SUMMARYThe method of polarized traces provides the first documented algorithm with truly scalable complexity for the highfrequency Helmholtz equation, i.e., with a runtime sublinear in the number of volume unknowns in a parallel environment. However, previous versions of this method were either restricted to a low order of accuracy, or suffered from computationally unfavorable boundary reduction to O(p) interfaces in the p-th order case. In this note we rectify this issue by proposing a high-order method of polarized traces with compact reduction to two, rather than O(p), interfaces. This method is based on a primal Hybridizable Discontinuous Galerkin (HDG) discretization in a domain decomposition setting. In addition, HDG is a welcome upgrade for the method of polarized traces, since it can be made to work with flexible meshes that align with discontinuous coefficients, and it allows for adaptive refinement in h and p. High order of accuracy is very important for attenuation of the pollution error, even in settings when the medium is not smooth. We provide some examples to corroborate the convergence and complexity claims.
In this work we propose a hybridizable discontinuous Galerkin (hdG) discretization of the high-frequency Helmholtz equation in the presence of point sources and highly heterogeneous and discontinuous wave speed models. We show that it delivers solutions that are provably second-order accurate and do not suffer from the pollution error, as long as a slightly higher order hdG method is used where the polynomial degree is chosen such that p = O(log ω). These results hold even if the discontinuities in the wave speed are not resolved by the hdG mesh, as long as the integration procedure used in the assembly of the stiffness matrix respects the discontinuities. Further, we show that the associated linear systems can be solved using a modification of the method of polarized traces resulting in a method with linear complexity up to a poly-logarithmic factor, or sub-linear complexity in a parallel environment. To our knowledge and surprise, this note contains the first instance of a numerical method that is at the same time fast (O(N) runtime) and accurate (second-order, pollution-free) in the context of models of geophysical interest.
SUMMARYIn this abstract, we present a case study in the application of a time-stepping method, unconstrained by the CFL condition, for computational acoustic wave propagation in the context of full waveform inversion. The numerical scheme is a locally one-dimensional (LOD) variant of alternating dimension implicit (ADI) method. The LOD method has a maximum time step that is restricted only by the Nyquist sampling rate. The advantage over traditional explicit time-stepping methods occurs in the presence of high contrast media, low frequencies, and steep, narrow perfectly matched layers (PML). The main technical point of the note, from a numerical analysis perspective, is that the LOD scheme is adapted to the presence of a PML. A complexity study is presented and an application to full waveform inversion is shown.
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