We consider the Schrodinger equation with a logarithmic nonlinearity and a repulsive harmonic potential. Depending on the parameters of the equation, the solution may or may not be dispersive. When dispersion occurs, it does with an exponential rate in time. To control this, we change the unknown function through a generalized lens transform. This approach neutralizes the possible boundary effects, and could be used in the case of the Schrodinger equation without potential. We then employ standard splitting methods on the new equation via a nonuniform grid, after the logarithmic nonlinearity has been regularized. We also discuss the case of a power nonlinearity and give some results concerning the error estimates of the first-order Lie-Trotter splitting method for both cases of nonlinearities. Finally extensive numerical experiments are reported to investigate the dynamics of the equations.

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

Consider the image restoration from incomplete noisy frequency data with total variation and sparsity regularizing penalty terms. Firstly, we establish an unconstrained optimization model with dierent smooth approximations on the regularizing terms. Then, to weaken the amount of computations for cost functional with total variation term, the alternating iterative scheme is developed to obtain the exact solution through shrinkage thresholding in inner loop, while the nonlinear Euler equation is appropriately linearized at each iteration in exterior loop, yielding a linear system with diagonal coefficient matrix in frequency domain. Finally the linearized iteration is proven to be convergent in generalized sense for suitable regularizing parameters, and the error between the linearized iterative solution and the one gotten from the exact nonlinear Euler equation is rigorously estimated, revealing the essence of the proposed alternative iteration scheme. Numerical tests for different configurations show the validity of the proposed scheme, compared with some existing algorithms.

We consider polynomial approximation on the unit sphere $\mathbb{S}^2=\{(x,y,z)\in \mathbb{R}^3:x^2+y^2+z^2=1\}$ by a class of regularized discrete least squares methods with novel choices for the regularization operator and the point sets of the discretization. We allow different kinds of rotationally invariant regularization operators, including the zero operator (in which case the approximation includes interpolation, quasi-interpolation, and hyperinterpolation); powers of the negative Laplace--Beltrami operator (which can be suitable when there are data errors); and regularization operators that yield filtered polynomial approximations. As node sets we use spherical $t$-designs, which are point sets on the sphere which when used as equal-weight quadrature rules integrate all spherical polynomials up to degree $t$ exactly. More precisely, we use well conditioned spherical $t$-designs obtained in a previous paper by maximizing the determinants of the Gram matrices subject to the spherical design constraint. For $t\ge 2L$ and an approximating polynomial of degree $L$ it turns out that there is no linear algebra problem to be solved and the approximation in some cases recovers known polynomial approximation schemes, including interpolation, hyperinterpolation, and filtered hyperinterpolation. For $t \in [L, 2L)$ the linear system needs to be solved numerically. Finally, we give numerical examples to illustrate the theoretical results and show that well chosen regularization operator and well conditioned spherical $t$-designs can provide good polynomial approximation on the sphere, with or without the presence of data errors.

In this paper, a novel penalty method based on the immersed boundary formulation is proposed for simulating the transient Stokes flow with an inextensible interface enclosing a suspended solid particle. The main idea of this approach relies on the penalty techniques by modifying the constitutive equation of Stokes flow to weaken the incompressibility condition, relating the surface divergence to the elastic tension $\sigma$ to relax the interface's inextensibility, and connecting the particle surface-velocity with the particle surface force ${\mbf F}$ to regularize the particle's rigid motion. The advantage of these regularized governing equations is that when they are discretized by the standard centered difference scheme on a staggered grid, the resulting linear system can easily be reduced by eliminating the unknowns $p_h$, $\sigma_h$ and ${\mbf F}_h$ directly, so that we just need to solve a smaller linear system of the velocity approximation ${\mbf u}_h$. This advantage is preserved and even enhanced when such approach is applied to the transient Stokes flow with multiple compound vesicles. Moreover, this smaller linear system is symmetric and negative-definite, which enables us to use efficient linear solvers. Another important feature of the proposed method is that the discretization scheme is unconditionally stable in the sense that an appropriately defined energy functional associated with the discrete system is decreasing and hence bounded in time. We numerically test the accuracy and stability of the immersed boundary discretization scheme. The tank-treading and tumbling motions of inextensible interface with a suspended solid particle in the simple shear flow will be studied extensively. The simulation of the motion of multiple compound vesicles will be performed as well. Numerical results illustrate the superior performance of the proposed penalty method.