Abstract: We address Bayesian inverse problems governed by autonomous dynamical systems, and in particular linear time-invariant (LTI) systems. Our focus is on large-scale source inversion problems in which real-time solution and uncertainty quantification are critical, and hyperbolic or nearly-hyperbolic forward PDEs govern the dynamics (e.g., high frequency acoustic, elastic, or electromagnetic wave propagation or advection-dominated transport). Such PDEs do not readily lend themselves to classical surrogate or reduced order representations due to large Kolmogorov n-widths. We show that the parameter-to-observable (p2o) operator inherits the autonomous structure of the forward problem, in particular the time shift invariance. Upon discretization, this leads to a block Toeplitz matrix, permitting compact storage, FFT diagonalization, and fast GPU implementation. Thus evaluation of the p2o map using this representation can be carried out exactly (up to rounding errors) many orders of magnitude faster than the PDE-based p2o map. The fast p2o map computation can be exploited to compute the Bayesian solution of the source inversion problem in real time. We present results for a tsunami early warning inverse and posterior prediction problem with one billion inversion parameters representing the earthquake-induced seafloor motion. The observational data come from acoustic pressure sensor data. The inverse solution and subsequent tsunami wave height forecasts can be computed online in a fraction of a second on a 512-GPU cluster (compared to ~50 years using the full wave propagation PDEs on the same cluster). Moreover, we exploit this capability for fast Bayesian inversion to determine the sensor placement that maximizes the expected information gain from the data using a greedy algorithm. This work is joint with Sreeram Venkat (UT Austin), Stefan Henneking (UT Austin), and Alice Gabriel (UCSD).