Abstract. In this paper, we develop a sparse grid discontinuous Galerkin (DG) scheme for transport equations and applied it to kinetic simulations. The method uses the weak formulations of traditional Runge-Kutta DG (RKDG) schemes for hyperbolic problems and is proven to be L 2 stable and convergent. A major advantage of the scheme lies in its low computational and storage cost due to the employed sparse finite element approximation space. This attractive feature is explored in simulating Vlasov and Boltzmann transport equations. Good performance in accuracy and conservation is verified by numerical tests in up to four dimensions.Key words. discontinuous Galerkin methods; sparse grid; high-dimensional transport equations; Vlasov equation; Boltzmann equation.1. Introduction. In this paper, we develop a sparse grid DG method for high-dimensional transport equations. High-dimensional transport problems are ubiquitous in science and engineering, and most evidently in kinetic simulations where it is necessary to track the evolution of probability density functions of particles. Deterministic kinetic simulations are very demanding due to the large computational and storage cost. To make the schemes more attractive comparing with the alternative probabilistic methods, an appealing approach is to explore the sparse grid techniques [6,15] with the aim of breaking the curse of dimensionality . In the context of wavelets or sparse grid methods for kinetic transport equations, we mention the work of using wavelet-MRA methods for Vlasov equations , the combination technique for linear gyrokinetics , sparse adaptive finite element method , sparse discrete ordinates method  and sparse tensor spherical harmonics  for radiative transfer, among many others. This paper focuses on the DG method , which is a class of finite element methods using discontinuous approximation space for the numerical solution and the test functions. The RKDG scheme  developed in a series of papers for hyperbolic equations became very popular due to its provable convergence, excellent conservation properties and accommodation for adaptivity and parallel implementations. Recent years have seen great growth in the interest of applying DG methods to kinetic systems (see for example [3,18,20,9, 10]) because of the conservation properties and long time performance of the resulting simulations. However, the DG method is still deemed too costly in a realistic setting, often requiring more degrees of freedom than other high order numerical calculations.Recently, we developed a sparse grid DG method for high-dimensional elliptic problems . A sparse DG finite element space has been constructed, reducing the degrees of freedom from the standardwhere h is the uniform mesh size in each dimension. The resulting scheme retains main properties of standard DG methods while making the computational cost tangebile for high-dimensional simulations. This motivates the current work for the transport equations, and we use kinetic problems as a...
This paper constitutes our initial effort in developing sparse grid discontinuous Galerkin (DG) methods for high-dimensional partial differential equations (PDEs). Over the past few decades, DG methods have gained popularity in many applications due to their distinctive features. However, they are often deemed too costly because of the large number of degrees of freedom of the approximation space, which are the main bottleneck for simulations in high dimensions. In this paper, we develop sparse grid DG methods for elliptic equations with the aim of breaking the curse of dimensionality. Using a hierarchical basis representation, we construct a sparse finite element approximation space, reducing the degrees of freedom from the standardproblems, where h is the uniform mesh size in each dimension. Our method, based on the interior penalty (IP) DG framework, can achieve accuracy of O(h k | log 2 h| d−1 ) in the energy norm, where k is the degree of polynomials used. Error estimates are provided and confirmed by numerical tests in multi-dimensions.
Abstract. In this paper, we develop a class of high order conservative semi-Lagrangian (SL) discontinuous Galerkin (DG) methods for solving multi-dimensional linear transport equations. The methods rely on a characteristic Galerkin weak formulation, leading to L 2 stable discretizations for linear problems. Unlike many existing SL methods, the high order accuracy and mass conservation of the proposed methods are realized in a non-splitting manner. Thus, the detrimental splitting error, which is known to significantly contaminate long term transport simulations, will be not incurred. One key ingredient in the scheme formulation, borrowed from CSLAM [Lauritzen, Nair & Ullrich, 2010], is the use of Green's theorem which allows us to convert volume integrals into a set of line integrals. The resulting line integrals are much easier to approximate with high order accuracy, hence facilitating the implementation. Another novel ingredient is the construction of quadratic curves in approximating sides of upstream cell, leading to quadratic-curved quadrilateral upstream cells. Formal third order accuracy is obtained by such a construction. The desired positivity-preserving property is further attained by incorporating a high order bound-preserving filter. To assess the performance of the proposed methods, we test and compare the numerical schemes with a variety of configurations for solving several benchmark transport problems with both smooth and nonsmooth solutions. The efficiency and efficacy are numerically verified.
Daily acquisition of large amounts of aerial and satellite images has facilitated subsequent automatic interpretations of these images. One such interpretation is object detection. Despite the great progress made in this domain, the detection of multi-scale objects, especially small objects in high resolution satellite (HRS) images, has not been adequately explored. As a result, the detection performance turns out to be poor. To address this problem, we first propose a unified multi-scale convolutional neural network (CNN) for geospatial object detection in HRS images. It consists of a multi-scale object proposal network and a multi-scale object detection network, both of which share a multi-scale base network. The base network can produce feature maps with different receptive fields to be responsible for objects with different scales. Then, we use the multi-scale object proposal network to generate high quality object proposals from the feature maps. Finally, we use these object proposals with the multi-scale object detection network to train a good object detector. Comprehensive evaluations on a publicly available remote sensing object detection dataset and comparisons with several state-of-the-art approaches demonstrate the effectiveness of the presented method. The proposed method achieves the best mean average precision (mAP) value of 89.6%, runs at 10 frames per second (FPS) on a GTX 1080Ti GPU.
The discontinuous Galerkin (DG) methods designed for hyperbolic problems arising from a wide range of applications are known to enjoy many computational advantages. DG methods coupled with strong-stability-preserving explicit Runge–Kutta discontinuous Galerkin (RKDG) time discretizations provide a robust numerical approach suitable for geoscience applications including atmospheric modeling. However, a major drawback of the RKDG method is its stringent Courant–Friedrichs–Lewy (CFL) stability restriction associated with explicit time stepping. To address this issue, the authors adopt a dimension-splitting approach where a semi-Lagrangian (SL) time-stepping strategy is combined with the DG method. The resulting SLDG scheme employs a sequence of 1D operations for solving multidimensional transport equations. The SLDG scheme is inherently conservative and has the option to incorporate a local positivity-preserving filter for tracers. A novel feature of the SLDG algorithm is that it can be used for multitracer transport for global models employing spectral-element grids, without using an additional finite-volume grid system. The quality of the proposed method is demonstrated via benchmark tests on Cartesian and cubed-sphere geometry, which employs nonorthogonal, curvilinear coordinates.
In this paper, we develop an adaptive multiresolution discontinuous Galerkin (DG) scheme for time-dependent transport equations in multi-dimensions. The method is constructed using multiwavlelets on tensorized nested grids. Adaptivity is realized by error thresholding based on the hierarchical surplus, and the Runge-Kutta DG (RKDG) scheme is employed as the reference time evolution algorithm. We show that the scheme performs similarly to a sparse grid DG method when the solution is smooth, reducing computational cost in multi-dimensions. When the solution is no longer smooth, the adaptive algorithm can automatically capture fine local structures. The method is therefore very suitable for deterministic kinetic simulations. Numerical results including several benchmark tests, the Vlasov-Poisson (VP) and oscillatory VP systems are provided.
Despite the uniform mortality in pancreatic adenocarcinoma (PDAC), clinical disease heterogeneity exists with limited genomic differences. A highly aggressive tumor subtype termed ‘basal-like’ was identified to show worse outcomes and higher inflammatory responses. Here, we focus on the microbial effect in PDAC progression and present a comprehensive analysis of the tumor microbiome in different PDAC subtypes with resectable tumors using metagenomic sequencing. We found distinctive microbial communities in basal-like tumors and identified an increasing abundance of Acinetobacter, Pseudomonas and Sphingopyxis to be highly associated with carcinogenesis. Functional characterization of microbial genes suggested the potential to induce pathogen-related inflammation. Host-microbiota interplay analysis provided new insights into the tumorigenic role of specific microbiome compositions and demonstrated the influence of host genetics in shaping the tumor microbiome. Taken together, these findings indicated that the tumor microbiome is closely related to PDAC oncogenesis and the induction of inflammation. Additionally, our data revealed the microbial basis of PDAC heterogeneity and proved the predictive value of the microbiome, which will contribute to the intervention and treatment of disease.
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