A covariance based framework for the propagation of uncertainty through inverse problems with an application to force identification

Inverse problems are widely encountered in fields as diverse as physics, geophysics, engineering and finance. In the present paper, a covariance based framework for the estimation of their uncertainty is presented and applied to the problem of inverse force identification. A key step in its application involves the propagation of frequency response function (FRF) uncertainty through a matrix inversion, for example, between mobility and impedance. To this end a linearised inverse propagation relation is derived. This relation may be considered a generalisation of work presented in the particle physics literature, where we consider both complex valued and non-square matrices through a bivariate description of their uncertainty. Results are illustrated, first, through a numerical simulation where force and moment pairs are applied to a free-free beam model. An experimental study then illustrates the in-situ determination of blocked forces and their subsequent use in the prediction of an operational response. The uncertainties predicted by the proposed framework are in agreement with those acquired through Monte-Carlo (MC) methods for small input variance but are obtained at much lower computational cost, and with improved insight. In the process illustrating the propagation framework, matrix condition number, often taken as an indicator of uncertainty, is shown to relate poorly to a more rigorous uncertainty estimate, leaving open the question as to whether condition number is an appropriate indicator of uncertainty.