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A robust nonconforming mixed finite element method is developed for a strain gradient elasticity (SGE) model. In two and three dimensional cases, a lower order $C^0$-continuous $H^2$-nonconforming finite element is constructed for the displacement field through enriching the quadratic Lagrange element with bubble functions. This together with the linear Lagrange element is exploited to discretize a mixed formulation of the SGE model. The robust discrete inf-sup condition is established. The sharp and uniform error estimates with respect to both the small size parameter and the Lam\'{e} coefficient are achieved, which is also verified by numerical results. In addition, the uniform regularity of the SGE model is derived under two reasonable assumptions.

The unified gas-kinetic wave-particle method (UGKWP) has been developed for the multiscale gas, plasma, and multiphase flow transport processes for the past years. In this work, we propose an implicit unified gas-kinetic wave-particle (IUGKWP) method to remove the CFL time step constraint. Based on the local integral solution of the radiative transfer equation (RTE), the particle transport processes are categorized into the long-$\lambda$ streaming process and the short-$\lambda$ streaming process comparing to a local physical characteristic time $t_p$. In the construction of the IUGKWP method, the long-$\lambda$ streaming process is tracked by the implicit Monte Carlo (IMC) method; the short-$\lambda$ streaming process is evolved by solving the implicit moments equations; and the photon distribution is closed by a local integral solution of RTE. In the IUGKWP method, the multiscale flux of radiation energy and the multiscale closure of photon distribution are constructed based on the local integral solution. The IUGKWP method preserves the second-order asymptotic expansion of RTE in the optically thick regime and adapts its computational complexity to the flow regime. The numerical dissipation is well controlled, and the teleportation error is significantly reduced in the optically thick regime. The computational complexity of the IUGKWP method decreases exponentially as the Knudsen number approaches zero, and the computational efficiency is remarkably improved in the optically thick regime. The IUGKWP is formulated on a generalized unstructured mesh, and multidimensional 2D and 3D algorithms are developed. Numerical tests are presented to validate the capability of IUGKWP in capturing the multiscale photon transport process. The algorithm and code will apply in the engineering applications of inertial confinement fusion (ICF).

Recently established, directed dependence measures for pairs $(X,Y)$ of random variables build upon the natural idea of comparing the conditional distributions of $Y$ given $X=x$ with the marginal distribution of $Y$. They assign pairs $(X,Y)$ values in $[0,1]$, the value is $0$ if and only if $X,Y$ are independent, and it is $1$ exclusively for $Y$ being a function of $X$. Here we show that comparing randomly drawn conditional distributions with each other instead or, equivalently, analyzing how sensitive the conditional distribution of $Y$ given $X=x$ is on $x$, opens the door to constructing novel families of dependence measures $\Lambda_\varphi$ induced by general convex functions $\varphi: \mathbb{R} \rightarrow \mathbb{R}$, containing, e.g., Chatterjee's coefficient of correlation as special case. After establishing additional useful properties of $\Lambda_\varphi$ we focus on continuous $(X,Y)$, translate $\Lambda_\varphi$ to the copula setting, consider the $L^p$-version and establish an estimator which is strongly consistent in full generality. A real data example and a simulation study illustrate the chosen approach and the performance of the estimator. Complementing the afore-mentioned results, we show how a slight modification of the construction underlying $\Lambda_\varphi$ can be used to define new measures of explainability generalizing the fraction of explained variance.

We consider a sharp interface formulation for the multi-phase Mullins-Sekerka flow. The flow is characterized by a network of curves evolving such that the total surface energy of the curves is reduced, while the areas of the enclosed phases are conserved. Making use of a variational formulation, we introduce a fully discrete finite element method. Our discretization features a parametric approximation of the moving interfaces that is independent of the discretization used for the equations in the bulk. The scheme can be shown to be unconditionally stable and to satisfy an exact volume conservation property. Moreover, an inherent tangential velocity for the vertices on the discrete curves leads to asymptotically equidistributed vertices, meaning no remeshing is necessary in practice. Several numerical examples, including a convergence experiment for the three-phase Mullins-Sekerka flow, demonstrate the capabilities of the introduced method.

This work presents a systematic methodology for describing the transient dynamics of coarse-grained molecular systems inferred from all-atom simulated data. We suggest Langevin-type dynamics where the coarse-grained interaction potential depends explicitly on time to efficiently approximate the transient coarse-grained dynamics. We apply the path-space force matching approach at the transient dynamics regime to learn the proposed model parameters. In particular, we parameterize the coarse-grained potential both with respect to the pair distance of the CG particles and the time, and we obtain an evolution model that is explicitly time-dependent. Moreover, we follow a data-driven approach to estimate the friction kernel, given by appropriate correlation functions directly from the underlying all-atom molecular dynamics simulations. To explore and validate the proposed methodology we study a benchmark system of a moving particle in a box. We examine the suggested model's effectiveness in terms of the system's correlation time and find that the model can approximate well the transient time regime of the system, depending on the correlation time of the system. As a result, in the less correlated case, it can represent the dynamics for a longer time interval. We present an extensive study of our approach to a realistic high-dimensional water molecular system. Posing the water system initially out of thermal equilibrium we collect trajectories of all-atom data for the, empirically estimated, transient time regime. Then, we infer the suggested model and strengthen the model's validity by comparing it with simplified Markovian models.

Serial femtosecond crystallography at X-ray free electron laser facilities opens a new era for the determination of crystal structure. However, the data processing of those experiments is facing unprecedented challenge, because the total number of diffraction patterns needed to determinate a high-resolution structure is huge. Machine learning methods are very likely to play important roles in dealing with such a large volume of data. Convolutional neural networks have made a great success in the field of pattern classification, however, training of the networks need very large datasets with labels. Th is heavy dependence on labeled datasets will seriously restrict the application of networks, because it is very costly to annotate a large number of diffraction patterns. In this article we present our job on the classification of diffraction pattern by weakly supervised algorithms, with the aim of reducing as much as possible the size of the labeled dataset required for training. Our result shows that weakly supervised methods can significantly reduce the need for the number of labeled patterns while achieving comparable accuracy to fully supervised methods.

We give a short survey of recent results on sparse-grid linear algorithms of approximate recovery and integration of functions possessing a unweighted or weighted Sobolev mixed smoothness based on their sampled values at a certain finite set. Some of them are extended to more general cases.

Ordinary state-based peridynamic (OSB-PD) models have an unparalleled capability to simulate crack propagation phenomena in solids with arbitrary Poisson's ratio. However, their non-locality also leads to prohibitively high computational cost. In this paper, a fast solution scheme for OSB-PD models based on matrix operation is introduced, with which, the graphics processing units (GPUs) are used to accelerate the computation. For the purpose of comparison and verification, a commonly used solution scheme based on loop operation is also presented. An in-house software is developed in MATLAB. Firstly, the vibration of a cantilever beam is solved for validating the loop- and matrix-based schemes by comparing the numerical solutions to those produced by a FEM software. Subsequently, two typical dynamic crack propagation problems are simulated to illustrate the effectiveness of the proposed schemes in solving dynamic fracture problems. Finally, the simulation of the Brokenshire torsion experiment is carried out by using the matrix-based scheme, and the similarity in the shapes of the experimental and numerical broken specimens further demonstrates the ability of the proposed approach to deal with 3D non-planar fracture problems. In addition, the speed-up of the matrix-based scheme with respect to the loop-based scheme and the performance of the GPU acceleration are investigated. The results emphasize the high computational efficiency of the matrix-based implementation scheme.

We study solute-laden flow through permeable geological formations with a focus on advection-dominated transport and volume reactions. As the fluid flows through the permeable medium, it reacts with the medium, thereby changing the morphology and properties of the medium; this in turn, affects the flow conditions and chemistry. These phenomena occur at various lengths and time scales, and makes the problem extremely complex. Multiscale modeling addresses this complexity by dividing the problem into those at individual scales, and systematically passing information from one scale to another. However, accurate implementation of these multiscale methods are still prohibitively expensive. We present a methodology to overcome this challenge that is computationally efficient and quantitatively accurate. We introduce a surrogate for the solution operator of the lower scale problem in the form of a recurrent neural operator, train it using one-time off-line data generated by repeated solutions of the lower scale problem, and then use this surrogate in application-scale calculations. The result is the accuracy of concurrent multiscale methods, at a cost comparable to those of classical models. We study various examples, and show the efficacy of this method in understanding the evolution of the morphology, properties and flow conditions over time in geological formations.