Efficient construction of tensor-train approximations for operator-value tensors

Develop an efficient method to obtain a tensor-train approximation of the tensor of parametric operator evaluations arising from the tensor interpolant, while avoiding the many sequential evaluations of the operator required by TT-cross and the restrictive structural assumptions required by ALS-cross.

Background

The paper discusses approximating the coefficient functions of an operator by tensor-product interpolation after truncating the input to its first finitely many dimensions. The resulting tensor of operator values has exponentially large storage requirements in the number of retained dimensions, motivating compression in the tensor-train format, whose storage and evaluation costs can be much lower when the tensor-train rank is moderate.

The paper identifies TT-cross as a standard construction but notes that it requires many sequential evaluations of the underlying operator, which may be computationally slow. ALS-cross can improve efficiency, but it requires a specific structure of the operator and therefore has limited applicability. The text consequently identifies the efficient construction of a tensor-train approximation in the general setting as an unresolved methodological question.

References

A key question is how to efficiently obtain a TT approximation of the tensor in eqref{eq:gitensor}. A standard algorithm is so-called TT-cross cite{oseledets10}. A drawback is that it requires many sequential evaluations of $G$, which can make the process slow in practice. For this reason the ALS-cross algorithm was developed in the context of parametric PDEs cite{dolgov18}; while it also relies on sequential evaluations of $G$, it can be significantly more efficient. The downside is that it requires a specific structure for the operator $G$, thereby limiting its applicability.

— Operator Surrogates  (2610.18210 - Schwab et al., 13 Oct 2026) in Section 3, subsection “Low-rank Tensor Approximations”