Multi parameter identification in the nonlinear periodic Westervelt equation

TL;DR

This paper develops a rigorous mathematical framework and iterative method for simultaneously identifying multiple parameters in a nonlinear periodic Westervelt equation, relevant for advanced nonlinear ultrasound imaging.

Contribution

It establishes differentiability, uniqueness, and a Newton-type reconstruction scheme for inverse parameter problems in the nonlinear Westervelt equation.

Findings

Proved Fréchet differentiability of the forward operator.

Established linearized uniqueness without reference states.

Demonstrated the effectiveness of the iterative reconstruction through numerical simulations.

Abstract

Nonlinear ultrasound imaging leverages harmonic wave generation to enhance contrast and spatial resolution beyond the capabilities of conventional linear techniques. This behavior is commonly modeled by the Westervelt equation, which captures finite-amplitude acoustic wave propagation in heterogeneous media. In this work, we investigate an inverse problem for a periodic nonlinear Westervelt equation in $\mathbb{R}^d$, where $d\in\{2,3\}$ with spatially varying coefficients and Robin-type boundary conditions. The objective is to simultaneously reconstruct the sound speed, diffusivity, and nonlinearity parameters from (partial) boundary measurements. We first establish the Fr\'echet differentiability of the forward solution operator with respect to the unknown parameters, providing a rigorous analytical foundation for parameter identification. To address uniqueness, we introduce a reference-state framework and prove linearized uniqueness of an all-at-once forward operator without requiring the reference states to satisfy the governing equation. Building on these results, we develop an iterative reconstruction scheme based on a frozen Newton-type method, supported by an exact range invariance property. Numerical simulations are presented to illustrate the feasibility and performance of the proposed approach.