On the identification of the nonlinearity parameter in the Westervelt equation from boundary measurements

Barbara Kaltenbacher; William Rundell

doi:10.3934/ipi.2021020

journal article Open Access Jan 01, 2021

On the identification of the nonlinearity parameter in the Westervelt equation from boundary measurements

Barbara Kaltenbacher

William Rundell

Inverse Problems and Imaging Vol. 15 No. 5 pp. 865-891 · American Institute of Mathematical Sciences (AIMS)

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References

53

[1]

[2]

10.1016/0041-5553(84)90253-2

[3]

10.1016/0041-624x(86)90102-2

[4]

D. T. Blackstock, Approximate equations governing finite-amplitude sound in thermoviscous fluids, Tech Report, GD/E Report, GD-1463-52, General Dynamics Corp., Rochester, NY, 1963.

[5]

10.1093/imanum/17.3.421

[6]

J. M. Burgers, The Nonlinear Diffusion Equation, Springer, Netherlands, 1974. 10.1007/978-94-010-1745-9

[7]

[8]

C. A. Cain, Ultrasonic reflection mode imaging of the nonlinear parameter B/A: A theoretical basis, IEEE 1985 Ultrasonics Symposium, San Francisco, CA, USA, 1985. 10.1109/ultsym.1985.198640

[9]

10.1515/jiip-2018-0026

[10]

10.1051/cocv/2010003

[11]

10.1146/annurev.fl.11.010179.000303

[12]

H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Mathematics and its Applications, 375, Kluwer Academic Publishers Group, Dordrecht, 1996. 10.1007/978-94-009-1740-8

[13]

10.1088/0266-5611/5/4/007

[14]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998.

[15]

M. F. Hamilton and D. T. Blackstock, Nonlinear Acoustics, Vol. 1, Academic Press, San Diego, 1998.

[16]

10.1088/0266-5611/13/1/007

[17]

10.1007/s002110050158

[18]

10.1137/s0036142998341246

[19]

10.1088/0266-5611/23/3/009

[20]

10.1088/0266-5611/13/5/012

[21]

S. Hubmer and R. Ramlau, Nesterov's accelerated gradient method for nonlinear ill-posed problems with a locally convex residual functional, Inverse Problems, 34 (2018), 30pp. 10.1088/1361-6420/aacebe

[22]

10.1177/016173468300500401

[23]

O. Y. Imanuvilov and M. Yamamoto, Carleman estimate and an inverse source problem for the Kelvin-Voigt model for viscoelasticity, Inverse Problems, 35 (2019), 45pp. 10.1088/1361-6420/ab323e

[24]

V. Isakov, Inverse Problems for Partial Differential Equations, Applied Mathematical Sciences, 127, Springer, New York, 2006.

[25]

[26]

10.1007/s00211-014-0682-5

[27]

Mathematics of nonlinear acoustics

Barbara Kaltenbacher

Evolution Equations and Control Theory 10.3934/eect.2015.4.447

[28]

B. Kaltenbacher, Periodic solutions and multiharmonic expansions for the Westervelt equation, to appear, Evol. Equ. Control Theory.

[29]

Global existence and exponential decay rates for the Westervelt equation

Barbara Kaltenbacher, Irena Lasiecka

Discrete and Continuous Dynamical Systems - S 10.3934/dcdss.2009.2.503

[30]

B. Kaltenbacher and A. Klassen, On convergence and convergence rates for Ivanov and Morozov regularisation and application to some parameter identification problems in elliptic PDEs, Inverse Problems, 34 (2018), 24pp. 10.1088/1361-6420/aab739

[31]

B. Kaltenbacher, A. Neubauer and O.Scherzer, Iterative Regularization Methods for Nonlinear Ill-Posed Problems, Radon Series on Computational and Applied Mathematics, 6, Walter de Gruyter GmbH & Co. KG, Berlin, 2008. 10.1515/9783110208276

[32]

[33]

10.1017/s0022112068000303

[34]

M. J. Lighthill, Viscosity effects in sound waves of finite amplitude, in Surveys in Mechanics, Cambridge, at the University Press, 1956, 250–351.

[35]

D. Lorenz and N. Worliczek, Necessary conditions for variational regularisation schemes, Inverse Problems, 29 (2013), 19pp. 10.1088/0266-5611/29/7/075016

[36]

10.1007/s00245-011-9138-9

[37]

[38]

10.3934/eect.2019010

[39]

10.1515/jiip-2016-0060

[40]

10.1016/0021-9045(88)90025-1

[41]

10.4171/zaa/679

[42]

H. Ockendon and J. R. Ockendon, Waves and Compressible Flow, Texts in Applied Mathematics, 47, Springer-Verlag, New York, 2004. 10.1115/1.1849177

[43]

10.1137/0317035

[44]

10.1007/pl00005448

[45]

10.1090/s0025-5718-1992-1106979-0

[46]

10.1007/s002459900081

[47]

10.1088/0266-5611/5/2/008

[48]

10.1109/tuffc.2011.1933

[49]

Parametric Acoustic Array

Peter J. Westervelt

The Journal of the Acoustical Society of America 10.1121/1.1918525

[50]

M. Yamamoto and B. Kaltenbacher, An inverse source problem related to acoustic nonlinearity parameter imaging, to appear, Time-Dependent Problems in Imaging and Parameter Identification, Springer, 2021. 10.1007/978-3-030-57784-1_14

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Published: Jan 01, 2021
Vol/Issue: 15(5)
Pages: 865-891

Authors

Department of Mathematics, Texas A&M University, Texas 77843, USA

Cite This Article

Barbara Kaltenbacher, William Rundell (2021). On the identification of the nonlinearity parameter in the Westervelt equation from boundary measurements. Inverse Problems and Imaging, 15(5), 865-891. https://doi.org/10.3934/ipi.2021020

On the identification of the nonlinearity parameter in the Westervelt equation from boundary measurements

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