Fractional time stepping and adjoint based gradient computation in an inverse problem for a fractionally damped wave equation

Inthispaperweconsidertheinverseproblemofidentifyingtheinitialdatainafractionallydampedwaveequationfromtimetracemeasurementsonasurface,asrelevantinphotoacousticorthermoacoustictomography.Wederiveandanalyzeatimesteppingmethodforthenumericalsolutionofthecorrespondingforwardproblem.Moreover,toefficientlyobtainreconstructionsbyminimizingaTikhonovregularizationfunctional(oralternatively,bycomputingtheMAPestimatorinaBayesianapproach),wedevelopanadjointbasedschemeforgradientcomputation.Numericalreconstructionsintwospacedimensionsillustratetheperformanceofthedevisedmethods. (c) 2021ElsevierInc.Allrightsreserved.

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This paper investigates the inverse problem of identifying the initial data in a fractionally damped wave equation from time trace measurements on a surface, relevant in photoacoustic or thermoacoustic tomography. The authors derive and analyze a time stepping method for the numerical solution of the corresponding forward problem, and develop an adjoint-based scheme for gradient computation to efficiently obtain reconstructions by minimizing a Tikhonov regularization functional. Numerical reconstructions in two space dimensions demonstrate the performance of the devised methods.