Three-dimensional frequency-domain full waveform inversion of fixed-spread data can be efficiently performed in the viscoacoustic approximation when seismic modeling is based on a sparse direct solver. We present a new algebraic Block Low-Rank (BLR) multifrontal solver which provides an approximate solution of the time-harmonic wave equation with a reduced operation count, memory demand and volume of communication relative to the full-rank solver. We show some preliminary simulations in the 3D SEG/EAGE overthrust model, that give some insights on the memory and time complexities of the low-rank solver for frequencies of interest in fullwaveform inversion (FWI) applications.
We introduce a novel approach to exploit mixed precision arithmetic for low-rank approximations. Our approach is based on the observation that singular vectors associated with small singular values can be stored in lower precisions while preserving high accuracy overall. We provide an explicit criterion to determine which level of precision is needed for each singular vector. We apply this approach to block low-rank (BLR) matrices, most of whose off-diagonal blocks have low rank. We propose a new BLR LU factorization algorithm that exploits the mixed precision representation of the blocks. We carry out the rounding error analysis of this algorithm and prove that the use of mixed precision arithmetic does not compromise the numerical stability of the BLR LU factorization. Moreover, our analysis determines which level of precision is needed for each floating-point operation (flop), and therefore guides us toward an implementation that is both robust and efficient. We evaluate the potential of this new algorithm on a range of matrices coming from real-life problems in industrial and academic applications. We show that a large fraction of the entries in the LU factors and flops to perform the BLR LU factorization can be safely switched to lower precisions, leading to significant reductions of the storage and expected time costs, of up to a factor three using fp64, fp32, and bfloat16 arithmetics.
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