  Calibration of large-scale differential equation models to observational or
experimental data is a widespread challenge throughout applied sciences and
engineering. A crucial bottleneck in state-of-the art calibration methods is
the calculation of local sensitivities, i.e. derivatives of the loss function
with respect to the estimated parameters, which often necessitates several
numerical solves of the underlying system of partial or ordinary differential
equations. In this paper we present a new probabilistic approach to computing
local sensitivities. The proposed method has several advantages over classical
methods. Firstly, it operates within a constrained computational budget and
provides a probabilistic quantification of uncertainty incurred in the
sensitivities from this constraint. Secondly, information from previous
sensitivity estimates can be recycled in subsequent computations, reducing the
overall computational effort for iterative gradient-based calibration methods.
The methodology presented is applied to two challenging test problems and
compared against classical methods.
