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Dynamical low-rank tensor approximations to high-dimensional parabolic problems: existence and convergence of spatial discretizations

Markus Bachmayr, Henrik Eisenmann, André Uschmajew

TL;DR

The paper develops a rigorous, geometric framework for dynamical low-rank tensor approximations of high-dimensional parabolic PDEs by constraining solutions to tensor manifolds and projecting the evolution onto the corresponding tangent spaces. It proves existence, uniqueness, and stability of continuous DLRA solutions within a Gelfand-triple setting, and establishes convergence of spatial semidiscretizations to the continuous solution. A general theory of low-rank tensor manifolds in Hilbert spaces is introduced, including tangent-space projections and curvature estimates, and applied to tensor train and Tucker formats. The results provide a solid foundation for reliable, scalable DLRA-based simulations of multidimensional parabolic problems, with direct implications for TT/Tucker-based numerical methods.

Abstract

We consider dynamical low-rank approximations to parabolic problems on higher-order tensor manifolds in Hilbert spaces. In addition to existence of solutions and their stability with respect to perturbations to the problem data, we show convergence of spatial discretizations. Our framework accommodates various standard low-rank tensor formats for multivariate functions, including tensor train and hierarchical tensors.

Dynamical low-rank tensor approximations to high-dimensional parabolic problems: existence and convergence of spatial discretizations

TL;DR

The paper develops a rigorous, geometric framework for dynamical low-rank tensor approximations of high-dimensional parabolic PDEs by constraining solutions to tensor manifolds and projecting the evolution onto the corresponding tangent spaces. It proves existence, uniqueness, and stability of continuous DLRA solutions within a Gelfand-triple setting, and establishes convergence of spatial semidiscretizations to the continuous solution. A general theory of low-rank tensor manifolds in Hilbert spaces is introduced, including tangent-space projections and curvature estimates, and applied to tensor train and Tucker formats. The results provide a solid foundation for reliable, scalable DLRA-based simulations of multidimensional parabolic problems, with direct implications for TT/Tucker-based numerical methods.

Abstract

We consider dynamical low-rank approximations to parabolic problems on higher-order tensor manifolds in Hilbert spaces. In addition to existence of solutions and their stability with respect to perturbations to the problem data, we show convergence of spatial discretizations. Our framework accommodates various standard low-rank tensor formats for multivariate functions, including tensor train and hierarchical tensors.
Paper Structure (21 sections, 21 theorems, 197 equations)

This paper contains 21 sections, 21 theorems, 197 equations.

Table of Contents

  1. Introduction
  2. Dynamical low-rank approximation for parabolic problems
  3. Contributions and outline
  4. Abstract formulation
  5. Basic assumptions and existence of solutions
  6. A stability estimate
  7. Convergence of spatial discretizations
  8. Properties of low-rank tensor manifolds in Hilbert space
  9. Low-rank tensor manifolds in function space
  10. Manifold structure
  11. Tangent space projection
  12. Distance to boundary
  13. Curvature estimates
  14. Application to tensor train manifolds
  15. Application to the model problem
  16. ...and 6 more sections

Key Result

Theorem 2.2

Let the Assumptions property:regularity--property:splitting hold and let $u_0$ have positive $\mathcal{H}$-distance from $\overline{\mathcal{M}}^w \setminus \mathcal{M}$. There exist $T^* \in (0,T]$ and $u \in W(0,T^*;\mathcal{V},\mathcal{H}) \cap L_\infty(0,T^*;\mathcal{V})$ such that $u$ solves Pr In either case, $u$ is the unique solution of Problem problem 1 in $W(0,T^*;\mathcal{V},\mathcal{H}

Theorems & Definitions (41)

  • Theorem 2.2
  • Remark 2.3
  • Remark 2.4
  • Theorem 2.5
  • proof
  • Theorem 3.2
  • proof
  • Theorem 4.1
  • Proposition 4.2
  • proof
  • ...and 31 more