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