Tucker Tensor Train Taylor Series
This work addresses a computational bottleneck in modeling high-dimensional systems, such as those in partial differential equations, by providing a scalable method for surrogate model construction, though it is incremental in its approach to tensor approximation.
The authors tackled the intractability of constructing Taylor series for high-dimensional mappings by introducing a Tucker tensor train Taylor series (T4S) surrogate model, which approximates derivative tensors using Tucker decompositions and tensor trains, and demonstrated its effectiveness through numerical evidence.
We present methods for constructing Taylor series surrogate models for covariance preconditioned high dimensional mappings that depend implicitly on the solution of a system of nonlinear equations, e.g., the solution of a partial differential equation. Taylor series are traditionally considered intractable for such mappings because the derivative tensors are enormous, and are only accessible through ``probing'' (contraction of the tensor with vectors in all but one index). We overcome these challenges using a ``Tucker tensor train Taylor series'' (T4S) surrogate model, in which each derivative tensor is approximated by a Tucker decomposition composed with a tensor train. After an initial dimension reduction, Tucker tensor trains are fit to directionally symmetric tensor probes using Riemannian manifold optimization within a rank continuation scheme. The optimization is enabled by fast sweeping methods for applying the Riemannian Jacobian (the Jacobian for the Tucker tensor train fitting problem) and its transpose to vectors. We justify the T4S model theoretically, and provide numerical evidence for the effectiveness of the proposed methods.