On an inverse problem of nonlinear imaging with fractional damping

Kaltenbacher, Barbara; Rundell, William

doi:10.1090/mcom/3683

Cited by 15 publications

(17 citation statements)

References 32 publications

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“…3.1 in Kaltenbacher and Rundell. 14 However, this obviously requires higher differentiability of f .…”

Section: Proposition 31 Under the Conditions Of Corollary 31 The Oper...mentioning

confidence: 99%

“…For doing this with Newton, the optimal basis would probably have to be piecewise linear, as for example in Kaltenbacher and Rundell. 14 To avoid a mismatch between the actual location of the non-differentiability points and the nodes of the basis functions requires a large number of basis functions in general, unless these points are a priori known. Thus, without this knowledge, the computational complexity of calculating the Jacobian would make (frozen) Newton much slower than the accelerated fixed-point schemes.…”

Section: Reconstructionsmentioning

confidence: 99%

“…Therefore, one would have to work with the pressure formulation () analogously to sect. 3.1 in Kaltenbacher and Rundell 14 . However, this obviously requires higher differentiability of f .…”

Section: Analysis Of the Forward Problemmentioning

confidence: 99%

“…In Figure 3, we therefore show the results of the fixed‐point scheme and its Anderson accelerated version using time trace data. For doing this with Newton, the optimal basis would probably have to be piecewise linear, as for example in Kaltenbacher and Rundell 14 . To avoid a mismatch between the actual location of the non‐differentiability points and the nodes of the basis functions requires a large number of basis functions in general, unless these points are a priori known.…”

Section: Reconstructionsmentioning

confidence: 99%

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Determining the nonlinearity in an acoustic wave equation

Kaltenbacher

Rundell

2021

Math Methods in App Sciences

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We consider an undetermined coefficient inverse problem for a nonlinear partial differential equation describing high‐intensity ultrasound propagation as widely used in medical imaging and therapy. The usual nonlinear term in the standard model using the Westervelt equation in pressure formulation is of the form ppt. However, this should be considered as a low‐order approximation to a more complex physical model where higher order terms will be required. Here we assume a more general case where the form taken is f(p) pt and f is unknown and must be recovered from data measurements. Corresponding to the typical measurement setup, the overposed data consist of time trace observations of the acoustic pressure at a single point or on a one‐dimensional set Σ representing the receiving transducer array at a fixed time. Additionally to an analysis of well‐posedness of the resulting pde, we show injectivity of the linearized forward map from f to the overposed data and use this as motivation for several iterative schemes to recover f. Numerical simulations will also be shown to illustrate the efficiency of the methods.

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“…3.1 in Kaltenbacher and Rundell. 14 However, this obviously requires higher differentiability of f .…”

Section: Proposition 31 Under the Conditions Of Corollary 31 The Oper...mentioning

confidence: 99%

Section: Reconstructionsmentioning

confidence: 99%

Section: Analysis Of the Forward Problemmentioning

confidence: 99%

Section: Reconstructionsmentioning

confidence: 99%

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Determining the nonlinearity in an acoustic wave equation

Kaltenbacher

Rundell

2021

Math Methods in App Sciences

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“…The value of µ depends on the tissue [9,Chapter 4.3] and is used to model the frequency dependence of attenuation [2,Chapter 3]. See also the recent [15], which includes other choices of nonlocal attenuation operators L. The case β = β B is of interest in modelling viscoelastic materials [16,25].…”

Section: Introductionmentioning

confidence: 99%

Numerical analysis of a time-stepping method for the Westervelt equation with time-fractional damping

Baker¹,

Banjai²,

Ptashnyk³

2022

Preprint

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We develop a numerical method for the Westervelt equation, an important equation in nonlinear acoustics, in the form where the attenuation is represented by a class of non-local in time operators. A semi-discretisation in time based on the trapezoidal rule and A-stable convolution quadrature is stated and analysed. Existence and regularity analysis of the continuous equations informs the stability and error analysis of the semi-discrete system. The error analysis includes the consideration of the singularity at t = 0 which is addressed by the use of a correction in the numerical scheme. Extensive numerical experiments confirm the theory.

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The Kuznetsov and Blackstock Equations of Nonlinear Acoustics with Nonlocal-in-Time Dissipation

Kaltenbacher,

Meliani,

Nikolić

2024

Appl Math Optim

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In ultrasonics, nonlocal quasilinear wave equations arise when taking into account a class of heat flux laws of Gurtin–Pipkin type within the system of governing equations of sound motion. The present study extends previous work by the authors to incorporate nonlocal acoustic wave equations with quadratic gradient nonlinearities which require a new approach in the energy analysis. More precisely, we investigate the Kuznetsov and Blackstock equations with dissipation of fractional type and identify a minimal set of assumptions on the memory kernel needed for each equation. In particular, we discuss the physically relevant examples of Abel and Mittag–Leffler kernels. We perform the well-posedness analysis uniformly with respect to a small parameter on which the kernels depend and which can be interpreted as the sound diffusivity or the thermal relaxation time. We then analyze the limiting behavior of solutions with respect to this parameter, and how it is influenced by the specific class of memory kernels at hand. Through such a limiting study, we relate the considered nonlocal quasilinear equations to their limiting counterparts and establish the convergence rates of the respective solutions in the energy norm.

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On an inverse problem of nonlinear imaging with fractional damping

Cited by 15 publications

References 32 publications

Determining the nonlinearity in an acoustic wave equation

Determining the nonlinearity in an acoustic wave equation

Numerical analysis of a time-stepping method for the Westervelt equation with time-fractional damping

The Kuznetsov and Blackstock Equations of Nonlinear Acoustics with Nonlocal-in-Time Dissipation

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