Homotopy continuation is an efficient tool for solving polynomial systems. Its efficiency relies on utilizing adaptive stepsize and adaptive precision path tracking, and endgames. In this article, we apply homotopy continuation to solve steady state problems of hyperbolic conservation laws. A third-order accurate finite difference weighted essentially non-oscillatory (WENO) scheme with Lax-Friedrichs flux splitting is utilized to derive the difference equation. This new approach is free of the CFL condition constraint. Extensive numerical examples in both scalar and system test problems in one and two dimensions demonstrate the efficiency and robustness of the new method.

In this paper we derive new fractional error bounds for polynomial systems with exponents explicitly determined by the dimension of the underlying space and the number/degree of the involved polynomials. Our major result extends the existing error bounds from the system involving only a single polynomial to a general polynomial system and do not require any regularity assumptions. In this way we resolve, in particular, some open questions posed in the literature. The developed techniques are largely based on variational analysis and generalized differentiation, which allow us to establish, e.g., a nonsmooth extension of the seminal Łojasiewicz’s gradient inequality to maxima of polynomials with explicitly determined exponents. Our major applications concern quantitative Hölderian stability of solution maps for parameterized polynomial optimization problems and nonlinear complementarity systems with polynomial data as well as high-order semismooth properties of the eigenvalues of symmetric tensors.

With the rapid development of internet economy, transparent logistics is stepping into a prosperity period with massive transportation data generated and collected every day. In this paper, we focus on the segmentation of GPS trajectory data generated in logistics transportation to analyze the vehicle behaviors and extract business affair information according to the vehicle behavior characteristics, which is challenging due to the complexity of trajectory data and unavailability of road information. We extract the stopping points from the trajectory data sequence based on the duration of nonmovement, and construct business time window and electronic fence by analyzing the driving habits of vehicles. Furthermore, we propose a probabilistic logic based data segmentation method (PLDSM) which not only helps finding all the business points but also assists in inferring the business affair categories. An efficient

In this paper, an analysis of the superconvergence property of the semidiscrete discontinuous Galerkin (DG) method applied to one-dimensional time-dependent nonlinear scalar conservation laws is carried out. We prove that the error between the DG solution and a particular projection of the exact solution achieves $\left(k + \frac32\right)$-th order superconvergence when upwind fluxes are used. The results hold true for arbitrary nonuniform regular meshes and for piecewise polynomials of degree $k$ ($k \ge 1$), under the condition that $|f'(u)|$ possesses a uniform positive lower bound. Numerical experiments are provided to show that the superconvergence property actually holds true for nonlinear conservation laws with general flux functions, indicating that the restriction on $f(u)$ is artificial.

In this paper, we give an integral approximation for the elliptic operators with anisotropic coecients on smooth manifold. Using the integral approximation, the elliptic equation is tranformed to an integral equation. The integral approximation preserves the symmetry and coercivity of the original elliptic operator. Based on these good properties, we prove the convergence between the solutions of the integral equation and the original elliptic equation.