Tucker Tensor Train Taylor Series Approximation of High-Dimensional Implicit Mappings

Abstract: We present an efficient method for constructing high-order Taylor series surrogate models for high-dimensional mappings that depend implicitly on the solution of a system of nonlinear equations, e.g., a partial differential equation.

 

High-order Taylor series are traditionally considered intractable here because the derivative tensors are extremely large and are only accessible through multilinear actions on vectors. We overcome these challenges using a ​“Tucker tensor train Taylor series” surrogate model, in which linear (Tucker) dimension reduction is performed on the input and output spaces, and the derivative tensors are approximated by tensor trains in the reduced subspaces. The Tucker bases are constructed using

randomized sketching, and the tensor trains are fit to directionally symmetric action data using a Riemannian manifold Newton method. We justify the model theoretically and provide numerical evidence for the effectiveness of the proposed method.

 

Bio: Nick Alger is a researcher in the Oden Institute at UT Austin working on numerical methods for solving inverse problems governed by PDEs. He received his Ph.D. and M.S. in computational science from UT Austin and his B.S. in physics from Harvey Mudd College.

 

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