proximal decomposition on the graph of a maximal monotone operator we present an algorithm to solve find x y in a times a bot such that y in tx where a is a subspace and t is a maximal monotone operator the algorithm is based on the proximal decomposition on the graph of a monotone operator and we show how to recover spingarn s decomposition method we give a proof of convergence that does not use the concept of partial inverse and show how to choose a scaling factor to accelerate the convergence in the strongly monotone case numerical results performed on quadratic problems confirm the robust behaviour of the algorithm

pade numerical schemes for the sine gordon equation abstract the sine gordon equation turn up in several problems in science and engineering although it is integrable in practical applications its numerical solution is powerful and versatile four novel implicit finite difference methods based on q s pade approximations with q s th order in space have been developed and analyzed for this equation all share the same treatment for the nonlinearity and integration in time concretely 0 4 2 2 2 4 and 4 4 pade methods additionally the energy conserving strauss vazquez scheme has been considered in a 0 2 pade implementation these methods have been compared among them for both the kink antikink and breather solutions in terms of global error computational cost and energy conservation the 0 4 and 2 4 pade methods are the most cost effective ones for small and large global error respectively our results indicate that spatial order of accuracy is more relevant to effectiveness of a method than energy conservation even in very long time integrations

analysis of four numerical schemes for a nonlinear klein gordon equation we compare the properties of four explicit finite difference schemes used to integrate the nonlinear klein gordon equation f t t f x x f f 0 it turns out that the energy conserving scheme is the most suitable to study the long time behavior of the solutions the same result is found for the special case of no spatial dependence

duality and statistical tests of optimality for two stage stochastic programs we present alternative methods for verifying the quality of a proposed solution to a two stage stochastic program with recourse our methods revolve around implications of a dual problem in which dual multipliers on the nonanticipativity constraints play a critical role using randomly sampled observations of the stochastic elements we introduce notions of statistical dual feasibility and sampled error bounds additionally we use the nonanticipativity multipliers to develop connections to reduced gradient methods finally we propose a statistical test based on directional derivatives we illustrate the applicability of these tests via some examples

the c 3 theorem and a d 2 algorithm for large scale stochastic mixed integer programming set convexification this paper considers the two stage stochastic integer programming problem with an emphasis on instances in which integer variables appear in the second stage drawing heavily on the theory of disjunctive programming we characterize convexifications of the second stage problem and develop a decomposition based algorithm for the solution of such problems in particular we verify that problems with fixed recourse are characterized by scenario dependent second stage convexifications that have a great deal in common we refer to this characterization as the c3 common cut coefficients theorem based on the c3 theorem we develop a decomposition algorithm which we refer to as disjunctive decomposition d2 in this new class of algorithms we work with master and subproblems that result from convexifications of two coupled disjunctive programs we show that when the second stage consists of 0 1 milp problems we can obtain accurate second stage objective function estimates after finitely many steps this result implies the convergence of the d2 algorithm

robust optimization for unconstrained simulation based problems in engineering design an optimized solution often turns out to be suboptimal when errors are encountered although the theory of robust convex optimization has taken significant strides over the past decade all approaches fail if the underlying cost function is not explicitly given it is even worse if the cost function is nonconvex in this work we present a robust optimization method that is suited for unconstrained problems with a nonconvex cost function as well as for problems based on simulations such as large partial differential equations pde solver response surface and kriging metamodels moreover this technique can be employed for most real world problems because it operates directly on the response surface and does not assume any specific structure of the problem we present this algorithm along with the application to an actual engineering problem in electromagnetic multiple scattering of aperiodically arranged dielectrics relevant to nanophotonic design the corresponding objective function is highly nonconvex and resides in a 100 dimensional design space starting from an optimized design we report a robust solution with a significantly lower worst case cost while maintaining optimality we further generalize this algorithm to address a nonconvex optimization problem under both implementation errors and parameter uncertainties

a proximal projection bundle method for lagrangian relaxation including semidefinite programming we give a proximal bundle method for minimizing a convex function f over a convex set c it requires evaluating f and its subgradients with a fixed but possibly unknown accuracy epsilon 0 each iteration involves solving an unconstrained proximal subproblem and projecting a certain point onto c the method asymptotically finds points that are epsilon optimal in lagrangian relaxation of convex programs it allows for e accurate solutions of lagrangian subproblems and finds e optimal primal solutions for semidefinite programming problems it extends the highly successful spectral bundle method to the case of inexact eigenvalue computations

two energy conserving numerical schemes for the sine gordon equation two explicit conservative numerical schemes for the sine gordon equation are proposed their stability and convergence are proved numerical simulation of a sine gordon soliton shows that the new schemes are very accurate and fast

duality gaps in nonconvex stochastic optimization we consider multistage stochastic optimization models containing nonconvex constraints e g due to logical or integrality requirements we study three variants of lagrangian relaxations and of the corresponding decomposition schemes namely scenario nodal and geographical decomposition based on convex equivalents for the lagrangian duals we compare the duality gaps for these decomposition schemes the first main result states that scenario decomposition provides a smaller or equal duality gap than nodal decomposition the second group of results concerns large stochastic optimization models with loosely coupled components the results provide conditions implying relations between the duality gaps of geographical decomposition and the duality gaps for scenario and nodal decomposition respectively

a finite branch and bound algorithm for two stage stochastic integer programs this paper addresses a general class of two stage stochastic programs with integer recourse and discrete distributions we exploit the structure of the value function of the second stage integer problem to develop a novel global optimization algorithm the proposed scheme departs from those in the current literature in that it avoids explicit enumeration of the search space while guaranteeing finite termination computational experiments on standard test problems indicate superior performance of the proposed algorithm in comparison to those in the existing literature

preserving energy resp dissipation in numerical pdes using the average vector field method we give a systematic method for discretizing hamiltonian partial differential equations pdes with constant symplectic structure while preserving their energy exactly the same method applied to pdes with constant dissipative structure also preserves the correct monotonic decrease of energy the method is illustrated by many examples in the hamiltonian case these include the sine gordon korteweg de vries nonlinear schrodinger linear time dependent schrodinger and maxwell equations in the dissipative case the examples are the allen cahn cahn hilliard ginzburg landau and heat equations

a parallelizable augmented lagrangian method applied to large scale non convex constrained optimization problems we contribute improvements to a lagrangian dual solution approach applied to large scale optimization problems whose objective functions are convex continuously differentiable and possibly nonlinear while the non relaxed constraint set is compact but not necessarily convex such problems arise for example in the split variable deterministic reformulation of stochastic mixed integer optimization problems we adapt the augmented lagrangian method framework to address the presence of nonconvexity in the non relaxed constraint set and to enable efficient parallelization the development of our approach is most naturally compared with the development of proximal bundle methods and especially with their use of serious step conditions however deviations from these developments allow for an improvement in efficiency with which parallelization can be utilized pivotal in our modification to the augmented lagrangian method is an integration of the simplicial decomposition method and the nonlinear block gauss seidel method an adaptation of a serious step condition associated with proximal bundle methods allows for the approximation tolerance to be automatically adjusted under mild conditions optimal dual convergence is proven and we report computational results on test instances from the stochastic optimization literature we demonstrate improvement in parallel speedup over a baseline parallel approach

the solution of the two dimensional sine gordon equation using the method of lines the method of lines is used to transform the initial boundary value problem associated with the two dimensional sine gordon equation in two space variables into a second order initial value problem the finite difference methods are developed by replacing the matrix exponential term in a recurrence relation with rational approximants the resulting finite difference methods are analyzed for local truncation error stability and convergence to avoid solving the nonlinear system a predictor corrector scheme using the explicit method as predictor and the implicit as corrector is applied numerical solutions for cases involving the most known from the bibliography line and ring solitons are given

an approximation scheme for stochastic integer programs arising in capacity expansion planning for capacity expansion forms a crucial part of the strategic level decision making in many applications consequently quantitative models for economic capacity expansion planning have been the subject of intense research however much of the work in this area has been restricted to linear cost models and or limited degree of uncertainty to make the problems analytically tractable this paper addresses a stochastic capacity expansion problem where the economies of scale in expansion costs are handled via fixed charge cost functions and forecast uncertainties in the problem parameters are explicitly considered by specifying a set of scenarios the resulting formulation is a multistage stochastic integer program we develop a fast linear programming based approximation scheme that exploits the decomposable structure and is guaranteed to produce feasible solutions for this problem through a probabilistic analysis we prove that the optimality gap of the heuristic solution almost surely vanishes asymptotically as the problem size increases

multi symplectic runge kutta collocation methods for hamiltonian wave equations a number of conservative pdes like various wave equations allow for a multi symplectic formulation which can be viewed as a generalization of the symplectic structure of hamiltonian odes we show that gauss legendre collocation in space and time leads to multi symplectic integrators i e to numerical methods that preserve a symplectic conservation law similar to the conservation of symplecticity under a symplectic method for hamiltonian odes we also discuss the issue of conservation of energy and momentum since time discretization by a gauss legendre method is computational rather expensive we suggest several semi explicit multi symplectic methods based on gauss legendre collocation in space and explicit or linearly implicit symplectic discretizations in time

solving lagrangian variational inequalities with applications to stochastic programming lagrangian variational inequalities feature both primal and dual elements in expressing first order conditions for optimality in a wide variety of settings where multipliers in a very general sense need to be brought in their stochastic version relates to problems of stochastic programming and covers not only classical formats with inequality constraints but also composite models with nonsmooth objectives the progressive hedging algorithm as a means of solving stochastic programming problems has however focused so far only on optimality conditions that correspond to variational inequalities in primal variables alone here that limitation is removed by appealing to a recent extension of progressive hedging to multistage stochastic variational inequalities in general

schwarz maps of algebraic linear ordinary differential equations abstract a linear ordinary differential equation is called algebraic if all its solution are algebraic over its field of definition in this paper we solve the problem of finding closed form solution to algebraic linear ordinary differential equations in terms of standard equations furthermore we obtain a method to compute all algebraic linear ordinary differential equations with rational coefficients by studying their associated schwarz map through the picard vessiot theory

an augmented lagrangian method for distributed optimization we propose a novel distributed method for convex optimization problems with a certain separability structure the method is based on the augmented lagrangian framework we analyze its convergence and provide an application to two network models as well as to a two stage stochastic optimization problem the proposed method compares favorably to two augmented lagrangian decomposition methods known in the literature as well as to decomposition methods based on the ordinary lagrangian function

multisymplectic geometry variational integrators and nonlinear pdes this paper presents a geometric variational approach to continuous and discrete mechanics and field theories using multisymplectic geometry we show that the existence of the fundamental geometric structures as well as their preservation along solutions can be obtained directly from the variational principle in particular we prove that a unique multisymplectic structure is obtained by taking the derivative of an action function and use this structure to prove covariant generalizations of conservation of symplecticity and noether s theorem natural discretization schemes for pdes which have these important preservation properties then follow by choosing a discrete action functional in the case of mechanics we recover the variational symplectic integrators of veselov type while for pdes we obtain covariant spacetime integrators which conserve the corresponding discrete multisymplectic form as well as the discrete momentum mappings corresponding to symmetries we show that the usual notion of symplecticity along an infinite dimensional space of fields can be naturally obtained by making a spacetime split all of the aspects of our method are demonstrated with a nonlinear sine gordon equation including computational results and a comparison with other discretization schemes

proximal minimization methods with generalized bregman functions we consider methods for minimizing a convex function f that generate a sequence xk by taking xk 1 to be an approximate minimizer of f x dh x xk ck where ck 0 and dh is the d function of a bregman function h extensions are made to b functions that generalize bregman functions and cover more applications convergence is established under criteria amenable to implementation applications are made to nonquadratic multiplier methods for nonlinear programs

lagrangian solution techniques and bounds for loosely coupled mixed integer stochastic programs many production problems involve facility setups that lead to integer variables production decisions that are continuous and demands that are likely to be random while these problems can be quite difficult to solve we propose a model and an efficient solution technique for this basic class of stochastic mixed integer programs we use a set of scenarios to reflect uncertainty the resulting mathematical model is solved using lagrangian relaxation we show that the duality gap of our relaxation is bounded above by a constant that depends on the cost function and the number of branching points in the scenario tree we apply our technique to the problem of generating electric power numerical results indicate significant savings when the stochastic model is used instead of a deterministic one

convergence of proximal like algorithms we analyze proximal methods based on entropy like distances for the minimization of convex functions subject to nonnegativity constraints we prove global convergence results for the methods with approximate minimization steps and an ergodic convergence result for the case of finding a zero of a maximal monotone operator we also consider linearly constrained convex problems and establish a quadratic convergence rate result for linear programs our analysis allows us to simplify and extend the available convergence results for these methods

numerical solution of one dimensional sine gordon equation using high accuracy multiquadric quasi interpolation abstract in this paper we propose a numerical scheme to solve one dimensional sine gordon equation related to many scientific research topics by using high accuracy multiquadric quasi interpolation we use the derivatives of a multiquadric quasi interpolant to approximate the spatial derivatives and a finite difference to approximate the temporal derivative the advantages of the scheme are that it is meshfree and in each time step only a multiquadric quasi interpolant is employed so that the algorithm is very easy to implement the accuracy of our scheme is demonstrated by some test problems

an alternating linearization bundle method for convex optimization and nonlinear multicommodity flow problems we give a bundle method for minimizing the sum of two convex functions one of them being known only via an oracle of arbitrary accuracy each iteration involves solving two subproblems in which the functions are alternately represented by their linearizations our approach is motivated by applications to nonlinear multicommodity flow problems encouraging numerical experience on large scale problems is reported

numerical simulation of quasi periodic solutions of the sine gordon equation abstract analytic solutions of the sine gordon equation corresponding to periodic boundary conditions can be complicated especially if initial values are chosen in the vicinity of homoclinic orbits for such initial values it has been demonstrated that numerical solutions develop instabilities and may become chaotic because of the analytical complexities one cannot easily calculate the accuracy of the numerical solutions and therefore compare different numerical schemes in a straightforward manner in this note we evaluate numerical methods in terms of the nonlinear spectrum in particular it provides one with a means of comparing symplectic and nonsymplectic integrators for integrable infinite dimensional hamiltonian systems

applications of the method of partial inverses to convex programming decomposition a primal dual decomposition method is presented to solve the separable convex programming problem convergence to a solution and lagrange multiplier vector occurs from an arbitrary starting point the method is equivalent to the proximal point algorithm applied to a certain maximal monotone multifunction in the nonseparable case it specializes to a known method the proximal method of multipliers conditions are provided which guarantee linear convergence

linear differential equations with finite differential galois group abstract for a finite irreducible subgroup h psl c n and an irreducible h invariant curve z p c n such that c z h c t a standard differential operator l s t c t d d t is constructed for n 2 this is essentially klein s work for n 2 an actual calculation of l s t is done by computing an evaluation of invariants c x 1 x n h c t and applying a scalar form of a theorem of e compoint in a procedure also in some cases where z is unknown evaluations are produced this new method is tested for n 2 and for three irreducible subgroups of sl 3 this supplements 18 the theory developed here relates to and continues classical work of h a schwarz g fano f klein and a hurwitz

finite difference schemes for nonlinear wave equation that inherit energy conservation property abstract we propose two general finite difference schemes that inherit energy conservation property from nonlinear wave equations such as the nonlinear klein gordon equation nlkge one of proposed schemes is implicit and another is explicit many studies exist on fdss that inherit energy conservation property from nlkge and we can derive all of their schemes from the proposed general schemes in this paper the most important feature of our procedure is a rigorous discretization of variational derivatives using summation by parts which implies that the inherited properties are satisfied exactly because of this the derived schemes are expected to be numerically stable and yield solutions converging to pde solutions we make new fdss for fermi pasta ulam equation string vibration equation shimoji kawai equation ske and ebihara equation and verify numerically the inheritance of the energy conservation property for nlkge and ske

a decomposition procedure based on approximate newton directions the efficient solution of large scale linear and nonlinear optimization problems may require exploiting any special structure in them in an efficient manner we describe and analyze some cases in which this special structure can be used with very little cost to obtain search directions from decomposed subproblems we also study how to correct these directions using decomposable preconditioned conjugate gradient methods to ensure local convergence in all cases the choice of appropriate preconditioners results in a natural manner from the structure in the problem finally we conduct computational experiments to compare the resulting procedures with direct methods

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

information relaxations and duality in stochastic dynamic programs we describe a general technique for determining upper bounds on maximal values or lower bounds on minimal costs in stochastic dynamic programs in this approach we relax the nonanticipativity constraints that require decisions to depend only on the information available at the time a decision is made and impose a penalty that punishes violations of nonanticipativity in applications the hope is that this relaxed version of the problem will be simpler to solve than the original dynamic program the upper bounds provided by this dual approach complement lower bounds on values that may be found by simulating with heuristic policies we describe the theory underlying this dual approach and establish weak duality strong duality and complementary slackness results that are analogous to the duality results of linear programming we also study properties of good penalties finally we demonstrate the use of this dual approach in an adaptive inventory control problem with an unknown and changing demand distribution and in valuing options with stochastic volatilities and interest rates these are complex problems of significant practical interest that are quite difficult to solve to optimality in these examples our dual approach requires relatively little additional computation and leads to tight bounds on the optimal values

fast alternating linearization methods for minimizing the sum of two convex functions we present in this paper alternating linearization algorithms based on an alternating direction augmented lagrangian approach for minimizing the sum of two convex functions our basic methods require at most o 1 epsilon iterations to obtain an epsilon optimal solution while our accelerated i e fast versions of them require at most o 1 sqrt epsilon iterations with little change in the computational effort required at each iteration for both types of methods we present one algorithm that requires both functions to be smooth with lipschitz continuous gradients and one algorithm that needs only one of the functions to be so algorithms in this paper are gauss seidel type methods in contrast to the ones proposed by goldfarb and ma in fast multiple splitting algorithms for convex optimization columbia university 2009 where the algorithms are jacobi type methods numerical results are reported to support our theoretical conclusions and demonstrate the practical potential of our algorithms

invariant conserving finite difference algorithms for the nonlinear klein gordon equation a formalism for systematicaiiy deriving second order accurate finite difference algorithms which conserve certain invariant quantities in the original nonlinear pdes is presented three algorithms are derived for the nonlinear klein gordon equation nlkge based on the proposed formalism the local conservation laws of the nlkge form the basic starting point in our derivation which hinges essentially on the commutativity of certain finite difference operators such commutativity in the discrete approximations allows a preservation of the derivation properties of the continuous counterparts at the pde level with appropriate boundary conditions the proposed algorithms preserve in the discrete sense either the total system energy or the system s linear momentum several variants of the present algorithms and their relation to previously proposed algorithms are discussed an analysis of the accuracy and stability is conducted to compare the different variants of the proposed algorithms the preservation of energy of the present algorithms for the nlkge can also be viewed as providing a method of stabilization for conditionally stable algo thms for the linear wave equation the computer impiementation of the proposed algorithms with the treatment of the boundary conditions is presented in detail numerical examples are given concerning soliton collisions in the sine gordon equation the double sine gordon equation and the cb phi four equation the numerical results demonstrate that the present algorithms can preserve accurately up to 10 decimal digits the total system energy for a very coarse grid reliable algorithms for josephson junction models which contain dissipation damping mechanisms and driving bias current are obtained as direct by products of the proposed invariant conserving algorithms for the nlkge even though presented mainly for the 1 d case the proposed algorithms are generalizable to the 2 d and 3 d cases and to the case of complex valued nlkge

derivation of the discrete conservation laws for a family of finite difference schemes abstract we present conservative schemes in finite differences applied to nonlinear odes and nonlinear klein gordon schrodinger and dirac field equations a systematic approach is used and an explicit derivation of the corresponding conservation laws is given

distributional efficiency in multiobjective stochastic linear programming several concepts of distributional efficiency are proposed for the multiobjective stochastic linear programming mslp problem in contexts where the probability distribution of random parameters is known and the decision maker dm has an unknown multi attribute utility function belonging to a given glass u we present a general efficient set the u admissible solutions and two subsets the u unanimous and u advocated solutions the latter being particularly relevant to the case of a single dm we show how advocated solutions can be generated and or tested when u is the class of non decreasing additive concave functions

asynchronous level bundle methods in this paper we consider nonsmooth convex optimization problems with additive structure featuring independent oracles black boxes working in parallel existing methods for solving these distributed problems in a general form are synchronous in the sense that they wait for the responses of all the oracles before performing a new iteration in this paper we propose level bundle methods handling asynchronous oracles these methods require original upper bounds using upper models or scarce coordinations to deal with asynchronicity we prove their convergence using variational analysis techniques and illustrate their practical performance on a lagrangian decomposition problem

a fourth order avf method for the numerical integration of sine gordon equation abstract in this paper a new scheme which has energy preserving property is proposed for solving the sine gordon equation with periodic boundary conditions it is obtained by the fourier pseudo spectral method and the fourth order average vector field method in numerical experiments the new high order energy preserving scheme is compared with a number of existing numerical schemes for the one dimensional sine gordon equation the new high order energy preserving scheme for the two dimensional sine gordon equation is also investigated numerical results are addressed to further illustrate the conservation of energy and the evolutional behaviors of solitons

complexity of stochastic dual dynamic programming stochastic dual dynamic programming is a cutting plane type algorithm for multi stage stochastic optimization originated about 30 years ago in spite of its popularity in practice there does not exist any analysis on the convergence rates of this method in this paper we first establish the number of iterations i e iteration complexity required by a basic dual dynamic programming method for solving single scenario multi stage optimization problems by introducing novel mathematical tools including the saturation of search points we then refine these basic tools and establish the iteration complexity for an explorative dual dynamic programing method proposed herein and the classic stochastic dual dynamic programming method for solving more general multi stage stochastic optimization problems under the standard stage wise independence assumption our results indicate that the complexity of these methods mildly increases with the number of stages t in fact linearly dependent on t for discounted problems therefore they are efficient for strategic decision making which involves a large number of stages but with a relatively small number of decision variables in each stage without explicitly discretizing the state and action spaces these methods might also be pertinent to the related reinforcement learning and stochastic control areas

energy conservation issues in the numerical solution of the semilinear wave equation in this paper we discuss energy conservation issues related to the numerical solution of the semilinear wave equation as is well known this problem can be cast as a hamiltonian system that may be autonomous or not depending on the prescribed boundary conditions we relate the conservation properties of the original problem to those of its semi discrete version obtained by the method of lines subsequently we show that the very same properties can be transferred to the solutions of the fully discretized problem obtained by using energy conserving methods in the hbvms hamiltonian boundary value methods class similar arguments hold true for different types of hamiltonian partial differential equations e g the nonlinear schrodinger equation

superconvergence of the local discontinuous galerkin method for the sine gordon equation in one space dimension abstract in this paper we present superconvergence results for the local discontinuous galerkin ldg method for the sine gordon nonlinear hyperbolic equation in one space dimension we identify a special numerical flux and a suitable projection of the initial conditions for the ldg scheme for which the l 2 norm of the ldg solution and its spatial derivative are of order p 1 when piecewise polynomials of degree at most p are used our numerical experiments demonstrate optimal order of convergence we further prove superconvergence toward particular projections of the exact solutions more precisely we prove that the ldg solution and its spatial derivative are o h p 3 2 super close to particular projections of the exact solutions while computational results show higher o h p 2 convergence rate our analysis is valid for arbitrary regular meshes and for p p polynomials with arbitrary p 1 numerical experiments validating these theoretical results are presented

alternating linearization for structured regularization problems we adapt the alternating linearization method for proximal decomposition to structured regularization problems in particular to the generalized lasso problems the method is related to two well known operator splitting methods the douglas rachford and the peaceman rachford method but it has descent properties with respect to the objective function this is achieved by employing a special update test which decides whether it is beneficial to make a peaceman rachford step any of the two possible douglas rachford steps or none the convergence mechanism of the method is related to that of bundle methods of nonsmooth optimization we also discuss implementation for very large problems with the use of specialized algorithms and sparse data structures finally we present numerical results for several synthetic and real world examples including a three dimensional fused lasso problem which illustrate the scalability efficacy and accuracy of the method

on the numerical solution of the sine gordon equation in this the first of two papers on the numerical solution of the sine gordon equation we investigate the numerical behavior of a double discrete completely integrable discretization of the sine gordon equation for certain initial values in the vicinity of homoclinic manifolds this discretization admits an instability in the form of grid scale oscillations we clarify the nature of the instability through an analytical investigation supported by numerical experiments in particular a perturbation analysis of the associated linear spectral problem shows that the initial values used for the numerical experiments lie exponentially close to a homoclinic manifold this paves the way for the second paper where we use the non linear spectrum as a basis for comparing different numerical schemes

discrete singular convolution for the sine gordon equation abstract this paper explores the utility of a discrete singular convolution dsc algorithm for the integration of the sine gordon equation the initial values are chosen close to a homoclinic manifold for which previous methods have encountered significant numerical difficulties such as numerically induced spatial and temporal chaos a number of new initial values are considered including a case where the initial value is exactly on the homoclinic orbit the present algorithm performs extremely well in terms of accuracy efficiency simplicity stability and reliability

scenarios and policy aggregation in optimization under uncertainty a common approach in coping with multiperiod optimization problems under uncertainty where statistical information is not really enough to support a stochastic programming model has been to set up and analyze a number of scenarios the aim then is to identify trends and essential features on which a robust decision policy can be based this paper develops for the first time a rigorous algorithmic procedure for determining such a policy in response to any weighting of the scenarios the scenarios are bundled at various levels to reflect the availability of information and iterative adjustments are made to the decision policy to adapt to this structure and remove the dependence on hindsight

lagrange distributed approximating functionals lagrange distributed approximating functionals ldafs are proposed as the basis for a new collocation type method for accurately approximating functions and their derivatives both on and off discrete grids example applications are presented to illustrate the use of ldafs for solving the schr umlt o dinger equation and fokker planck equation ldafs are constructed by combining the daf concept with the lagrange interpolation scheme copyright ital 1997 ital the american physical society

bfc a branch and fix coordination algorithmic framework for solving some types of stochastic pure and mixed 0 1 programs we present a framework for solving some types of 0 1 multi stage scheduling planning problems under uncertainty in the objective function coe cients and the right hand side a scenario analysis scheme with full recourse is used the solution o ered for each scenario group at each stage takes into account all scenarios but without subordinating to any of them the constraints are modelled by a splitting variables representation via scenarios so a 0 1 model for each scenario is consid ered plus the non anticipativity constraints that equate the 0 1 variables from the same group of scenarios in each stage the mathematical representation of the model is very amenable for the proposed framework to deal with the 0 1 character of the variables a branch and fix coordi nation approach is introduced for coordinating the selection of the branching nodes and branching variables in the scenario subproblems to be jointly optimized some computational experience is reported for di erent types of problems

fast multiple splitting algorithms for convex optimization we present in this paper two different classes of general multiple splitting algorithms for solving finite dimensional convex optimization problems under the assumption that the function being minimized can be written as the sum of k convex functions each of which has a lipschitz continuous gradient we prove that the number of iterations needed by the first class of algorithms to obtain an epsilon optimal solution is o k 1 l epsilon where l is an upper bound on all of the lipschitz constants the algorithms in the second class are accelerated versions of those in the first class where the complexity result is improved to o sqrt k 1 l epsilon while the computational effort required at each iteration is almost unchanged to the best of our knowledge the complexity results presented in this paper are the first ones of this type that have been given for splitting and alternating direction type methods moreover all algorithms proposed in this paper are parallelizable which makes the

a decomposition method and its application to convex programming a method is proposed for minimizing a function of the form q x d f1 x fm x where q is definite quadratic and fi are proper closed convex the key feature of the method is its capability of reducing the problem to a sequence of subproblems of the form min q x z fi x the method is an extension of the successive projection method given in han s p 1988 a successive projection method math programming 40 1 14 and has some similar features as the partial inverse method of spingarn spingarn j e 1983 partial inverse of a monotone operator appl math optim 10 247 265 in combination with the proximal point algorithm martinet b 1970 regularisation d inequations variationelles par approximations successives rev francaise inform rech oper 154 159 martinet b 1972 determination approchee d un point fixe d une application pseudo contractante c r acad sci paris 274 163 165 rockafellar r t 1976 monotone operators and the proximal point algorithm siam j control

the sine gordon equation in the finite line a rather complete numerical study of the perturbed and nonperturbed sine gordon equation in one dimensional cartesian coordinates and finite domains is presented the numerical study is based on five three point linearly implicit finite difference methods of different spatial accuracy for the unperturbed sge the accuracy of these five methods is found to be very sensitive to the implicitness parameter less sensitive to the spatial order of accuracy and almost no sensitive to the time step for second order accurate temporal discretizations the largest errors of the five methods were found to occur at the front of the soliton solution of the unperturbed sge but the techniques were very accurate for long time computations of the unperturbed sge in both infinite domains and in finite domains upon many collisions of the kinks and antikinks with the boundaries despite the fact that they do not preserve a discrete energy it is shown that the largest values of the kinetic and total energies and the smallest values of the strain and potential energies occur when solitons and antisolitons strike on the boundaries subject to homogeneous neumann conditions and the four energies remain constant between collisions with the boundaries for the malomed s perturbation one of the fronts of the soliton soliton doublet and soliton antisoliton doublet is trapped at one boundary and the other one undergoes a change in amplitude and its front describes a curved trajectory the soliton antisoliton breather oscillates in a damped manner and slowly drifts towards one boundary the soliton antisoliton soliton soliton doublet and soliton antisoliton doublet of the unperturbed sge are robust under kivshar and malomed s perturbations whereas the soliton antisoliton breather loses its integrity becomes a kink antikink pair and then yields a chaotic behaviour in both space and time small amplifications proportional to the first order temporal derivative of the amplitude do not alter substantially the initial dynamics of the soliton antisoliton soliton soliton doublet and soliton antisoliton doublet but their amplitudes exhibit oscillatory behaviour of increasing amplitude at large times however small amplifications cause first a transition from the soliton antisoliton breather to the kink antikink solution a linearized stability analysis of the spatially homogeneous solutions of the unperturbed sge is performed and analytical solutions are obtained by means of the method of separation of variables for both homogeneous dirichlet and neumann boundary conditions and the results show that the solution corresponding to a zero amplitude is oscillatory in both space and time whereas that corresponding to an amplitude equal to p may exhibit exponential or linear growth depending on the length of the domain if either of these growths is present it is shown that the evolution of small initial perturbations may result in homoclinic crossings and chaotic behaviour in both space and time in the absence of perturbations in the sge and the existence of an attractor when perturbations of the malomed s type are present in either case the solution may exhibit certain spatial symmetries or antisymmetries at large times finally the results presented in this paper indicate that the accurate computation of homoclinic orbits of the sge is not only controlled by the spatial accuracy of the numerical method but also by its energy preserving characteristics and perhaps by the number of grid points used in the numerical technique

finite difference calculus invariant structure of a class of algorithms for the nonlinear klein gordon equation in a previous work the authors have presented a formalism for deriving systematically invariant symmetric finite difference algorithms for nonlinear evolution differential equations that admit conserved quantities this formalism is herein cast in the context of exact finite difference calculus the algorithms obtained from the proposed formalism are shown to derive exactly from discrete scalar potential functions using finite difference calculus in the same sense as that of the corresponding differential equation being derivable from its associated energy function a conserved quantity a clear ramification of this result is that the derived algorithms preserve certain discrete invariant quantities which are the consistent counterpart of the invariant quantities in the continuous case results on the nonlinear stability of a class of algorithms that are derived using the proposed formalism and that preserve energy or linear momentum are discussed in the context of finite difference calculus some

on linear differential equations with reductive galois group abstract given a connection on a meromorphic vector bundle over a compact riemann surface with reductive differential galois group we associate to it a projective variety connections such that their associated projective variety is curves can be classified up to projective equivalence using ruled surfaces in particular such a meromorphic connection is the pullback of a standard connection this extends a similar result by klein for second order ordinary linear differential equations to a broader class of equations

sympletic finite difference approximations of the nonlinear klein gordon equation we analyze three finite difference approximations of the nonlinear klein gordon equation and show that they are directly related to symplectic mappings two of the schemes the perring skyrme and ablowitz kruskal ladik are long established and the third is a new higher order accurate scheme we test the schemes on traveling wave and periodic breather problems over long time intervals and compare their accuracy and computational costs with those of symplectic and nonsymplectic method of lines approximations and a nonsymplectic energy conserving method

exact solutions of semiclassical non characteristic cauchy problems for the sine gordon equation abstract the use of the sine gordon equation as a model of magnetic flux propagation in josephson junctions motivates studying the initial value problem for this equation in the semiclassical limit in which the dispersion parameter e tends to zero assuming natural initial data having the profile of a moving 2 kink at time zero we analytically calculate the scattering data of this completely integrable cauchy problem for all e 0 sufficiently small and further we invert the scattering transform to calculate the solution for a sequence of arbitrarily small e this sequence of exact solutions is analogous to that of the well known n soliton or higher order soliton solutions of the focusing nonlinear schrodinger equation we then use plots obtained from a careful numerical implementation of the inverse scattering algorithm for reflectionless potentials to study the asymptotic behavior of solutions in the semiclassical limit in the limit e 0 one observes the appearance of nonlinear caustics i e curves in space time that are independent of e but vary with the initial data and that separate regions in which the solution is expected to have different numbers of nonlinear phases in the appendices we give a self contained account of the cauchy problem from the perspectives of both inverse scattering and classical analysis picard iteration specifically appendix a contains a complete formulation of the inverse scattering method for generic l 1 sobolev initial data and appendix b establishes the well posedness for l p sobolev initial data which in particular completely justifies the inverse scattering analysis in appendix a

proximal decomposition via alternating linearization a new approximate proximal point method for minimizing the sum of two convex functions is introduced it replaces the original problem by a sequence of regularized subproblems in which the functions are alternately represented by linear models the method updates the linear models and the prox center as well as the prox coefficient it is monotone in terms of the objective values and converges to a solution of the problem if any a dual version of the method is derived and analyzed applications of the methods to multistage stochastic programming problems are discussed and preliminary numerical experience is presented

an engineer s guide to soliton phenomena application of the finite element method abstract the paper attempts an elementary survey of the physical and mathematical background appertaining to solitons and discusses in particular the numerical solution of three types of dispersive nonlinear partial differential equations exhibiting soliton type solutions namely the korteweg de vries equation the nonlinear schrodinger equation and the sine gordon equation throughout this study a semidiscrete galerkin method is applied using a finite element discretization in space and a step by step time integration of the resulting system of nonlinear ordinary differential equations depending upon the special type of the evolutionary equation the application of a petrov galerkin procedure may increase significantly the numerical stability accuracy and effectivity of the different approaches are demonstrated on a series of computer plots

a proximal based decomposition method for convex minimization problems this paper presents a decomposition method for solving convex minimization problems at each iteration the algorithm computes two proximal steps in the dual variables and one proximal step in the primal variables we derive this algorithm from rockafellar s proximal method of multipliers which involves an augmented lagrangian with an additional quadratic proximal term the algorithm preserves the good features of the proximal method of multipliers with the additional advantage that it leads to a decoupling of the constraints and is thus suitable for parallel implementation we allow for computing approximately the proximal minimization steps and we prove that under mild assumptions on the problem s data the method is globally convergent and at a linear rate the method is compared with alternating direction type methods and applied to the particular case of minimizing a convex function over a finite intersection of closed convex sets

stochastic dual dynamic integer programming multistage stochastic integer programming msip combines the difficulty of uncertainty dynamics and non convexity and constitutes a class of extremely challenging problems a common formulation for these problems is a dynamic programming formulation involving nested cost to go functions in the linear setting the cost to go functions are convex polyhedral and decomposition algorithms such as nested benders decomposition and its stochastic variant stochastic dual dynamic programming sddp which proceed by iteratively approximating these functions by cuts or linear inequalities have been established as effective approaches however it is difficult to directly adapt these algorithms to msip due to the nonconvexity of integer programming value functions in this paper we propose an extension to sddp called stochastic dual dynamic integer programming sddip for solving msip problems with binary state variables the crucial component of the algorithm is a new reformulation of the subproblems in each stage and a new class of cuts termed lagrangian cuts derived from a lagrangian relaxation of a specific reformulation of the subproblems in each stage where local copies of state variables are introduced we show that the lagrangian cuts satisfy a tightness condition and provide a rigorous proof of the finite convergence of sddip with probability one we show that under fairly reasonable assumptions an msip problem with general state variables can be approximated by one with binary state variables to desired precision with only a modest increase in problem size thus our proposed sddip approach is applicable to very general classes of msip problems extensive computational experiments on three classes of real world problems namely electric generation expansion financial portfolio management and network revenue management show that the proposed methodology is very effective in solving large scale multistage stochastic integer optimization problems

the dual reciprocity boundary element method drbem for two dimensional sine gordon equation this paper presents the dual reciprocity boundary element method drbem for solving two dimensional sine gordon sg equation the integral equation formulation employs the fundamental solution of the laplace equation and hence a domain integral arises in the boundary integral equation furthermore the time derivatives are approximated by the time stepping method and the domain integral also appears from these approximations the domain integral is transformed into boundary integral by using the dual reciprocity method drm the linear radial basis function rbf is employed for drm the dynamics of line solitons and ring solitons of circular and elliptic shapes are studied numerical results are presented for some problems involving line and ring solitons to demonstrate the usefulness and accuracy of this approach

evaluation of mixed crank nicolson scheme and tau method for the solution of klein gordon equation abstract numerical method based on the crank nicolson scheme and the tau method is proposed for solving nonlinear klein gordon equation nonlinear klein gordon equation is reduced by crank nicolson scheme to the system of ordinary differential equations then tau method is used to solve this system by using interpolating scaling functions and operational matrix of derivative the order of convergence is proposed and some numerical examples are included to demonstrate the validity and applicability of the technique the method is easy to implement and produces accurate results

stopping rules for a class of sampling based stochastic programming algorithms monte carlo sampling based algorithms hold much promise for solving stochastic programs with many scenarios a critical component of such algorithms is a stopping criterion to ensure the quality of the solution in this paper we develop a stopping rule theory for a class of algorithms that estimate bounds on the optimal objective function value by sampling we provide rules for selecting sample sizes and terminating the algorithm under which asymptotic validity of confidence intervals for the quality of the proposed solution can be verified empirical coverage results are given for a simple example

numerical solution of the sine gordon equation two new difference schemes are proposed for the sine gordon equation they have the advantage that there is a discrete energy which is conserved their stability and convergence is proved also we show some numerical computations using one of these schemes

progressive hedging as a meta heuristic applied to stochastic lot sizing abstract in a great many situations the data for optimization problems cannot be known with certainty and furthermore the decision process will take place in multiple time stages as the uncertainties are resolved this gives rise to a need for stochastic programming sp methods that create solutions that are hedged against future uncertainty the progressive hedging algorithm pha of rockafellar and wets is a general method for sp we cast the pha in a meta heuristic framework with the sub problems generated for each scenario solved heuristically rather than using an approximate search algorithm for the exact problem as is typically the case in the meta heuristic literature we use an algorithm for sub problems that is exact in its usual context but serves as a heuristic for our meta heuristic computational results reported for stochastic lot sizing problems demonstrate that the method is effective

numerical inverse spectral transform for the periodic sine gordon equation theta function solutions and their linearized stability abstract the inverse spectral transform for integrable nonlinear ordinary and partial differential equations such as the toda lattice korteweg de vries sine gordon and nonlinear schrodinger equations provides explicit algorithms to generate exact solutions under periodic or quasiperiodic boundary conditions these oscillatory wavetrains may be prescribed a priori to consist of a nonlinear superposition of n phases j x t j x j t 0 j j 1 n where the wave is 2 periodic independently in each phase this paper exhibits the numerical implementation of the inverse spectral solution of the sine gordon equation the general construction is outlined and then implemented for n 1 2 and 3 we compute 1 the exact theta function solutions 2 the floquet spectrum of x periodic solutions 3 the labelling of linearized instabilities of n phase solutions in terms of spectral data and 4 the linearized growth rate in each unstable mode the associated surfaces q n x t are displayed to illustrate a variety of spatial and dynamical phenomena in the oscillatory solution space of this integrable system