Multiscale mimetic reduced-order models for spectrally accurate wavefield simulations
Mikhail Zaslavsky, Vladimir Druskin, Alexander V. Mamonov · 2015
Summary We have developed a novel approach for discretizing spatial operator in acoustic and elastic wavefield simulations. The method is built with high-performance computing (HPC) implementation in mind. It implies a high level of parallelism and greatly reduced communication requirements compared to the traditional finite-difference time-domain (FDTD) methods. We split the reference fine grid model into multiple subdomains (coarse cells). Compared to high-order and spectral methods, these cells can be arbitrarily chosen and are fully model-independent. The interaction of coarse cells with their neighbors is determined by a Neumann-to-Dirichlet (NtD) map. We construct a low-dimensional spectrally accurate approximant of the NtD map for each cell using reduced order modeling (ROM) techniques and then sparsify the approximant via transformation to Stieltjes continued fraction (s-fraction). These steps constitute the off-line part of the approach, which is embarassingly parallel. It is performed just once for the entire time-domain simulation for all sources. Similar to multi-scale finite volume methods, to construct the discretization of the full spatial operator, the coarse cells are conjugated via continuity conditions. In the on-line part of the approach, the discretized operator is employed in a time-stepping scheme. Thanks to low-dimensionality and to s-fraction transformation of the coarse cells ROMs, the communication costs in the on-line part are significantly reduced. Further performance improvement has been achieved by increasing the size of the time step. Indeed, properly chosen ROMs enables relaxing the Courant-Friedrichs-Lewy (CFL) condition. It potentially allows the CFL time step to approach the Nyquist limit, which is typically unattainable with traditional schemes that have the CFL time step much smaller than the Nyquist sampling rate. The constructed approach allows simulating wavefields in media with unlimited complexity (inhomogeneuity, anisotropy) and achieving spectral accuracy even on regular model-independent grids. Our method also perfectly fits for multi-CPU and multi-GPU computations. Numerical results show the advantages of our approach.