# A Parallel Two-Scale Solution Algorithm for the Elastic Wave Equation

Susan Minkoff, David Trott, Sean Griffith, Shiming Yang, Department of Mathematics and Statistics, UMBC, Tanya Vdovina, Computational and Applied Math, Rice University

Inverse problems in seismic imaging commonly involve manipulating terabytes of data. Being able to model seismic wave propagation efficiently in higher dimensions is of much interest, since solution of the inverse problem requires multiple forward simulations. Operator based upscaling is a two-scale algorithm that speeds up the solution of the forward problem by producing a coarse grid solution which incorporates much of the local fine-scale solution information. We present the first implementation of operator upscaling for the elastic wave equation. By using the velocity-displacement formulation of the three-dimensional elastic wave equation, basis functions that are linear in all three directions, and applying mass-lumping, the subgrid solve (first stage of the two-step algorithm) reduces to solving explicit difference equations. At the second stage of the algorithm, we upscale both velocity and displacement by using the local subgrid information to formulate the coarse-grid problem. The coarse grid system matrix is independent of time, sparse, and banded. This paper explores the serial and parallel implementations of the method. The main simplifying assumption of the method (special zero boundary conditions imposed on coarse blocks in the first stage of the algorithm) leads to an easily parallelizable algorithm because no ghost cell communication is required between processors.