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In this paper we study some theoretical and numerical issues of the Boussinesq/Full dispersion system. This is a a three-parameter system of pde's that models the propagation of internal waves along the interface of two-fluid layers with rigid lid condition for the upper layer, and under a Boussinesq regime for the upper layer and a full dispersion regime for the lower layer. We first discretize in space the periodic initial-value problem with a Fourier-Galerkin spectral method and prove error estimates for several ranges of values of the parameters. Solitary waves of the model systems are then studied numerically in several ways. The numerical generation is analyzed by approximating the ode system with periodic boundary conditions for the solitary-wave profiles with a Fourier spectral scheme, implemented in a collocation form, and solving iteratively the corresponding algebraic system in Fourier space with the Petviashvili method accelerated with the minimal polynomial extrapolation technique. Motivated by the numerical results, a new result of existence of solitary waves is proved. In the last part of the paper, the dynamics of these solitary waves is studied computationally, To this end, the semidiscrete systems obtained from the Fourier-Galerkin discretization in space are integrated numerically in time by a Runge-Kutta Composition method of order four. The fully discrete scheme is used to explore numerically the stability of solitary waves, their collisions, and the resolution of other initial conditions into solitary waves.

This paper offers a new approach to address the model uncertainty in (potentially) divergent-dimensional single-index models (SIMs). We propose a model-averaging estimator based on cross-validation, which allows the dimension of covariates and the number of candidate models to increase with the sample size. We show that when all candidate models are misspecified, our model-averaging estimator is asymptotically optimal in the sense that its squared loss is asymptotically identical to that of the infeasible best possible averaging estimator. In a different situation where correct models are available in the model set, the proposed weighting scheme assigns all weights to the correct models in the asymptotic sense. We also extend our method to average regularized estimators and propose pre-screening methods to deal with cases with high-dimensional covariates. We illustrate the merits of our method via simulations and two empirical applications.

We develop a systematic information-theoretic framework for quantification and mitigation of error in probabilistic Lagrangian (i.e., path-based) predictions which are obtained from dynamical systems generated by uncertain (Eulerian) vector fields. This work is motivated by the desire to improve Lagrangian predictions in complex dynamical systems based either on analytically simplified or data-driven models. We derive a hierarchy of general information bounds on uncertainty in estimates of statistical observables $\mathbb{E}^{\nu}[f]$, evaluated on trajectories of the approximating dynamical system, relative to the "true'' observables $\mathbb{E}^{\mu}[f]$ in terms of certain $\varphi$-divergences, $\mathcal{D}_\varphi(\mu\|\nu)$, which quantify discrepancies between probability measures $\mu$ associated with the original dynamics and their approximations $\nu$. We then derive two distinct bounds on $\mathcal{D}_\varphi(\mu\|\nu)$ itself in terms of the Eulerian fields. This new framework provides a rigorous way for quantifying and mitigating uncertainty in Lagrangian predictions due to Eulerian model error.

We study synchronous Q-learning with Polyak-Ruppert averaging (a.k.a., averaged Q-leaning) in a $\gamma$-discounted MDP. We establish asymptotic normality for the averaged iteration $\bar{\boldsymbol{Q}}_T$. Furthermore, we show that $\bar{\boldsymbol{Q}}_T$ is actually a regular asymptotically linear (RAL) estimator for the optimal Q-value function $\boldsymbol{Q}^*$ with the most efficient influence function. It implies the averaged Q-learning iteration has the smallest asymptotic variance among all RAL estimators. In addition, we present a non-asymptotic analysis for the $\ell_{\infty}$ error $\mathbb{E}\|\bar{\boldsymbol{Q}}_T-\boldsymbol{Q}^*\|_{\infty}$, showing it matches the instance-dependent lower bound as well as the optimal minimax complexity lower bound. As a byproduct, we find the Bellman noise has sub-Gaussian coordinates with variance $\mathcal{O}((1-\gamma)^{-1})$ instead of the prevailing $\mathcal{O}((1-\gamma)^{-2})$ under the standard bounded reward assumption. The sub-Gaussian result has potential to improve the sample complexity of many RL algorithms. In short, our theoretical analysis shows averaged Q-Leaning is statistically efficient.

In this paper, we study deep neural networks (DNNs) for solving high-dimensional evolution equations with oscillatory solutions. Different from deep least-squares methods that deal with time and space variables simultaneously, we propose a deep adaptive basis Galerkin (DABG) method which employs the spectral-Galerkin method for time variable by tensor-product basis for oscillatory solutions and the deep neural network method for high-dimensional space variables. The proposed method can lead to a linear system of differential equations having unknown DNNs that can be trained via the loss function. We establish a posterior estimates of the solution error which is bounded by the minimal loss function and the term $O(N^{-m})$, where $N$ is the number of basis functions and $m$ characterizes the regularity of the equation, and show that if the true solution is a Barron-type function, the error bound converges to zero as $M=O(N^p)$ approaches to infinity where $M$ is the width of the used networks and $p$ is a positive constant. Numerical examples including high-dimensional linear parabolic and hyperbolic equations, and nonlinear Allen-Cahn equation are presented to demonstrate the performance of the proposed DABG method is better than that of existing DNNs.

We study Hibridizable Discontinuous Galerkin (HDG) discretizations for a class of non-linear interior elliptic boundary value problems posed in curved domains where both the source term and the diffusion coefficient are non-linear. We consider the cases where the non-linear diffusion coefficient depends on the solution and on the gradient of the solution. To sidestep the need for curved elements, the discrete solution is computed on a polygonal subdomain that is not assumed to interpolate the true boundary, giving rise to an unfitted computational mesh. We show that, under mild assumptions on the source term and the computational domain, the discrete systems are well posed. Furthermore, we provide a priori error estimates showing that the discrete solution will have optimal order of convergence as long as the distance between the curved boundary and the computational boundary remains of the same order of magnitude as the mesh parameter.

The paper concerns convergence and asymptotic statistics for stochastic approximation driven by Markovian noise: $$ \theta_{n+1}= \theta_n + \alpha_{n + 1} f(\theta_n, \Phi_{n+1}) \,,\quad n\ge 0, $$ in which each $\theta_n\in\Re^d$, $ \{ \Phi_n \}$ is a Markov chain on a general state space X with stationary distribution $\pi$, and $f:\Re^d\times \text{X} \to\Re^d$. In addition to standard Lipschitz bounds on $f$, and conditions on the vanishing step-size sequence $\{\alpha_n\}$, it is assumed that the associated ODE is globally asymptotically stable with stationary point denoted $\theta^*$, where $\bar f(\theta)=E[f(\theta,\Phi)]$ with $\Phi\sim\pi$. Moreover, the ODE@$\infty$ defined with respect to the vector field, $$ \bar f_\infty(\theta):= \lim_{r\to\infty} r^{-1} \bar f(r\theta) \,,\qquad \theta\in\Re^d, $$ is asymptotically stable. The main contributions are summarized as follows: (i) The sequence $\theta$ is convergent if $\Phi$ is geometrically ergodic, and subject to compatible bounds on $f$. The remaining results are established under a stronger assumption on the Markov chain: A slightly weaker version of the Donsker-Varadhan Lyapunov drift condition known as (DV3). (ii) A Lyapunov function is constructed for the joint process $\{\theta_n,\Phi_n\}$ that implies convergence of $\{ \theta_n\}$ in $L_4$. (iii) A functional CLT is established, as well as the usual one-dimensional CLT for the normalized error $z_n:= (\theta_n-\theta^*)/\sqrt{\alpha_n}$. Moment bounds combined with the CLT imply convergence of the normalized covariance, $$ \lim_{n \to \infty} E [ z_n z_n^T ] = \Sigma_\theta, $$ where $\Sigma_\theta$ is the asymptotic covariance appearing in the CLT. (iv) An example is provided where the Markov chain $\Phi$ is geometrically ergodic but it does not satisfy (DV3). While the algorithm is convergent, the second moment is unbounded.

This work studies the $K$-user broadcast channel with $\Lambda$ caches, when the association between users and caches is random, i.e., for the scenario where each user can appear within the coverage area of -- and subsequently is assisted by -- a specific cache based on a given probability distribution. Caches are subject to a cumulative memory constraint that is equal to $t$ times the size of the library. We provide a scheme that consists of three phases: the storage allocation phase, the content placement phase, and the delivery phase, and show that an optimized storage allocation across the caches together with a modified uncoded cache placement and delivery strategy alleviates the adverse effect of cache-load imbalance by significantly reducing the multiplicative performance deterioration due to randomness. In a nutshell, our work provides a scheme that manages to substantially mitigate the impact of cache-load imbalance in stochastic networks, as well as -- compared to the best-known state-of-the-art -- the well-known subpacketization bottleneck by showing its applicability in deterministic settings for which it achieves the same delivery time -- which was proven to be close to optimal for bounded values of $t$ -- with an exponential reduction in the subpacketization.

In this paper, we want to clarify the Gibbs phenomenon when continuous and discontinuous finite elements are used to approximate discontinuous or nearly discontinuous PDE solutions from the approximation point of view. For a simple discontinuous function, we explicitly compute its continuous and discontinuous piecewise constant or linear projections on discontinuity matched or non-matched meshes. For the simple discontinuity-aligned mesh case, piecewise discontinuous approximations are always good. For the general non-matched case, we explain that the piecewise discontinuous constant approximation combined with adaptive mesh refinements is a good choice to achieve accuracy without overshoots. For discontinuous piecewise linear approximations, non-trivial overshoots will be observed unless the mesh is matched with discontinuity. For continuous piecewise linear approximations, the computation is based on a "far-away assumption", and non-trivial overshoots will always be observed under regular meshes. We calculate the explicit overshoot values for several typical cases. Numerical tests are conducted for a singularly-perturbed reaction-diffusion equation and linear hyperbolic equations to verify our findings in the paper. Also, we discuss the $L^1$-minimization based methods and do not recommend such methods due to their similar behavior as $L^2$-based methods and more complicated implementations.

The recent statistical finite element method (statFEM) provides a coherent statistical framework to synthesise finite element models with observed data. Through embedding uncertainty inside of the governing equations, finite element solutions are updated to give a posterior distribution which quantifies all sources of uncertainty associated with the model. However to incorporate all sources of uncertainty, one must integrate over the uncertainty associated with the model parameters, the known forward problem of uncertainty quantification. In this paper, we make use of Langevin dynamics to solve the statFEM forward problem, studying the utility of the unadjusted Langevin algorithm (ULA), a Metropolis-free Markov chain Monte Carlo sampler, to build a sample-based characterisation of this otherwise intractable measure. Due to the structure of the statFEM problem, these methods are able to solve the forward problem without explicit full PDE solves, requiring only sparse matrix-vector products. ULA is also gradient-based, and hence provides a scalable approach up to high degrees-of-freedom. Leveraging the theory behind Langevin-based samplers, we provide theoretical guarantees on sampler performance, demonstrating convergence, for both the prior and posterior, in the Kullback-Leibler divergence, and, in Wasserstein-2, with further results on the effect of preconditioning. Numerical experiments are also provided, for both the prior and posterior, to demonstrate the efficacy of the sampler, with a Python package also included.