Title: An inverse problem for the nonlinear Gao beam

Authors: Kimberly M. Levere

Addresses: School of Engineering, University of Guelph, 50 Stone Road East, Guelph, Ontario, N1G 2W1, Canada

Abstract: A goal of many inverse problems is to find unknown parameter values, λ ∈ Λ, so that given observed data u_true agrees well with solution data produced using these parameters u_λ. That is, we wish to solve the minimisation problem min_λ∈Λ || u_true - u_λ ||; where || • || is some appropriate norm. Unfortunately finding u_λ in terms of the parameters of the problem may be a difficult or even impossible task. Further, the objective function may be a complicated function of the parameters λ ∈ Λ and may require complex minimisation techniques. In recent literature, the collage coding approach to solving inverse problems has emerged. This approach avoids the aforementioned difficulties by bounding the approximation error above by a more readily minimisable distance, thus making the approximation error small. In this paper, we apply a collage-based method to a hyperbolic problem that models the 'Gao beam'; a nonlinear beam model that incorporates the possibility of buckling of a beam under a load. We explore an inverse problem that seeks the flexural rigidity of the beam and present and discuss the results.

Keywords: inverse problems; parameter estimation; collage theorem; weak solution theory; nonlinear Gao beam; partial differential equations; PDEs; hyperbolic problem; functional analysis; optimisation; numerical analysis; beam buckling; flexural rigidity.

DOI: 10.1504/IJANS.2014.061007

International Journal of Applied Nonlinear Science, 2014 Vol.1 No.2, pp.122 - 135

Received: 05 Nov 2012
Accepted: 29 Jan 2013
Published online: 12 Jul 2014 *

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