We present a solver for the 2D high-frequency Helmholtz equation in heterogeneous acoustic media, with online parallel complexity that scales optimally as @math , where @math is the number of volume unknowns, and @math is the number of processors, as long as @math grows at most like a small fractional power of @math . The solver decomposes the domain into layers, and uses transmission conditions in boundary integral form to explicitly define "polarized traces", i.e., up- and down-going waves sampled at interfaces. Local direct solvers are used in each layer to precompute traces of local Green's functions in an embarrassingly parallel way (the offline part), and incomplete Green's formulas are used to propagate interface data in a sweeping fashion, as a preconditioner inside a GMRES loop (the online part). Adaptive low-rank partitioning of the integral kernels is used to speed up their application to interface data. The method uses second-order finite differences. The complexity scalings are empirical but motivated by an analysis of ranks of off-diagonal blocks of oscillatory integrals. They continue to hold in the context of standard geophysical community models such as BP and Marmousi 2, where convergence occurs in 5 to 10 GMRES iterations.
Chen and Xiang proposed another instance of efficient domain decomposition where the emphasis is on transferring sources from one subdomain to another @cite_92 @cite_64 ;
@cite_92 @cite_64
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In this paper we extend the source transfer domain decomposition method (STDDM) introduced by the authors to solve the Helmholtz problems in two-layered media, the Helmholtz scattering problems with bounded scatterer, and Helmholtz prob- lems in 3D unbounded domains. The STDDM is based on the decomposition of the domain into non-overlapping layers and the idea of source transfer which transfers the sources equivalently layer by layer so that the solution in the final layer can be solved using a PML method defined locally outside the last two layers. The details of STDDM is given for each extension. Numerical results are presented to demonstrate the efficiency of STDDM as a preconditioner for solving the discretization problem of the Helmholtz problems considered in the paper. Given an entity in a source domain, finding its matched entities from another (target) domain is an important task in many applications. Traditionally, the problem was usually addressed by first extracting major keywords corresponding to the source entity and then query relevant entities from the target domain using those keywords. However, the method would inevitably fails if the two domains have less or no overlapping in the content. An extreme case is that the source domain is in English and the target domain is in Chinese. In this paper, we formalize the problem as entity matching across heterogeneous sources and propose a probabilistic topic model to solve the problem. The model integrates the topic extraction and entity matching, two core subtasks for dealing with the problem, into a unified model. Specifically, for handling the text disjointing problem, we use a cross-sampling process in our model to extract topics with terms coming from all the sources, and leverage existing matching relations through latent topic layers instead of at text layers. Benefit from the proposed model, we can not only find the matched documents for a query entity, but also explain why these documents are related by showing the common topics they share. Our experiments in two real-world applications show that the proposed model can extensively improve the matching performance (+19.8 and +7.1 in two applications respectively) compared with several alternative methods.