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Approximation in the extended functional tensor train format

Christoph Strössner, Bonan Sun, Daniel Kressner

TL;DR

The paper tackles the challenge of high-dimensional function approximation by introducing the extended functional tensor train (EFTT), a data-efficient surrogate that blends tensorized Chebyshev interpolation with a Tucker-based low-rank representation and subsequent TT compression. The core approach proceeds via adaptive Tucker factorization of the Chebyshev evaluation tensor using fiber-based ACA with randomized pivots, then uses discrete empirical interpolation to form a compact core, and finally applies a rank-adaptive TT-cross to obtain the EFTT representation, enabling efficient evaluation and storage. Empirical results show EFTT reduces function evaluations by up to 96% and storage by over 90% compared to existing FTT-based methods, while maintaining or improving accuracy across a wide range of benchmark and Genz functions and an uncertainty-quantification PDE example. The work demonstrates that adaptive, multi-stage tensor decompositions can significantly improve efficiency in multivariate function approximation, with direct implications for surrogate modeling in uncertainty quantification, computational chemistry, and physics.

Abstract

This work proposes the extended functional tensor train (EFTT) format for compressing and working with multivariate functions on tensor product domains. Our compression algorithm combines tensorized Chebyshev interpolation with a low-rank approximation algorithm that is entirely based on function evaluations. Compared to existing methods based on the functional tensor train format, the adaptivity of our approach often results in reducing the required storage, sometimes considerably, while achieving the same accuracy. In particular, we reduce the number of function evaluations required to achieve a prescribed accuracy by up to over 96% compared to the algorithm from [Gorodetsky, Karaman and Marzouk, Comput. Methods Appl. Mech. Eng., 347 (2019)].

Approximation in the extended functional tensor train format

TL;DR

The paper tackles the challenge of high-dimensional function approximation by introducing the extended functional tensor train (EFTT), a data-efficient surrogate that blends tensorized Chebyshev interpolation with a Tucker-based low-rank representation and subsequent TT compression. The core approach proceeds via adaptive Tucker factorization of the Chebyshev evaluation tensor using fiber-based ACA with randomized pivots, then uses discrete empirical interpolation to form a compact core, and finally applies a rank-adaptive TT-cross to obtain the EFTT representation, enabling efficient evaluation and storage. Empirical results show EFTT reduces function evaluations by up to 96% and storage by over 90% compared to existing FTT-based methods, while maintaining or improving accuracy across a wide range of benchmark and Genz functions and an uncertainty-quantification PDE example. The work demonstrates that adaptive, multi-stage tensor decompositions can significantly improve efficiency in multivariate function approximation, with direct implications for surrogate modeling in uncertainty quantification, computational chemistry, and physics.

Abstract

This work proposes the extended functional tensor train (EFTT) format for compressing and working with multivariate functions on tensor product domains. Our compression algorithm combines tensorized Chebyshev interpolation with a low-rank approximation algorithm that is entirely based on function evaluations. Compared to existing methods based on the functional tensor train format, the adaptivity of our approach often results in reducing the required storage, sometimes considerably, while achieving the same accuracy. In particular, we reduce the number of function evaluations required to achieve a prescribed accuracy by up to over 96% compared to the algorithm from [Gorodetsky, Karaman and Marzouk, Comput. Methods Appl. Mech. Eng., 347 (2019)].
Paper Structure (24 sections, 1 theorem, 35 equations, 3 figures, 6 tables, 5 algorithms)

This paper contains 24 sections, 1 theorem, 35 equations, 3 figures, 6 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Functional low-rank approximation via tensorized interpolation
  3. Tensorized Chebyshev interpolation
  4. Functional tensor train (FTT) approximation
  5. Extended functional tensor train (EFTT) approximation
  6. Approximation algorithm
  7. Factor matrices
  8. Fiber approximations.
  9. Tucker approximation
  10. TT cores
  11. EFTT approximation algorithm
  12. Numerical experiments
  13. Comparison to a direct TT approximation
  14. Benchmark functions.
  15. Genz functions
  16. ...and 9 more sections

Key Result

Lemma 1

Let $\tilde{f}$ be defined as in eq:ChebyshevInterpolantForm. For $\hat{\mathcal{T}} \in \mathbb R^{(n_1+1) \times \dots \times (n_d+1)}$, consider the polynomial with $\hat{\mathcal{A}} = \hat{\mathcal{T}} \times_{1} F^{(1)}\times_2 \dots\times_d F^{(d)}$. Then where $\lVert \cdot \rVert_\infty$ denotes the uniform norm for functions and the maximum norm for tensors.

Figures (3)

  • Figure 1: Tensor network representation Orus14 of the coefficient tensor $\hat{\mathcal{A}}$ in \ref{['eq:CoeffTensorExtendedFunTT']} corresponding to EFTT format \ref{['eq:ExtendedFunctionalTT']}. The colored boxes mark the subtensors corresponding to the approximation of the evaluation tensor $\hat{\mathcal{T}}$ as in \ref{['eq:HatF']} and the approximation of the core tensor $\hat{\mathcal{C}}$ as in \ref{['eq:TTofCore']}.
  • Figure 2: We apply Algorithm \ref{['alg:Approximation']} (EFTT) and an approximation of the evaluation tensor using Algorithm \ref{['alg:TTcross']} (DirectTT) to approximate the Genz functions (see Appendix \ref{['appendix:Genz']}) for dimension $d \in \{20,50,100,200,300,400,500\}$. Each row corresponds to one test function. Left: We plot the $L^2$ error for EFTT and DirectTT. Right: We plot the number of function evaluations required to compute the approximation for EFTT and DirectTT. Throughout this figure, we sample $30$ different parameters for the Genz functions as in Bigoni16 and display the geometric mean of the error and the arithmetic mean for the number of function evaluations.
  • Figure 3: We apply Algorithm \ref{['alg:Approximation']} (EFTT) and the algorithm in the c3py package to approximate $f(x_1,\dots,x_d) = \sin(x_1 + x_2 + \dots + x_d)$ by functional low-rank approximation with Legendre polynomial basis functions \ref{['eq:LegendreBasis']} as in Gorodetsky17. Left: We plot the relative error of the integral of the approximations. Right: We plot the number of function evaluations required to compute the approximation.

Theorems & Definitions (2)

  • Lemma 1
  • Remark 1