  In this paper, we investigate tensor based nonintrusive reduced-order models
(ROMs) for parametric cross-diffusion equations. The full-order model (FOM)
consists of ordinary differential equations (ODEs) in matrix or tensor form
resulting from finite-difference discretization of the differential operators
by taking the advantage of Kronecker structure. The matrix/tensor differential
equations are integrated in time with the implicit-explicit (IMEX) Euler
method. The reduced bases, relying on a finite sample set of parameter values,
are constructed in form of a two-level approach by applying higher-order
singular value decomposition (HOSVD) to the space-time snapshots in tensor
form, which leads to a large amount of computational and memory savings. The
nonintrusive reduced approximations for an arbitrary parameter value are
obtained through tensor product of the reduced basis by the parameter dependent
core tensor that contains the reduced coefficients. The reduced coefficients
for new parameter values are computed using radial basis function (RBF)
interpolation. The efficiency of the proposed method is illustrated through
numerical experiments for two-dimensional Schnakenberg and three-dimensional
Brusselator cross-diffusion equations. The spatiotemporal patterns are
accurately predicted by the reduced-order models with speed-up factors of
orders two and three over the full-order models.
