High spatial resolution is a central aspect of tomographic imaging. In photoacoustic tomography, this requires accurate models for acoustic wave propagation and corresponding efficient image reconstruction methods. In this article we consider such models accounting for frequency dependent attenuation according to a wide class of attenuation laws that may include memory. We formulate the inverse problem of photoacoustic tomography in attenuating medium as ill-posed operator equation in a Hilbert space framework that is tackled by iterative regularization methods. For that purpose we derive explicit expressions for the adjoint problem that can efficiently be implemented. Our approach comes with a clear convergence analysis that is not shared by existing approaches for the considered model class. In contrast to time reversal the employed adjoint wave equation is again damping and thus has a stable solution. This stability property can be clearly seen in our numerical results. Indeed, if the attenuation is not too strong then the reconstruction results are even better than in the absence of attenuation. Moreover, the presented numerical results clearly demonstrate the efficiency and accuracy of the derived iterative reconstruction algorithms in various situations including the limited view case, where alternative methods cannot be directly applied.
A different class of algorithms extends the time reversal technique to the attenuated case (see @cite_48 @cite_24 @cite_38 @cite_3 @cite_37 @cite_69 @cite_4 @cite_0 ). Note that the time reversal of the attenuated wave equation yields a noise amplifying equation. Therefore regularization methods have to be incorporated in its numerical solution. Opposed to the time reversal, the adjoint wave equation used in our approach is again damping and no regularization is required for its stable solution. This yields a clear convergence analysis for our method by using standard regularization theory @cite_58 @cite_14 @cite_57 . We are not aware of similar existing results for PAT in attenuating acoustic media. The approaches which are closest to our work seem @cite_74 @cite_67 . In @cite_74 discrete iterative methods are considered, where the problem is first discretized and the adjoint is computed from the discretized problem. Further, both works @cite_74 @cite_67 use attenuation models based on the fractional Laplacian (see @cite_29 @cite_61 ) which yields an equation that is non-local in space. It is not obvious how to extend these approaches to model which can also include memory.
@cite_61 @cite_38 @cite_37 @cite_69 @cite_4 @cite_14 @cite_67 @cite_48 @cite_29 @cite_3 @cite_24 @cite_0 @cite_57 @cite_74 @cite_58
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We consider a mathematical model of thermoacoustic tomography and other multi-wave imaging techniques with variable sound speed and attenuation. We find that a Neumann series reconstruction algorithm, previously studied under the assumption of zero attenuation, still converges if attenuation is sufficiently small. With complete boundary data, we show the inverse problem has a unique solution, and modified time reversal provides a stable reconstruction. We also consider partial boundary data, and in this case study those singularities that can be stably recovered. In this work, we show how to incorporate attenuation into the optoacoustic tomography (OAT) imaging equation and develop a strategy for compensating for this attenuation during image reconstruction. In OAT, one exposes a sample to pulses of electromagnetic radiation that cause small amounts of heating in the specimen. The heating engenders thermal expansion which, in turn, gives rise to acoustic waves. The resulting acoustic pressure signal is generally measured by transducers arrayed around the object, and these data may be used to reconstruct images of the original electromagnetic absorption. Frequency-dependent absorption of the acoustic waves can lead to blurring and distortion in reconstructed images. We show that in the temporal frequency domain, the optoacoustic wave equation incorporating attenuation is equivalent to the inhomogeneous Helmholtz equation with a complex wave number. While some work has been done in other fields on directly solving Helmholtz equations with complex wave numbers, these are generally computationally intensive numerical approaches. We pursue a different approach, deriving an integral equation that relates the temporal optoacoustic signal at a given transducer location in the presence of attenuation to the ideal signal that would have been obtained in the absence of attenuation. This equation is readily discretized and the resulting linear system of equations involves a matrix that need only be inverted once, at which point the inverse can be used to correct all of the measured time signals prior to reconstruction by conventional methods. SUMMARY In this paper, we address the solution of three-dimensional heterogeneous Helmholtz problems discretized with second-order finite difference methods with application to acoustic waveform inversion in geophysics. In this setting, the numerical simulation of wave propagation phenomena requires the approximate solution of possibly very large indefinite linear systems of equations. For that purpose, we propose and analyse an iterative two-grid method acting on the Helmholtz operator where the coarse grid problem is solved inaccurately. A cycle of a multigrid method applied to a complex shifted Laplacian operator is used as a preconditioner for the approximate solution of this coarse problem. A single cycle of the new method is then used as a variable preconditioner of a flexible Krylov subspace method. We analyse the properties of the resulting preconditioned operator by Fourier analysis. Numerical results demonstrate the effectiveness of the algorithm on three-dimensional applications. The proposed numerical method allows us to solve three-dimensional wave propagation problems even at high frequencies on a reasonable number of cores of a distributed memory computer. Copyright © 2012 John Wiley & Sons, Ltd. A new PDE solver was introduced recently, in Part I of this two-paper sequence, on the basis of two main concepts: the well-known Alternating Direction Implicit (ADI) approach, on one hand, and a certain ''Fourier Continuation'' (FC) method for the resolution of the Gibbs phenomenon, on the other. Unlike previous alternating direction methods of order higher than one, which only deliver unconditional stability for rectangular domains, the new high-order FC-AD (Fourier-Continuation Alternating-Direction) algorithm yields unconditional stability for general domains-at an O(Nlog(N)) cost per time-step for an N point spatial discretization grid. In the present contribution we provide an overall theoretical discussion of the FC-AD approach and we extend the FC-AD methodology to linear hyperbolic PDEs. In particular, we study the convergence properties of the newly introduced FC(Gram) Fourier Continuation method for both approximation of general functions and solution of the alternating-direction ODEs. We also present (for parabolic PDEs on general domains, and, thus, for our associated elliptic solvers) a stability criterion which, when satisfied, ensures unconditional stability of the FC-AD algorithm. Use of this criterion in conjunction with numerical evaluation of a series of singular values (of the alternating-direction discrete one-dimensional operators) suggests clearly that the fifth-order accurate class of parabolic and elliptic FC-AD solvers we propose is indeed unconditionally stable for all smooth spatial domains and for arbitrarily fine discretizations. To illustrate the FC-AD methodology in the hyperbolic PDE context, finally, we present an example concerning the Wave Equation-demonstrating sixth-order spatial and fourth-order temporal accuracy, as well as a complete absence of the debilitating ''dispersion error'', also known as ''pollution error'', that arises as finite-difference and finite-element solvers are applied to solution of wave propagation problems.