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An efficient approximate version of implicit Taylor methods for initial-value problems of systems of ordinary differential equations (ODEs) is introduced. The approach, based on an approximate formulation of Taylor methods, produces a method that requires less evaluations of the function that defines the ODE and its derivatives than the usual version. On the other hand, an efficient numerical solution of the equation that arises from the discretization by means of Newton's method is introduced for an implicit scheme of any order. Numerical experiments illustrate that the resulting algorithm is simpler to implement and has better performance than its exact counterpart.

We develop a sparse spectral method for a class of fractional differential equations, posed on $\mathbb{R}$, in one dimension. These equations can include sqrt-Laplacian, Hilbert, derivative and identity terms. The numerical method utilizes a basis consisting of weighted Chebyshev polynomials of the second kind in conjunction with their Hilbert transforms. The former functions are supported on $[-1,1]$ whereas the latter have global support. The global approximation space can contain different affine transformations of the basis, mapping $[-1,1]$ to other intervals. Remarkably, not only are the induced linear systems sparse, but the operator decouples across the different affine transformations. Hence, the solve reduces to solving $K$ independent sparse linear systems of size $\mathcal{O}(n)\times \mathcal{O}(n)$, with $\mathcal{O}(n)$ nonzero entries, where $K$ is the number of different intervals and $n$ is the highest polynomial degree contained in the sum space. This results in an $\mathcal{O}(n)$ complexity solve. Applications to fractional heat and wave equations are considered.

The stochastic heat equation on the sphere driven by additive isotropic Wiener noise is approximated by a spectral method in space and forward and backward Euler-Maruyama schemes in time. The spectral approximation is based on a truncation of the series expansion with respect to the spherical harmonic functions. Optimal strong convergence rates for a given regularity of the initial condition and driving noise are derived for the Euler-Maruyama methods. Besides strong convergence, convergence of the expectation and second moment is shown, where the approximation of the second moment converges with twice the strong rate. Numerical simulations confirm the theoretical results.

For stochastic wave equation, when the dissipative damping is a non-globally Lipschitz function of the velocity, there are few results on the long-time dynamics, in particular, the exponential ergodicity and strong law of large numbers, for the equation and its numerical discretization to our knowledge. Focus on this issue, the main contributions of this paper are as follows. First, based on constructing novel Lyapunov functionals, we show the unique invariant measure and exponential ergodicity of the underlying equation and its full discretization. Second, the error estimates of invariant measures both in Wasserstein distance and in the weak sense are obtained. Third, the strong laws of large numbers of the equation and the full discretization are obtained, which states that the time averages of the exact and numerical solutions are shown to converge to the ergodic limit almost surely.

We provide a Lyapunov convergence analysis for time-inhomogeneous variable coefficient stochastic differential equations (SDEs). Three typical examples include overdamped, irreversible drift, and underdamped Langevin dynamics. We first formula the probability transition equation of Langevin dynamics as a modified gradient flow of the Kullback-Leibler divergence in the probability space with respect to time-dependent optimal transport metrics. This formulation contains both gradient and non-gradient directions depending on a class of time-dependent target distribution. We then select a time-dependent relative Fisher information functional as a Lyapunov functional. We develop a time-dependent Hessian matrix condition, which guarantees the convergence of the probability density function of the SDE. We verify the proposed conditions for several time-inhomogeneous Langevin dynamics. For the overdamped Langevin dynamics, we prove the $O(t^{-1/2})$ convergence in $L^1$ distance for the simulated annealing dynamics with a strongly convex potential function. For the irreversible drift Langevin dynamics, we prove an improved convergence towards the target distribution in an asymptotic regime. We also verify the convergence condition for the underdamped Langevin dynamics. Numerical examples demonstrate the convergence results for the time-dependent Langevin dynamics.

We revisit the question of whether the strong law of large numbers (SLLN) holds uniformly in a rich family of distributions, culminating in a distribution-uniform generalization of the Marcinkiewicz-Zygmund SLLN. These results can be viewed as extensions of Chung's distribution-uniform SLLN to random variables with uniformly integrable $q^\text{th}$ absolute central moments for $0 < q < 2;\ q \neq 1$. Furthermore, we show that uniform integrability of the $q^\text{th}$ moment is both sufficient and necessary for the SLLN to hold uniformly at the Marcinkiewicz-Zygmund rate of $n^{1/q - 1}$. These proofs centrally rely on distribution-uniform analogues of some familiar almost sure convergence results including the Khintchine-Kolmogorov convergence theorem, Kolmogorov's three-series theorem, a stochastic generalization of Kronecker's lemma, and the Borel-Cantelli lemmas. The non-identically distributed case is also considered.

An adaptive method for parabolic partial differential equations that combines sparse wavelet expansions in time with adaptive low-rank approximations in the spatial variables is constructed and analyzed. The method is shown to converge and satisfy similar complexity bounds as existing adaptive low-rank methods for elliptic problems, establishing its suitability for parabolic problems on high-dimensional spatial domains. The construction also yields computable rigorous a posteriori error bounds for such problems. The results are illustrated by numerical experiments.

We propose a new numerical domain decomposition method for solving elliptic equations on compact Riemannian manifolds. One advantage of this method is its ability to bypass the need for global triangulations or grids on the manifolds. Additionally, it features a highly parallel iterative scheme. To verify its efficacy, we conduct numerical experiments on some $4$-dimensional manifolds without and with boundary.

Partial differential equations (PDEs) have become an essential tool for modeling complex physical systems. Such equations are typically solved numerically via mesh-based methods, such as finite element methods, with solutions over the spatial domain. However, obtaining these solutions are often prohibitively costly, limiting the feasibility of exploring parameters in PDEs. In this paper, we propose an efficient emulator that simultaneously predicts the solutions over the spatial domain, with theoretical justification of its uncertainty quantification. The novelty of the proposed method lies in the incorporation of the mesh node coordinates into the statistical model. In particular, the proposed method segments the mesh nodes into multiple clusters via a Dirichlet process prior and fits Gaussian process models with the same hyperparameters in each of them. Most importantly, by revealing the underlying clustering structures, the proposed method can provide valuable insights into qualitative features of the resulting dynamics that can be used to guide further investigations. Real examples are demonstrated to show that our proposed method has smaller prediction errors than its main competitors, with competitive computation time, and identifies interesting clusters of mesh nodes that possess physical significance, such as satisfying boundary conditions. An R package for the proposed methodology is provided in an open repository.

We introduce a predictor-corrector discretisation scheme for the numerical integration of a class of stochastic differential equations and prove that it converges with weak order 1.0. The key feature of the new scheme is that it builds up sequentially (and recursively) in the dimension of the state space of the solution, hence making it suitable for approximations of high-dimensional state space models. We show, using the stochastic Lorenz 96 system as a test model, that the proposed method can operate with larger time steps than the standard Euler-Maruyama scheme and, therefore, generate valid approximations with a smaller computational cost. We also introduce the theoretical analysis of the error incurred by the new predictor-corrector scheme when used as a building block for discrete-time Bayesian filters for continuous-time systems. Finally, we assess the performance of several ensemble Kalman filters that incorporate the proposed sequential predictor-corrector Euler scheme and the standard Euler-Maruyama method. The numerical experiments show that the filters employing the new sequential scheme can operate with larger time steps, smaller Monte Carlo ensembles and noisier systems.