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Multiscale interpolative construction of quantized tensor trains

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arxiv 2311.12554 v3 pith:7YPDRF2H submitted 2023-11-21 math.NA cs.NA

Michael Lindsey This is my paper
classification math.NAcs.NA
keywords qttsconstructionfunctionfunctionsmultiscalenumericalperspectivequantized
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Quantized tensor trains (QTTs) have recently emerged as a framework for the numerical discretization of continuous functions, with the potential for widespread applications in numerical analysis. However, the theory of QTT approximation is not fully understood. In this work, we advance this theory from the point of view of multiscale polynomial interpolation. This perspective clarifies why QTT ranks decay with increasing depth, quantitatively controls QTT rank in terms of smoothness of the target function, and explains why certain functions with sharp features and poor quantitative smoothness can still be well approximated by QTTs. The perspective also motivates new practical and efficient algorithms for the construction of QTTs from function evaluations on multiresolution grids.

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Cited by 11 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. A tensor-train multidimensional inverse Laplace transform

    math.NA 2026-06 unverdicted novelty 7.0 of 10

    Tensor-train formulation reduces multidimensional inverse Laplace transform cost from exponential to polynomial under low-rank assumptions.

  2. Linear-Scaling Tensor Train Sketching

    math.NA 2026-03 accept novelty 7.0 of 10

    TTStack achieves oblivious subspace embedding and injection for tensor trains with sample complexity linear in order d and subspace dimension r, yielding quasi-optimal randomized TT rounding.

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    physics.comp-ph 2026-02 unverdicted novelty 7.0 of 10

    An adaptive patching method exploits block-sparse QTT structures to reduce computational costs for tensor contractions and enables efficient evaluation of bubble diagrams and Bethe-Salpeter equations.

  4. Stable full-field simulation of a multiscale elliptic equation by means of Quantized Tensor Trains

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    A QTT-based solver with Helmholtz-Leray penalization in Fourier space stably computes solutions and gradients for multiscale elliptic PDEs on meshes with up to 10^37 virtual degrees of freedom in 3D.

  5. Stable full-field simulation of a multiscale elliptic equation by means of Quantized Tensor Trains

    math.NA 2026-05 unverdicted novelty 6.0 of 10

    A QTT-based solver for multiscale elliptic equations achieves full-field solutions and gradients on meshes up to 10^37 DoFs in 3D via a penalized Helmholtz-Leray formulation solved in Fourier space, with claimed uncon...

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    Derives rigorous entanglement scaling laws in MPS for smooth real or complex functions and applies them via tensor cross interpolation to construct and test shallow quantum encoding circuits on up to 156 qubits.

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    Tensor-Train Interpolation (TTI) refines a coarse QTT to arbitrary resolution by appending constant-rank polynomial-kernel cores, with error controlled by the coarse grid spacing.

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    SeeMPS is a Python MPS/TT library offering a BLAS/LAPACK-style API for compressed linear algebra, from DMRG and time evolution to PDE solving and Fourier transforms.

  11. A practical investigation on time integration in the quantized tensor train format

    physics.comp-ph 2026-05 unverdicted novelty 3.0 of 10

    The paper investigates the effects of time integrator selection, numerical dissipation, and problem representation on the efficiency and stability of quantized tensor train simulations for advection-dominated test problems.

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