亚洲男人的天堂2018av,欧美草比,久久久久久免费视频精选,国色天香在线看免费,久久久久亚洲av成人片仓井空

We propose an Extended Hybrid High-Order scheme for the Poisson problem with solution possessing weak singularities. Some general assumptions are stated on the nature of this singularity and the remaining part of the solution. The method is formulated by enriching the local polynomial spaces with appropriate singular functions. Via a detailed error analysis, the method is shown to converge optimally in both discrete and continuous energy norms. Some tests are conducted in two dimensions for singularities arising from irregular geometries in the domain. The numerical simulations illustrate the established error estimates, and show the method to be a significant improvement over a standard Hybrid High-Order method.

Peskin's Immersed Boundary (IB) model and method are among the most popular modeling tools and numerical methods. The IB method has been known to be first order accurate in the velocity. However, almost no rigorous theoretical proof can be found in the literature for Stokes equations with a prescribed velocity boundary condition. In this paper, it has been shown that the pressure of the Stokes equation has convergence order $O(\sqrt{h})$ in the $L^2$ norm while the velocity has $O(h)$ convergence in the infinity norm in two-dimensions (2D). The proofs are based on the idea of the immersed interface method, and the convergence proof of the IB method for elliptic interface problems \cite{li:mathcom}. The proof is intuitive and the conclusion can apply to different boundary conditions as long as the problem is well-posed. The proof process also provides an efficient way to decouple the system into three Helmholtz/Poisson equations without affecting the accuracy. A non-trivial numerical example is also provided to confirm the theoretical analysis.

This paper proposes a level set-based topology optimization method for designing acoustic structures with viscous and thermal boundary layers in perspective. It is known that acoustic waves propagating in a narrow channel are damped by viscous and thermal boundary layers. To estimate these viscothermal effects, we first introduce a sequential linearized Navier-Stokes model based on three weakly coupled Helmholtz equations for viscous, thermal, and acoustic pressure fields. Then, the optimization problem is formulated, where a sound-absorbing structure comprising air and an isothermal rigid medium is targeted, and its sound absorption coefficient is set as an objective function. The adjoint variable method and the concept of the topological derivative are used to obtain design sensitivity. A level set-based topology optimization method is used to solve the optimization problem. Two-dimensional numerical examples are provided to support the validity of the proposed method. In addition, the mechanisms that lead to the high absorption coefficient of the optimized design are discussed.

Partially recorded data are frequently encountered in many applications and usually clustered by first removing incomplete cases or features with missing values, or by imputing missing values, followed by application of a clustering algorithm to the resulting altered dataset. Here, we develop clustering methodology through a model-based approach using the marginal density for the observed values, assuming a finite mixture model of multivariate $t$ distributions. We compare our approximate algorithm to the corresponding full expectation-maximization (EM) approach that considers the missing values in the incomplete data set and makes a missing at random (MAR) assumption, as well as case deletion and imputation methods. Since only the observed values are utilized, our approach is computationally more efficient than imputation or full EM. Simulation studies demonstrate that our approach has favorable recovery of the true cluster partition compared to case deletion and imputation under various missingness mechanisms, and is at least competitive with the full EM approach, even when MAR assumptions are violated. Our methodology is demonstrated on a problem of clustering gamma-ray bursts and is implemented at //github.com/emilygoren/MixtClust.

We investigate the use of models from the theory of regularity structure as features in machine learning tasks. A model is a multi-linear function of a space-time signal designed to well-approximate solutions to partial differential equations (PDEs), even in low regularity regimes. Models can be seen as natural multi-dimensional generalisations of signatures of paths; our work therefore aims to extend the recent use of signatures in data science beyond the context of time-ordered data. We provide a flexible definition of a model feature vector associated to a space-time signal, along with two algorithms which illustrate ways in which these features can be combined with linear regression. We apply these algorithms in several numerical experiments designed to learn solutions to PDEs with a given forcing and boundary data. Our experiments include semi-linear parabolic and wave equations with forcing, and Burgers' equation with no forcing. We find an advantage in favour of our algorithms when compared to several alternative methods. Additionally, in the experiment with Burgers' equation, we noticed stability in the prediction power when noise is added to the observations.

We investigate analytic properties of the double Fourier sphere (DFS) method, which transforms a function defined on the two-dimensional sphere to a function defined on the two-dimensional torus. Then the resulting function can be written as a Fourier series yielding an approximation of the original function. We show that the DFS method preserves smoothness: it continuously maps spherical H\"older spaces into the respective spaces on the torus, but it does not preserve spherical Sobolev spaces in the same manner. Furthermore, we prove sufficient conditions for the absolute convergence of the resulting series expansion on the sphere as well as results on the speed of convergence.

Graph convolutional networks (GCNs) have shown promising results in processing graph data by extracting structure-aware features. This gave rise to extensive work in geometric deep learning, focusing on designing network architectures that ensure neuron activations conform to regularity patterns within the input graph. However, in most cases the graph structure is only accounted for by considering the similarity of activations between adjacent nodes, which in turn degrades the results. In this work, we augment GCN models by incorporating richer notions of regularity by leveraging cascades of band-pass filters, known as geometric scatterings. The produced graph features incorporate multiscale representations of local graph structures, while avoiding overly smooth activations forced by previous architectures. Moreover, inspired by skip connections used in residual networks, we introduce graph residual convolutions that reduce high-frequency noise caused by joining together information at multiple scales. Our hybrid architecture introduces a new model for semi-supervised learning on graph-structured data, and its potential is demonstrated for node classification tasks on multiple graph datasets, where it outperforms leading GCN models.

It is not until recently that graph neural networks (GNNs) are adopted to perform graph representation learning, among which, those based on the aggregation of features within the neighborhood of a node achieved great success. However, despite such achievements, GNNs illustrate defects in identifying some common structural patterns which, unfortunately, play significant roles in various network phenomena. In this paper, we propose GraLSP, a GNN framework which explicitly incorporates local structural patterns into the neighborhood aggregation through random anonymous walks. Specifically, we capture local graph structures via random anonymous walks, powerful and flexible tools that represent structural patterns. The walks are then fed into the feature aggregation, where we design various mechanisms to address the impact of structural features, including adaptive receptive radius, attention and amplification. In addition, we design objectives that capture similarities between structures and are optimized jointly with node proximity objectives. With the adequate leverage of structural patterns, our model is able to outperform competitive counterparts in various prediction tasks in multiple datasets.

An attributed network enriches a pure network by encoding a part of widely accessible node auxiliary information into node attributes. Learning vector representation of each node a.k.a. Network Embedding (NE) for such an attributed network by considering both structure and attribute information has recently attracted considerable attention, since each node embedding is simply a unified low-dimension vector representation that makes downstream tasks e.g. link prediction more efficient and much easier to realize. Most of previous works have not considered the significant case of a network with incomplete structure information, which however, would often appear in our real-world scenarios e.g. the abnormal users in a social network who intentionally hide their friendships. And different networks obviously have different levels of incomplete structure information, which imposes more challenges to balance two sources of information. To tackle that, we propose a robust NE method called Attributed Biased Random Walks (ABRW) to employ attribute information for compensating incomplete structure information by using transition matrices. The experiments of link prediction and node classification tasks on real-world datasets confirm the robustness and effectiveness of our method to the different levels of the incomplete structure information.

Many resource allocation problems in the cloud can be described as a basic Virtual Network Embedding Problem (VNEP): finding mappings of request graphs (describing the workloads) onto a substrate graph (describing the physical infrastructure). In the offline setting, the two natural objectives are profit maximization, i.e., embedding a maximal number of request graphs subject to the resource constraints, and cost minimization, i.e., embedding all requests at minimal overall cost. The VNEP can be seen as a generalization of classic routing and call admission problems, in which requests are arbitrary graphs whose communication endpoints are not fixed. Due to its applications, the problem has been studied intensively in the networking community. However, the underlying algorithmic problem is hardly understood. This paper presents the first fixed-parameter tractable approximation algorithms for the VNEP. Our algorithms are based on randomized rounding. Due to the flexible mapping options and the arbitrary request graph topologies, we show that a novel linear program formulation is required. Only using this novel formulation the computation of convex combinations of valid mappings is enabled, as the formulation needs to account for the structure of the request graphs. Accordingly, to capture the structure of request graphs, we introduce the graph-theoretic notion of extraction orders and extraction width and show that our algorithms have exponential runtime in the request graphs' maximal width. Hence, for request graphs of fixed extraction width, we obtain the first polynomial-time approximations. Studying the new notion of extraction orders we show that (i) computing extraction orders of minimal width is NP-hard and (ii) that computing decomposable LP solutions is in general NP-hard, even when restricting request graphs to planar ones.