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We target time-dependent partial differential equations (PDEs) with coefficients that are arbitrarily rough in both 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.

We propose local space-time approximation spaces for parabolic problems that are optimal in the sense of Kolmogorov and may be employed in multiscale and domain decomposition methods. The diffusion coefficient can be arbitrarily rough in space and time. To construct local approximation spaces we consider a compact transfer operator that acts on the space of local solutions and covers the full time dimension. The optimal local spaces are then given by the left singular vectors of the transfer operator. To proof compactness of the latter we combine a suitable parabolic Caccioppoli inequality with the compactness theorem of Aubin-Lions. In contrast to the elliptic setting [I. Babuška and R. Lipton, Multiscale Model. Simul., 9 (2011), pp. 373-406] we need an additional regularity result to combine the two results. Furthermore, we employ the generalized finite element method to couple local spaces and construct an approximation of the global solution. Since our approach yields reduced space-time bases, the computation of the global approximation does not require a time stepping method and is thus computationally efficient. Moreover, we derive rigorous local and global a priori error bounds. In detail, we bound the global approximation error in a graph norm by the local errors in the L 2 (H 1 )-norm, noting that the space the transfer operator maps to is equipped with this norm. Numerical experiments demonstrate an exponential decay of the singular values of the transfer operator and the local and global approximation errors for problems with high contrast or multiscale structure regarding space and time.