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Stable determination of the nonlinear parameter in the non-diffusive Westervelt equation from the Dirichlet-to-Neumann map

Mike Wendels

Abstract

The Westervelt equation models the propagation of nonlinear acoustic waves in a regime well-suited for applications such as medical ultrasound imaging. In this work, we prove that the nonlinear parameter, as well as the sound speed, can be stably recovered from the Dirichlet-to-Neumann map associated with the non-diffusive Westervelt equation in (1+3)-dimensions. This result is essential for the feasibility of reconstruction methods. The Dirichlet-to-Neumann map encodes boundary measurements by associating a prescribed pressure profile on the boundary with the resulting pressure fluctuations. We prove stability provided the sound speed is a priori known to be close to a reference sound speed and under certain geometrical conditions. We also verify the result through numerical experiments.

Stable determination of the nonlinear parameter in the non-diffusive Westervelt equation from the Dirichlet-to-Neumann map

Abstract

The Westervelt equation models the propagation of nonlinear acoustic waves in a regime well-suited for applications such as medical ultrasound imaging. In this work, we prove that the nonlinear parameter, as well as the sound speed, can be stably recovered from the Dirichlet-to-Neumann map associated with the non-diffusive Westervelt equation in (1+3)-dimensions. This result is essential for the feasibility of reconstruction methods. The Dirichlet-to-Neumann map encodes boundary measurements by associating a prescribed pressure profile on the boundary with the resulting pressure fluctuations. We prove stability provided the sound speed is a priori known to be close to a reference sound speed and under certain geometrical conditions. We also verify the result through numerical experiments.

Paper Structure

This paper contains 18 sections, 14 theorems, 159 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Setting
  3. Main result
  4. Preliminaries
  5. Foliation condition
  6. Estimates for the acoustic wave equation
  7. Higher-order linearization method
  8. Construction of Gaussian beams for the linear acoustic wave equation
  9. Fermi coordinates
  10. WKB approximation
  11. Construction phase function
  12. Construction amplitude function
  13. Proof of Proposition \ref{['thm:stabb1']}
  14. Proof of Theorem \ref{['thm:stabb2']}
  15. Stability of geodesic flow, Jacobi fields, and Gaussian beams
  16. ...and 3 more sections

Key Result

Proposition 1.1

Let $\Omega \subset \mathbb{R}^3$ be a compact, connected domain with smooth boundary $\partial \Omega$, and let $\delta$ be such that inequality eq:DNdelta holds for all $f \in H_{0,\epsilon_0}^{s+3}(\Sigma)$. Suppose that $c, \beta_\ell \in \mathcal{C}^{\infty}(\Omega), \, c > 0$ with $\lVert\beta for any $q < l$, where $0 < \mu < 1$ is dependent on $s,l,q$.

Figures (3)

  • Figure 1: Sketch of the setting of uhlmann2016inverse to locally invert a weighted geodesic ray transform around some point $p \in \partial \Omega$ for which the boundary is locally strictly convex with respect to the metric.
  • Figure 2: A log-log plot of $\lVert D_{\nu} (\tilde{u}_1 - \tilde{u}_2)\rVert_{L^2(\Sigma)}$ with $f(t,x) = \frac{1}{10} e^{-t^{-2}}$ for $\beta_1,\beta_2 \in [10^{-4},1], \, c \equiv 1$ (left) and for $c_1,c_2 \in [1,1.8], \, \beta \equiv 0$ (right), where $\Omega = [-1,1]^3, \, \Sigma = [0,3.48] \times \partial \Omega, \, \Delta x = 2^{-4}, \, \Delta t = 0.03$. The results are restricted to limited values for $\beta_2,c_2$ for clarity of presentation.
  • Figure 3: A log-log plot of $\lVert D_{\nu} (\tilde{u}_1 - \tilde{u}_2)\rVert_{L^2(\Sigma)}$ with $f(t,x) = \frac{1}{10} e^{-t^{-2}}$ and $c_{\alpha}(x)$ as in equation \ref{['eq:c_herglotz']} for $\beta_1,\beta_2 \in [10^{-4},0.34], \, \alpha = \frac{3}{2}$ (left) and for $\alpha_1,\alpha_2 \in [1,\frac{3}{2}], \, \beta \equiv 0$ (right), where $\Omega = [-1,1]^3, \, \Sigma = [0,3.48] \times \partial \Omega, \, \Delta x = 2^{-4}, \, \Delta t = 0.03$. The results are restricted to limited values for $\beta_2,\alpha_2$ for clarity of presentation.

Theorems & Definitions (26)

  • Proposition 1.1
  • Theorem 1.1
  • Definition 2.1: stefanov2016boundary
  • Lemma 2.1: kachalov2001inverse,lassas2020uniqueness
  • Lemma 2.2: kachalov2001inverse
  • Corollary 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 16 more