No abstract uploaded!

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.

In this paper, we propose a structured bisection method with adaptive randomized sampling for finding selected or all of the eigenvalues of certain real symmetric matrices A. For A with a low-rank property, we construct a hierarchically semiseparable (HSS) approximation and show how to quickly evaluate and update its inertia in the bisection method. Unlike some existing randomized HSS constructions, the methods here do not require the knowledge of the off-diagonal (numerical) ranks in advance. Moreover, for A with a weak rank property or slowly decaying off-diagonal singular values, we show an idea for aggressive low-rank inertia evaluation, which means that a compact HSS approximation can preserve the inertia for certain shifts. This is analytically justified for a special case, and numerically shown for more general ones. A generalized LDL factorization of the HSS approximation is then designed for the fast evaluation of the inertia. A significant advantage over standard LDL factorizations is that the HSS LDL factorization (and thus the inertia) of A − sI can be quickly updated with multiple shifts s in bisection. The factorization with each new shift can reuse about 60% of the work. As an important application, the structured eigensolver can be applied to symmetric Toeplitz matrices, and the cost to find one eigenvalue is nearly linear in the order of the matrix. The numerical examples demonstrate the efficiency and the accuracy of our methods, especially the benefit of low-rank inertia evaluations. The ideas and methods can be potentially adapted to other HSS computations where shifts are involved and to more problems without a significant low-rank property.

Image segmentation is an important task in the domain of computer vision and medical imaging. In natural and medical images, intensity inhomogeneity, i.e. the varying image intensity, occurs often and it poses considerable challenges for image segmentation. In this paper, we propose an efficient variational method for segmenting images with intensity inhomogeneity. The method is inspired by previous works on two-stage segmentation and variational Retinex. Our method consists of two stages. In the first stage, we decouple the image into reflection and illumination parts by solving a convex energy minimization model with either total variation or tight-frame regularisation. In the second stage, we segment the original image by thresholding on the reflection part, and the inhomogeneous intensity is estimated by the smoothly varying illumination part. We adopt a primal dual algorithm to solve the convex model in the first stage, and the convergence is guaranteed. Numerical experiments clearly show that our method is robust and efficient to segment both natural and medical images.

We propose a class of simple and efficient numerical scheme for incompressible fluid equations with coordinate symmetry. By introducing a generalized vorticity-stream formulation, the divergence free constraints are automatically satisfied. In addition, with explicit treatment of the nonlinear terms and local vorticity boundary condition, the Navier–Stokes (MHD, respectively) equation essentially decouples into 2 (4, respectively) scalar equation and thus the scheme is very efficient. Moreover, with proper discretization of the nonlinear terms, the scheme preserves both energy and helicity identities numerically. This is achieved by recasting the nonlinear terms (convection, vorticity stretching, geometric source, Lorentz force and electro-motive force) in terms of Jacobians. This conservative property is valid even in the presence of the pole singularity for axisymmetric flows. The exact conservation of energy and helicity has effectively eliminated excessive numerical viscosity. Numerical examples have demonstrated both accuracy and efficiency of the scheme. Finally, local mesh refinement near the boundary can also be easily incorporated into the scheme without extra cost.