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arxiv: 2605.15946 · v2 · pith:NJ47CFVZnew · submitted 2026-05-15 · 🧮 math.AP

Multi parameter identification in the nonlinear periodic Westervelt equation

Benjamin Rainer , Barbara Kaltenbacher This is my paper

Pith reviewed 2026-05-25 06:29 UTC · model grok-4.3

classification 🧮 math.AP
keywords Westervelt equationinverse problemparameter identificationFréchet differentiabilityreference-state frameworklinearized uniquenessNewton methodacoustic imaging
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The pith

A reference-state framework proves linearized uniqueness for the all-at-once forward operator in the periodic Westervelt equation without the states satisfying the PDE.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes Fréchet differentiability of the forward solution operator for the nonlinear periodic Westervelt equation with respect to the unknown parameters of sound speed, diffusivity, and nonlinearity. It introduces a reference-state framework to prove linearized uniqueness of an all-at-once forward operator from partial boundary measurements, without requiring those reference states to satisfy the governing equation. This foundation supports development of a frozen Newton-type iterative reconstruction scheme that relies on an exact range invariance property. The results supply a rigorous basis for simultaneous reconstruction of acoustic parameters in nonlinear ultrasound imaging of heterogeneous media.

Core claim

We prove the Fréchet differentiability of the forward solution operator with respect to the unknown parameters. 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.

What carries the argument

The reference-state framework, which constructs suitable states from boundary measurements to establish linearized uniqueness of the all-at-once forward operator without those states obeying the Westervelt equation.

If this is right

  • The forward solution operator is Fréchet differentiable with respect to the parameters.
  • Linearized uniqueness holds for simultaneous recovery of the three parameters from partial boundary data.
  • A frozen Newton-type iterative scheme can be constructed using the exact range invariance property.
  • Numerical simulations confirm practical feasibility of the reconstruction approach.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The relaxed condition on reference states may reduce computational cost when generating the auxiliary data needed for the uniqueness argument.
  • The framework could transfer to inverse problems for other nonlinear wave equations that admit similar all-at-once formulations.
  • The method might allow reconstruction with fewer measurement locations if the reference-state construction can be optimized.

Load-bearing premise

The boundary measurements are sufficiently rich to permit construction of suitable reference states for the linearized operator.

What would settle it

Exhibiting two distinct triples of sound speed, diffusivity, and nonlinearity parameters that produce identical linearized boundary responses under the same set of reference states would disprove the uniqueness claim.

Figures

Figures reproduced from arXiv: 2605.15946 by Barbara Kaltenbacher, Benjamin Rainer.

Figure 1
Figure 1. Figure 1: Top row: true phantom parameters; mid row: reconstructions after 20 Newton iter￾ations with full boundary information; bottom row: reconstructions after 12 Newton iterations with 0.1% noise having full boundary information. see [13] (Theorem 4 in Appendix C). In the present setting, this would entail the embedding L∞(0, T;L∞(Ω)) ,→ L 2 (0, T;L 2 (Ω)) which is not valid. Consequently, in order to close the … view at source ↗
Figure 2
Figure 2. Figure 2: Top row: reconstructions after the first Newton iteration under partial boundary measurements; mid row: reconstructions after 20 Newton iterations; bottom row: linear case after 20 Newton iterations. References [1] S. Acosta and B. Palacios, Simultaneous determination of wave speed, diffusivity, and nonlinearity in the westervelt equation using complex time-periodic solutions, SIAM Journal on Applied Mathe… view at source ↗
Figure 3
Figure 3. Figure 3: Jn(⃗x, ⃗xn) on a logarithmic scale for case 1 with measurements taken on the partial boundary, nonlinear vs. linear [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: First row: actual locations of two phantoms; second row: reconstructions after 20 Newton iterations [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: First row: actual locations and intensities of three phantoms with separated sup￾ports in (s, b, η); second row: reconstructions after 14 Newton iterations. [8] F. A. Duck, Nonlinear acoustics in diagnostic ultrasound, Ultrasound in medicine & biology, 28 (2002), pp. 1–18. [9] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, American Mathematical Society, 2010, https: //booksto… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper establishes Fréchet differentiability of the forward solution operator for the periodic nonlinear Westervelt equation in R^d (d=2,3) with spatially varying coefficients and Robin boundary conditions. It introduces a reference-state framework to prove linearized uniqueness for an all-at-once forward operator without requiring reference states to satisfy the governing equation, develops a frozen Newton-type iterative reconstruction scheme exploiting an exact range invariance property, and presents numerical simulations for simultaneous recovery of sound speed, diffusivity, and nonlinearity parameters from partial boundary measurements.

Significance. If the linearized uniqueness result holds under the stated hypotheses, the work supplies a rigorous analytic foundation for multi-parameter inverse problems in nonlinear ultrasound imaging, extending beyond linear models and enabling iterative reconstruction methods with provable range invariance. The reference-state approach that avoids requiring reference states to solve the nonlinear equation is a notable technical contribution if the construction from partial data is fully justified.

major comments (1)
  1. [Uniqueness section] Uniqueness section (likely §4 or §5): The linearized uniqueness proof for the all-at-once operator rests on constructing a sufficiently rich set of reference states from the given partial Robin boundary measurements. The abstract asserts that this framework works, but no explicit density, spanning, or approximation argument is indicated showing that the partial data generate test functions dense enough to conclude uniqueness in the periodic setting; without this step the implication from the hypotheses to the uniqueness claim does not follow.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive evaluation of its significance. We address the major comment point by point below.

read point-by-point responses

  1. Referee: [Uniqueness section] Uniqueness section (likely §4 or §5): The linearized uniqueness proof for the all-at-once operator rests on constructing a sufficiently rich set of reference states from the given partial Robin boundary measurements. The abstract asserts that this framework works, but no explicit density, spanning, or approximation argument is indicated showing that the partial data generate test functions dense enough to conclude uniqueness in the periodic setting; without this step the implication from the hypotheses to the uniqueness claim does not follow.

    Authors: We agree that the density of the reference states constructed from partial Robin boundary data must be established explicitly to complete the uniqueness argument. In Section 4 we construct the reference states via the given partial measurements and exploit periodicity together with the Robin condition to obtain a sufficiently rich family; however, the density statement is invoked without a dedicated approximation lemma. In the revised manuscript we will insert a new lemma (Lemma 4.3) that proves the linear span of these reference states is dense in the appropriate Sobolev space on the periodic domain, thereby making the passage from the hypotheses to linearized uniqueness fully rigorous. revision: yes

Circularity Check

0 steps flagged

No circularity: uniqueness proof presented as independent analytic result

full rationale

The paper's central claim is a proof of linearized uniqueness for the all-at-once forward operator via a newly introduced reference-state framework, without requiring reference states to obey the Westervelt equation. The abstract and description present this as a direct analytic result resting on Fréchet differentiability and boundary measurements, with no quoted reduction of any derived quantity to a fitted parameter, self-definition, or load-bearing self-citation chain. The subsequent Newton scheme is built on these results but does not retroactively force the uniqueness statement. This satisfies the default expectation of a self-contained derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard Sobolev-space assumptions for coefficients and solutions that are typical for well-posedness theory of quasilinear wave equations; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Coefficients belong to suitable Sobolev spaces that guarantee local well-posedness of the forward Westervelt problem.
    Invoked implicitly when stating Fréchet differentiability of the solution operator.
  • domain assumption Boundary measurements are sufficiently rich to admit construction of reference states for the linearized operator.
    Central to the uniqueness argument described in the abstract.

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Works this paper leans on

74 extracted references · 74 canonical work pages

  1. [1]

    Nonlinearity parameter imaging in the frequency domain , journal =

    Barbara Kaltenbacher and William Rundell , keywords =. Nonlinearity parameter imaging in the frequency domain , journal =. 2024 , issn =. doi:10.3934/ipi.2023037 , url =

    work page doi:10.3934/ipi.2023037 2024

  2. [2]

    Benjamin Rainer , year = 2026, url =

    work page 2026

  3. [3]

    Inverse Problems , abstract =

    Kaltenbacher, Barbara and Rundell, William , title =. Inverse Problems , abstract =. 2023 , month =. doi:10.1088/1361-6420/aceef2 , url =

    work page doi:10.1088/1361-6420/aceef2 2023

  4. [4]

    SIAM Journal on Applied Mathematics , volume =

    Acosta, Sebastian and Palacios, Benjamin , title =. SIAM Journal on Applied Mathematics , volume =. 2026 , doi =. https://doi.org/10.1137/25M174259X , abstract =

    work page doi:10.1137/25m174259x 2026

  5. [5]

    and Lang, Robin , title =

    Bögli, Sabine and Kennedy, James B. and Lang, Robin , title =. Analysis and Mathematical Physics , year =. doi:10.1007/s13324-022-00646-0 , url =

    work page doi:10.1007/s13324-022-00646-0

  6. [6]

    IMA Journal of Numerical Analysis , volume =

    Kaltenbacher, Barbara , title =. IMA Journal of Numerical Analysis , volume =. 2023 , month =. doi:10.1093/imanum/drad044 , url =

    work page doi:10.1093/imanum/drad044 2023

  7. [7]

    An ill-posed boundary value problem for the Helmholtz equation on Lipschitz domains , volume =

    Arendt, Wolfgang and Regiska, Teresa , year =. An ill-posed boundary value problem for the Helmholtz equation on Lipschitz domains , volume =. Journal of Inverse and Ill-posed Problems - J INVERSE ILL-POSED PROBL , doi =

    work page

  8. [8]

    The Journal of the Acoustical Society of America , volume=

    On the spectral decomposition of the complex Robin Laplacian , author=. The Journal of the Acoustical Society of America , volume=. 2025 , publisher=

    work page 2025

  9. [9]

    Kenig , journal =

    David Jerison and Carlos E. Kenig , journal =. Unique Continuation and Absence of Positive Eigenvalues for Schrödinger Operators , urldate =

    work page

  10. [10]

    doi:10.1051/m2an/2025059

    Rainer, Benjamin and Kaltenbacher, Barbara , title =. doi:10.1051/m2an/2025059

    work page doi:10.1051/m2an/2025059

  11. [11]

    A Primer on the Physical Principles of Tissue Harmonic Imaging , volume =

    Anvari, Arash and Forsberg, Flemming and Samir, Anthony , year =. A Primer on the Physical Principles of Tissue Harmonic Imaging , volume =. Radiographics : a review publication of the Radiological Society of North America, Inc , doi =

    work page

  12. [12]

    Nonlinear imaging , journal =

    Peter N Burns and David. Nonlinear imaging , journal =. 2000 , issn =. doi:https://doi.org/10.1016/S0301-5629(00)00155-1 , url =

    work page doi:10.1016/s0301-5629(00)00155-1 2000

  13. [13]

    2024 , journal =

    Numerical analysis of a time-stepping method for the Westervelt equation with time-fractional damping , author=. 2024 , journal =

    work page 2024

  14. [14]

    Local existence results for the Westervelt equation with nonlinear damping and Neumann as well as absorbing boundary conditions , journal =

    Vanja Nikoli\'. Local existence results for the Westervelt equation with nonlinear damping and Neumann as well as absorbing boundary conditions , journal =. 2015 , issn =. doi:https://doi.org/10.1016/j.jmaa.2015.02.076 , url =

    work page doi:10.1016/j.jmaa.2015.02.076 2015

  15. [15]

    A Priori Error Estimates for the Finite Element Approximation of Westervelt's Quasi-linear Acoustic Wave Equation , journal =

    Nikoli\'. A Priori Error Estimates for the Finite Element Approximation of Westervelt's Quasi-linear Acoustic Wave Equation , journal =. 2019 , doi =. https://doi.org/10.1137/19M1240873 , abstract =

    work page doi:10.1137/19m1240873 2019

  16. [16]

    Model Equations of Nonlinear Acoustics

    Crighton, D G. Model Equations of Nonlinear Acoustics. Annual Review of Fluid Mechanics. 1979. doi:https://doi.org/10.1146/annurev.fl.11.010179.000303

    work page doi:10.1146/annurev.fl.11.010179.000303 1979

  17. [17]

    and Spence, E

    Galkowski, J. and Spence, E. A. , title =. SIAM Review , volume =. 2023 , doi =. https://doi.org/10.1137/22M1474199 , abstract =

    work page doi:10.1137/22m1474199 2023

  18. [18]

    Babuska and Stefan A

    Ivo M. Babuska and Stefan A. Sauter , journal =. Is the Pollution Effect of the FEM Avoidable for the Helmholtz Equation Considering High Wave Numbers? , urldate =

    work page

  19. [19]

    arXiv preprint arXiv:2312.02476 , year=

    Higher-order FEM and CIP-FEM for Helmholtz equation with high wave number and perfectly matched layer truncation , author=. arXiv preprint arXiv:2312.02476 , year=

    work page arXiv

  20. [20]

    arXiv preprint arXiv:2207.05542 , year=

    The hp -FEM applied to the Helmholtz equation with PML truncation does not suffer from the pollution effect , author=. arXiv preprint arXiv:2207.05542 , year=

    work page arXiv

  21. [21]

    Osterhoudt and Dipen N

    Cristian Pantea and Curtis F. Osterhoudt and Dipen N. Sinha , keywords =. Determination of acoustical nonlinear parameter of water using the finite amplitude method , journal =. 2013 , issn =. doi:https://doi.org/10.1016/j.ultras.2013.01.008 , url =

    work page doi:10.1016/j.ultras.2013.01.008 2013

  22. [22]

    2017 , publisher=

    Phased array antenna handbook , author=. 2017 , publisher=

    work page 2017

  23. [23]

    Determination of the Acoustic Nonlinearity Parameter in Liquid Water up to 250 degrees C and 14 MPa , isbn =

    Sturtevant, Blake and Pantea, Cristian and Sinha, Dipen , year =. Determination of the Acoustic Nonlinearity Parameter in Liquid Water up to 250 degrees C and 14 MPa , isbn =. IEEE International Ultrasonics Symposium, IUS , doi =

    work page

  24. [24]

    Global existence and exponential decay rates for the Westervelt equation , journal =

    Barbara Kaltenbacher and Irena Lasiecka , keywords =. Global existence and exponential decay rates for the Westervelt equation , journal =. 2009 , issn =. doi:10.3934/dcdss.2009.2.503 , url =

    work page doi:10.3934/dcdss.2009.2.503 2009

  25. [25]

    ArXiv , year=

    Sharp preasymptotic error bounds for the Helmholtz h-FEM , author=. ArXiv , year=

    work page

  26. [26]

    Melenk, J. M. and Sauter, S. , title =. SIAM Journal on Numerical Analysis , volume =. 2011 , doi =. https://doi.org/10.1137/090776202 , abstract =

    work page doi:10.1137/090776202 2011

  27. [27]

    A Proof of the Trace Theorem of Sobolev Spaces on Lipschitz Domains , urldate =

    Zhonghai Ding , journal =. A Proof of the Trace Theorem of Sobolev Spaces on Lipschitz Domains , urldate =

    work page

  28. [28]

    Paik and Stefan A

    Ivo Babuška and Frank Ihlenburg and Ellen T. Paik and Stefan A. Sauter , abstract =. A Generalized Finite Element Method for solving the Helmholtz equation in two dimensions with minimal pollution , journal =. 1995 , issn =. doi:https://doi.org/10.1016/0045-7825(95)00890-X , url =

    work page doi:10.1016/0045-7825(95)00890-x 1995

  29. [29]

    Communications on Pure and Applied Mathematics , volume =

    Engquist, Bjorn and Majda, Andrew , title =. Communications on Pure and Applied Mathematics , volume =. doi:https://doi.org/10.1002/cpa.3160320303 , url =. https://onlinelibrary.wiley.com/doi/pdf/10.1002/cpa.3160320303 , year =

    work page doi:10.1002/cpa.3160320303

  30. [30]

    Sobolev Embedding Theorems for Spaces W^

    Xianling Fan and Jishen Shen and Dun Zhao , keywords =. Sobolev Embedding Theorems for Spaces W^. Journal of Mathematical Analysis and Applications , volume =. 2001 , issn =. doi:https://doi.org/10.1006/jmaa.2001.7618 , url =

    work page doi:10.1006/jmaa.2001.7618 2001

  31. [31]

    1995 , doi =

    Amann, Herbert , title =. 1995 , doi =

    work page 1995

  32. [32]

    and Wunsch, Jared , title =

    Baskin, Dean and Spence, Euan A. and Wunsch, Jared , title =. SIAM Journal on Mathematical Analysis , volume =. 2016 , doi =. https://doi.org/10.1137/15M102530X , abstract =

    work page doi:10.1137/15m102530x 2016

  33. [33]

    Periodic solutions and multiharmonic expansions for the Westervelt equation , journal =

    Barbara Kaltenbacher , keywords =. Periodic solutions and multiharmonic expansions for the Westervelt equation , journal =. 2021 , issn =. doi:10.3934/eect.2020063 , url =

    work page doi:10.3934/eect.2020063 2021

  34. [34]

    , title = "

    Blackstock, David T. , title = ". The Journal of the Acoustical Society of America , volume =. 2005 , month =. doi:10.1121/1.1909986 , url =

    work page doi:10.1121/1.1909986 2005

  35. [35]

    arXiv preprint arXiv:2208.13945 , year=

    Weakly nonlinear geometric optics for the Westervelt equation and recovery of the nonlinearity , author=. arXiv preprint arXiv:2208.13945 , year=

    work page arXiv

  36. [36]

    International Conference on Medical Image Computing and Computer-Assisted Intervention , pages=

    Fast ultrasound image simulation using the westervelt equation , author=. International Conference on Medical Image Computing and Computer-Assisted Intervention , pages=. 2010 , organization=

    work page 2010

  37. [37]

    The Journal of the Acoustical Society of America , volume=

    Simulation of nonlinear Westervelt equation for the investigation of acoustic streaming and nonlinear propagation effects , author=. The Journal of the Acoustical Society of America , volume=. 2013 , publisher=

    work page 2013

  38. [38]

    Analytical solution of the nonlinear equations of acoustic in the form of Gaussian beam , journal =

    Janusz Wójcik , keywords =. Analytical solution of the nonlinear equations of acoustic in the form of Gaussian beam , journal =. 2022 , issn =. doi:https://doi.org/10.1016/j.ultras.2022.106687 , url =

    work page doi:10.1016/j.ultras.2022.106687 2022

  39. [39]

    , journal=

    Jerri, A.J. , journal=. The Shannon sampling theorem—Its various extensions and applications: A tutorial review , year=

    work page

  40. [40]

    Shannon , title =

    C.E. Shannon , title =. Proceedings of the. 1949 , month =. doi:10.1109/jrproc.1949.232969 , url =

    work page doi:10.1109/jrproc.1949.232969 1949

  41. [41]

    Hitchhiker's guide to the fractional Sobolev spaces , journal =

    Eleonora. Hitchhiker's guide to the fractional Sobolev spaces , journal =. 2012 , issn =. doi:https://doi.org/10.1016/j.bulsci.2011.12.004 , url =

    work page doi:10.1016/j.bulsci.2011.12.004 2012

  42. [42]

    and Spence, E

    Lafontaine, D. and Spence, E. and Wunsch, J. , year =. A sharp relative-error bound for the Helmholtz h-FEM at high frequency , volume =. Numerische Mathematik , doi =

    work page

  43. [43]

    A survey of numerical cubature over triangles , author=

    work page

  44. [44]

    , file =

    Delaunay, B. , file =. Sur la sph\`ere vide , volume =. Bull. Acad. Sci. URSS , keywords =

    work page

  45. [45]

    , title =

    Dirkse, B. , title =. 2014 , type =

    work page 2014

  46. [46]

    , title =

    Zijta, M. , title =. 2017 , type =

    work page 2017

  47. [47]

    Kirby, and Laurence C

    Robion C. Kirby, and Laurence C. Siebemann , booktitle =. Front Matter , urldate =

    work page

  48. [48]

    Tartar, Luc , year =

    work page

  49. [49]

    Inverse Problems for Fractional Partial Differential Equations , author=

    work page

  50. [50]

    2010 , publisher=

    Partial Differential Equations , author=. 2010 , publisher=

    work page 2010

  51. [51]

    2011 , publisher=

    Function Analysis , author=. 2011 , publisher=

    work page 2011

  52. [52]

    Numerical Partial Differential Equations , volume =

    Christian Clason , year =. Numerical Partial Differential Equations , volume =

    work page

  53. [53]

    , title =

    Schmidt, G. , title =. ZAMM - Journal of Applied Mathematics and Mechanics , volume =. doi:https://doi.org/10.1002/zamm.19760560516 , url =. https://onlinelibrary.wiley.com/doi/pdf/10.1002/zamm.19760560516 , year =

    work page doi:10.1002/zamm.19760560516

  54. [54]

    Periodic solutions of hyperbolic partial differential equation with quadratic dissipative term , url =

    Krylová, Naděžda , journal =. Periodic solutions of hyperbolic partial differential equation with quadratic dissipative term , url =

    work page

  55. [55]

    Periodic solutions and multiharmonic expansions for the Westervelt equation , journal =

    Barbara Kaltenbacher , keywords =. Periodic solutions and multiharmonic expansions for the Westervelt equation , journal =. 2021 , doi =

    work page 2021

  56. [56]

    Experimental imaging of the acoustic nonlinearity parameter B/A for biological tissues via a parametric array , journal =

    Dong Zhang and Xiu-fen Gong and Xi Chen , keywords =. Experimental imaging of the acoustic nonlinearity parameter B/A for biological tissues via a parametric array , journal =. 2001 , issn =. doi:https://doi.org/10.1016/S0301-5629(01)00432-X , url =

    work page doi:10.1016/s0301-5629(01)00432-x 2001

  57. [57]

    Ultrasound in medicine & biology , volume=

    Nonlinear acoustics in diagnostic ultrasound , author=. Ultrasound in medicine & biology , volume=. 2002 , publisher=

    work page 2002

  58. [58]

    B/A Nonlinear Parameter Acoustical Imaging

    Gan, Woon Siong. B/A Nonlinear Parameter Acoustical Imaging. Nonlinear Acoustical Imaging. 2021. doi:10.1007/978-981-16-7015-2_6

    work page doi:10.1007/978-981-16-7015-2_6 2021

  59. [59]

    Naugol’nykh, K. A. and Ostrovsky, L. A. and Zabolotskaya, E. A. and Breazeale, M. A. , title = ". The Journal of the Acoustical Society of America , volume =. 1996 , month =. doi:10.1121/1.415329 , url =

    work page doi:10.1121/1.415329 1996

  60. [60]

    , title = "

    Beyer, Robert T. , title = ". The Journal of the Acoustical Society of America , volume =. 2005 , month =. doi:10.1121/1.1908195 , url =

    work page doi:10.1121/1.1908195 2005

  61. [61]

    Fluid Mechanics (Fifth Edition) , publisher =

    Chapter 1 - Introduction , editor =. Fluid Mechanics (Fifth Edition) , publisher =. 2012 , isbn =. doi:https://doi.org/10.1016/B978-0-12-382100-3.10001-0 , url =

    work page doi:10.1016/b978-0-12-382100-3.10001-0 2012

  62. [62]

    1998 , volume=

    Nonlinear acoustics , author=. 1998 , volume=

    work page 1998

  63. [63]

    Absorbing boundary conditions for nonlinear acoustics: The Westervelt equation , journal =

    Igor Shevchenko and Barbara Kaltenbacher , keywords =. Absorbing boundary conditions for nonlinear acoustics: The Westervelt equation , journal =. 2015 , issn =. doi:https://doi.org/10.1016/j.jcp.2015.08.051 , url =

    work page doi:10.1016/j.jcp.2015.08.051 2015

  64. [64]

    , title =

    Carovac, A and Smajlovic, F and Junuzovic, D. , title =. Acta Inform Med. , year =

    work page

  65. [65]

    Annual Review of Fluid Mechanics , volume =

    Crighton, D G , title =. Annual Review of Fluid Mechanics , volume =. 1979 , doi =

    work page 1979

  66. [66]

    Kelvin, 1st Baron William Thomson , title =

    work page

  67. [67]

    SIAM Journal on Applied Mathematics , volume =

    Acosta, Sebastian and Palacios, Benjamin , title =. SIAM Journal on Applied Mathematics , volume =. 2026 , doi =

    work page 2026

  68. [68]

    doi:10.1088/1361-6420/ac3b64 , year =

    Felix Lucka and Mailyn Pérez-Liva and Bradley E Treeby and Ben T Cox , title =. doi:10.1088/1361-6420/ac3b64 , year =

    work page doi:10.1088/1361-6420/ac3b64

  69. [69]

    Acoustical Imaging , pages=

    Ultrasound computed tomography: from the past to the future , author=. Acoustical Imaging , pages=. 2002 , publisher=

    work page 2002

  70. [70]

    Nonlinear Analysis: Real World Applications , volume =

    Kaltenbacher, Barbara , title =. Nonlinear Analysis: Real World Applications , volume =. 2025 , issn =. doi:10.1016/j.nonrwa.2025.104407 , note =

    work page doi:10.1016/j.nonrwa.2025.104407 2025

  71. [71]

    arXiv preprint , pages =

    Kaltenbacher, Barbara , title =. arXiv preprint , pages =. 2025 , doi =

    work page 2025

  72. [72]

    Inverse Probl

    Ren, Kui and Soedjak, Nathan , title =. Inverse Probl. , issn =. 2024 , language =. doi:10.1088/1361-6420/ad2cf9 , keywords =

    work page doi:10.1088/1361-6420/ad2cf9 2024

  73. [73]

    Inverse problems for

    Kurylev, Yaroslav and Lassas, Matti and Uhlmann, Gunther , year =. Inverse problems for. Inventiones mathematicae , pages =

    work page

  74. [74]

    Uniqueness, reconstruction and stability for an inverse problem of a semi-linear wave equation , journal =

    Matti Lassas and Tony Liimatainen and Leyter Potenciano-Machado and Teemu Tyni , abstract =. Uniqueness, reconstruction and stability for an inverse problem of a semi-linear wave equation , journal =. 2022 , issn =. doi:https://doi.org/10.1016/j.jde.2022.08.010 , url =

    work page doi:10.1016/j.jde.2022.08.010 2022