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LR-WaveHoltz: A Low-Rank Helmholtz Solver

Andreas Granath, Daniel Appelö, Siyang Wang

TL;DR

The paper introduces LR-WaveHoltz, a low-rank solver for the Helmholtz equation that combines WaveHoltz with multi-block SBP-SAT discretization and low-rank representations (SVD in 2D, tensor trains in 3D). Step-truncation and low-rank Anderson acceleration control rank growth during time stepping, enabling efficient fixed-point iterations and scalable 3D simulations. Numerical experiments across open and layered 2D/3D geometries demonstrate substantial compression in 3D and competitive performance in 2D, with robust convergence for constant and piecewise-constant speeds. The work points toward hybrid near-source/far-field strategies to further enhance efficiency in underwater acoustics applications.

Abstract

We propose a low-rank method for solving the Helmholtz equation. Our approach is based on the WaveHoltz method, which computes Helmholtz solutions by applying a time-domain filter to the solution of a related wave equation. The wave equation is discretized by high-order multiblock summation-by-parts finite differences. In two dimensions we use the singular value decomposition and in three dimensions we use tensor trains to compress the numerical solution. To control rank growth we use step-truncation during time stepping and a low-rank Anderson acceleration for the WaveHoltz fixed point iteration. We have carried out extensive numerical experiments demonstrating the convergence and efficacy of the iterative scheme for free- and half-space problems in two and three dimensions with constant and piecewise constant wave speeds.

LR-WaveHoltz: A Low-Rank Helmholtz Solver

TL;DR

The paper introduces LR-WaveHoltz, a low-rank solver for the Helmholtz equation that combines WaveHoltz with multi-block SBP-SAT discretization and low-rank representations (SVD in 2D, tensor trains in 3D). Step-truncation and low-rank Anderson acceleration control rank growth during time stepping, enabling efficient fixed-point iterations and scalable 3D simulations. Numerical experiments across open and layered 2D/3D geometries demonstrate substantial compression in 3D and competitive performance in 2D, with robust convergence for constant and piecewise-constant speeds. The work points toward hybrid near-source/far-field strategies to further enhance efficiency in underwater acoustics applications.

Abstract

We propose a low-rank method for solving the Helmholtz equation. Our approach is based on the WaveHoltz method, which computes Helmholtz solutions by applying a time-domain filter to the solution of a related wave equation. The wave equation is discretized by high-order multiblock summation-by-parts finite differences. In two dimensions we use the singular value decomposition and in three dimensions we use tensor trains to compress the numerical solution. To control rank growth we use step-truncation during time stepping and a low-rank Anderson acceleration for the WaveHoltz fixed point iteration. We have carried out extensive numerical experiments demonstrating the convergence and efficacy of the iterative scheme for free- and half-space problems in two and three dimensions with constant and piecewise constant wave speeds.

Paper Structure

This paper contains 30 sections, 1 theorem, 53 equations, 20 figures, 1 table, 6 algorithms.

Table of Contents

  1. Introduction
  2. Model problem and the WaveHoltz method
  3. The model problem
  4. The WaveHoltz method
  5. Anderson acceleration
  6. Full rank wave solver in matrix form
  7. SBP operators
  8. The full rank formulation in 2D
  9. The step-truncated LR-SBP method
  10. Explicit step-truncation
  11. The two dimensional SVD representation
  12. The three dimensional tensor train representation
  13. The low-rank WaveHoltz method
  14. Numerical Examples
  15. Problems in 2D
  16. ...and 15 more sections

Key Result

Lemma A.1

Let ${\mathbf a},{\mathbf b},{\mathbf c},{\mathbf d}\in\mathbb{R}^n$ satisfy ${\mathbf a}^T{\mathbf b}={\mathbf c}^T{\mathbf d}=0$, $R\in\mathbb{R}^{n\times n}$ a matrix with the SVD representation $USV^T$, $U,V\in\mathbb{R}^{n\times r}$, $S\in\mathbb{R}^{r\times r}$ and $\epsilon>0$ the truncation where $A=\frac{1}{2}I_n+{\mathbf a}{\mathbf a}^T+{\mathbf b}{\mathbf b}^T$ and $B=-\frac{1}{2}-{\ma

Figures (20)

  • Figure 1: (a) An illustration of a multiblock partitioning of $\Omega$, highlighting the multiblocks contributing terms to the discrete Laplacian $L_{ij}$ via interface conditions on $\partial\Omega_{ij}$. (b) An illustration of a corner block, indicating the boundaries (dashed) contributing to the coefficient matrices in the Sylvester equation \ref{['eq:FRSylvester2']} via the boundary conditions. We also highlight the internal edges where interface conditions are imposed (solid).
  • Figure 2: The free-space Green's function $G(x,y)$ of the Helmholtz equation centered at $(-0.1,0.5)$ throughout the domain $[0,5]\times[0,1]$.
  • Figure 3: Ranks of the truncated Greens function $\mathcal{T}_\epsilon(G(X,Y))$ in each of the $5\times 1$ blocks using the the truncation tolerances $\epsilon=10^{-3}h^{-1},\epsilon=10^{-4}h^{-1}$ and $\epsilon=10^{-5}h^{-1}$ for varying number of points per wavelength.
  • Figure 4: Time to simulate hundred time steps in each block using full- and low-rank methods (top) discrete errors between analytical and numerical time steps as a function of points per wavelength using the low-rank discretization with truncation tolerances $\epsilon=10^{-3}h^{-1},\epsilon=10^{-4}h^{-1}$ and $\epsilon=10^{-5}h^{-1}$.
  • Figure 5: Residuals obtained using the LRWH and $\operatorname{LRAA}(4)$ methods for a free-space problem and problem with two corners, respectively. This is obtained using the stopping tolerance $\epsilon^\star=10^{-3}$ and scheduling.
  • ...and 15 more figures

Theorems & Definitions (3)

  • Definition 3.1
  • Lemma A.1
  • proof