validation analysis of mirror descent stochastic approximation method the main goal of this paper is to develop accuracy estimates for stochastic programming problems by employing stochastic approximation sa type algorithms to this end we show that while running a mirror descent stochastic approximation procedure one can compute with a small additional effort lower and upper statistical bounds for the optimal objective value we demonstrate that for a certain class of convex stochastic programs these bounds are comparable in quality with similar bounds computed by the sample average approximation method while their computational cost is considerably smaller

a hybrid discontinuous galerkin method for computing the ground state solution of bose einstein condensates a numerical method for computing the ground state solution of bose einstein condensates modeled by the gross pitaevskii equation is presented in this method the three dimensional computational domain is divided into hexahedral elements in which the solution is approximated by a sum of basis functions both polynomial and plane wave bases are considered for this purpose and lagrange multipliers are introduced to weakly enforce the interelement continuity of the solution the ground state is computed by an iterative procedure for minimizing the energy the performance results obtained for several numerical experiments demonstrate that the proposed method is more computationally efficient than similar solution approaches based on the standard higher order finite element method

exact converging bounds for stochastic dual dynamic programming via fenchel duality the stochastic dual dynamic programming sddp algorithm has become one of the main tools to address convex multistage stochastic optimal control problem recently a large amount of work has been devoted to improve the convergence speed of the algorithm through cut selection and regularization or to extend the field of applications to non linear integer or risk averse problems however one of the main downside of the algorithm remains the difficulty to give an upper bound of the optimal value usually estimated through monte carlo methods and therefore difficult to use in the algorithm stopping criterion in this paper we present a dual sddp algorithm that yields a converging exact upper bound for the optimal value of the optimization problem incidently we show how to compute an alternative control policy based on an inner approximation of bellman value functions instead of the outer approximation given by the standard sddp algorithm we illustrate the approach on an energy production problem involving zones of production and transportation links between the zones the numerical experiments we carry out on this example show the effectiveness of the method

stochastic lipschitz dynamic programming we propose a new algorithm for solving multistage stochastic mixed integer linear programming milp problems with complete continuous recourse in a similar way to cutting plane methods we construct nonlinear lipschitz cuts to build lower approximations for the non convex cost to go functions an example of such a class of cuts are those derived using augmented lagrangian duality for milps the family of lipschitz cuts we use is milp representable so that the introduction of these cuts does not change the class of the original stochastic optimization problem we illustrate the application of this algorithm on two case studies comparing our approach with the convex relaxation of the problems for which we can apply sddp and for a discretized approximation applying sddip

boundary behavior of interior point algorithms in linear programming this paper studies the boundary behavior of some interior point algorithms for linear programming the algorithms considered are karmarkar s projective rescaling algorithm the linear rescaling algorithm which was proposed as a variation on karmarkar s algorithm and the logarithmic barrier technique the study includes both the continuous trajectories of the vector fields induced by these algorithms and also the discrete orbits it is shown that although the algorithms are defined on the interior of the feasible polyhedron they actually determine differentiable vector fields on the closed polyhedron conditions are given under which a vector field gives rise to trajectories that each visit the neighborhoods of all the vertices of the klee minty cube the linear rescaling algorithm satisfies these conditions thus limits of such trajectories obtained when a starting point is pushed to the boundary may have an exponential number of breakpoints it is shown that limits of projective rescaling trajectories may have a linear number of such breakpoints however projective rescaling trajectories may visit the neighborhoods of linearly many vertices the behavior of the linear rescaling algorithm near vertices is analyzed it is proved that all the trajectories have a unique asymptotic direction of convergence to the optimum

hermite spectral methods with a time dependent scaling for parabolic equations in unbounded domains hermite spectral methods are investigated for linear diffusion equations and nonlinear convection diffusion equations in unbounded domains when the solution domain is unbounded the diffusion operator no longer has a compact resolvent which makes the hermite spectral methods unstable to overcome this difficulty a time dependent scaling factor is employed in the hermite expansions which yields a positive bilinear form as a consequence stability and spectral convergence can be established for this approach the present method plays a similar role in the stability of the similarity transformation technique proposed by funaro and kavian math comp 57 1991 pp 597 619 however since coordinate transformations are not required the present approach is more efficient and is easier to implement in fact with the time dependent scaling the resulting discretization system is of the same form as that associated with the classical straightforward but unstable hermite spectral method numerical experiments are carried out to support the theoretical stability and convergence results

efficient spectral computation of the stationary states of rotating bose einstein condensates by preconditioned nonlinear conjugate gradient methods we propose a preconditioned nonlinear conjugate gradient method coupled with a spectral spatial dis cretization scheme for computing the ground states gs of rotating bose einstein condensates bec modeled by the gross pitaevskii equation gpe we first start by reviewing the classical gradient flow also known as imaginary time imt method which considers the problem from the pde standpoint leading to numerically solve a dissipative equation based on this imt equation we analyze the forward euler fe crank nicolson cn and the classical backward euler be schemes for linear problems and recognize classical power iterations allowing us to derive convergence rates by considering the alternative point of view of minimization problems we propose the preconditioned gradient pg and conjugate gradient pcg methods for the gs computation of the gpe we investigate the choice of the preconditioner which plays a key role in the acceleration of the convergence process the performance of the new algorithms is tested in 1d 2d and 3d we conclude that the pcg method outperforms all the previous methods most particularly for 2d and 3d fast rotating becs while being simple to implement

a new generalized laguerre spectral approximation and its applications a new family of generalized laguerre polynomials is introduced various orthogonal projections are investigated some approximation results are established as an example of their important applications the mixed spherical harmonic generalized laguerre approximation is developed a mixed spectral scheme is proposed for a three dimensional model problem its convergence is proved numerical results demonstrate the high accuracy of this new spectral method

how good are interior point methods klee minty cubes tighten iteration complexity bounds by refining a variant of the klee minty example that forces the central path to visit all the vertices of the klee minty n cube we exhibit a nearly worst case example for path following interior point methods namely while the theoretical iteration complexity upper bound is o 2 n n frac 5 2 we prove that solving this n dimensional linear optimization problem requires at least 2 n 1 iterations

energy minimization related to the nonlinear schrodinger equation in this the window of the sobolev gradient technique to the problem of minimizing a schrodinger functional associated with a nonlinear schrodinger equation we show that gradients act in a suitably chosen sobolev space sobolev gradients can be used in finite difference and finite element settings in a computationally efficient way to find minimum energy states of schrodinger functionals

stochastic decomposition an algorithm for two stage linear programs with recourse we present a cutting plane algorithm for two stage stochastic linear programs with recourse motivated by benders decomposition our method uses randomly generated observations of random variables to construct statistical estimates of supports of the objective function in general the resulting piecewise linear approximations do not agree with the objective function in finite time however certain subsequences of the estimated supports are shown to accumulate at supports of the objective function with probability one from this we establish the convergence of the algorithm under relatively mild assumptions

approximating time evolution related to ginzburg landau functionals via sobolev gradient methods in a finite element setting sobolev gradients have previously 1 been used to approximate time evolution related to a model a functional in a finite difference setting in this journal here a related approach in a finite element setting is discussed

stability analysis and continuation for the coupled gross pitaevskii equations abstract we study linear stability analysis for the coupled gross pitaevskii equations cgpes which are used as the governing equations for rotating two component bose einstein condensates bec and ultrarapidly rotating two component rydberg dressed bec the cgpes describe the coexistence of superfluidity and crystallization of bec simultaneously we show numerically that the linear stability of discrete steady state solutions is related to the angular velocity next we describe some continuation algorithms for computing the ground state solutions of the cgpes specifically one of them can efficiently handle the case when long range intra component interactions of the two components are different for ultrarapid rotation our numerical experiments demonstrate a competition between the supersolid crystal structure and the rotating induced vortex lattice which gives rise to new phases

midas a mixed integer dynamic approximation scheme mixed integer dynamic approximation scheme midas is a new sampling based algorithm for solving finite horizon stochastic dynamic programs with monotonic bellman functions midas approximates these value functions using step functions leading to stage problems that are mixed integer programs we provide a general description of midas and prove its almost sure convergence to a 2t varepsilon optimal policy for problems with t stages when the bellman functions are known to be monotonic and the sampling process satisfies standard assumptions

analysis and approximation of the ginzburg landau model of superconductivity the authors consider the ginzburg landau model for superconductivity first some well known features of superconducting materials are reviewed and then various results concerning the model the resultant differential equations and their solution on bounded domains are derived then finite element approximations of the solutions of the ginzburg landau equations are considered and error estimates of optimal order are derived

analysis of stochastic dual dynamic programming method abstract in this paper we discuss statistical properties and convergence of the stochastic dual dynamic programming sddp method applied to multistage linear stochastic programming problems we assume that the underline data process is stagewise independent and consider the framework where at first a random sample from the original true distribution is generated and consequently the sddp algorithm is applied to the constructed sample average approximation saa problem then we proceed to analysis of the sddp solutions of the saa problem and their relations to solutions of the true problem finally we discuss an extension of the sddp method to a risk averse formulation of multistage stochastic programs we argue that the computational complexity of the corresponding sddp algorithm is almost the same as in the risk neutral case

multi stage stochastic optimization applied to energy planning this paper presents a methodology for the solution of multistage stochastic optimization problems based on the approximation of the expected cost to go functions of stochastic dynamic programming by piecewise linear functions no state discretization is necessary and the combinatorial explosion with the number of states the well known curse of dimensionality of dynamic programming is avoided the piecewise functions are obtained from the dual solutions of the optimization problem at each stage and correspond to benders cuts in a stochastic multistage decomposition framework a case study of optimal stochastic scheduling for a 39 reservoir system is presented and discussed

a newton method with adaptive finite elements for solving phase change problems with natural convection we present a new numerical system using finite elements with mesh adaptivity for the simulation of solid liquid phase change systems in the liquid phase the natural convection flow is simulated by solving the incompressible navier stokes equations with boussinesq approximation a variable viscosity model allows the velocity to progressively vanish in the solid phase through an intermediate mushy region the phase change is modeled by introducing an implicit enthalpy source term in the heat equation the final system of equations describing the liquid solid system by a single domain approach is solved using a newton iterative algorithm the space discretization is based on a p2 p1 taylor hood finite elements and mesh adaptivity by metric control is used to accurately track the solid liquid interface or the density inversion interface for water flows the numerical method is validated against classical benchmarks that progressively add strong non linearities in the system of equations natural convection of air natural convection of water melting of a phase change material and water freezing very good agreement with experimental data is obtained for each test case proving the capability of the method to deal with both melting and solidification problems with convection the presented numerical method is easy to implement using freefem software using a syntax close to the mathematical formulation

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

fourier splitting methods for the dynamics of rotating bose einstein condensates abstract we present a new method to propagate rotating bose einstein condensates subject to explicitly time dependent trapping potentials using algebraic techniques we combine magnus expansions and splitting methods to yield any order methods for the multivariate and nonautonomous quadratic part of the hamiltonian that can be computed using only fourier transforms at the cost of solving a small system of polynomial equations the resulting scheme solves the challenging component of the nonlinear hamiltonian and can be combined with optimized splitting methods to yield efficient algorithms for rotating bose einstein condensates the method is particularly efficient for potentials that can be regarded as perturbed rotating and trapped condensates e g for small nonlinearities since it retains the near integrable structure of the problem for large nonlinearities the method remains highly efficient if higher order p 2 is sought furthermore we show how it can be adapted to the presence of dissipation terms numerical examples illustrate the performance of the scheme

on solving multistage stochastic programs with coherent risk measures we consider a class of multistage stochastic linear programs in which at each stage a coherent risk measure of future costs is to be minimized a general computational approach based on dynamic programming is derived that can be shown to converge to an optimal policy by computing an inner approximation to future cost functions we can evaluate an upper bound on the cost of an optimal policy and an outer approximation delivers a lower bound the approach we describe is particularly useful in sampling based algorithms and a numerical example is provided to show the efficacy of the methodology when used in conjunction with stochastic dual dynamic programming

spectral methods for the navier stokes equations with one infinite and two periodic directions abstract two numerical methods were designed to solve the time dependent three dimensional incompressible navier stokes equations in boundary layers method a semi infinite domain and mixing layers or wakes method b fully infinite domain their originality lies in the use of rapidly decaying spectral basis functions to approximate the vertical dependence of the solutions combined with one method a or two method b slowly decaying extra functions for each wave vector that exactly represent the irrotational component of the solution at large distances both methods eliminate the pressure term as part of the formulation thus avoiding fractional step time integration they yield rapid convergence and are free of spurious modes in the orr sommerfeld spectra they are also efficient although the operation count is of order n 2 n is the number of modes in the infinite direction these methods have been used for extensive direct numerical simulations of transition and turbulence a new time integration scheme with low storage requirements and good stability properties is also described

convergence analysis of high order time splitting pseudo spectral methods for rotational gross pitaevskii equations a convergence analysis of time splitting pseudo spectral methods adapted for time dependent gross pitaevskii equations with additional rotation term is given for the time integration high order exponential operator splitting methods are studied and the space discretization relies on the generalized laguerre fourier spectral method with respect to the x y x y variables as well as the hermite spectral method in the z z direction essential ingredients in the stability and error analysis are a general functional analytic framework of abstract nonlinear evolution equations fractional power spaces defined by the principal linear part a sobolev type inequality in a curved rectangle and results on the asymptotical distribution of the nodes and weights associated with gau laguerre quadrature the obtained global error estimate ensures that the nonstiff convergence order of the time integrator and the spectral accuracy of the spatial discretization are retained provided that the problem data satisfy suitable regularity requirements a numerical example confirms the theoretical convergence estimate

evaluating policies in risk averse multi stage stochastic programming we consider a risk averse multi stage stochastic program using conditional value at risk as the risk measure the underlying random process is assumed to be stage wise independent and a stochastic dual dynamic programming sddp algorithm is applied we discuss the poor performance of the standard upper bound estimator in the risk averse setting and propose a new approach based on importance sampling which yields improved upper bound estimators modest additional computational effort is required to use our new estimators our procedures allow for significant improvement in terms of controlling solution quality in sddp style algorithms in the risk averse setting we give computational results for multi stage asset allocation using a log normal distribution for the asset returns

computing ground states of spin 1 bose einstein condensates by the normalized gradient flow in this paper we propose an efficient and accurate numerical method for computing the ground state of spin 1 bose einstein condensates becs by using the normalized gradient flow or imaginary time method the key idea is to find a third projection or normalization condition based on the relation between the chemical potentials so that the three projection parameters used in the projection step of the normalized gradient flow are uniquely determined by this condition as well as the other two physical conditions given by the conservation of total mass and total magnetization this allows us to successfully extend the most popular and powerful normalized gradient flow or imaginary time method for computing the ground state of a single component bec to compute the ground state of spin 1 becs an efficient and accurate discretization scheme the backward forward euler sine pseudospectral method is proposed to discretize the normalized gradient flow extensive numerical results on ground states of spin 1 becs with ferromagnetic antiferromagnetic interaction and harmonic optical lattice potential in one three dimensions are reported to demonstrate the efficiency of our new numerical method

on the convergence of decomposition methods for multistage stochastic convex programs we prove the almost sure convergence of a class of sampling based nested decomposition algorithms for multistage stochastic convex programs in which the stage costs are general convex functions of the decisions and uncertainty is modelled by a scenario tree as special cases our results imply the almost sure convergence of stochastic dual dynamic programming cutting plane and partial sampling cupps algorithm and dynamic outer approximation sampling algorithms when applied to problems with general convex cost functions

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

numerical approximations of the ginzburg landau models for superconductivity in this paper we review various methods for the numerical approximations of the ginzburg landau models of superconductivity particular attention is given to the different treatment of gauge invariance in both the finite element finite difference and finite volume settings representative theoretical results typical numerical simulations and computational challenges are presented generalizations to other relevant models are also discussed

two level discretization techniques for ground state computations of bose einstein condensates this work presents a new methodology for computing ground states of bose einstein condensates based on finite element discretizations on two different scales of numerical resolution in a preprocessing step a low dimensional coarse generalized finite element space is constructed it is based on a local orthogonal decomposition of the solution space and exhibits high approximation properties the nonlinear eigenvalue problem that characterizes the ground state is solved by some suitable iterative solver exclusively in this low dimensional space without significant loss of accuracy when compared with the solution of the full fine scale problem the preprocessing step is independent of the types and numbers of bosons a postprocessing step further improves the accuracy of the method we present rigorous a priori error estimates that predict convergence rates h 3 for the ground state eigenfunction and h 4 for the corresponding eigenvalue without pre asymptotic effects h being the coarse scale discretization parameter numerical experiments indicate that these high rates may still be pessimistic

a generalized laguerre fourier hermite pseudospectral method for computing the dynamics of rotating bose einstein condensates a time splitting generalized laguerre fourier hermite pseudospectral method is proposed for computing the dynamics of rotating bose einstein condensates becs in two and three dimensions the new numerical method is based on the following i the use of a time splitting technique for decoupling the nonlinearity ii the adoption of polar coordinates in two dimensions and cylindrical coordinates in three dimensions such that the angular rotation term becomes constant coefficient and iii the construction of eigenfunctions for the linear operator by properly scaling the generalized laguerre fourier and hermite functions the new method enjoys the following properties i it is explicit time reversible and time transverse invariant ii it conserves the position density and is spectrally accurate in space and second order or fourth order accurate in time and iii it solves the problem in the original whole space instead of in a truncated artificial computational domain the method is also extend

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

sobolev gradients and the ginzburg landau functional we describe a sobolev gradient method for finding minima of the ginzburg landau functional for superconductivity this method leads to a particularly simple algorithm which avoids consideration of the nonlinear boundary conditions associated with the ginzburg landau equations

an analysis of schwarz waveform relaxation domain decomposition methods for the imaginary time linear schrodinger and gross pitaevskii equations the aim of this paper is to derive and numerically validate some asymptotic estimates of the convergence rate of classical and quasi optimal schwarz waveform relaxation swr domain decomposition methods applied to the computation of the stationary states of the one dimensional linear and nonlinear schrodinger equations with a space dependent potential although swr methods are currently used for efficiently solving high dimensional partial differential equations their convergence analysis and most particularly obtaining expressions of their convergence rate remains largely open even in one dimension except in simple cases in this work we tackle this problem for linear and nonlinear one dimensional schrodinger equations by developing techniques which can be extended to higher dimensional problems and other types of pdes the approach combines the method developed in gander and halpern siam j num anal 45 2 666 697 2007 for the linear advection reaction diffusion equation and the theory of inhomogeneous pseudodifferential operators in conjunction with the associated symbolic asymptotic expansions for computing the stationary states we consider the imaginary time formulation of the schrodinger equation based on the normalized gradient flow ngf method and use a semi implicit euler scheme for the discretization some numerical results in the one dimensional case illustrate the analysis for both the linear schrodinger and gross pitaevskii equations

efficient numerical methods for computing ground states of spin 1 bose einstein condensates based on their characterizations in this paper we propose efficient numerical methods for computing ground states of spin 1 bose einstein condensates becs with without the ioffe pritchard magnetic field b x when b x 0 a numerical method is introduced to compute the ground states and it is also applied to study properties of ground states numerical results suggest that the densities of m f 1 components in ground states are identical for any nonzero b x in particular if b x b 0 is a constant the ground states satisfy the single mode approximation when b x 0 efficient and simpler numerical methods are presented to solve the ground states of spin 1 becs based on their ferromagnetic antiferromagnetic characterizations numerical simulations show that our methods are more efficient than those in the literature in addition some conjectures are made from our numerical observations

a new sobolev gradient method for direct minimization of the gross pitaevskii energy with rotation in this paper we improve traditional steepest descent methods for the direct minimization of the gross pitaevskii gp energy with rotation at two levels we first define a new inner product to equip the sobolev space h 1 and derive the corresponding gradient second for the treatment of the mass conservation constraint we use a projection method that avoids more complicated approaches based on modified energy functionals or traditional normalization methods the descent method with these two new ingredients is studied theoretically in a hilbert space setting and we give a proof of the global existence and convergence in the asymptotic limit to a minimizer of the gp energy the new method is implemented in both finite difference and finite element two dimensional settings and is used to compute various complex configurations with vortices of rotating bose einstein condensates the new sobolev gradient method shows better numerical performances compared to classical l 2 or h 1 gradient methods especially when high rotation rates are considered

using krylov subspace and spectral methods for solving complementarity problems in many body contact dynamics simulation summary many body dynamics problems are expected to handle millions of unknowns when for instance investigating the three dimensional flow of granular material unfortunately the size of the problems tractable by existing numerical solution techniques is severely limited on convergence grounds this is typically the case when the equations of motion embed a differential variational inequality problem that captures contact and possibly frictional interactions between rigid and or flexible bodies as the size of the physical system increases the speed and or the quality of the numerical solution decreases this paper describes three methods the gradient projected minimum residual method the preconditioned spectral projected gradient with fallback method and the modified proportioning with reduced gradient projection method that demonstrate better scalability than the projected jacobi and gauss seidel methods commonly used to solve contact problems that draw on a differential variational inequality based modeling approach copyright 2013 john wiley sons ltd

the use of generalized laguerre polynomials in spectral methods for nonlinear differential equations the expansion of products of generalized laguerre polynomials l n x in terms of a series of generalized laguerre polynomials is considered the expansion coefficients which are equal to triple product integrals of generalized laguerre polynomials are expressed in terms of a three index recurrence relation this is reduced to a one index relation which facilitates computation of the expansion coefficients the results are useful in the solution of nonlinear differential equations when it is desired to express products of generalized laguerre polynomials as a linear series of these functions as an application we use the results to compute a spectral solution of a nonlinear boundary value problem namely the blasius equation on a semi infinite interval by using a truncated series containing the first eight polynomials l1 2n x a solution is obtained within 4 accuracy

stable and efficient spectral methods in unbounded domains using laguerre functions stable and efficient spectral methods using laguerre functions are proposed and analyzed for model elliptic equations on regular unbounded domains it is shown that spectral galerkin approximations based on laguerre functions are stable and convergent with spectral accuracy in the usual not weighted sobolev spaces efficient accurate and well conditioned algorithms using laguerre functions are developed and implemented numerical results indicating the spectral convergence rate and effectiveness of these algorithms are presented

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

laguerre galerkin method for nonlinear partial differential equations on a semi infinite interval summary a laguerre galerkin method is proposed and analyzed for the burgers equation and benjamin bona mahony bbm equation on a semi infinite interval by reformulating these equations with suitable functional transforms it is shown that the laguerre galerkin approximations are convergent on a semi infinite interval with spectral accuracy an efficient and accurate algorithm based on the laguerre galerkin approximations to the transformed equations is developed and implemented numerical results indicating the high accuracy and effectiveness of this algorithm are presented

a fourth order time splitting laguerre hermite pseudospectral method for bose einstein condensates a fourth order time splitting laguerre hermite pseudospectral method is introduced for bose einstein condensates becs in three dimensions with cylindrical symmetry the method is explicit time reversible and time transverse invariant it conserves the position density and is spectral accurate in space and fourth order accurate in time moreover the new method has two other important advantages i it reduces a three dimensional 3 d problem with cylindrical symmetry to an effective two dimensional 2 d problem ii it solves the problem in the whole space instead of in a truncated artificial computational domain the method is applied to vector gross pitaevskii equations vgpes for multicomponent becs extensive numerical tests are presented for the one dimensional 1 d gpe the 2 d gpe with radial symmetry the 3 d gpe with cylindrical symmetry as well as 3 d vgpes for two component becs to show the efficiency and accuracy of the new numerical method

a generalized laguerre hermite pseudospectral method for computing symmetric and central vortex states in bose einstein condensates a generalized laguerre hermite pseudospectral method is proposed for computing symmetric and central vortex states in bose einstein condensates becs in three dimensions with cylindrical symmetry the new method is based on the properly scaled generalized laguerre hermite functions and a normalized gradient flow it enjoys three important advantages i it reduces a three dimensional 3d problem with cylindrical symmetry into an effective two dimensional 2d problem ii it solves the problem in the whole space instead of in a truncated artificial computational domain and iii it is spectrally accurate extensive numerical results for computing symmetric and central vortex states in becs are presented for one dimensional 1d bec 2d bec with radial symmetry and 3d bec with cylindrical symmetry

the hermite spectral method for gaussian type functions although hermite functions were widely used for many practical problems numerical experiments with the standard normalized hermite functions psi n v worked poorly in the sense that too many hermite functions are required to solve differential equations in order to obtain accurate numerical solutions it is necessary to choose a scaling factor alpha and use psi n alpha v as the basis functions in this paper the scaling factors are given for functions that are of gaussian type which have finite supports m m the scaling factor used is max 0 leqq j leqq n gamma j m where gamma j j 0 n are the roots of psi n 1 v and n 1 is the number of the truncated terms used the numerical results show that after using this scaling factor only reasonable numbers of the hermite functions are required to solve differential equations

a finite element method with mesh adaptivity for computing vortex states in fast rotating bose einstein condensates numerical computations of stationary states of fast rotating bose einstein condensates require high spatial resolution due to the presence of a large number of quantized vortices in this paper we propose a low order finite element method with mesh adaptivity by metric control as an alternative approach to the commonly used high order finite difference or spectral approximation methods the mesh adaptivity is used with two different numerical algorithms to compute stationary vortex states an imaginary time propagation method and a sobolev gradient descent method we first address the basic issue of the choice of the variable used to compute new metrics for the mesh adaptivity and show that refinement using simultaneously the real and imaginary part of the solution is successful mesh refinement using only the modulus of the solution as adaptivity variable fails for complicated test cases then we suggest an optimized algorithm for adapting the mesh during the evolution of the solution towards the equilibrium state considerable computational time saving is obtained compared to uniform mesh computations the new method is applied to compute difficult cases relevant for physical experiments large nonlinear interaction constant and high rotation rates

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

efficient and spectrally accurate numerical methods for computing ground and first excited states in bose einstein condensates in this paper we present two efficient and spectrally accurate numerical methods for computing the ground and first excited states in bose einstein condensates becs we begin with a review on the gradient flow with discrete normalization gfdn for computing stationary states of a nonconvex minimization problem and show how to choose initial data effectively for the gfdn for discretizing the gradient flow we use sine pseudospectral method for spatial derivatives and either backward euler scheme besp or backward forward euler schemes for linear nonlinear terms bfsp for temporal derivatives both besp and bfsp are spectral order accurate for computing the ground and first excited states in bec of course they have their own advantages i for linear case besp is energy diminishing for any time step size where bfsp is energy diminishing under a constraint on the time step size ii at every time step the linear system in bfsp can be solved directly via fast sine transform fst and thus it is extremely efficient and in besp it needs to be solved iteratively via fst by introducing a stabilization term and thus it could be efficient too comparisons between besp and bfsp as well as other existing numerical methods are reported in terms of accuracy and total computational time our numerical results show that both besp and bfsp are much more accurate and efficient than those existing numerical methods in the literature finally our new numerical methods are applied to compute the ground and first excited states in bec in one dimension 1d 2d and 3d with a combined harmonic and optical lattice potential for demonstrating their efficiency and high resolution