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

The recently proposed Uniswap v3 replaces the fungible liquidity provider token (LP token) into non-fungible ones, making the design for liquidity mining more difficult. In this paper, we propose a flexible liquidity mining scheme that realizes the overall liquidity distribution through the fine control of local rewards. From the liquidity provider's point of view, the liquidity provision strategy forms a multiplayer zero-sum game. We analyze the Nash Equilibrium and the corresponding strategy, approximately, deploying the liquidity proportional to the reward distribution, in some special cases and use it to guide the general situations. Based on the strategic response above, such a scheme allows the mining rewards provider to optimize the distribution of liquidity for the purpose such as low slippage and price stabilization.

In this paper, we propose and analyze a temporally second-order accurate, fully discrete finite element method for the magnetohydrodynamic (MHD) equations. A modified Crank--Nicolson method is used to discretize the model and appropriate semi-implicit treatments are applied to the fluid convection term and two coupling terms. These semi-implicit approximations result in a linear system with variable coefficients for which the unique solvability can be proved theoretically. In addition, we use a decoupling projection method of the Van Kan type \cite{vankan1986} in the Stokes solver, which computes the intermediate velocity field based on the gradient of the pressure from the previous time level, and enforces the incompressibility constraint via the Helmholtz decomposition of the intermediate velocity field. The energy stability of the scheme is theoretically proved, in which the decoupled Stokes solver needs to be analyzed in details. Optimal-order convergence of $\mathcal{O} (\tau^2+h^{r+1})$ in the discrete $L^\infty(0,T;L^2)$ norm is proved for the proposed decoupled projection finite element scheme, where $\tau$ and $h$ are the time stepsize and spatial mesh size, respectively, and $r$ is the degree of the finite elements. Existing error estimates of second-order projection methods of the Van Kan type \cite{vankan1986} were only established in the discrete $L^2(0,T;L^2)$ norm for the Navier--Stokes equations. Numerical examples are provided to illustrate the theoretical results.

Stabilized explicit methods are particularly efficient for large systems of stiff stochastic differential equations (SDEs) due to their extended stability domain. However, they loose their efficiency when a severe stiffness is induced by very few "fast" degrees of freedom, as the stiff and nonstiff terms are evaluated concurrently. Therefore, inspired by [A. Abdulle, M. J. Grote, and G. Rosilho de Souza, Preprint (2020), arXiv:2006.00744] we introduce a stochastic modified equation whose stiffness depends solely on the "slow" terms. By integrating this modified equation with a stabilized explicit scheme we devise a multirate method which overcomes the bottleneck caused by a few severely stiff terms and recovers the efficiency of stabilized schemes for large systems of nonlinear SDEs. The scheme is not based on any scale separation assumption of the SDE and therefore it is employable for problems stemming from the spatial discretization of stochastic parabolic partial differential equations on locally refined grids. The multirate scheme has strong order 1/2, weak order 1 and its stability is proved on a model problem. Numerical experiments confirm the efficiency and accuracy of the scheme.

We consider the problem of approximating a function in general nonlinear subsets of $L^2$ when only a weighted Monte Carlo estimate of the $L^2$-norm can be computed. Of particular interest in this setting is the concept of sample complexity, the number of samples that are necessary to recover the best approximation. Bounds for this quantity have been derived in a previous work and depend primarily on the model class and are not influenced positively by the regularity of the sought function. This result however is only a worst-case bound and is not able to explain the remarkable performance of iterative hard thresholding algorithms that is observed in practice. We reexamine the results of the previous paper and derive a new bound that is able to utilize the regularity of the sought function. A critical analysis of our results allows us to derive a sample efficient algorithm for the model set of low-rank tensors. The viability of this algorithm is demonstrated by recovering quantities of interest for a classical high-dimensional random partial differential equation.

By improving the trace finite element method, we developed another higher-order trace finite element method by integrating on the surface with exact geometry description. This method restricts the finite element space on the volume mesh to the surface accurately, and approximates Laplace-Beltrami operator on the surface by calculating the high-order numerical integration on the exact surface directly. We employ this method to calculate the Laplace-Beltrami equation and the Laplace-Beltrami eigenvalue problem. Numerical error analysis shows that this method has an optimal convergence order in both problems. Numerical experiments verify the correctness of the theoretical analysis. The algorithm is more accurate and easier to implement than the existing high-order trace finite element method.

We introduce a simple, rigorous, and unified framework for solving nonlinear partial differential equations (PDEs), and for solving inverse problems (IPs) involving the identification of parameters in PDEs, using the framework of Gaussian processes. The proposed approach: (1) provides a natural generalization of collocation kernel methods to nonlinear PDEs and IPs; (2) has guaranteed convergence for a very general class of PDEs, and comes equipped with a path to compute error bounds for specific PDE approximations; (3) inherits the state-of-the-art computational complexity of linear solvers for dense kernel matrices. The main idea of our method is to approximate the solution of a given PDE as the maximum a posteriori (MAP) estimator of a Gaussian process conditioned on solving the PDE at a finite number of collocation points. Although this optimization problem is infinite-dimensional, it can be reduced to a finite-dimensional one by introducing additional variables corresponding to the values of the derivatives of the solution at collocation points; this generalizes the representer theorem arising in Gaussian process regression. The reduced optimization problem has the form of a quadratic objective function subject to nonlinear constraints; it is solved with a variant of the Gauss--Newton method. The resulting algorithm (a) can be interpreted as solving successive linearizations of the nonlinear PDE, and (b) in practice is found to converge in a small number of iterations (2 to 10), for a wide range of PDEs. Most traditional approaches to IPs interleave parameter updates with numerical solution of the PDE; our algorithm solves for both parameter and PDE solution simultaneously. Experiments on nonlinear elliptic PDEs, Burgers' equation, a regularized Eikonal equation, and an IP for permeability identification in Darcy flow illustrate the efficacy and scope of our framework.

Plasmon-induced transparency (PIT) displays complex nonlinear dynamics that find critical phenomena in areas such as nonlinear waves. However, such a nonlinear solution depends sensitively on the selection of parameters and different potentials in the Schr\"odinger equation. Despite this complexity, the machine learning community has developed remarkable efficiencies in predicting complicated datasets by regression. Here, we consider a recurrent neural network (RNN) approach to predict the complex propagation of nonlinear solitons in plasmon-induced transparency metamaterial systems with applied potentials bypassing the need for analytical and numerical approaches of a guiding model. We demonstrate the success of this scheme on the prediction of the propagation of the nonlinear solitons solely from a given initial condition and potential. We prove the prominent agreement of results in simulation and prediction by long short-term memory (LSTM) artificial neural networks. The framework presented in this work opens up a new perspective for the application of RNN in quantum systems and nonlinear waves using Schr\"odinger-type equations, for example, the nonlinear dynamics in cold-atom systems and nonlinear fiber optics.

Quadrotors are among the most agile flying robots. However, planning time-optimal trajectories at the actuation limit through multiple waypoints remains an open problem. This is crucial for applications such as inspection, delivery, search and rescue, and drone racing. Early works used polynomial trajectory formulations, which do not exploit the full actuator potential because of their inherent smoothness. Recent works resorted to numerical optimization but require waypoints to be allocated as costs or constraints at specific discrete times. However, this time allocation is a priori unknown and renders previous works incapable of producing truly time-optimal trajectories. To generate truly time-optimal trajectories, we propose a solution to the time allocation problem while exploiting the full quadrotor's actuator potential. We achieve this by introducing a formulation of progress along the trajectory, which enables the simultaneous optimization of the time allocation and the trajectory itself. We compare our method against related approaches and validate it in real-world flights in one of the world's largest motion-capture systems, where we outperform human expert drone pilots in a drone-racing task.

We revisit the minimum dominating set problem on graphs with arboricity bounded by $\alpha$. Bansal and Umboh [BU17] gave an $O(\alpha)$-approximation LP rounding algorithm, which also translates into a near-linear time algorithm using general-purpose approximation results for explicit mixed packing and covering or pure covering LPs [KY14, You14, AZO19, Qua10]. Moreover, [BU17] showed that it is NP-hard to achieve an asymptotic improvement for the approximation factor. On the other hand, the previous two non-LP-based algorithms, by Lenzen and Wattenhofer [LW10], and Jones et al. [JLR+13], achieve an approximation factor of $O(\alpha^2)$ in linear time. There is a similar situation in the distributed setting: While there is an $O(\log^2 n)$-round LP-based $O(\alpha)$-approximation algorithm implied in [KMW06], the best non-LP-based algorithm by Lenzen and Wattenhofer [LW10] is an implementation of their centralized algorithm, providing an $O(\alpha^2)$-approximation within $O(\log n)$ rounds. We address the questions of whether one can achieve an $O(\alpha)$-approximation algorithm that is not LP-based, either in the centralized setting or in the distributed setting. We resolve both questions in the affirmative, and en route achieve algorithms that are faster than the state-of-the-art LP-based algorithms. More specifically, our contribution is two-fold: 1. In the centralized setting, we provide a surprisingly simple combinatorial algorithm that is asymptotically optimal in terms of both approximation factor and running time: an $O(\alpha)$-approximation in linear time. 2. Based on our centralized algorithm, we design a distributed combinatorial $O(\alpha)$-approximation algorithm in the CONGEST model that runs in $O(\alpha\log n )$ rounds with high probability.

Interactive recommendation that models the explicit interactions between users and the recommender system has attracted a lot of research attentions in recent years. Most previous interactive recommendation systems only focus on optimizing recommendation accuracy while overlooking other important aspects of recommendation quality, such as the diversity of recommendation results. In this paper, we propose a novel recommendation model, named \underline{D}iversity-promoting \underline{D}eep \underline{R}einforcement \underline{L}earning (D$^2$RL), which encourages the diversity of recommendation results in interaction recommendations. More specifically, we adopt a Determinantal Point Process (DPP) model to generate diverse, while relevant item recommendations. A personalized DPP kernel matrix is maintained for each user, which is constructed from two parts: a fixed similarity matrix capturing item-item similarity, and the relevance of items dynamically learnt through an actor-critic reinforcement learning framework. We performed extensive offline experiments as well as simulated online experiments with real world datasets to demonstrate the effectiveness of the proposed model.