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In this work the Lippmann-Schwinger equation is used to model seismic wavesin strongly scattering acoustic media. We consider the Helmholtz equation,which is the scalar wave equation in the frequency domain with constant densityand variable velocity, and transform it to an integral equation of theLippmann-Schwinger type. To directly solve the discretized problem with matrixinversion is time-consuming, therefore we use iterative methods. The Bornseries is a well-known scattering series which gives the solution withrelatively small cost, but it has limited use as it only converges for smallscattering potentials. There exist other scattering series with preconditionersthat have been shown to converge for any contrast, but the methods mightrequire many iterations for models with high contrast. Here we develop newpreconditioners based on randomized matrix approximations and hierarchicalmatrices which can make the scattering series converge for any contrast with alow number of iterations. We describe two different preconditioners; one isbest for lower frequencies and the other for higher frequencies. We use thefast Fourier transform both in the construction of the preconditioners and inthe iterative solution, and this makes the methods efficient. The performanceof the methods are illustrated by numerical experiments on two 2D models.

Iterative solution of the Lippmann-Schwinger equation in strongly scattering acoustic media by randomized construction of preconditioners