  We present a framework enabling variational data assimilation for gradient
flows in general metric spaces, based on the minimizing movement (or
Jordan-Kinderlehrer-Otto) approximation scheme. After discussing stability
properties in the most general case, we specialise to the space of probability
measures endowed with the Wasserstein distance. This setting covers many
non-linear partial differential equations (PDEs), such as the porous medium
equation or general drift-diffusion-aggregation equations, which can be treated
by our methods independent of their respective properties (such as finite speed
of propagation or blow-up). We then focus on the numerical implementation of
our approach using an primal-dual algorithm. The strength of our approach lies
in the fact that by simply changing the driving functional, a wide range of
PDEs can be treated without the need to adopt the numerical scheme. We conclude
by presenting detailed numerical examples.
