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Efficient approximation of high-frequency Helmholtz solutions by Gaussian coherent states

T. Chaumont-Frelet, V. Dolean, M. Ingremeau

TL;DR

The paper tackles efficient numerical approximation of high-frequency Helmholtz solutions with smooth coefficients by constructing finite-dimensional spaces spanned by Gaussian coherent states, exploiting their localisation in phase space. It develops a semiclassical framework with a phase-space Gaussian frame and proves two central approximability results: for general RHS the DOFs scale like $(kR)^d$, while for plane-wave scattering RHS the DOFs scale like $(kR)^{d-1/2+oldsymbol{ abla}}$ with super-algebraic convergence, under micro-localisation assumptions. A least-squares discretization using these spaces is analyzed and verified in 1D, with the matrix entries shown to decay super-algebraically away from the diagonal. The approach provides a frequency-aware discretisation that significantly reduces degrees of freedom in high-frequency scattering and offers a viable route for efficient solvers in smooth heterogeneous media, with clear paths for extension to higher dimensions and Galerkin-type schemes.

Abstract

We introduce new finite-dimensional spaces specifically designed to approximate the solutions to high-frequency Helmholtz problems with smooth variable coefficients in dimension $d$. These discretization spaces are spanned by Gaussian coherent states, that have the key property to be localised in phase space. We carefully select the Gaussian coherent states spanning the approximation space by exploiting the (known) micro-localisation properties of the solution. For a large class of source terms (including plane-wave scattering problems), this choice leads to discrete spaces that provide a uniform approximation error for all wavenumber $k$ with a number of degrees of freedom scaling as $k^{d-1/2}$, which we rigorously establish. In comparison, for discretization spaces based on (piecewise) polynomials, the number of degrees of freedom has to scale at least as $k^d$ to achieve the same property. These theoretical results are illustrated by one-dimensional numerical examples, where the proposed discretization spaces are coupled with a least-squares variational formulation.

Efficient approximation of high-frequency Helmholtz solutions by Gaussian coherent states

TL;DR

The paper tackles efficient numerical approximation of high-frequency Helmholtz solutions with smooth coefficients by constructing finite-dimensional spaces spanned by Gaussian coherent states, exploiting their localisation in phase space. It develops a semiclassical framework with a phase-space Gaussian frame and proves two central approximability results: for general RHS the DOFs scale like , while for plane-wave scattering RHS the DOFs scale like with super-algebraic convergence, under micro-localisation assumptions. A least-squares discretization using these spaces is analyzed and verified in 1D, with the matrix entries shown to decay super-algebraically away from the diagonal. The approach provides a frequency-aware discretisation that significantly reduces degrees of freedom in high-frequency scattering and offers a viable route for efficient solvers in smooth heterogeneous media, with clear paths for extension to higher dimensions and Galerkin-type schemes.

Abstract

We introduce new finite-dimensional spaces specifically designed to approximate the solutions to high-frequency Helmholtz problems with smooth variable coefficients in dimension . These discretization spaces are spanned by Gaussian coherent states, that have the key property to be localised in phase space. We carefully select the Gaussian coherent states spanning the approximation space by exploiting the (known) micro-localisation properties of the solution. For a large class of source terms (including plane-wave scattering problems), this choice leads to discrete spaces that provide a uniform approximation error for all wavenumber with a number of degrees of freedom scaling as , which we rigorously establish. In comparison, for discretization spaces based on (piecewise) polynomials, the number of degrees of freedom has to scale at least as to achieve the same property. These theoretical results are illustrated by one-dimensional numerical examples, where the proposed discretization spaces are coupled with a least-squares variational formulation.
Paper Structure (24 sections, 20 theorems, 191 equations, 6 figures, 3 tables)

This paper contains 24 sections, 20 theorems, 191 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Setting and main results
  3. Notations
  4. Basic notation
  5. Key functional spaces
  6. The frame of Gaussian coherent states
  7. Settings and key assumptions
  8. Statement of the approximability result
  9. Proof of Theorem \ref{['theorem_approximability']}
  10. Preliminary results on Gaussian states
  11. Main proof
  12. Application to the Helmholtz equation
  13. Notation
  14. Model problem
  15. Perfectly matched layers
  16. ...and 9 more sections

Key Result

Theorem 2.4

Let $0 < \varepsilon < 1/2$ and $0 \leq \alpha < \delta_0/2$. For $\hbar \in \mathscr H$, consider the right-hand side where $F^\hbar \in \Lambda_{\hbar,\rm rhs}$ with Then, if $u_\hbar$ is the solution to $P_\hbar u_\hbar = f_\hbar$, we have for all $p \in \mathbb N$ with

Figures (6)

  • Figure 1: Representation of $\Lambda$ for $k=400$ in the homogeneous example. Top-left: $\delta = 0.1$. Top-right: $\delta = 0.4$. Bottom-left: $\delta = 0.6$. The horizontal and vertical distances between dots are $\sqrt{\pi/k}$.
  • Figure 2: Convergence histories in the homogeneous example
  • Figure 3: Computational cost for a fixed accuracy in the homogeneous example
  • Figure 4: Convergence histories in the heterogeneous example
  • Figure 5: Computational cost for a fixed accuracy in the heterogeneous example
  • ...and 1 more figures

Theorems & Definitions (36)

  • Remark 2.1: General expansions in the Gaussian frame
  • Remark 2.2: $\hbar$-dependent symbol
  • Remark 2.3: non-divergence form
  • Theorem 2.4: Approximability for Gaussian state right-hand sides
  • Corollary 2.5: Approximability for micro-localised right-hand sides
  • Proposition 3.1: Tight expansion
  • proof
  • Proposition 3.2: Quasi orthogonality
  • Proposition 3.3: Control of $(P_\hbar-p_\hbar)^L$
  • Lemma 3.4: Index sets
  • ...and 26 more