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 study algorithmic applications of a natural discretization for the hard-sphere model and the Widom-Rowlinson model in a region $\mathbb{V}\subset\mathbb{R}^d$. These models are used in statistical physics to describe mixtures of one or multiple particle types subjected to hard-core interactions. For each type, particles follow a Poisson point process with a type specific activity parameter (fugacity). The Gibbs distribution is characterized by the mixture of these point processes conditioned that no two particles are closer than a type-dependent distance threshold. A key part in better understanding the Gibbs distribution is its normalizing constant, called partition function. We give sufficient conditions that the partition function of a discrete hard-core model on a geometric graph based on a point set $X \subset \mathbb{V}$ closely approximates those of such continuous models. Previously, this was only shown for the hard-sphere model on cubic regions $\mathbb{V}=[0, \ell)^d$ when $X$ is exponential in the volume of the region $\nu(\mathbb{V})$, limiting algorithmic applications. In the same setting, our refined analysis only requires a quadratic number of points, which we argue to be tight. We use our improved discretization results to approximate the partition functions of the hard-sphere model and the Widom-Rowlinson efficiently in $\nu(\mathbb{V})$. For the hard-sphere model, we obtain the first quasi-polynomial deterministic approximation algorithm for the entire fugacity regime for which, so far, only randomized approximations are known. Furthermore, we simplify a recently introduced fully polynomial randomized approximation algorithm. Similarly, we obtain the best known deterministic and randomized approximation bounds for the Widom-Rowlinson model. Moreover, we obtain approximate sampling algorithms for the respective spin systems within the same fugacity regimes.

Singularity subtraction for linear weakly singular Fredholm integral equations of the second kind is generalized to nonlinear integral equations. Two approaches are presented: The Classical Approach discretizes the nonlinear problem, and uses some finite dimensional linearization process to solve numerically the discrete problem. Its convergence is proved under mild hypotheses on the nonlinearity and the quadrature rule of the singularity subtraction scheme. The New Approach is based on linearization of the problem in its infinite dimensional setting, and discretization of the sequence of linear problems by singularity subtraction. It is more efficient than the former, as two numerical experiments confirm.

In distributional reinforcement learning not only expected returns but the complete return distributions of a policy are taken into account. The return distribution for a fixed policy is given as the solution of an associated distributional Bellman equation. In this note we consider general distributional Bellman equations and study existence and uniqueness of their solutions as well as tail properties of return distributions. We give necessary and sufficient conditions for existence and uniqueness of return distributions and identify cases of regular variation. We link distributional Bellman equations to multivariate affine distributional equations. We show that any solution of a distributional Bellman equation can be obtained as the vector of marginal laws of a solution to a multivariate affine distributional equation. This makes the general theory of such equations applicable to the distributional reinforcement learning setting.

A key advantage of isogeometric discretizations is their accurate and well-behaved eigenfrequencies and eigenmodes. For degree two and higher, however, optical branches of spurious outlier frequencies and modes may appear due to boundaries or reduced continuity at patch interfaces. In this paper, we introduce a variational approach based on perturbed eigenvalue analysis that eliminates outlier frequencies without negatively affecting the accuracy in the remainder of the spectrum and modes. We then propose a pragmatic iterative procedure that estimates the perturbation parameters in such a way that the outlier frequencies are effectively reduced. We demonstrate that our approach allows for a much larger critical time-step size in explicit dynamics calculations. In addition, we show that the critical time-step size obtained with the proposed approach does not depend on the polynomial degree of spline basis functions.

Consider a set $P$ of $n$ points in $\mathbb{R}^d$. In the discrete median line segment problem, the objective is to find a line segment bounded by a pair of points in $P$ such that the sum of the Euclidean distances from $P$ to the line segment is minimized. In the continuous median line segment problem, a real number $\ell>0$ is given, and the goal is to locate a line segment of length $\ell$ in $\mathbb{R}^d$ such that the sum of the Euclidean distances between $P$ and the line segment is minimized. We show how to compute $(1+\epsilon\Delta)$- and $(1+\epsilon)$-approximations to a discrete median line segment in time $O(n\epsilon^{-2d}\log n)$ and $O(n^2\epsilon^{-d})$, respectively, where $\Delta$ is the spread of line segments spanned by pairs of points. While developing our algorithms, by using the principle of pair decomposition, we derive new data structures that allow us to quickly approximate the sum of the distances from a set of points to a given line segment or point. To our knowledge, our utilization of pair decompositions for solving minsum facility location problems is the first of its kind; it is versatile and easily implementable. We prove that it is impossible to construct a continuous median line segment for $n\geq3$ non-collinear points in the plane by using only ruler and compass. In view of this, we present an $O(n^d\epsilon^{-d})$-time algorithm for approximating a continuous median line segment in $\mathbb{R}^d$ within a factor of $1+\epsilon$. The algorithm is based upon generalizing the point-segment pair decomposition from the discrete to the continuous domain. Last but not least, we give an $(1+\epsilon)$-approximation algorithm, whose time complexity is sub-quadratic in $n$, for solving the constrained median line segment problem in $\mathbb{R}^2$ where an endpoint or the slope of the median line segment is given at input.

This paper proposes a regularization of the Monge-Amp\`ere equation in planar convex domains through uniformly elliptic Hamilton-Jacobi-Bellman equations. The regularized problem possesses a unique strong solution $u_\varepsilon$ and is accessible to the discretization with finite elements. This work establishes locally uniform convergence of $u_\varepsilon$ to the convex Alexandrov solution $u$ to the Monge-Amp\`ere equation as the regularization parameter $\varepsilon$ approaches $0$. A mixed finite element method for the approximation of $u_\varepsilon$ is proposed, and the regularized finite element scheme is shown to be locally uniformly convergent. Numerical experiments provide empirical evidence for the efficient approximation of singular solutions $u$.

A singularly perturbed parabolic problem of convection-diffusion type with a discontinuous initial condition is examined. An analytic function is identified which matches the discontinuity in the initial condition and also satisfies the homogenous parabolic differential equation associated with the problem. The difference between this analytical function and the solution of the parabolic problem is approximated numerically, using an upwind finite difference operator combined with an appropriate layer-adapted mesh. The numerical method is shown to be parameter-uniform. Numerical results are presented to illustrate the theoretical error bounds established in the paper.

In this paper, we propose GT-GDA, a distributed optimization method to solve saddle point problems of the form: $\min_{\mathbf{x}} \max_{\mathbf{y}} \{F(\mathbf{x},\mathbf{y}) :=G(\mathbf{x}) + \langle \mathbf{y}, \overline{P} \mathbf{x} \rangle - H(\mathbf{y})\}$, where the functions $G(\cdot)$, $H(\cdot)$, and the the coupling matrix $\overline{P}$ are distributed over a strongly connected network of nodes. GT-GDA is a first-order method that uses gradient tracking to eliminate the dissimilarity caused by heterogeneous data distribution among the nodes. In the most general form, GT-GDA includes a consensus over the local coupling matrices to achieve the optimal (unique) saddle point, however, at the expense of increased communication. To avoid this, we propose a more efficient variant GT-GDA-Lite that does not incur the additional communication and analyze its convergence in various scenarios. We show that GT-GDA converges linearly to the unique saddle point solution when $G(\cdot)$ is smooth and convex, $H(\cdot)$ is smooth and strongly convex, and the global coupling matrix $\overline{P}$ has full column rank. We further characterize the regime under which GT-GDA exhibits a network topology-independent convergence behavior. We next show the linear convergence of GT-GDA to an error around the unique saddle point, which goes to zero when the coupling cost ${\langle \mathbf y, \overline{P} \mathbf x \rangle}$ is common to all nodes, or when $G(\cdot)$ and $H(\cdot)$ are quadratic. Numerical experiments illustrate the convergence properties and importance of GT-GDA and GT-GDA-Lite for several applications.

Physical systems are usually modeled by differential equations, but solving these differential equations analytically is often intractable. Instead, the differential equations can be solved numerically by discretization in a finite computational domain. The discretized equation is reduced to a large linear system, whose solution is typically found using an iterative solver. We start with an initial guess, x_0, and iterate the algorithm to obtain a sequence of solution vectors, x_m. The iterative algorithm is said to converge to solution $x$ if and only if x_m converges to $x$. Accuracy of the numerical solutions is important, especially in the design of safety critical systems such as airplanes, cars, or nuclear power plants. It is therefore important to formally guarantee that the iterative solvers converge to the "true" solution of the original differential equation. In this paper, we first formalize the necessary and sufficient conditions for iterative convergence in the Coq proof assistant. We then extend this result to two classical iterative methods: Gauss-Seidel iteration and Jacobi iteration. We formalize conditions for the convergence of the Gauss--Seidel classical iterative method, based on positive definiteness of the iterative matrix. We then formally state conditions for convergence of Jacobi iteration and instantiate it with an example to demonstrate convergence of iterative solutions to the direct solution of the linear system. We leverage recent developments of the Coq linear algebra and mathcomp library for our formalization.

In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.