  We present a new methodology for computing sensitivities in evolutionary
systems using a model-driven low-rank approximation. To this end, we formulate
a variational principle that seeks to minimize the distance between the time
derivative of the reduced approximation and sensitivity dynamics. The
first-order optimality condition of the variational principle leads to a system
of closed-form evolution equations for an orthonormal basis and corresponding
sensitivity coefficients. This approach allows for the computation of
sensitivities with respect to a large number of parameters in an accurate and
tractable manner by extracting correlations between different sensitivities on
the fly. The presented method requires solving forward evolution equations,
sidestepping the restrictions imposed by forward/backward workflow of adjoint
sensitivities. For example, the presented method, unlike the adjoint equation,
does not impose any I/O load and can be used in applications in which real time
sensitivities are of interest. We demonstrate the utility of the method for
three test cases: (1) computing sensitivity with respect to model parameters in
the Rossler system (2) computing sensitivity with respect to an
infinite-dimensional forcing parameter in the chaotic Kuramoto-Sivashinsky
equation and (3) computing sensitivity with respect to reaction parameters for
species transport in a turbulent reacting flow.
