In this paper, local multiplicative and additive multilevel methods on adaptively refined meshes are considered for second-order elliptic problems with highly discontinuous coefficients. For the multilevel-preconditioned system, we study the distribution of its spectrum by using the abstract Schwarz theory. It is proved that, except for a few small eigenvalues, the spectrum of the preconditioned system is bounded quasi-uniformly with respect to the jumps of the coefficient and the mesh sizes. The convergence rate of multilevel-preconditioned conjugate gradient methods is shown to be quasi-optimal regarding the jumps and the meshes. Numerical experiments are presented to illustrate the theoretical findings.

Classical optical flow techniques were developed for computing virtual motion fields between two images of the same scene, assuming conservation of intensity and a smoothness of the flow field. If these assumptions are dropped, such techniques can be adapted to compute apparent flows between time snapshots of data that do not come from images, even if these flows are turbulent and divergent, as in the case of flows representing complex spatiotemporal dynamics. While imaging methods have been used to analyze dynamics in experimental applications, they are only beginning to be applied to dynamics computations in settings outside the laboratory, for example in the analysis of species population dynamics from satellite data. In this work we present a variational optical flow approach based on the continuity equation and total variation regularization for computing the flow fields between population densities generated from a two-species predator-prey model for phyto- and zooplankton interactions. Given the time-varying vector fields produced from the optical flow, computational methods from dynamical systems can be employed to study pseudo-barriers present in the species interaction. This method allows to measure the mixing of the species, as well as the transport of the populations throughout the domain.

In this paper, a new trust region algorithm for minimax optimization problems is proposed, which solves only one quadratic subproblem based on a new approximation model at each iteration. The approach is different from the traditional algorithms that usually require to solve two quadratic subproblems. Moreover, to avoid Maratos effect, the nonmonotone strategy is employed. The analysis shows that, under standard conditions, the algorithm has global and superlinear convergence. Preliminary numerical experiments are conducted to show the efficiency of the new method.

In this paper, the discrete-time static output feedback control design problem is considered. A nonlinear conjugate gradient method is analyzed and studied for solving an unconstrained matrix optimization problem that results from this optimal control problem. In addition, through certain parametrization to the optimization problem an initial stabilizing static output feedback gain matrix is not required to start the conjugate gradient method. Finally, the proposed algorithms are tested numerically through several test problems from the benchmark collection.

In this paper, we study the local discontinuous Galerkin (LDG) method for one-dimensional singularly perturbed convection-diffusion problems by an exponentially fitted technique. We prove that the method is uniformly first-order convergent in the energy norm with respect to the small diffusion parameter.

In this paper, by using multivariate divided differences to approximate the partial derivative and superposition, we extend the multivariate quasi-interpolation scheme based on dimension-splitting technique which can reproduce linear polynomials to the scheme quadric polynomials. Furthermore, we give the approximation error of the modified scheme. Our multivariate multiquadric quasi-interpolation scheme only requires information of location points but not that of the derivatives of approximated function. Finally, numerical experiments demonstrate that the approximation rate of our scheme is significantly improved which is consistent with the theoretical results.

We present a higher-order in-space characteristic method for the solution of the transient advection diffusion equations in two space dimensions. This method uses biquadratic trial and test functions within the framework of the Eulerian-Lagrangian localized Adjoint Methods (ELLAM). It therefore maintains the advantages of previous ELLAM schemes. Namely, it treats general boundary conditions naturally in a systematic manner, conserves mass, and symmetrizes the governing transport equations. Moreover, it generates accurate numerical solutions even if large time steps are used in the simulation. Numerical experiments are presented to illustrate the performance of this method and establish its order of convergence numerically.