh 11
)()()(
)( iTR tqsign
hS
61/79
RShh R RShh R
),()( RiRi tt hpp ),()( RiRi tt hpp
))(,()( RiiTiT ttqtq p))(,()( RiiTiT ttqtq p
)](),([)(ˆ iTiTiT tqtqtq
62/79
Truss structure example
63/79
Accuracy of sensitivity analysis method (5% uncertainty)
Accuracy in %
0 1,04E-02
0 0,00E+00
0,003855 0,00E+00
0 0,00E+00
0 0,00E+00
0 0,00E+00
0 1,89E-03
0 5,64E-01
0,026326 0,00E+00
0 4,87E-03
0 1,21E-03
0 0
18 – interval parameters
64/79
Taylor expansion method
m
iii
i
iii hh
huuu
10,
00
hhh
m
iii
i
iii hh
huuu
10,
00
hh
m
iii
i
iii hh
huuu
10,
00
hh
65/79
Accuracy
%100,
,
exactimidi
iexactii uu
uudu %100
,
,
midiexacti
exactiii uu
uudu
66/79
Accuracy of two methods of calculation (20% uncertainty)
67/79
Accuracy of two methods of calculation (50% uncertainty)
68/79
Comparison 50% uncertainty Sensitivity method [%] Taylor method [%] Comparison [%]
-0,03 -1,19 43,01 -48,34 143466,7 3962,185
-37,1 -0,39 -11,27 -46,95 69,62264 11938,46
-1,53 -0,24 28,41 -44,04 1956,863 18250
-0,25 -4,3 -41,91 21,75 16664 605,814
-0,29 -0,28 43,11 -47,35 14965,52 16810,71
-0,33 -0,04 -45,43 38,26 13666,67 95750
0 -1,97 31,88 -45,78 inf 2223,858
-13,59 -15,68 -32,33 -30,86 137,8955 96,81122
Si
SiTi
du
dudu
,
,, %100
Si
SiTi
du
dudu
,
,, %100Tidu ,
Sidu ,
Tidu ,
Sidu ,
69/79
Time of calculation(endpoints combination method)
70/79
Time of calculation(First order sensitivity analysis)
71/79
Time of calculation(First order Taylor expansion)
72/79
Comparison
Number of interval parameters Sensitivity Taylor %
105 2 0,02 9900410 452 1,22 36949,18915 15 208 16,64 91294,23
Time in seconds
73/79
APDL description N 1 0 0 N 2 1 0
MP 1 EX 210E9 F 3 FX 1000 R 1 0.0025
(description of the nodes)
(material characteristics)
(forces)
(other parameters – cross section)
74/79
Interval extension of APDL language
MP EX 1 5 F 3 FX 5 R 1 10
(material characteristics)
(forces)
(other parameters – cross section)
Uncertainty in percent
75/79
Web applications
http://zeus.polsl.gliwice.pl/~pownuk/interval_truss.htm
Endpoint combination method
Sensitivity analysis method
Taylor method
76/79
Sensitivity analysis method
Pownuk A., Numerical solutions of fuzzy partial differential equation and its application in computational mechanics,
Fuzzy Partial Differential Equations and Relational Equations: Reservoir Characterization and Modeling (M. Nikravesh, L. Zadeh and V. Korotkikh, eds.), Studies in Fuzziness and Soft Computing,
Physica-Verlag, 2004, pp. 308-347
77/79
Monotonicity tests Taylor expansion of derivative Control of the gradient Interval methods
78/79
Conclusions In cases where data is limited and pdfs for
uncertain variables are unavailable, it is better to use imprecise probability (interval) rather than pure probabilistic methods.
Using interval methods we can create mathematical model of the reservoir which is based on very uncertain information.
79/79
Presented algorithm is efficient when compared to other methods which model uncertainty, and can be applied to nonlinear problems of reservoirs simulations.
Sensitivity analysis method gives very accurate results.
Taylor expansion method is more efficient than sensitivity analysis method but less accurate.
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