An inverse problem for the nonlinear Gao beam (Q902961)

Summary: A goal of many inverse problems is to find unknown parameter values, \({\lambda} \in {\Lambda}\), so that given observed data \(u_{true}\) agrees well with solution data produced using these parameters \(u_{\lambda}\). That is, we wish to solve the minimisation problem \(\min_{\lambda \in \Lambda}\| u_{true} - u_{\lambda} \|\); where \(\| \cdot \|\) is some appropriate norm. Unfortunately finding \(u_{\lambda}\) 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 \(\lambda \in \Lambda\) 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.