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

In this work, we use the integral definition of the fractional Laplace operator and study a sparse optimal control problem involving a fractional, semilinear, and elliptic partial differential equation as state equation; control constraints are also considered. We establish the existence of optimal solutions and first and second order optimality conditions. We also analyze regularity properties for optimal variables. We propose and analyze two finite element strategies of discretization: a fully discrete scheme, where the control variable is discretized with piecewise constant functions, and a semidiscrete scheme, where the control variable is not discretized. For both discretization schemes, we analyze convergence properties and a priori error bounds.

In this paper, we propose a generic approach to perform global sensitivity analysis (GSA) for compartmental models based on continuous-time Markov chains (CTMC). This approach enables a complete GSA for epidemic models, in which not only the effects of uncertain parameters such as epidemic parameters (transmission rate, mean sojourn duration in compartments) are quantified, but also those of intrinsic randomness and interactions between the two. The main step in our approach is to build a deterministic representation of the underlying continuous-time Markov chain by controlling the latent variables modeling intrinsic randomness. Then, model output can be written as a deterministic function of both uncertain parameters and controlled latent variables, so that it becomes possible to compute standard variance-based sensitivity indices, e.g. the so-called Sobol' indices. However, different simulation algorithms lead to different representations. We exhibit in this work three different representations for CTMC stochastic compartmental models and discuss the results obtained by implementing and comparing GSAs based on each of these representations on a SARS-CoV-2 epidemic model.

We present a discontinuous Galerkin method for moist atmospheric dynamics, with and without warm rain. By considering a combined density for water vapour and cloud water, we avoid the need to model and compute a source term for condensation. We recover the vapour and cloud densities by solving a pointwise non-linear problem each time step. Consequently, we enforce the requirement for the water vapour not to be supersaturated implicitly. Together with an explicit time-stepping scheme, the method is highly parallelisable and can utilise high-performance computing hardware. Furthermore, the discretisation works on structured and unstructured meshes in two and three spatial dimensions. We illustrate the performance of our approach using several test cases in two and three spatial dimensions. In the case of a smooth, exact solution, we illustrate the optimal higher-order convergence rates of the method.

We consider the estimation of generalized additive models using basis expansions coupled with Bayesian model selection. Although Bayesian model selection is an intuitively appealing tool for regression splines, its use has traditionally been limited to Gaussian additive regression because of the availability of a tractable form of the marginal model likelihood. We extend the method to encompass the exponential family of distributions using the Laplace approximation to the likelihood. Although the approach exhibits success with any Gaussian-type prior distribution, there remains a lack of consensus regarding the best prior distribution for nonparametric regression through model selection. We observe that the classical unit information prior distribution for variable selection may not be well-suited for nonparametric regression using basis expansions. Instead, our investigation reveals that mixtures of g-priors are more suitable. We consider various mixtures of g-priors to evaluate the performance in estimating generalized additive models. Furthermore, we conduct a comparative analysis of several priors for knots to identify the most practically effective strategy. Our extensive simulation studies demonstrate the superiority of model selection-based approaches over other Bayesian methods.

In this paper we prove that rectified deep neural networks do not suffer from the curse of dimensionality when approximating McKean--Vlasov SDEs in the sense that the number of parameters in the deep neural networks only grows polynomially in the space dimension $d$ of the SDE and the reciprocal of the accuracy $\epsilon$.

This research focuses on trajectory planning problems for autonomous vehicles utilizing numerical optimal control techniques. The study reformulates the constrained optimization problem into a nonlinear programming problem, incorporating explicit collision avoidance constraints. We present three novel, exact formulations to describe collision constraints. The first formulation is derived from a proposition concerning the separation of a point and a convex set. We prove the separating proposition through De Morgan's laws. Then, leveraging the hyperplane separation theorem we propose two efficient reformulations. Compared with the existing dual formulations and the first formulation, they significantly reduce the number of auxiliary variables to be optimized and inequality constraints within the nonlinear programming problem. Finally, the efficacy of the proposed formulations is demonstrated in the context of typical autonomous parking scenarios compared with state of the art. For generality, we design three initial guesses to assess the computational effort required for convergence to solutions when using the different collision formulations. The results illustrate that the scheme employing De Morgan's laws performs equally well with those utilizing dual formulations, while the other two schemes based on hyperplane separation theorem exhibit the added benefit of requiring lower computational resources.

In this study, we present a precise anisotropic interpolation error estimate for the Morley finite element method (FEM) and apply it to fourth-order elliptical equations. We did not impose a shape-regularity mesh condition for the analysis. Therefore, anisotropic meshes can be used. The main contributions of this study include providing new proof of the consistency term. This enabled us to obtain an anisotropic consistency error estimate. The core idea of the proof involves using the relationship between the Raviart--Thomas and Morley finite element spaces. Our results show optimal convergence rates and imply that the modified Morley FEM may be effective for errors.

In this article, we propose and study a stochastic preconditioned Douglas-Rachford splitting method to solve saddle-point problems which have separable dual variables. We prove the almost sure convergence of the iteration sequences in Hilbert spaces for a class of convexconcave and nonsmooth saddle-point problems. We also provide the sublinear convergence rate for the ergodic sequence with respect to the expectation of the restricted primal-dual gap functions. Numerical experiments show the high efficiency of the proposed stochastic preconditioned Douglas-Rachford splitting methods.

Deep learning methods are emerging as popular computational tools for solving forward and inverse problems in traffic flow. In this paper, we study a neural operator framework for learning solutions to nonlinear hyperbolic partial differential equations with applications in macroscopic traffic flow models. In this framework, an operator is trained to map heterogeneous and sparse traffic input data to the complete macroscopic traffic state in a supervised learning setting. We chose a physics-informed Fourier neural operator ($\pi$-FNO) as the operator, where an additional physics loss based on a discrete conservation law regularizes the problem during training to improve the shock predictions. We also propose to use training data generated from random piecewise constant input data to systematically capture the shock and rarefied solutions. From experiments using the LWR traffic flow model, we found superior accuracy in predicting the density dynamics of a ring-road network and urban signalized road. We also found that the operator can be trained using simple traffic density dynamics, e.g., consisting of $2-3$ vehicle queues and $1-2$ traffic signal cycles, and it can predict density dynamics for heterogeneous vehicle queue distributions and multiple traffic signal cycles $(\geq 2)$ with an acceptable error. The extrapolation error grew sub-linearly with input complexity for a proper choice of the model architecture and training data. Adding a physics regularizer aided in learning long-term traffic density dynamics, especially for problems with periodic boundary data.

Human-in-the-loop aims to train an accurate prediction model with minimum cost by integrating human knowledge and experience. Humans can provide training data for machine learning applications and directly accomplish some tasks that are hard for computers in the pipeline with the help of machine-based approaches. In this paper, we survey existing works on human-in-the-loop from a data perspective and classify them into three categories with a progressive relationship: (1) the work of improving model performance from data processing, (2) the work of improving model performance through interventional model training, and (3) the design of the system independent human-in-the-loop. Using the above categorization, we summarize major approaches in the field, along with their technical strengths/ weaknesses, we have simple classification and discussion in natural language processing, computer vision, and others. Besides, we provide some open challenges and opportunities. This survey intends to provide a high-level summarization for human-in-the-loop and motivates interested readers to consider approaches for designing effective human-in-the-loop solutions.