## On an inverse problem of nonlinear imaging with fractional damping.(English)Zbl 1479.35951

Summary: This paper considers the attenuated Westervelt equation in pressure formulation. The attenuation is by various models proposed in the literature and characterised by the inclusion of non-local operators that give power law damping as opposed to the exponential of classical models. The goal is the inverse problem of recovering a spatially dependent coefficient in the equation, the parameter of nonlinearity $$\kappa (x)$$, in what becomes a nonlinear hyperbolic equation with non-local terms. The overposed measured data is a time trace taken on a subset of the domain or its boundary. We shall show injectivity of the linearised map from $$\kappa$$ to the overposed data and from this basis develop and analyse Newton-type schemes for its effective recovery.

### MSC:

 35R30 Inverse problems for PDEs 35R11 Fractional partial differential equations 35L20 Initial-boundary value problems for second-order hyperbolic equations 35L72 Second-order quasilinear hyperbolic equations 78A46 Inverse problems (including inverse scattering) in optics and electromagnetic theory
Full Text:

### References:

 [1] Alikhanov, A. A., A priori estimates for solutions of boundary value problems for equations of fractional order, Differ. Uravn.. Differ. Equ., 46 46, 5, 660-666 (2010) · Zbl 1208.35161 [2] Bj{\o }rn{\o }, L., Characterization of biological media by means of their non-linearity (1986) [3] Burov, V.; Gurinovich, I.; Rudenko, O.; Tagunov, E., Reconstruction of the spatial distribution of the nonlinearity parameter and sound velocity in acoustic nonlinear tomography, Acoust. Phys., 40, 816- 823 (1994) [4] Cai, Wei; Chen, Wen; Fang, Jun; Holm, Sverre, A survey on fractional derivative modeling of power-law frequency-dependent viscous dissipative and scattering attenuation in acoustic wave propagation (2018) [5] Cain, Charles A., Ultrasonic reflection mode imaging of the nonlinear parameter {{B/A}}: I. a theoretical basis, J. Acoust. Soc. Amer., 80, 1, 28- 32 (1986) [6] Caputo, Michele, Linear models of dissipation whose {$$Q$$} is almost frequency independent – {II}, Geophys. J. Int., 13, 5, 529- 539 (1967) [7] Chen, W.; Holm, S., Fractional laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power-law dependency, J. Acoust. Soc. Amer., 115, 4, 1424- 1430 (2004) [8] Djrbashian, Mkhitar M., Integralnye preobrazovaniya i predstavleniya funktsiiv kompleksnoi oblasti (1966), Izdat. “Nauka”, Moscow [9] Djrbashian, Mkhitar M., Harmonic analysis and boundary value problems in the complex domain, Operator Theory: Advances and Applications 65, xiv+256 pp. (1993), Birkh\"{a}user Verlag, Basel · Zbl 0798.43001 [10] Edelman, Alan; Murakami, H., Polynomial roots from companion matrix eigenvalues, Math. Comp., 64, 210, 763-776 (1995) · Zbl 0833.65041 [11] Eggermont, Paul P. B., On Galerkin methods for Abel-type integral equations, SIAM J. Numer. Anal., 25, 5, 1093-1117 (1988) · Zbl 0665.65096 [12] Grebenkov, D. S.; Nguyen, B.-T., Geometrical structure of Laplacian eigenfunctions, SIAM Rev., 55, 4, 601-667 (2013) · Zbl 1290.35157 [13] Hanke, Martin, A regularizing Levenberg-Marquardt scheme, with applications to inverse groundwater filtration problems, Inverse Problems, 13, 1, 79-95 (1997) · Zbl 0873.65057 [14] Holm, Sverre; N\"asholm, Sven Peter, A causal and fractional all-frequency wave equation for lossy media, J. Acoust. Soc. Amer., 130, 4, 2195- 2202 (2011) [15] Ichida, Nobuyuki; Sato, Takuso; Linzer, Melvin, Imaging the nonlinear ultrasonic parameter of a medium, Ultrasonic Imaging, 5, 4, 295- 299 (1983) [16] Jin, Bangti; Rundell, William, A tutorial on inverse problems for anomalous diffusion processes, Inverse Problems, 31, 3, 035003, 40 pp. (2015) · Zbl 1323.34027 [17] Kaltenbacher, Barbara, Mathematics of nonlinear acoustics, Evol. Equ. Control Theory, 4, 4, 447-491 (2015) · Zbl 1339.35003 [18] Kaltenbacher, Barbara; Neubauer, Andreas; Scherzer, Otmar, Iterative regularization methods for nonlinear ill-posed problems, Radon Series on Computational and Applied Mathematics 6, viii+194 pp. (2008), Walter de Gruyter GmbH & Co. KG, Berlin · Zbl 1145.65037 [19] Kaltenbacher, Barbara; Klassen, Andrej; Previatti de Souza, Mario Luiz, The Ivanov regularized Gauss-Newton method in Banach space with an a posteriori choice of the regularization radius, J. Inverse Ill-Posed Probl., 27, 4, 539-557 (2019) · Zbl 1418.65055 [20] Kaltenbacher, Barbara; Nikoli\'c, Vanja, Time-fractional Moore-Gibson-Thompson equations (2021) · Zbl 1475.35282 [21] Kaltenbacher, Barbara; Rundell, William, On the identification of the nonlinearity parameter in the {W}estervelt equation from boundary measurements, Inverse Probl. Imaging, 15, 865-891 (2021) · Zbl 1472.35453 [22] Kaltenbacher, Barbara; Rundell, William, Some inverse problems for wave equations with fractional derivative attenuation, Inverse Problems, 37, 4, 045002, 28 pp. (2021) · Zbl 1459.35396 [23] Mainardi, Francesco, Fractional calculus and waves in linear viscoelasticity, xx+347 pp. (2010), Imperial College Press, London · Zbl 1210.26004 [24] Mainardi, Francesco; Gorenflo, Rudolf, On Mittag-Leffler-type functions in fractional evolution processes, J. Comput. Appl. Math., 118, 1-2, 283-299 (2000) · Zbl 0970.45005 [25] Oparnica, Ljubica; S\"{u}li, Endre, Well-posedness of the fractional Zener wave equation for heterogeneous viscoelastic materials, Fract. Calc. Appl. Anal., 23, 1, 126-166 (2020) · Zbl 1439.35544 [26] Rieder, Andreas, On convergence rates of inexact Newton regularizations, Numer. Math., 88, 2, 347-365 (2001) · Zbl 0990.65061 [27] Rundell, William; Zhang, Zhidong, On the identification of source term in the heat equation from sparse data, SIAM J. Math. Anal., 52, 2, 1526- 1548 (2020) · Zbl 1433.65198 [28] Sakamoto, Kenichi; Yamamoto, Masahiro, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382, 1, 426-447 (2011) · Zbl 1219.35367 [29] Samko, Stefan G.; Kilbas, Anatoly A.; Marichev, Oleg I., Fractional integrals and derivatives, xxxvi+976 pp. (1993), Gordon and Breach Science Publishers, Yverdon · Zbl 0818.26003 [30] Szabo, Thomas L., Time domain wave equations for lossy media obeying a frequency power law, J. Acoust. Soc. Amer., 96, 1, 491- 500 (1994) [31] Temam, Roger, Infinite-dimensional dynamical systems in mechanics and physics, Applied Mathematical Sciences 68, xvi+500 pp. (1988), Springer-Verlag, New York · Zbl 0662.35001 [32] Treeby, Bradley E.; Cox, B. T., Modeling power law absorption and dispersion for acoustic propagation using the fractional laplacian, J. Acoust. Soc. Amer., 127, 5, 2741- 2748 (2010) [33] Varray, Fran\c{c}ois; Basset, Olivier; Tortoli, Piero; Cachard, Christian, Extensions of nonlinear {B/A} parameter imaging methods for echo mode, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 58, 1232-1244 (201106) [34] V\"{o}geli, Urs; Nedaiasl, Khadijeh; Sauter, Stefan A., A fully discrete Galerkin method for Abel-type integral equations, Adv. Comput. Math., 44, 5, 1601-1626 (2018) · Zbl 1402.65185 [35] Wismer, Margaret G., Finite element analysis of broadband acoustic pulses through inhomogenous media with power law attenuation, J. Acoust. Soc. Amer., 120, 6, 3493- 3502 (2006) [36] Zhang, Dong; Chen, Xi; Gong, Xiu-fen, Acoustic nonlinearity parameter tomography for biological tissues via parametric array from a circular piston source-theoretical analysis and computer simulations, J. Acoust. Soc. Amer., 109, 3, 1219- 1225 (2001) [37] Zhang, Dong; Gong, Xiufen; Ye, Shigong, Acoustic nonlinearity parameter tomography for biological specimens via measurements of the second harmonics, J. Acoust. Soc. Amer., 99, 4, 2397- 2402 (1996)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.