Low-rank cross approximation of function-valued tensors for reduced-order modeling of parametric PDEs

The paper considers function-valued tensors, viewed as multidimensional arrays with entries in an abstract Hilbert space. Despite the absence of the algebraic structure of a field, the geometric inner-product structure suffices to introduce the Tucker rank, higher-order SVD, and Tucker-cross decomposition for function-valued tensors. An adaptive cross-approximation algorithm is developed to compute low-rank approximations of such tensors. The framework is motivated by, and applied to, model order reduction of the parameter-to-solution map for a parametric PDE. The resulting reduced-order model can be interpreted as an encoder-decoder scheme with a nonlinear encoder and a multilinear decoder. The performance of the proposed non-intrusive approximation method is demonstrated in numerical examples for two nonlinear parametric PDE systems.