We target time-dependent partial differential equations (PDEs) with
heterogeneous coefficients in space and time. To tackle these problems, we
construct reduced basis/ multiscale ansatz functions defined in space that can
be combined with time stepping schemes within model order reduction or
multiscale methods. To that end, we propose to perform several simulations of
the PDE for few time steps in parallel starting at different, randomly drawn
start points, prescribing random initial conditions; applying a singular value
decomposition to a subset of the so obtained snapshots yields the reduced
basis/ multiscale ansatz functions. This facilitates constructing the reduced
basis/ multiscale ansatz functions in an embarrassingly parallel manner. In
detail, we suggest using a data-dependent probability distribution based on the
data functions of the PDE to select the start points. Each local in time
simulation of the PDE with random initial conditions approximates a local
approximation space in one time point that is optimal in the sense of
Kolmogorov. The derivation of these optimal local approximation spaces which
are spanned by the left singular vectors of a compact transfer operator that
maps arbitrary initial conditions to the solution of the PDE in a later point
of time, is one other main contribution of this paper. By solving the PDE
locally in time with random initial conditions, we construct local ansatz
spaces in time that converge provably at a quasi-optimal rate and allow for
local error control. Numerical experiments demonstrate that the proposed method
can outperform existing methods like the proper orthogonal decomposition even
in a sequential setting and is well capable of approximating
advection-dominated problems