Tucker Tensor Train Taylor Series

TL;DR

This paper introduces a novel Tucker tensor train Taylor series (T4S) surrogate modeling approach that efficiently constructs high-dimensional Taylor series for complex mappings, overcoming traditional intractability issues.

Contribution

The paper develops a Tucker tensor train-based surrogate model for high-dimensional Taylor series, combining tensor decompositions with Riemannian optimization for efficient computation.

Findings

T4S effectively approximates derivatives in high dimensions.

Numerical experiments demonstrate the method's accuracy and efficiency.

Theoretical justification supports the approach's validity.

Abstract

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.