1. Interval-based Inverse Problems with Uncertainties Francesco Fedele 1,2 and Rafi L. Muhanna 1 1 School of Civil and Environmental Engineering 2 School of Electrical and Computer Engineering Georgia Institute of Technology, Atlanta, GA 30332-0355, USA fedele@gatech.edufedele@gatech.edu / rafi.muhanna@gtsav.gatech.edurafi.muhanna@gtsav.gatech.edu REC2012, June 13-15, 2012, Brno, Czech Republic

2. Outline Introduction Introduction Measurements Uncertainty Measurements Uncertainty Inverse Problem Inverse Problem Interval Arithmetic Interval Arithmetic Interval Finite Elements Interval Finite Elements Examples Examples Conclusions Conclusions

3.  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

4. Available Information Information Introduction- Measurements Uncertainty Interval Device Tolerance

5. 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

6. 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

7. Introducing the associated Lagrangian with we get Inverse Problem in Elastostatics

8. 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

9. 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

10.  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 Interval Approach

11.  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

12. Interval arithmetic Interval number represents a range of possible values within a closed set

13. Properties of Interval Arithmetic Let 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

14. Sharp Results– Overestimation 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}

15. Sharp Results– Overestimation Sharp Results – Overestimation Let a, b, c and d be independent variables, each with interval [1, 3]

16. Finite Elements Finite Element Methods (FEM) are numerical method that provide approximate solutions to differential equations (ODE and PDE)

17. 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

18. Multiple occurrences – element level Multiple occurrences – element level Coupling – assemblage process Coupling – assemblage process Transformations – local to global and back Transformations – local to global and back Solvers – tightest enclosure Solvers – tightest enclosure Derived quantities – function of primary Derived quantities – function of primary Overestimation in IFEM

19. Interval FEM In 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 C 1 U=V and C 2 U = ε

20. New Formulation Invoking the stationarity of  *, that is  * = 0, and using C 1 U=0 and bold for intervalswe obtain or

21. Numerical ex ample Bar truss 25 elements Initial guess for E is 60×10 6 kN/m 2 for all elements Target E×10 -6 kN/m 2 = 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

22. Numerical ex ample  5% measurements uncertainty   Deterministic/interval approach  Containment stopping criterion

23. 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