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Interval-based Inverse Problems with Uncertainties
Francesco Fedele1,2 and Rafi L. Muhanna1
1 School of Civil and Environmental Engineering2School of Electrical and Computer Engineering
Georgia Institute of Technology, Atlanta, GA 30332-0355, USA
fedele@gatech.edu / rafi.muhanna@gtsav.gatech.edu
REC2012, June 13-15, 2012, Brno, Czech Republic
Outline Introduction Measurements Uncertainty Inverse Problem Interval Arithmetic Interval Finite Elements Examples Conclusions
Inverse problems in science and engineering aim at estimating model parameters of a physical system using observations of the model’s response Variational least square type approaches are typically
adopted Solving the forward model Comparing the calculated data with the actual measured
data Data mismatch is minimized and the process is iterated
until the best match is achieved
Introduction- Inverse Problem
Available Information
Information
Introduction- Measurements Uncertainty
Interval
Device Tolerance
Consider an elastic bar of length L subject to distributed tensional forces f (x). The differential equation
with prescribed boundary conditions
and
E(x): Young’s Modulus, A(x): Cross-sectional Area
Inverse Problem in Elastostatics
,0,0 Lxf
dx
du
dx
d
, ,)0( 00 Qdx
duuu
)()()( xAxEx
To solve for α, the problem becomes the following constrained optimization
: error function (mismatch b-t measured
and predicted u)
: the differential equation
Inverse Problem in Elastostatics
0),,,( subject to
)~,,,( minimize
fuxg
uuxh
)~,,,( uuxh u~
0),,,( fuxg
Introducing the associated Lagrangian
with we get
Inverse Problem in Elastostatics
N
jjj uxuuuxh
1
2)~)((2
1)~,,,(
L
dxfuxg wuuxhwuuF0
),,,()~,,,(),~,,(
LN
jjj dxf
dx
du
dx
dwuxuwuuF
01
2)~)((2
1),~,,(
To find the optimal α that minimizes the Lagrangian F we introduce an imaginary time that rules the evolution/ convergence of the initial guess for α toward the minimal solution.
We wish to find the rate ά = dα / dt so that F always decreases (i.e. F´ < 0 )
Inverse Problem in Elastostatics
dx
dw
dx
du
If we approximate the time derivative of α and use FEM discretization, the deterministic inverse algorithm can be introduced as
K: stiffness matrix
Du, Dw: first derivative of u and w respectively
Δt: scale multiplier
Inverse Problem in Elastostatics
tDwDu
uuwK
PuK
iiii
iii
ii
1
)~()(
)(
Only range of information (tolerance) is available
Represents an uncertain quantity by giving a range of possible values
How to define bounds on the possible ranges of uncertainty? experimental data, measurements, expert knowledge
0t t
0 0[ , ]t t t
Interval Approach
Simple and elegant Conforms to practical tolerance concept Describes the uncertainty that can not be appropriately
modeled by probabilistic approach
Computational basis for other uncertainty approaches
Introduction- Why Interval?
Provides guaranteed enclosures
Interval arithmetic Interval number represents a range of possible
values within a closed set
}|{:],[ xxxRxxx x
Properties of Interval ArithmeticLet x, y and z be interval numbers
1. Commutative Law
x + y = y + x
xy = yx
2. Associative Law
x + (y + z) = (x + y) + z
x(yz) = (xy)z
3. Distributive Law does not always hold, but
x(y + z) xy + xz
Sharp Results – Overestimation
The DEPENDENCY problem arises when one or several variables occur more than once in an interval expression
f (x) = x (1 1) f (x) = 0 f (x) = { f (x) = x x | x x}
f (x) = x x , x = [1, 2] f (x) = [1 2, 2 1] = [1, 1] 0 f (x, y) = { f (x, y) = x y | x x, y y}
Sharp Results – Overestimation Let a, b, c and d be independent variables, each with
interval [1, 3]
B ,
dc
baB
]22[]22[
]22[]22[,
11
11
,,
,,AA
bbbb
bbbbB
bb
bbB
][][
][][,,
11
11physphys AA
00
00,
11
11,
11
11 **physphys ABA B b
Finite Elements
Finite Element Methods (FEM) are
numerical method that provide
approximate solutions to differential
equations (ODE and PDE)
Interval Finite Elements (IFEM) Follows conventional FEM Loads, geometry and material property are expressed as
interval quantities System response is a function of the interval variables
and therefore varies in an interval Computing the exact response range is proven NP-hard The problem is to estimate the bounds on the unknown
exact response range based on the bounds of the parameters
Multiple occurrences – element level Coupling – assemblage process Transformations – local to global and back Solvers – tightest enclosure Derived quantities – function of primary
Overestimation in IFEM
Interval FEMIn steady-state analysis, the variational formulation for a discrete structural model within the context of Finite Element Method (FEM) is given in the following form of the total potential energy functional when subjected
to the constraints C1 U=V and C2 U = ε
),()(
2
12211
* εUCVUCPUUKU TTT
c
T
New FormulationInvoking the stationarity of *, that is *= 0, and using C1 U=0 and bold for intervalswe obtain
or
PKU
0
0
0
000
00
000
0
1
1 ccTT
c
I
IB
C
BC P
λ
λ
UK
ε2
1
Numerical example Bar truss
25 elements Initial guess for E is 60×106 kN/m2 for all elements Target E×10-6 kN/m2 = 100, 105, 110, 115, 120, 120, 115,
110, 105, 100, 105, 110,115, 120, 130, 140, 150, 140, 130, 125, 120, 115, 105, 100, 90
B C
P
Numerical example
0 5 10 15 20 25
0.9
1
1.1
1.2
1.3
1.4
1.5
x 108
Number of Elements
E [
kN/m
2 ]
0 5 10 15 20 250
0.002
0.004
0.006
0.008
0.01
Number of nodes
u [m
]
5% measurements uncertainty Deterministic/interval approach Containment stopping criterion
Conclusions Interval-Based inverse problem solution is developed Measurements uncertainty are modeled as intervals
conforming with the tolerance concept Solution is based on the new deterministic/interval
strategy Containment is used as a new stopping criterion
which is intrinsic to intervals Applications in different fields
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