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A hybrid interpolation ACA accelerated method for parabolic boundary integral operators

Sivaram Ambikasaran, Ritesh Khan, Johannes Tausch, Sihao Wang

TL;DR

Addresses efficient solution of parabolic boundary-integral equations for the heat equation by leveraging a hybrid approach that couples Chebyshev-time interpolation for history terms with adaptive cross approximation (ACA) for spatial blocks. The method uses a hierarchical time-space discretization, resulting in a data-sparse representation and a block-forward-elimination solver with controllable accuracy. A rigorous error analysis links ACA tolerances to temporal-spatial granularity and provides storage/compute complexity estimates that grow only logarithmically with problem size. Numerical experiments on a unit-sphere geometry validate the expected convergence rates and demonstrate substantial compression, offering a scalable, kernel-agnostic framework for parabolic boundary-element methods and potential extensions to other parabolic problems like transient Stokes flow.

Abstract

We consider piecewise polynomial discontinuous Galerkin discretizations of boundary integral reformulations of the heat equation. The resulting linear systems are dense and block-lower triangular and hence can be solved by block forward elimination. For the fast evaluation of the history part, the matrix is subdivided into a family of sub-matrices according to the temporal separation. Separated blocks are approximated by Chebyshev interpolation of the heat kernel in time. For the spatial variable, we propose an adaptive cross approximation (ACA) framework to obtain a data-sparse approximation of the entire matrix. We analyse how the ACA tolerance must be adjusted to the temporal separation and present numerical results for a benchmark problem to confirm the theoretical estimates.

A hybrid interpolation ACA accelerated method for parabolic boundary integral operators

TL;DR

Addresses efficient solution of parabolic boundary-integral equations for the heat equation by leveraging a hybrid approach that couples Chebyshev-time interpolation for history terms with adaptive cross approximation (ACA) for spatial blocks. The method uses a hierarchical time-space discretization, resulting in a data-sparse representation and a block-forward-elimination solver with controllable accuracy. A rigorous error analysis links ACA tolerances to temporal-spatial granularity and provides storage/compute complexity estimates that grow only logarithmically with problem size. Numerical experiments on a unit-sphere geometry validate the expected convergence rates and demonstrate substantial compression, offering a scalable, kernel-agnostic framework for parabolic boundary-element methods and potential extensions to other parabolic problems like transient Stokes flow.

Abstract

We consider piecewise polynomial discontinuous Galerkin discretizations of boundary integral reformulations of the heat equation. The resulting linear systems are dense and block-lower triangular and hence can be solved by block forward elimination. For the fast evaluation of the history part, the matrix is subdivided into a family of sub-matrices according to the temporal separation. Separated blocks are approximated by Chebyshev interpolation of the heat kernel in time. For the spatial variable, we propose an adaptive cross approximation (ACA) framework to obtain a data-sparse approximation of the entire matrix. We analyse how the ACA tolerance must be adjusted to the temporal separation and present numerical results for a benchmark problem to confirm the theoretical estimates.
Paper Structure (9 sections, 4 theorems, 66 equations, 7 figures, 2 tables, 1 algorithm)

This paper contains 9 sections, 4 theorems, 66 equations, 7 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Discretization of Thermal Layer Potentials
  3. Hybrid ACA Method for the Discrete Thermal Layer Potential
  4. ACA compression for the spatial blocks
  5. Continuous spatial discretizations
  6. Analysis
  7. Numerical Results
  8. Example Problem
  9. Conclusions

Key Result

Lemma 1

\newlabellem:blocksparse0 If $A = [B_{k \ell}]_{k,\ell}$, where $B_{k \ell}\in \mathbb{R}^{m_k \times n_\ell}$, then the following estimate holds

Figures (7)

  • Figure 1: \newlabelfig:H-matrix0 Partitioning of ${\bf V}$ for the case of four temporal levels.
  • Figure 1: Triangulation of the unit sphere and its refinements.
  • Figure 2: Setup time (sec) versus the $N_s N_t$. The quadrature order is mentioned in the parenthesis for $a=1$.
  • Figure 3: Solution time (sec) versus the $N_s N_t$.
  • Figure 4: $L_2$-norm of the absolute error in the solution versus the $N_s N_t$.
  • ...and 2 more figures

Theorems & Definitions (6)

  • Lemma 1
  • Proof 1
  • Lemma 2
  • Lemma 3
  • Proof 2
  • Theorem 4