High performance cluster based low-rank solution of helmholtz problem for applying in full waveform inversion

We describe a MPI-based multifrontal direct solver for Helmholtz problem in 3D heterogeneous media. To reduce memory consumption and computational time at the factorization step, intermediate data are compressed with help of Low-Rank approximation and Hierarchically Semi-Separable format. The cluster version is based on performance efficient approach: the factorization of various parts of matrix are distributed through cluster nodes in advance. It allows us highly parallelize the major jobs of factorization process. i.e. low-rank approximation and computing the Schur complement. Such improvements make it possible to factorize in acceptable time (∼1 hour) a system of more than 10 ∧ 8 equations corresponding to a realistic geophysical velocity model on ∼1200km ∧ 2 square. The inversion of factorized system takes less than 1 second per one right hand side. It allows us to use this solver in solving series of forward problems in Full Waveform Inversion (FWI) process where the main step is solution of Systems of Linear Algebraic Equations (SLAE) of big size. Numerical experiments show well scalability on moderate number of cluster nodes and demonstrate high performance and memory compressibility both on homogeneous and high contrast heterogeneous velocity models for various frequencies.