  In this study, we introduce a new approach, the inverse Kalman filter (IKF),
which enables accurate matrix-vector multiplication between a covariance matrix
from a dynamic linear model and any real-valued vector with linear
computational cost. We incorporate the IKF with the conjugate gradient
algorithm, which substantially accelerates the computation of matrix inversion
for a general form of covariance matrices, whereas other approximation
approaches may not be directly applicable. We demonstrate the scalability and
efficiency of the IKF approach through distinct applications, including
nonparametric estimation of particle interaction functions and predicting
incomplete lattices of correlated data, using both simulation and real-world
observations, including cell trajectory and satellite radar interferogram.
