The Picone identity: A device to get optimal uniqueness results and global dynamics in Population Dynamics

This paper infers from a generalized Picone identity the uniqueness of the stable positive solution for a class of semilinear equations of superlinear indefinite type, as well as the uniqueness and global attractivity of the coexistence state in two generalized diffusive prototypes of the symbiotic and competing species models of Lotka–Volterra. The optimality of these uniqueness theorems reveals the tremendous strength of the Picone identity.     

A posteriori error analysis of a quadratic finite volume method for nonlinear elliptic problems

In this article, we construct and analyze a residual-based a posteriori error estimator for a quadratic finite volume method (FVM) for solving nonlinear elliptic partial differential equations with homogeneous Dirichlet boundary conditions. We shall prove that the a posteriori error estimator yields the global upper and local lower bounds for the norm error of the FVM. So that the a posteriori error estimator is equivalent to the true error in a certain sense. Numerical experiments are performed to illustrate the theoretical results.     

On the Local Uniqueness of Solutions of Variational Inequalities Under H-Differentiability

In this paper, we give some sufficient conditions for the local uniqueness of solutions to nonsmooth variational inequalities where the underlying functions are H-differentiable and the underlying set is a closed convex set/polyhedral set/box/polyhedral cone. We show how the solution of a linearized variational inequality is related to the solution of the variational inequality. These results extend/unify various similar results proved for C ¹ and locally Lipschitzian variational inequality problems. When specialized to the nonlinear complementarity problem, our results extend/unify those of C ² and C ¹ nonlinear complementarity problems.     

Efficient Collocation Schemes for Singular Boundary Value Problems

We discuss an error estimation procedure for the global error of collocation schemes applied to solve singular boundary value problems with a singularity of the first kind. This a posteriori estimate of the global error was proposed by Stetter in 1978 and is based on the idea of Defect Correction, originally due to Zadunaisky. Here, we present a new, carefully designed modification of this error estimate which not only results in less computational work but also appears to perform satisfactorily for singular problems. We give a full analytical justification for the asymptotical correctness of the error estimate when it is applied to a general nonlinear regular problem. For the singular case, we are presently only able to provide computational evidence for the full convergence order, the related analysis is still work in progress. This global estimate is the basis for a grid selection routine in which the grid is modified with the aim to equidistribute the global error. This procedure yields meshes suitable for an efficient numerical solution. Most importantly, we observe that the grid is refined in a way reflecting only the behavior of the solution and remains unaffected by the unsmooth direction field close to the singular point.     

An Improved Gradient Projection-based Decomposition Technique for Support Vector Machines

In this paper we propose some improvements to a recent decomposition technique for the large quadratic program arising in training support vector machines. As standard decomposition approaches, the technique we consider is based on the idea to optimize, at each iteration, a subset of the variables through the solution of a quadratic programming subproblem. The innovative features of this approach consist in using a very effective gradient projection method for the inner subproblems and a special rule for selecting the variables to be optimized at each step. These features allow to obtain promising performance by decomposing the problem into few large subproblems instead of many small subproblems as usually done by other decomposition schemes. We improve this technique by introducing a new inner solver and a simple strategy for reducing the computational cost of each iteration. We evaluate the effectiveness of these improvements by solving large-scale benchmark problems and by comparison with a widely used decomposition package.     

The Neumann problem for some degenerate elliptic equations

In the paper we study the equation L _u = f, where L is a degenerate elliptic operator, with Neumann boundary condition in a bounded open set μ. We prove existence and uniqueness of solutions in the space H(μ) for the Neumann problem.     

变系数非线性方程的Jacobi椭圆函数展开解

把Jacobi椭圆函数展开法扩展并应用到求解变系数的非线性演化方程,比较方便地得到新的解析解 关键词： Jacobi椭圆函数 变系数非线性方程 类椭圆余弦波解 类孤子解     

Mathematical modeling of flows through inelastic porous solids

We propose a framework, based on classical mixture theory, to describe the isothermal flow of an incompressible fluid through a deformable inelastic porous solid. The modeling of the behavior of the inelastic solid takes into account changes in the elastic response due to evolution in the microstructure of the material. We apply the model to a compression layer problem. The mathematical problem generated by the model is a free boundary problem.