Interval-based Inverse Problems with Uncertainties

Interval-based Inverse Problems with Uncertainties Francesco Fedele1,2 and Rafi L. Muhanna1 1 School of Civil and Environmental Engineering 2School 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

Introduction- Inverse Problem • 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- Measurements Uncertainty Information Available Information Device Tolerance Interval

Inverse Problem in Elastostatics 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 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 Introducing the associated Lagrangian with we get

Inverse Problem in Elastostatics To find the optimal α that minimizes the LagrangianF we introduce an imaginary time that rules the evolution/ convergence of the initial guess for αtoward the minimal solution. We wish to find the rate ά = dα / dtso that F always decreases (i.e. F´ < 0 )

Inverse Problem in Elastostatics 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

Interval Approach • 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

Introduction- Why Interval? • 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 • Provides guaranteed enclosures

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

Properties of Interval Arithmetic Let x, yand 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  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} • f (x) = x (1 1) f (x) = 0 • f (x) = { f (x) =x x | x x}

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

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

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 Interval Finite Elements (IFEM)

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

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 C1 U=Vand C2 U = ε

New Formulation Invoking the stationarity of *, that is *= 0, and using C1 U=0 and bold for intervalswe obtain or

Numerical example • Bar truss • 25 elements • Initial guess for E is 60×106kN/m2 for all elements • Target E×10-6kN/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

Numerical example • 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

Interval-based Inverse Problems with Uncertainties