Approximate inversion of the wave-equation Hessian via randomized matrix probing

We present a method for approximately inverting the Hessian of full waveform inversion as a dip-dependent and scale-dependent amplitude correction. The terms in the expansion of this correction are determined by least-squares fitting from a handful of applications of the Hessian to random models — a procedure called matrix probing. We show numerical indications that randomness is important for generating a robust preconditioner, i.e., one that works regardless of the model to be corrected. To be successful, matrix probing requires an accurate determination of the nullspace of the Hessian, which we propose to implement as a local dip-dependent mask in curvelet space. Numerical experiments show that the novel preconditioner fits 70% of the inverse Hessian (in Frobenius norm) for the 1-parameter acoustic 2D Marmousi model.National Science Foundation (U.S.); Alfred P. Sloan Foundatio

Un document réformateur des usages familiaux de la communauté grecque orthodoxe de Beyrouth en 1873

Dans les fonds anciens de la Bibliotheque Orientale de Beyrouth, nous avons retrouve un exemplaire d'une lettre pastorale publiee en 1873 par Ghoufra' il (Gabriel) Chatila, I'archeveque grec orthodoxe de cette ville. Intitulee [slah al- 'awayd Ii-fa 'ifat al-roum al-orthodoksiyin fi Bayrout (Reforme des coutumes de la communaute grecque orthodoxe de Beyrouth), cette circulaire vise it informer les membres de la communaute orthodoxe beyrouthine de la reforme des usages familiaux et sociaux telle que nouvellement statuee par I ' archeveche. Nous ne connaissions pas I'existence de ce document de 1873. Seules des versions tardives, mais qui ne sont pas parfaitement identiques, ont ete publiees et n'y font aucunement reference. La premiere est rectigee par I'eveque Malatius de Lattaquie, et diffusee une dizaine d'annee plus tard par I'archidiacre Gerassimos Messarra dans la revue al-Hadiyyaf (Messarra 1887 : 283-284, 293-295, 300-302)

Approximate Multi-Parameter Inverse Scattering Using Pseudodifferential Scaling

I propose a computationally efficient method to approximate the inverse of the normal operator arising in the multi-parameter linearized inverse problem for reflection seismology in two and three spatial dimensions. Solving the inverse problem using direct matrix methods like Gaussian elimination is computationally infeasible. In fact, the application of the normal operator requires solving large scale PDE problems. However, under certain conditions, the normal operator is a matrix of pseudodifferential operators. This manuscript shows how to generalize Cramer's rule for matrices to approximate the inverse of a matrix of pseudodifferential operators. Approximating the solution to the normal equations proceeds in two steps: (1) First, a series of applications of the normal operator to specific permutations of the right hand side. This step yields a phase-space scaling of the solution. Phase space scalings are scalings in both physical space and Fourier space. Second, a correction for the phase space scaling. This step requires applying the normal operator once more. The cost of approximating the inverse is a few applications of the normal operator (one for one parameter, two for two parameters, six for three parameters). The approximate inverse is an adequately accurate solution to the linearized inverse problem when it is capable of fitting the data to a prescribed precision. Otherwise, the approximate inverse of the normal operator might be used to precondition Krylov subspace methods in order to refine the data fit. I validate the method on a linearized version of the Marmousi model for constant density acoustics for the one-parameter problem. For the two parameter problem, the inversion of a variable density acoustics layered model corroborates the success of the proposed method. Furthermore, this example details the various steps of the method. I also apply the method to a 1D section of the Marmousi model to test the behavior of the method on complex two-parameter layered models