a mode elimination technique to improve convergence of iteration methods for finding solitary waves we extend the key idea behind the generalized petviashvili method of t i lakoba j yang a generalized petviashvili iteration method for scalar and vector hamiltonian equations with arbitrary form of nonlinearity j comput phys this issue doi 10 1016 j jcp 2007 06 009 by proposing a novel technique based on a similar idea this technique systematically eliminates from the iteratively obtained solution a mode that is responsible either for the divergence or the slow convergence of the iterations we demonstrate theoretically and with examples that this mode elimination technique can be used both to obtain some nonfundamental solitary waves and to considerably accelerate convergence of various iteration methods as a collateral result we compare the linearized iteration operators for the generalized petviashvili method and the well known imaginary time evolution method and explain how their different structures account for the differences in the convergence rates of these two methods

numerical generation of periodic traveling wave solutions of some nonlinear dispersive wave systems in this paper a numerical procedure to generate approximations to periodic traveling wave profiles of some nonlinear dispersive wave systems is introduced the method is based on a suitable modification of a fixed point algorithm of petviashvili type and solves several drawbacks of some previous algorithms presented in the literature by way of illustration the method is applied to generate numerically periodic traveling waves of two problems of interest the fractional kdv type equations and the extended boussinesq system we propose a method to compute periodic traveling waves the method can be applied to some nonlinear dispersive systems it is checked in fractional kdv equations and boussinesq systems

newton conjugate gradient methods for solitary wave computations in this paper the newton conjugate gradient methods are developed for solitary wave computations these methods are based on newton iterations coupled with conjugate gradient iterations to solve the resulting linear newton correction equation when the linearization operator is self adjoint the preconditioned conjugate gradient method is proposed to solve this linear equation if the linearization operator is non self adjoint the preconditioned biconjugate gradient method is proposed to solve the linear equation the resulting methods are applied to compute both the ground states and excited states in a large number of physical systems such as the two dimensional nls equations with and without periodic potentials the fifth order kdv equation and the fifth order kp equation numerical results show that these proposed methods are faster than the other leading numerical methods often by orders of magnitude in addition these methods are very robust and always converge in all the examples being tested furthermore they are very easy to implement it is also shown that the nonlinear conjugate gradient methods are not robust and inferior to the proposed methods

vertex similarity in networks we consider methods for quantifying the similarity of vertices in networks we propose a measure of similarity based on the concept that two vertices are similar if their immediate neighbors in the network are themselves similar this leads to a self consistent matrix formulation of similarity that can be evaluated iteratively using only a knowledge of the adjacency matrix of the network we test our similarity measure on computer generated networks for which the expected results are known and on a number of real world networks

self similar scaling of density in complex real world networks abstract despite their diverse origin networks of large real world systems reveal a number of common properties including small world phenomena scale free degree distributions and modularity recently network self similarity as a natural outcome of the evolution of real world systems has also attracted much attention within the physics literature here we investigate the scaling of density in complex networks under two classical box covering renormalizations network coarse graining and also different community based renormalizations the analysis on over 50 real world networks reveals a power law scaling of network density and size under adequate renormalization technique yet irrespective of network type and origin the results thus advance a recent discovery of a universal scaling of density among different real world networks p j laurienti k e joyce q k telesford j h burdette s hayasaka universal fractal scaling of self organized networks physica a 390 20 2011 3608 3613 and imply an existence of a scale free density also within among different self similar scales of complex real world networks the latter further improves the comprehension of self similar structure in large real world networks with several possible applications

finding community structure in networks using the eigenvectors of matrices we consider the problem of detecting communities or modules in networks groups of vertices with a higher than average density of edges connecting them previous work indicates that a robust approach to this problem is the maximization of the benefit function known as modularity over possible divisions of a network here we show that this maximization process can be written in terms of the eigenspectrum of a matrix we call the modularity matrix which plays a role in community detection similar to that played by the graph laplacian in graph partitioning calculations this result leads us to a number of possible algorithms for detecting community structure as well as several other results including a spectral measure of bipartite structure in networks and a centrality measure that identifies vertices that occupy central positions within the communities to which they belong the algorithms and measures proposed are illustrated with applications to a variety of real world complex networks

a modified forward backward splitting method for maximal monotone mappings we consider the forward backward splitting method for finding a zero of the sum of two maximal monotone mappings this method is known to converge when the inverse of the forward mapping is strongly monotone we propose a modification to this method in the spirit of the extragradient method for monotone variational inequalities under which the method converges assuming only the forward mapping is lipschitz continuous on some closed convex subset of its domain the modification entails an additional forward step and a projection step at each iteration applications of the modified method to decomposition in convex programming and monotone variational inequalities are discussed

nonlinear proximal point algorithms using bregman functions with applications to convex programming a bregman function is a strictly convex differentiable function that induces a well behaved distance measure or d function on euclidean space this paper shows that for every bregman function there exists a nonlinear version of the proximal point algorithm and presents an accompanying convergence theory applying this generalization of the proximal point algorithm to convex programming one obtains the d function proximal minimization algorithm of censor and zenios and a wide variety of new multiplier methods these multiplier methods are different from those studied by kort and bertsekas and include nonquadratic variations on the proximal method of multipliers

energy dissipating flows for solving nonlinear eigenpair problems abstract this work is concerned with computing nonlinear eigenpairs which model solitary waves and various other physical phenomena we aim at solving nonlinear eigenvalue problems of the general form t u q u in our setting t is a variational derivative of a convex functional such as the laplacian operator with respect to the dirichlet energy q is an arbitrary bounded nonlinear operator and is an unknown real eigenvalue we introduce a flow that numerically generates an eigenpair solution by its steady state analysis for the general case is performed showing a monotone decrease in the convex functional throughout the flow when t is the laplacian operator a complete discretized version is presented and anlalyzed we implement our algorithm on korteweg and de vries kdv and nonlinear schrodinger nls equations in one and two dimensions the proposed approach is very general and can be applied to a large variety of models moreover it is highly robust to noise and to perturbations in the initial conditions compared to classical petiashvili based methods

approximation solvability of a class of nonlinear set valued variational inclusions involving a η monotone mappings abstract a new class of nonlinear set valued variational inclusions involving a monotone mappings in a hilbert space setting is introduced and then based on the generalized resolvent operator technique associated with a monotonicity the existence and approximation solvability of solutions using an iterative algorithm is investigated

enhanced collective influence a paradigm to optimize network disruption abstract the function of complex networks typically relies on the integrity of underlying structure sometimes practical applications need to attack networks function namely inactivate and fragment networks underlying structure to effectively dismantle complex networks and regulate the function of them a centrality measure named ci morone and makse 2015 was proposed for node ranking we observe that the performance of ci centrality in network disruption problem may deteriorate when it is used in networks with different topology properties specifically the structural features of local network topology are overlooked in ci centrality even though the local network topology of the nodes with a fixed ci value may have very different organization to improve the ranking accuracy of ci this paper proposes a variant eci to ci by considering loop density and degree diversity of local network topology and the proposed eci centrality would degenerate into ci centrality with the reduction of the loop density and the degree diversity level by comparing eci with ci and classical centrality measures in both synthetic and real networks the experimental results suggest that eci can largely improve the performance of ci for network disruption based on the results we analyze the correlation between the improvement and the properties of the networks we find that the performance of eci is positively correlated with assortative coefficient and community modularity and negatively correlated with degree inequality of networks which can be used as guidance for practical applications

conjugate gradient method for finding fundamental solitary waves abstract the conjugate gradient method cgm is known to be the fastest generic iterative method for solving linear systems with symmetric sign definite matrices in this paper we modify this method so that it can find fundamental solitary waves of nonlinear hamiltonian equations the main obstacle that such a modified cgm overcomes is that the operator of the equation linearized about a solitary wave is not sign definite instead it has a finite number of eigenvalues on the opposite side of zero than the rest of its spectrum we present versions of the modified cgm that can find solitary waves with prescribed values of either the propagation constant or power we also extend these methods to handle multi component nonlinear wave equations convergence conditions of the proposed methods are given and their practical implications are discussed we demonstrate that our modified cgms converge much faster than say petviashvili s or similar methods especially when the latter converge slowly

boussinesq peregrine water wave models and their numerical approximation abstract in this paper we consider the numerical solution of boussinesq peregrine type systems by the application of the galerkin finite element method the structure of the boussinesq systems is explained and certain alternative nonlinear and dispersive terms are compared a detailed study of the convergence properties of the standard galerkin method using various finite element spaces on unstructured triangular grids is presented along with the study of the peregrine system a new boussinesq system of bbm bbm type is derived the new system has the same structure in its momentum equation but differs slightly in the mass conservation equation compared to the peregrine system further the finite element method applied to the new system has better convergence properties when used for its numerical approximation due to the lack of analytical formulas for solitary wave solutions for the systems under consideration a galerkin finite element method combined with the petviashvili iteration is proposed for the numerical generation of accurate approximations of line solitary waves various numerical experiments related to the propagation of solitary and periodic waves over variable bottom topography and their interaction with the boundaries of the domains are presented we conclude that both systems have similar accuracy when approximate long waves of small amplitude while the galerkin finite element method is more efficient when applied to bbm bbm type systems

convergence of petviashvili s iteration method for numerical approximation of stationary solutions of nonlinear wave equations we analyze a heuristic numerical method suggested by v i petviashvili in 1976 for approximation of stationary solutions of nonlinear wave equations the method is used to construct numerically the solitary wave solutions such as solitons lumps and vortices in a space of one and higher dimensions assuming that the stationary solution exists we find conditions when the iteration method converges to the stationary solution and when the rate of convergence is the fastest the theory is illustrated with examples of physical interest such as generalized korteweg de vries benjamin ono zakharov kuznetsov kadomtsev petviashvili and klein gordon equations

predicting missing links via effective paths recently in complex network link prediction has brought a surge of researches among which similarity based link prediction outstandingly gains considerable success especially similarity in terms of paths in investigation of paths based similarity we find that the effective influence of endpoints and strong connectivity make paths contribute more similarity between two unconnected endpoints leading to a more accurate link prediction accordingly we propose a so called effective path index ep in this paper to leverage effective influence of endpoints and strong connectivity in similarity calculation for demonstrating excellence of our index the comparisons with six mainstream indices are performed on experiments in 15 real datasets and results show a great improvement of performance via our index

new approach to the η proximal point algorithm and nonlinear variational inclusion problems abstract a new approach using the over relaxed proximal point algorithm in the context of solving a class of inclusion problems based on the notion of maximal monotonicity is developed and examined convergence analysis seems to be reasonable and finally some specializations are included furthermore the model developed in this communication seems to be appropriate to the yosida approximation in the sense that it can be applied to first order evolution equations inclusions as well

finite termination of the proximal point algorithm this paper concerns the notion of a sharp minimum on a set and its relationship to the proximal point algorithm we give several equivalent definitions of the property and use the notion to prove finite termination of the proximal point algorithm

mixed equilibrium problems sensitivity analysis and algorithmic aspect abstract the aim of this paper is twofold first it is to extend the sensitivity analysis framework developed recently for variational inequalities to mixed equilibrium problems the second is to propose iterative methods for solving this kind of problems in the process we establish an equivalence between an extended version of wiener hopf equations and the given problems relying on a generalization of the yosida approximation notion our results generalize results obtained for optimization variational inequalities complementarity problems and problems of nash equilibria

a proximal point algorithm for finding a common zero of a finite family of maximal monotone operators in the presence of computational errors abstract in a hilbert space we study the convergence of a proximal point method to a common zero of a finite family of maximal monotone operators under the presence of computational errors most results known in the literature establish the convergence of proximal point methods when computational errors are summable in the present paper the convergence of the method is established for nonsummable computational errors we show that the proximal point method generates a good approximate solution if the sequence of computational errors is bounded from above by a constant

petviashvili type methods for traveling wave computations i analysis of convergence in this paper a family of fixed point algorithms for the numerical resolution of some systems of nonlinear equations is designed and analyzed the family introduced here generalizes the petviashvili method and can be applied to the numerical generation of traveling waves in some nonlinear dispersive systems conditions for the local convergence are derived and numerical comparisons between different elements of the family are carried out

the extragradient method for finding a common solution of a finite family of variational inequalities and a finite family of fixed point problems in the presence of computational errors abstract in a hilbert space we study the convergence of the subgradient method to a common solution of a finite family of variational inequalities and of a finite family of fixed point problems under the presence of computational errors most results known in the literature establish the convergence of algorithms when computational errors are summable in the present paper the convergence of the subgradient method is established for nonsummable computational errors we show that the subgradient method generates a good approximate solution if the sequence of computational errors is bounded from above by a constant

optimizing schrodinger functionals using sobolev gradients applications to quantum mechanics and nonlinear optics in this paper we study the application of the sobolev gradients technique to the problem of minimizing several schrodinger functionals related to timely and difficult nonlinear problems in quantum mechanics and nonlinear optics we show that these gradients act as preconditioners over traditional choices of descent directions in minimization methods and show a computationally inexpensive way to obtain them using a discrete fourier basis and a fast fourier transform we show that the sobolev preconditioning provides a great convergence improvement over traditional techniques for finding solutions with minimal energy as well as stationary states and suggest a generalization of the method using arbitrary linear operators

a generalized petviashvili iteration method for scalar and vector hamiltonian equations with arbitrary form of nonlinearity the petviashvili s iteration method has been known as a rapidly converging numerical algorithm for obtaining fundamental solitary wave solutions of stationary scalar nonlinear wave equations with power law nonlinearity mu up 0 where m is a positive definite self adjoint operator and p const in this paper we propose a systematic generalization of this method to both scalar and vector hamiltonian equations with arbitrary form of nonlinearity and potential functions for scalar equations our generalized method requires only slightly more computational effort than the original petviashvili method

applications of splitting algorithm to decomposition in convex programming and variational inequalities recently han and lou proposed a highly parallelizable decomposition algorithm for minimizing a strongly convex cost over the intersection of closed convex sets it is shown that their algorithm is in fact a special case of a splitting algorithm analyzed by gabay for finding a zero of the sum of two maximal monotone operators gabay s convergence analysis for the splitting algorithm is sharpened and new applications of this algorithm to variational inequalities convex programming and the solution of linear complementarily problems are proposed for convex programs with a certain separable structure a multiplier method that is closely related to the alternating direction method of multipliers of gabay mercier and of glowinski marrocco but which uses both ordinary and augmented lagrangians is obtained

smooth methods of multipliers for complementarity problems this paper describes several methods for solving nonlinear complementarity problems a general duality framework for pairs of monotone operators is developed and then applied to the monotone complementarity problem obtaining primal dual and primal dual formulations we derive bregman function based generalized proximal algorithms for each of these formulations generating three classes of complementarity algorithms the primal class is well known the dual class is new and constitutes a general collection of methods of multipliers or augmented lagrangian methods for complementarity problems in a special case it corresponds to a class of variational inequality algorithms proposed by gabay by appropriate choice of bregman function the augmented lagrangian subproblem in these methods can be made continuously differentiable the primal dual class of methods is entirely new and combines the best theoretical features of the primal and dual methods some preliminary computation shows that this class of algorithms is effective at solving many of the standard complementarity test problems

computing the ground state solution of bose einstein condensates by a normalized gradient flow in this paper we present a continuous normalized gradient flow cngf and prove its energy diminishing property which provides a mathematical justification of the imaginary time method used in the physics literature to compute the ground state solution of bose einstein condensates bec we also investigate the energy diminishing property for the discretization of the cngf two numerical methods are proposed for such discretizations one is the backward euler centered finite difference befd method the other is an explicit time splitting sine spectral tssp method energy diminishing for befd and tssp for the linear case and monotonicity for befd for both linear and nonlinear cases are proven comparison between the two methods and existing methods e g crank nicolson finite difference cnfd or forward euler finite difference fefd shows that befd and tssp are much better in terms of preserving the energy diminishing property of the cngf numerical results in one two and three dimensions with magnetic trap confinement potential as well as a potential of a stirrer corresponding to a far blue detuned gaussian laser beam are reported to demonstrate the effectiveness of befd and tssp methods furthermore we observe that the cngf and its befd discretization can also be applied directly to compute the first excited state solution in bec when the initial data is chosen as an odd function

asymptotic nonlinear and dispersive pulsatile flow in elastic vessels with cylindrical symmetry the asymptotic derivation of a new family of one dimensional weakly nonlinear and weakly dispersive equations that model the flow of an ideal fluid in an elastic vessel is presented dissipative effects due to the viscous nature of the fluid are also taken into account the new models validate by asymptotic reasoning other non dispersive systems of equations that are commonly used and improve other nonlinear and dispersive mathematical models derived to describe the blood flow in elastic vessels the new systems are studied analytically in terms of their basic characteristic properties such as the linear dispersion characteristics symmetries conservation laws and solitary waves unidirectional model equations are also derived and analysed in the case of vessels of constant radius the capacity of the models to be used in practical problems is being demonstrated by employing a particular system with favourable properties to study the blood flow in a large artery two different cases are considered a vessel with constant radius and a tapered vessel significant changes in the flow can be observed in the case of the tapered vessel

linear convergence of epsilon subgradient descent methods for a class of convex functions this paper establishes a linear convergence rate for a class of epsilon subgradient descent methods for minimizing certain convex functions currently prominent methods belonging to this class include the resolvent proximal point method and the bundle method in proximal form considered as a sequence of serious steps other methods such as the recently proposed descent proximal level method may also fit this framework depending on implementation the convex functions covered by the analysis are those whose conjugates have subdifferentials that are locally upper lipschitzian at the origin a class introduced by zhang and treiman we argue that this class is a natural candidate for study in connection with minimization algorithms

inexact variants of the proximal point algorithm without monotonicity this paper studies convergence properties of inexact variants of the proximal point algorithm when applied to a certain class of nonmonotone mappings the presented algorithms allow for constant relative errors in the line of the recently proposed hybrid proximal extragradient algorithm the main convergence result extends a recent work of the second author where exact solutions for the proximal subproblems were required we also show that the linear convergence property is preserved in the case when the inverse of the operator is locally lipschitz continuous near the origin as an application we give a convergence analysis for an inexact version of the proximal method of multipliers for a rather general family of problems which includes variational inequalities and constrained optimization problems

pseudofractal scale free web we find that scale free random networks are excellently modeled by simple deterministic graphs our graph has a discrete degree distribution degree is the number of connections of a vertex which is characterized by a power law with exponent gamma 1 ln 3 ln 2 properties of this compact structure are surprisingly close to those of growing random scale free networks with gamma in the most interesting region between 2 and 3 we succeed to find exactly and numerically with high precision all main characteristics of the graph in particular we obtain the exact shortest path length distribution for a large network ln n 1 the distribution tends to a gaussian of width approximately sqrt ln n centered at l approximately ln n we show that the eigenvalue spectrum of the adjacency matrix of the graph has a power law tail with exponent 2 gamma

alternating projection proximal methods for convex programming and variational inequalities we consider a mixed problem composed in part of finding a zero of a maximal monotone operator and in part of solving a monotone variational inequality problem we propose a solution method for this problem that alternates between a proximal step for the maximal monotone operator part and a projection type step for the monotone variational inequality part and analyze its convergence and rate of convergence this method extends a decomposition method of chen and teboulle math programming 64 1994 pp 81 101 for convex programming and yields as a by product new decomposition methods

composition duality and maximal monotonicity this paper shows how the attouch thera duality principle for operator inclusions can be extended to compositions of multifunctions so that the primal and dual inclusions may involve operators between different pairs of spaces we first present the extension and give an example from computational economics to demonstrate that it has practical utility then we consider a particular case often found in applications involving real hilbert spaces and maximal monotone operators we show that in this case our duality framework includes those previously developed by rockafellar mosco and gabay finally we demonstrate that in this case under very simple hypotheses the duality transformation preserves the maximal monotonicity of the operators involved this proof uses an apparently new criterion for maximal monotonicity of operators of the form l tl where t is maximal monotone and l is linear and continuous with adjoint l

a minimisation approach for computing the ground state of gross pitaevskii systems in this paper we present a minimisation method for computing the ground state of systems of coupled gross pitaevskii equations our approach relies on a spectral decomposition of the solution into hermite basis functions inserting the spectral representation into the energy functional yields a constrained nonlinear minimisation problem for the coefficients for its numerical solution we employ a newton like method with an approximate line search strategy we analyse this method and prove global convergence appropriate starting values for the minimisation process are determined by a standard continuation strategy numerical examples with two and three component two dimensional condensates are included these experiments demonstrate the reliability of our method and nicely illustrate the effect of phase segregation