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
2095956181 2079659743 2111516947 1905522558
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. We consider the partitioning of a problem on a domain with unequal work estimates in different subdomains in a way that balances the workload across multiple processors. Such a problem arises for example in solving partial differential equations using an adaptive method that places extra grid points in certain subregions of the domain. We use a binary decomposition of the domain to partition it into rectangles requiring equal computational effort. We then study the communication costs of mapping this partitioning onto different multiprocessors: a mesh- connected array, a tree machine, and a hypercube. The communication cost expressions can be used to determine the optimal depth of the above partitioning. We explore efficient domain adaptation for the task of statistical machine translation based on extracting sentences from a large general-domain parallel corpus that are most relevant to the target domain. These sentences may be selected with simple cross-entropy based methods, of which we present three. As these sentences are not themselves identical to the in-domain data, we call them pseudo in-domain subcorpora. These subcorpora -- 1 the size of the original -- can then used to train small domain-adapted Statistical Machine Translation (SMT) systems which outperform systems trained on the entire corpus. Performance is further improved when we use these domain-adapted models in combination with a true in-domain model. The results show that more training data is not always better, and that best results are attained via proper domain-relevant data selection, as well as combining in- and general-domain systems during decoding.