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

This paper introduces a differential dynamic programming (DDP) based framework for polynomial trajectory generation for differentially flat systems. In particular, instead of using a linear equation with increasing size to represent multiple polynomial segments as in literature, we take a new perspective from state-space representation such that the linear equation reduces to a finite horizon control system with a fixed state dimension and the required continuity conditions for consecutive polynomials are automatically satisfied. Consequently, the constrained trajectory generation problem (both with and without time optimization) can be converted to a discrete-time finite-horizon optimal control problem with inequality constraints, which can be approached by a recently developed interior-point DDP (IPDDP) algorithm. Furthermore, for unconstrained trajectory generation with preallocated time, we show that this problem is indeed a linear-quadratic tracking (LQT) problem (DDP algorithm with exact one iteration). All these algorithms enjoy linear complexity with respect to the number of segments. Both numerical comparisons with state-of-the-art methods and physical experiments are presented to verify and validate the effectiveness of our theoretical findings. The implementation code will be open-sourced,

We develop an efficient, non-intrusive, adaptive algorithm for the solution of elliptic partial differential equations with random coefficients. The sparse Fast Fourier Transform detects the most important frequencies in a given search domain and therefore adaptively generates a suitable Fourier basis corresponding to the approximately largest Fourier coefficients of the function. Our uniform sFFT does this w.r.t. the stochastic domain simultaneously for every node of a finite element mesh in the spatial domain and creates a suitable approximation space for all spatial nodes by joining the detected frequency sets. This strategy allows for a faster and more efficient computation, than just using the full sFFT algorithm for each node separately. We then test the usFFT for different examples using periodic, affine and lognormal random coefficients. The results are significantly better than when using given standard frequency sets and the algorithm does not require any a priori information about the solution.

The tube method or the volume-of-tube method approximates the tail probability of the maximum of a smooth Gaussian random field with zero mean and unit variance. This method evaluates the volume of a spherical tube about the index set, and then transforms it to the tail probability. In this study, we generalize the tube method to a case in which the variance is not constant. We provide the volume formula for a spherical tube with a non-constant radius in terms of curvature tensors, and the tail probability formula of the maximum of a Gaussian random field with inhomogeneous variance, as well as its Laplace approximation. In particular, the critical radius of the tube is generalized for evaluation of the asymptotic approximation error. As an example, we discuss the approximation of the largest eigenvalue distribution of the Wishart matrix with a non-identity matrix parameter. The Bonferroni method is the tube method when the index set is a finite set. We provide the formula for the asymptotic approximation error for the Bonferroni method when the variance is not constant.

In this paper, we consider the density estimation problem associated with the stationary measure of ergodic It\^o diffusions from a discrete-time series that approximate the solutions of the stochastic differential equations. To take an advantage of the characterization of density function through the stationary solution of a parabolic-type Fokker-Planck PDE, we proceed as follows. First, we employ deep neural networks to approximate the drift and diffusion terms of the SDE by solving appropriate supervised learning tasks. Subsequently, we solve a steady-state Fokker-Plank equation associated with the estimated drift and diffusion coefficients with a neural-network-based least-squares method. We establish the convergence of the proposed scheme under appropriate mathematical assumptions, accounting for the generalization errors induced by regressing the drift and diffusion coefficients, and the PDE solvers. This theoretical study relies on a recent perturbation theory of Markov chain result that shows a linear dependence of the density estimation to the error in estimating the drift term, and generalization error results of nonparametric regression and of PDE regression solution obtained with neural-network models. The effectiveness of this method is reflected by numerical simulations of a two-dimensional Student's t distribution and a 20-dimensional Langevin dynamics.

Insights into complex, high-dimensional data can be obtained by discovering features of the data that match or do not match a model of interest. To formalize this task, we introduce the "data selection" problem: finding a lower-dimensional statistic - such as a subset of variables - that is well fit by a given parametric model of interest. A fully Bayesian approach to data selection would be to parametrically model the value of the statistic, nonparametrically model the remaining "background" components of the data, and perform standard Bayesian model selection for the choice of statistic. However, fitting a nonparametric model to high-dimensional data tends to be highly inefficient, statistically and computationally. We propose a novel score for performing both data selection and model selection, the "Stein volume criterion", that takes the form of a generalized marginal likelihood with a kernelized Stein discrepancy in place of the Kullback-Leibler divergence. The Stein volume criterion does not require one to fit or even specify a nonparametric background model, making it straightforward to compute - in many cases it is as simple as fitting the parametric model of interest with an alternative objective function. We prove that the Stein volume criterion is consistent for both data selection and model selection, and we establish consistency and asymptotic normality (Bernstein-von Mises) of the corresponding generalized posterior on parameters. We validate our method in simulation and apply it to the analysis of single-cell RNA sequencing datasets using probabilistic principal components analysis and a spin glass model of gene regulation.

For a general third-order tensor $\mathcal{A}\in\mathbb{R}^{n\times n\times n}$ the paper studies two closely related problems, the SVD-like tensor decomposition and the (approximate) tensor diagonalization. We develop the alternating least squares Jacobi-type algorithm that maximizes the squares of the diagonal entries of $\mathcal{A}$. The algorithm works on $2\times2\times2$ subtensors such that in each iteration the sum of the squares of two diagonal entries is maximized. We show how the rotation angles are calculated and prove the convergence of the algorithm. Different initializations of the algorithm are discussed, as well as the special cases of symmetric and antisymmetric tensors. The algorithm can be generalized to work on the higher-order tensors.

In this note we prove that the version of Newton algorithm with line search we used in [2] converges quadratically.

There are plenty of applications and analysis for time-independent elliptic partial differential equations in the literature hinting at the benefits of overtesting by using more collocation conditions than the number of basis functions. Overtesting not only reduces the problem size, but is also known to be necessary for stability and convergence of widely used unsymmetric Kansa-type strong-form collocation methods. We consider kernel-based meshfree methods, which is a method of lines with collocation and overtesting spatially, for solving parabolic partial differential equations on surfaces without parametrization. In this paper, we extend the time-independent convergence theories for overtesting techniques to the parabolic equations on smooth and closed surfaces.

In this paper we generalize the polynomial time integration framework to additively partitioned initial value problems. The framework we present is general and enables the construction of many new families of additive integrators with arbitrary order-of-accuracy and varying degree of implicitness. In this first work, we focus on a new class of implicit-explicit polynomial block methods that are based on fully-implicit Runge-Kutta methods with Radau nodes. We show that the new integrators have improved stability compared to existing IMEX Runge-Kutta methods, while also being more computationally efficient due to recent developments in preconditioning techniques for solving the associated systems of nonlinear equations. For PDEs on periodic domains where the implicit component is trivial to invert, we will show how parallelization of the right-hand-side evaluations can be exploited to obtain significant speedup compared to existing serial IMEX Runge-Kutta methods. For parallel (in space) finite-element discretizations, the new methods obtain accuracy several orders of magnitude lower than existing IMEX Runge-Kutta methods, and/or obtain a given accuracy as much as 16 times faster in terms of computational runtime.

Interpretation of Deep Neural Networks (DNNs) training as an optimal control problem with nonlinear dynamical systems has received considerable attention recently, yet the algorithmic development remains relatively limited. In this work, we make an attempt along this line by reformulating the training procedure from the trajectory optimization perspective. We first show that most widely-used algorithms for training DNNs can be linked to the Differential Dynamic Programming (DDP), a celebrated second-order trajectory optimization algorithm rooted in the Approximate Dynamic Programming. In this vein, we propose a new variant of DDP that can accept batch optimization for training feedforward networks, while integrating naturally with the recent progress in curvature approximation. The resulting algorithm features layer-wise feedback policies which improve convergence rate and reduce sensitivity to hyper-parameter over existing methods. We show that the algorithm is competitive against state-ofthe-art first and second order methods. Our work opens up new avenues for principled algorithmic design built upon the optimal control theory.