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Abstract

This paper proposes two new algorithms related to the Tucker tensor format. The first method is a new cross approximation for Tucker tensors, which we call Cross$^2$-DEIM. Cross$^2$-DEIM is an iterative method that uses a fiber sampling strategy, sampling $O(r)$ fibers in each mode, where $r$ denotes the target rank. The fibers are selected based on the discrete empirical interpolation method (DEIM). Cross$^2$-DEIM resemblances the Fiber Sampling Tucker Decomposition (FSTD)2 approximation, and has favorable computational scaling compared to existing methods in the literature. We demonstrate good performance of Cross$^2$-DEIM in terms of iteration count and intermediate memory. First we design a fast direct Poisson solver based on Cross$^2$-DEIM and the fast Fourier transform. This solver can be used as a stand alone or as a preconditioner for low-rank solvers for elliptic problems. The second method is a low-rank solver for nonlinear tensor equation in Tucker format by Anderson acceleration (AA), which we call Tucker-AA. Tucker-AA is an extension of low-rank AA (lrAA) proposed in our prior work for low-rank solution to nonlinear matrix equation. We apply Cross$^2$-DEIM with warm-start in Tucker-AA to deal with the nonlinearity in the equation. We apply low-rank operations in AA, and by an appropriate rank truncation strategy, we are able to control the intermediate rank growth. We demonstrated the performance for Tucker-AA for approximate solutions nonlinear PDEs in 3D.