Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A priori analysis of a tensor ROM for parameter dependent parabolic problems

Alexander V. Mamonov, Maxim A. Olshanskii

TL;DR

The paper addresses the error analysis of the Galerkin LRTD-ROM for an abstract linear parabolic problem that depends on multiple physical parameters and proves an error estimate for the LRTd-ROM solution, which is uniform with respect to problem parameters and extends to parameter values not in a sampling/training set.

Abstract

A space-time-parameters structure of parametric parabolic PDEs motivates the application of tensor methods to define reduced order models (ROMs). Within a tensor-based ROM framework, the matrix SVD - a traditional dimension reduction technique - yields to a low-rank tensor decomposition (LRTD). Such tensor extension of the Galerkin proper orthogonal decomposition ROMs (POD-ROMs) benefits both the practical efficiency of the ROM and its amenability for rigorous error analysis when applied to parametric PDEs. The paper addresses the error analysis of the Galerkin LRTD-ROM for an abstract linear parabolic problem that depends on multiple physical parameters. An error estimate for the LRTD-ROM solution is proved, which is uniform with respect to problem parameters and extends to parameter values not in a sampling/training set. The estimate is given in terms of discretization and sampling mesh properties, and LRTD accuracy. The estimate depends on the local smoothness rather than on the Kolmogorov n-widths of the parameterized manifold of solutions. Theoretical results are illustrated with several numerical experiments.

A priori analysis of a tensor ROM for parameter dependent parabolic problems

TL;DR

The paper addresses the error analysis of the Galerkin LRTD-ROM for an abstract linear parabolic problem that depends on multiple physical parameters and proves an error estimate for the LRTd-ROM solution, which is uniform with respect to problem parameters and extends to parameter values not in a sampling/training set.

Abstract

A space-time-parameters structure of parametric parabolic PDEs motivates the application of tensor methods to define reduced order models (ROMs). Within a tensor-based ROM framework, the matrix SVD - a traditional dimension reduction technique - yields to a low-rank tensor decomposition (LRTD). Such tensor extension of the Galerkin proper orthogonal decomposition ROMs (POD-ROMs) benefits both the practical efficiency of the ROM and its amenability for rigorous error analysis when applied to parametric PDEs. The paper addresses the error analysis of the Galerkin LRTD-ROM for an abstract linear parabolic problem that depends on multiple physical parameters. An error estimate for the LRTD-ROM solution is proved, which is uniform with respect to problem parameters and extends to parameter values not in a sampling/training set. The estimate is given in terms of discretization and sampling mesh properties, and LRTD accuracy. The estimate depends on the local smoothness rather than on the Kolmogorov n-widths of the parameterized manifold of solutions. Theoretical results are illustrated with several numerical experiments.
Paper Structure (14 sections, 6 theorems, 99 equations, 3 figures)

This paper contains 14 sections, 6 theorems, 99 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Conventional POD
  4. Multi-linear algebra preliminaries and LRTD
  5. LRTD--ROM
  6. Approximation estimate
  7. $\boldsymbol{\alpha}$-uniform estimates for the FOM solution
  8. An interpolation result
  9. Gramian matrices and singular values
  10. Approximation estimate
  11. Error estimate
  12. Numerical examples
  13. Heat equation
  14. Advection-diffusion equation

Key Result

Lemma 1

\newlabelL:Phi_norm0 Assume the solution of eqn:GenericPDE$u$ is sufficiently regular such that UnifEst is satisfied. Then there holds with a constant $C$ independent of $h$, $\Delta t$, and ${\widehat{\mathcal{A}}}$.

Figures (3)

  • Figure 1: Domain $\Omega$ and the solution $u(T, \mathbf x)$ of \ref{['eqn:heat']}--\ref{['eqn:bcins']} corresponding to $\boldsymbol{\alpha} = (0.5, 0.9)^T$.
  • Figure 2: Error estimates $\widetilde{E}_{\max}$ (solid blue line with circles) and $\widetilde{E}_{\text{mean}}$ (dashed red line with circles) as functions of $\varepsilon$ (left), $\delta$ (middle) and $\Lambda_{l+1}$ (right). The slopes of dashed black lines represent the theoretical bounds from the right-hand side of \ref{['eqn:err1']}: $\varepsilon$ (left), $\delta^2$ (middle) and $\Lambda_{l+1}^{1/2}$ (right).
  • Figure 3: Error estimates $\widetilde{E}_{\max}$ (solid blue line with circles) and $\widetilde{E}_{\text{mean}}$ (dashed red line with circles) as functions of $\varepsilon$ (left), $\delta$ (middle) and $\Lambda_{l+1}$ (right). The slopes of dashed black lines represent the theoretical bounds from the right-hand side of \ref{['eqn:err1']}: $\varepsilon$ (left), $\delta^2$ (middle) and $\Lambda_{l+1}^{1/2}$ (right).

Theorems & Definitions (17)

  • Remark 1
  • Lemma 1
  • Proof 1
  • Remark 2
  • Remark 3: Implementation
  • Remark 4: Interpolation procedure
  • Remark 5: LRTD vs POD ROMs
  • Theorem 1
  • Lemma 2
  • Proof 2
  • ...and 7 more