A Petrov–Galerkin RBF method for diffusion equation on the unit sphere

This paper concerns a numerical solution for the diffusion equation on the unit sphere. The given method is based on the spherical basis function approximation and the Petrov–Galerkin test discretization. The method is meshless because spherical triangulation is not required neither for approximation nor for numerical integration. This feature is achieved through the spherical basis function approximation and the use of local weak forms instead of a global variational formulation. The local Petrov–Galerkin formulation allows to compute the integrals on small independent spherical caps without any dependence on a connected background mesh. Experimental results show the accuracy and the efficiency of the new method.     

Stability results for scattered‐data interpolation on Euclidean spheres

Let denote the unit sphere in and the geodesic distance in . A spherical‐basis function approximant is a function of the form , where are real constants, is a fixed function, and is a set of distinct points in . It is known that if is a strictly positive definite function in , then the interpolation matrix is positive definite, hence invertible, for every choice of distinct points and every positive integer M. The paper studies a salient subclass of such functions , and provides stability estimates for the associated interpolation matrices. This revised version was published online in June 2006 with corrections to the Cover Date.     

Improved estimates on majorizing sequences for the Newton–Kantorovich method

We approximate a locally unique solution of an equation in Banach space using the Newton–Kantorovich method. Motivated by our earlier works (see references [2–7] in the references list), optimization consideration, and the elegant studies by Cianciaruso with DePascale in (Numer. Funct. Anal. Optim. 27(5–6):529–538, 2006), and Cianciaruso in (Nonlinear Funct. Anal. Appl., 2009, to appear), we provide (by using more precise error estimates on the distances involved): finer error bounds; an at least as precise information on the location of the solution, and a larger convergence domain than in (Numer. Funct. Anal. Optim. 27(5–6):529–538, 2006). Finally, we provide numerical examples where our results can apply to solve equations, but earlier ones can not (see references [8–19]).     

Convergence of the method of fundamental solutions for Poisson’s equation on the unit sphere

Solutions of boundary value problems of the Laplace equation on the unit sphere are constructed by using the fundamental solution

Local well-posedness of the nonlinear Schrödinger equations on the sphere for data in modulation spaces

In this paper, the authors discuss a priori estimates derived from the energy method to the initial value problem for the cubic nonlinear Schrödinger on the sphere S². Exploring suitable a priori estimates, the authors prove the existence of solution for data whose regularity is s = 1/4.     

Non-uniform dependence on initial data for the μ − b equation

This paper is concerned with the non-uniform dependence on initial data for the μ?b equation on the circle. Using the approximate solution method, we construct two solution sequences to show that the data-to-solution map of the Cauchy problem of the μ?b equation is not uniformly continuous in ${H^s(\mathbb{S})}$ .     

A semismooth Newton method for SOCCPs based on a one-parametric class of SOC complementarity functions

In this paper, we present a detailed investigation for the properties of a one-parametric class of SOC complementarity functions, which include the globally Lipschitz continuity, strong semismoothness, and the characterization of their B-subdifferential. Moreover, for the merit functions induced by them for the second-order cone complementarity problem (SOCCP), we provide a condition for each stationary point to be a solution of the SOCCP and establish the boundedness of their level sets, by exploiting Cartesian P-properties. We also propose a semismooth Newton type method based on the reformulation of the nonsmooth system of equations involving the class of SOC complementarity functions. The global and superlinear convergence results are obtained, and among others, the superlinear convergence is established under strict complementarity. Preliminary numerical results are reported for DIMACS second-order cone programs, which confirm the favorable theoretical properties of the method.     

A predictor-corrector smoothing method for second-order cone programming

In this paper, the second order cone programming problem is studied. By introducing a parameter into the Fischer-Burmeister function, a predictor-corrector smoothing Newton method for solving the problem is presented. The proposed algorithm does neither have restrictions on its starting point nor need additional computation which keep the iteration sequence staying in the given neighborhood. Furthermore, the global and the local quadratic convergence of the algorithm are obtained, among others, the local quadratic convergence of the algorithm is established without strict complementarity. Preliminary numerical results indicate that the algorithm is effective.     

Holonomic gradient descent for the Fisher–Bingham distribution on the d-dimensional sphere

We propose an accelerated version of the holonomic gradient descent and apply it to calculating the maximum likelihood estimate (MLE) of the Fisher–Bingham distribution on a \(d\) -dimensional sphere. We derive a Pfaffian system (an integrable connection) and a series expansion associated with the normalizing constant with an error estimation. These enable us to solve some MLE problems up to dimension \(d=7\) with a specified accuracy.     

Continuous dependence on the boundary conditions for the Wentzell Laplacian

The solution u of the well-posed problem

A note on the event horizon for a processor sharing queue

In this note we identify a phenomenon for processor sharing queues that is unique to ones with time-varying rates. This property was discovered while correcting a proof in Hampshire, Harchol-Balter and Massey (Queueing Syst. 53(1–2), 19–30, 2006). If the arrival rate for a processor sharing queue has unbounded growth over time, then it is possible for the number of customers in a processor sharing queue to grow so quickly that a newly entering job never finishes. We define the minimum size for such a job to be the event horizon for a processor sharing queue. We discuss the use of such a concept and develop some of its properties. This short article serves both as errata for Hampshire, Harchol-Balter and Massey (Queueing Syst. 53(1–2), 19–30, 2006) and as documentation of a characteristic feature for some processor sharing queues with time varying rates.     

A cell functional minimization scheme for domain decomposition method on non-orthogonal and non-matching meshes

We are interested in a robust and accurate domain decomposition algorithm with interface conditions of Robin type on non-matching multiblock grids using a cell functional minimization scheme, which has a good performance on non-orthogonal meshes. In order to treat the non-matching grids at the interface, we introduce the \(L^2\) projection operator to ensure weak continuity of the primary unknown and of the normal flux across the non-matching interface. Furthermore, we prove the wellposedness of local and global problems and obtain as well an error estimate of first order in a discrete \(H^{1}\) -norm only using the \(L^{2}\) projection operator on the non-matching interface, as done in the matching case. Numerical results are presented in confirmation of the theoretical results.     

A hybrid simulation-optimization algorithm for the Hamiltonian cycle problem

In this paper, we propose a new hybrid algorithm for the Hamiltonian cycle problem by synthesizing the Cross Entropy method and Markov decision processes. In particular, this new algorithm assigns a random length to each arc and alters the Hamiltonian cycle problem to the travelling salesman problem. Thus, there is now a probability corresponding to each arc that denotes the probability of the event “this arc is located on the shortest tour.” Those probabilities are then updated as in cross entropy method and used to set a suitable linear programming model. If the solution of the latter yields any tour, the graph is Hamiltonian. Numerical results reveal that when the size of graph is small, say less than 50 nodes, there is a high chance the algorithm will be terminated in its cross entropy component by simply generating a Hamiltonian cycle, randomly. However, for larger graphs, in most of the tests the algorithm terminated in its optimization component (by solving the proposed linear program).     

A presentation for the singular part of the full transformation semigroup

The (full) transformation semigroup T_n\mathcal{T}_{n} is the semigroup of all functions from the finite set {1,…,n} to itself, under the operation of composition. The symmetric group S_n í T_n{\mathcal{S}_{n}\subseteq \mathcal{T}_{n}} is the group of all permutations on {1,…,n} and is the group of units of T_n\mathcal{T}_{n}. The complement T_n\S_n\mathcal{T}_{n}\setminus \mathcal{S}_{n} is a subsemigroup (indeed an ideal) of T_n\mathcal{T}_{n}. In this article we give a presentation, in terms of generators and relations, for T_n\S_n\mathcal{T}_{n}\setminus \mathcal{S}_{n}, the so-called singular part of T_n\mathcal{T}_{n}.     

The Collet–Eckmann condition for rational functions on the Riemann sphere

We show that the set of Collet–Eckmann maps has positive Lebesgue measure in the space of rational maps on the Riemann sphere for any fixed degree d ≥ 2.     

A method for calculating the Painlevé transcendents

A numerical method for solving the Cauchy problem for all the six Painlevé equations is proposed. The difficulty of solving these equations is that the unknown functions can have movable (that is, dependent on the initial data) singular points of the pole type. Moreover, the Painlevé III–VI equations may have singularities at points where the solution takes certain finite values. The positions of all these singularities are not a priori known and are determined in the process of solving the equation. The proposed method is based on the transition to auxiliary systems of differential equations in neighborhoods of the indicated points. The equations in these systems and their solutions have no singularities at the corresponding point and its neighborhood. Such auxiliary equations are derived for all Painlevé equations and for all types of singularities. Efficient criteria for transition to auxiliary systems are formulated, and numerical results illustrating the potentials of the method are presented.     

A linear programming algorithm for curve fitting in the L∞ norm

The L_∞ norm has been widely studied as a criterion for curve fitting problems. This paper presents an algorithm to solve discrete approximation problems in the L_∞ norm. The algorithm is a special-purpose linear programming method using the dual form of the problem, which employs a reduced basis and multiple pivots. Results of the computational experience with a computer code version of the algorithm are presented.     

Critical Landscape Topology for Optimization on the Symplectic Group

Optimization problems over compact Lie groups have been studied extensively due to their broad applications in linear programming and optimal control. This paper analyzes an optimization problem over a noncompact symplectic Lie group Sp(2N,ℝ), i.e., minimizing the Frobenius distance from a target symplectic transformation, which can be used to assess the fidelity function over dynamical transformations in classical mechanics and quantum optics. The topology of the set of critical points is proven to have a unique local minimum and a number of saddlepoint submanifolds, exhibiting the absence of local suboptima that may hinder the search for ultimate optimal solutions. Compared with those of previously studied problems on compact Lie groups, such as the orthogonal and unitary groups, the topology is more complicated due to the significant nonlinearity brought by the incompatibility of the Frobenius norm with the pseudo-Riemannian structure on the symplectic group.