new singular solutions of the nonlinear schrodinger equation we present numerical simulations of a new type of singular solutions of the critical nonlinear schrodinger equation nls that collapse with a quasi self similar ring profile at a square root blowup rate we find and analyze the equation of the ring profile we observe that the self similar ring profile is an attractor for a large class of radially symmetric initial conditions but is unstable under symmetry breaking perturbations the equation for the ring profile admits also multi ring solutions that give rise to collapsing self similar multi ring solutions but these solutions are unstable even in the radially symmetric case and eventually collapse with a single ring profile collapsing ring solutions are also observed in the supercritical nls

a constrained min max algorithm for rival models of the same economic system there are well established rival theories about the economy these have in turn led to the development of rival models purporting to represent the economic system the models are large systems of discrete time nonlinear dynamic equations observed data of the real system does not in general provide sufficient information for statistical methods to invalidate all but one of the rival models in such a circumstance there is uncertainty about which model to use in the formulation of policy prudent policy design would suggest that a model based policy should take into account all the rival models this is achieved as a pooling of the models the pooling that yields the policy which is robust to model choice is formulated as a constrained min max problem the minimization is over the decision variables and the maximization is over the rival models only equality constraints are considered

the sample average approximation method for stochastic discrete optimization in this paper we study a monte carlo simulation based approach to stochastic discrete optimization problems the basic idea of such methods is that a random sample is generated and the expected value function is approximated by the corresponding sample average function the obtained sample average optimization problem is solved and the procedure is repeated several times until a stopping criterion is satisfied we discuss convergence rates stopping rules and computational complexity of this procedure and present a numerical example for the stochastic knapsack problem

moving mesh methods in multiple dimensions based on harmonic maps in practice there are three types of adaptive methods using the finite element approach namely the h method p method and r method in the h method the overall method contains two parts a solution algorithm and a mesh selection algorithm these two parts are independent of each other in the sense that the change of the pdes will affect the first part only however in some of the existing versions of the r method also known as the moving mesh method these two parts are strongly associated with each other and as a result any change of the pdes will result in the rewriting of the whole code in this work we will propose a moving mesh method which also contains two parts a solution algorithm and a mesh redistribution algorithm our efforts are to keep the advantages of the r method e g keep the number of nodes unchanged and of the h method e g the two parts in the code are independent a framework for adaptive meshes based on the hamilton schoen yau theory was proposed by dvinsky in this work we will extend dvinsky s method to provide an efficient solver for the mesh redistribution algorithm the key idea is to construct the harmonic map between the physical space and a parameter space by an iteration procedure each iteration step is to move the mesh closer to the harmonic map this procedure is simple and easy to program and also enables us to keep the map harmonic even after long times of numerical integration the numerical schemes are applied to a number of test problems in two dimensions it is observed that the mesh redistribution strategy based on the harmonic maps adapts the mesh extremely well to the solution without producing skew elements for multi dimensional computations

distributionally robust convex optimization distributionally robust optimization is a paradigm for decision making under uncertainty where the uncertain problem data are governed by a probability distribution that is itself subject to uncertainty the distribution is then assumed to belong to an ambiguity set comprising all distributions that are compatible with the decision maker s prior information in this paper we propose a unifying framework for modeling and solving distributionally robust optimization problems we introduce standardized ambiguity sets that contain all distributions with prescribed conic representable confidence sets and with mean values residing on an affine manifold these ambiguity sets are highly expressive and encompass many ambiguity sets from the recent literature as special cases they also allow us to characterize distributional families in terms of several classical and or robust statistical indicators that have not yet been studied in the context of robust optimization we determine conditions under which distributionally robust optimization problems based on our standardized ambiguity sets are computationally tractable we also provide tractable conservative approximations for problems that violate these conditions

robust stochastic approximation approach to stochastic programming in this paper we consider optimization problems where the objective function is given in a form of the expectation a basic difficulty of solving such stochastic optimization problems is that the involved multidimensional integrals expectations cannot be computed with high accuracy the aim of this paper is to compare two computational approaches based on monte carlo sampling techniques namely the stochastic approximation sa and the sample average approximation saa methods both approaches the sa and saa methods have a long history current opinion is that the saa method can efficiently use a specific say linear structure of the considered problem while the sa approach is a crude subgradient method which often performs poorly in practice we intend to demonstrate that a properly modified sa approach can be competitive and even significantly outperform the saa method for a certain class of convex stochastic problems we extend the analysis to the case of convex concave stochastic saddle point problems and present in our opinion highly encouraging results of numerical experiments

minimizing piecewise concave functions over polyhedra we introduce an iterative method for solving linearly constrained optimization problems whose nonsmooth nonconvex objective function is defined as the pointwise maximum of finitely many concave functions such problems often arise from reformulations of certain constraint structures e g binary constraints finite max min constraints in diverse areas of optimization we state a local optimization strategy which exploits piecewise concavity of the objective function giving rise to a linearized model corrected by a proximity term in addition we introduce an approximate line search strategy based on a curvilinear model which similarly to bundle methods can return either a satisfactory descent or a null step declaration termination at a point satisfying an approximate stationarity condition is proved we embed the local minimization algorithm into a variable neighborhood search scheme and a coordinate direction search heuristic whose aim is to improve the current estimate of the global minimizer

stochastic programming approach to optimization under uncertainty in this paper we discuss computational complexity and risk averse approaches to two and multistage stochastic programming problems we argue that two stage say linear stochastic programming problems can be solved with a reasonable accuracy by monte carlo sampling techniques while there are indications that complexity of multistage programs grows fast with increase of the number of stages we discuss an extension of coherent risk measures to a multistage setting and in particular dynamic programming equations for such problems

models for minimax stochastic linear optimization problems with risk aversion we propose a semidefinite optimization sdp model for the class of minimax two stage stochastic linear optimization problems with risk aversion the distribution of second stage random variables belongs to a set of multivariate distributions with known first and second moments for the minimax stochastic problem with random objective we provide a tight sdp formulation the problem with random right hand side is np hard in general in a special case the problem can be solved in polynomial time explicit constructions of the worst case distributions are provided applications in a productiontransportation problem and a single facility minimax distance problem are provided to demonstrate our approach in our experiments the performance of minimax solutions is close to that of data driven solutions under the multivariate normal distribution and better under extremal distributions the minimax solutions thus guarantee to hedge against these worst possible distributions and provide a natural distribution to stress test stochastic optimization problems under distributional ambiguity

distributionally robust multi item newsvendor problems with multimodal demand distributions we present a risk averse multi dimensional newsvendor model for a class of products whose demands are strongly correlated and subject to fashion trends that are not fully understood at the time when orders are placed the demand distribution is known to be multimodal in the sense that there are spatially separated clusters of probability mass but otherwise lacks a complete description we assume that the newsvendor hedges against distributional ambiguity by minimizing the worst case risk of the order portfolio over all distributions that are compatible with the given modality information we demonstrate that the resulting distributionally robust optimization problem is mathrm np np hard but admits an efficient numerical solution in quadratic decision rules this approximation is conservative and computationally tractable moreover it achieves a high level of accuracy in numerical tests we further demonstrate that disregarding ambiguity or multimodality can lead to unstable solutions that perform poorly in stress test experiments

an l stability analysis for the finite difference solution of one dimensional linear convection diffusion equations on moving meshes the stability of three moving mesh finite difference schemes is studied in the l norm for one dimensional linear convection diffusion equations these schemes use central finite differences for spatial discretization and the q method for temporal discretization and they are based on conservative and non conservative forms of transformed partial differential equations the stability conditions obtained consist of the cfl condition and the mesh speed related conditions the cfl condition is independent of the mesh speed and has the same form as that for fixed meshes the mesh speed related conditions restrict how fast the mesh can move the conditions of this type obtained in this paper are weaker than those in the existing literature and can be satisfied when the mesh is sufficiently fine illustrative numerical results are presented

combined lp and quasi newton methods for minimax optimization we present an algorithm for minimax optimization that combines lp methods and quasi newton methods the quasi newton algorithm is used only if an irregular solution is detected in which case second order derivative information is needed in order to obtain a fast final rate of convergence we prove that the algorithm can converge only to a stationary point and that normally the final rate of convergence will be either quadratic or superlinear the performance is illustrated through some numerical examples

convexity and decomposition of mean risk stochastic programs traditional stochastic programming is risk neutral in the sense that it is concerned with the optimization of an expectation criterion a common approach to addressing risk in decision making problems is to consider a weighted mean risk objective where some dispersion statistic is used as a measure of risk we investigate the computational suitability of various mean risk objective functions in addressing risk in stochastic programming models we prove that the classical mean variance criterion leads to computational intractability even in the simplest stochastic programs on the other hand a number of alternative mean risk functions are shown to be computationally tractable using slight variants of existing stochastic programming decomposition algorithms we propose decomposition based parametric cutting plane algorithms to generate mean risk efficient frontiers for two particular classes of mean risk objectives

a smooth method for the finite minimax problem we consider unconstrained minimax problems where the objective function is the maximum of a finite number of smooth functions we prove that under usual assumptions it is possible to construct a continuously differentiable function whose minimizers yield the minimizers of the max function and the corresponding minimum values on this basis we can define implementable algorithms for the solution of the minimax problem which are globally convergent at a superlinear convergence rate preliminary numerical results are reported

computing bounds for stochastic programming problems by means of a generalized moment problem bounds on the expected value of a convex function are obtained by means of an approximating generalized moment problem numerical implementation is discussed in the context of stochastic programming problems

fixed width sequential stopping rules for a class of stochastic programs monte carlo sampling based methods are frequently used in stochastic programming when exact solution is not possible a critical component of monte carlo sampling based methods is determining when to stop sampling to ensure the desired quality of the solutions in this paper we develop stopping rules for sequential sampling procedures that depend on the width of an optimality gap confidence interval estimator the procedures solve a sequence of sampling approximations with increasing sample size to generate solutions and stop when the width of the confidence interval on the current solution s optimality gap plus an inflation factor fall below a prespecified value epsilon we first present a method that takes the schedule of sample sizes as an input and provide guidelines on the growth of sample sizes then we present a method that increases the sample sizes according to the current estimates of the optimality gap the larger the estimates the larger the increases in sample sizes we provide conditi

optimization of convex risk functions we consider optimization problems involving convex risk functions by employing techniques of convex analysis and optimization theory in vector spaces of measurable functions we develop new representation theorems for risk models and optimality and duality theory for problems with convex risk functions

an efficient adaptive mesh redistribution method for a non linear dirac equation this paper presents an efficient adaptive mesh redistribution method to solve a non linear dirac nld equation our algorithm is formed by three parts the nld evolution the iterative mesh redistribution of the coarse mesh and the local uniform refinement of the final coarse mesh at each time level the equidistribution principle is first employed to iteratively redistribute coarse mesh points and the scalar monitor function is subsequently interpolated on the coarse mesh in order to do one new iteration and improve the grid adaptivity after an adaptive coarse mesh is generated ideally and finally each coarse mesh interval is equally divided into some fine cells to give an adaptive fine mesh of the physical domain and then the solution vector is remapped on the resulting new fine mesh by an affine method the nld equation is finally solved by using a high resolution shock capturing method on the fixed non uniform fine mesh extensive numerical experiments demonstrate that the proposed adaptive mesh method gives the third order rate of convergence and yields an efficient and fast nld solver that tracks and resolves both small local and large solution gradients automatically

an adaptive ghost fluid finite volume method for compressible gas water simulations an adaptive ghost fluid finite volume method is developed for one and two dimensional compressible multi medium flows in this work it couples the real ghost fluid method gfm c w wang t g liu b c khoo a real ghost fluid method for the simulation of multi medium compressible flow siam j sci comput 28 2006 278 302 and the adaptive moving mesh method h z tang t tang moving mesh methods for one and two dimensional hyperbolic conservation laws siam j numer anal 41 2003 487 515 h z tang t tang p w zhang an adaptive mesh redistribution method for non linear hamilton jacobi equations in two and three dimensions j comput phys 188 2003 543 572 and thus combines their advantages this work shows that the local mesh clustering in the vicinity of the material interface can effectively reduce both numerical and conservative errors caused by the gfm around the material interface and other discontinuities besides the improvement of flow field resolution the adaptive gfm also largely increases the computational efficiency several numerical experiments are conducted to demonstrate robustness and efficiency of the current method they include several 1d and 2d gas water flow problems involving a large density gradient at the material interface and strong shock interface interactions the results show that our algorithm can capture the shock waves and the material interface accurately and is stable and robust even for solutions with large density and pressure gradients

adaptive zoning for singular problems in two dimensions abstract winslow s method for the automatic generation of computation meshes is extended to adaptively vary the zone sizes and orthogonality of grid lines in the resulting mesh through simple analysis and numerical examples the adaptive mesh is shown to give significant increases in accuracy in the computation of singular problems

an adaptive grid with directional control abstract a variational formulation of winslow s variable diffusion method is combined with a directional control functional to derive an adaptive grid generator for adaptive node movement the addition of directional control allows one to align mesh lines with a prescribed vector field while adapting the spacing to resolve the data it is shown that harmonic function theory can be applied to define the conditions under which unique solutions of the resulting elliptic equations exist the grid generator is applied to the complex problem of magnetic field reconnection driven by fluid instability the results demonstrate that the cost of a computation with an adaptive grid with directional control is only one tenth of the cost of either a calculation on an eulerian grid or on an adaptive grid without directional control

on deterministic reformulations of distributionally robust joint chance constrained optimization problems a joint chance constrained optimization problem involves multiple uncertain constraints i e constraints with stochastic parameters that are jointly required to be satisfied with probability exc

on balanced moving mesh methods moving mesh methods are a widely used approach in the numerical solution of pdes where the original pdes are transformed from a physical domain to a computational domain the objective is to utilize a uniform mesh in the computational domain to get a non uniform physical mesh that better captures the behavior of the solution the movement of the physical mesh points can be governed by a moving mesh pde associated with a corresponding monitor function and both the original pdes and the moving mesh pdes must be solved simultaneously the motivation for this paper is to study a balanced moving mesh method where the aim is to strike a balance between the properties of the solution of the original pde and that of the moving mesh pde we focus on particular choices of the monitor function that give both a well behaved mesh transformation and a well behaved solution in the computational domain both theoretical analysis and numerical experiments are presented as illustrations

a barrier function method for minimax problems this paper presents an algorithm based on barrier functions for solving semi infinite minimax problems which arise in an engineering design setting the algorithm bears a resemblance to some of the current interior penalty function methods used to solve constrained minimization problems global convergence is proven and numerical results are reported which show that the algorithm is exceptionally robust and that its performance is comparable while its structure is simpler than that of current first order minimax algorithms

approximations for viscosity solutions of hamilton jacobi equations with locally varying time and space grids a new monotone finite difference scheme is introduced that approximates viscosity solutions of first order nonlinear hamilton jacobi equations the main feature of the scheme is that it allows for locally varying time and space grids making it ideal for use with adaptive algorithms explicit a priori error estimates are given to establish convergence numerical examples including adaptive mesh refinement examples demonstrate the effectiveness of the proposed scheme

a moving mesh finite element algorithm for singular problems in two and three space dimensions a framework for adaptive meshes based on the hamilton schoen yau theory was proposed by dvinsky in a recent work 2001 j comput phys 170 562 588 we extended dvinsky s method to provide an efficient moving mesh algorithm which compared favorably with the previously proposed schemes in terms of simplicity and reliability in this work we will further extend the moving mesh methods based on harmonic maps to deal with mesh adaptation in three space dimensions in obtaining the variational mesh we will solve an optimization problem with some appropriate constraints which is in contrast to the traditional method of solving the euler lagrange equation directly the key idea of this approach is to update the interior and boundary grids simultaneously rather than considering them separately application of the proposed moving mesh scheme is illustrated with some two and three dimensional problems with large solution gradients the numerical experiments show that our methods can accurately resolve detail features of singular problems in 3d

generalized chebychev inequalities theory and applications in decision analysis in many decision analysis problems we have only limited information about the relevant probability distributions in problems like these it is natural to ask what conclusions can be drawn on the basis of this limited information for example in the early stages of analysis of a complex problem we may have only limited fractile information for the distributions in the problem what can we say about the optimal strategy or certainty equivalents given these few fractiles this paper describes a very general framework for analyzing these kinds of problems where given certain moments of a distribution we can compute bounds on the expected value of an arbitrary objective function by suitable choice of moment and objective functions we can formulate and solve many practical decision analysis problems we describe the general framework and theoretical results discuss computational strategies and provide specific results for examples in dynamic programming decision analysis with incomplete information bayesian statistics and option pricing

variational mesh adaptation methods for axisymmetrical problems we study variational mesh adaptation for axially symmetric solutions to two dimensional problems the study is focused on the relationship between the mesh density distribution and the monitor function and is carried out for a traditional functional that includes several widely used variational methods as special cases and a recently proposed functional that allows for a weighting between mesh isotropy or regularity and global equidistribution of the monitor function the main results are stated in theorems ref thm4 1 and ref thm4 2 for axially symmetric problems it is natural to choose axially symmetric mesh adaptation to this end it is reasonable to use the monitor function in the form g lambda 1 r mbox boldmath e r mbox boldmath e r t lambda 2 r mbox boldmath e theta mbox boldmath e theta t where mbox boldmath e r and mbox boldmath e theta are the radial and angular unit vectors it is shown that when higher mesh concentration at the origin is desired a choice of lambda 1 and lambda 2 satisfying lambda 1 0 0 by choosing lambda 1 and lambda 2 in contrast numerical results show that the new functional provides better control of the mesh concentration through the monitor function two dimensional numerical results are presented to support the analysis

optimization with reference based robust preference constraints we propose an optimization model using a novel robust preference relationship reference based almost stochastic dominance rsd the concept of rsd addresses the two problems in utility based dec

stochastic optimization problems with incomplete information on distribution functions the main purpose of this paper is to discuss numerical optimization procedures based on duality theory for stochastic extremal problems in which the distribution function is only partially known we formulate such problems as minimax problems in which the inner problem involves optimization with respect to probability measures the latter problem is solved using generalized linear programming techniques then we state the dual problem to the initial stochastic optimization problem numerical procedures that avoid the difficulties associated with solving the inner problem are proposed

singular standing ring solutions of nonlinear partial differential equations abstract we present a general framework for constructing singular solutions of nonlinear evolution equations that become singular on a d dimensional sphere where d 1 the asymptotic profile and blowup rate of these solutions are the same as those of solutions of the corresponding one dimensional equation that become singular at a point we provide a detailed numerical investigation of these new singular solutions for the following equations the nonlinear schrodinger equation i t t x 2 0 with 2 the biharmonic nonlinear schrodinger equation i t t x 2 2 0 with 4 the nonlinear heat equation t t x 2 0 with 0 and the nonlinear biharmonic heat equation t t x 2 2 0 with 0

conditional risk mappings we introduce an axiomatic definition of a conditional convex risk mapping and we derive its properties in particular we prove a representation theorem for conditional risk mappings in terms of conditional expectations we also develop dynamic programming relations for multistage optimization problems involving conditional risk mappings

stability of isotropic singularities for the nonlinear schro dinger equation abstract we describe a method by which the fully two and three dimensional cubic schrodinger equations can be accurately integrated numerically up to times very close to the formation of singularities in both cases anisotropic initial data collapse very rapidly towards isotropic singularities in three dimensions the solutions become self similar with a blowup rate t t 1 2 in two dimensions the self similarity is weakly broken and the blowup rate is t t ln ln 1 t t 1 2 the stability of the singular isotropic solutions is very firmly backed by the numerical results

moving mesh strategy based on a gradient flow equation for two dimensional problems in this paper we introduce a moving mesh method for solving pdes in two dimensions it can be viewed as a higher dimensional generalization of the moving mesh pde mmpde strategy developed in our previous work for one dimensional problems w huang y ren and r d russell siam j numer anal 31 1994 pp 709 730 the mmpde is derived from a gradient flow equation which arises using a mesh adaptation functional in turn motivated from the theory of harmonic maps geometrical interpretations are given for the gradient equation and functional and basic properties of this mmpde are discussed numerical examples are presented where the method is used both for mesh generation and for solving time dependent pdes the results demonstrate the potential of the mesh movement strategy to concentrate the mesh points so as to adapt to special problem features and to also preserve a suitable level of mesh orthogonality

digital signal restoration using fuzzy sets a new signal restoration method with considerable flexibility in incorporating a priori information is developed the method defines a fuzzy set for each piece of information to restrict the set of acceptable solutions using fuzzy sets makes it possible to model partially defined information as well as exact knowledge the intersection of all the fuzzy sets is the feasibility set the original signal is a member of this set with a high membership value and any high membership valued element of this set is a nonrejectable solution such solutions can be computed by using optimization techniques ideally the feasibility set contains only the original signal the chance of recovering the original signal decreases as the feasibility set gets larger thus the size of the feasibility set gives a quality measure for the solution the method generated successful results in many restorations for which the conventional techniques have failed and may be applied in image coding and tomography

polyhedral risk measures in stochastic programming we consider stochastic programs with risk measures in the objective and study stability properties as well as decomposition structures thereby we place emphasis on dynamic models i e multistage stochastic programs with multiperiod risk measures in this context we define the class of polyhedral risk measures such that stochastic programs with risk measures taken from this class have favorable properties polyhedral risk measures are defined as optimal values of certain linear stochastic programs where the arguments of the risk measure appear on the right hand side of the dynamic constraints dual representations for polyhedral risk measures are derived and used to deduce criteria for convexity and coherence as examples of polyhedral risk measures we propose multiperiod extensions of the conditional value at risk

applying the minimax criterion in stochastic recourse programs we consider an optimization problem in which some uncertain parameters are replaced by random variables the minimax approach to stochastic programming concerns the problem of minimizing the worst expected value of the objective function with respect to the set of probability measures that are consistent with the available information on the random data only very few practicable solution procedures have been proposed for this problem and the existing ones rely on simplifying assumptions in this paper we establish a number of stability results for the minimax stochastic program justifying in particular the approach of restricting attention to probability measures with support in some known finite set following this approach we elaborate solution procedures for the minimax problem in the setting of two stage stochastic recourse models considering the linear recourse case as well as the integer recourse case since the solution procedures are modifications of well known algorithms their efficacy is immediate from the computational testing of these procedures and we do not report results of any computational experiments

an algorithm for the inequality constrained discrete min max problem in this paper we discuss an approach using an augmented lagrangian formulation to directly solve the inequality constrained min max problem the algorithm involves a sequential quadratic programming subproblem an adaptive penalty parameter selection rule and a stepsize strategy convergent to unit steps that ensures progress toward optimality and feasibility of the inequality constraints it is shown that the penalty parameter does not grow indefinitely the convergence of the algorithm is established and its numerical effectiveness is demonstrated with test examples

moving finite elements i we present new and general numerical methods for dealing with problems whose solutions develop sharp transition layers or near shocks these methods allow many nodes automatically to concentrate in the critical regions and move with them for clarity of exposition we concentrate on the space of piecewise linear functions with movable nodes with burgers equation as our test equation but the generalization to much more general spaces and equations including even certain previous moving vorticity blobs of the first author and s doss for the navier stokes equations becomes clear in this paper we present the theoretical and computational details of our scheme along with computational trial runs for burgers equation showing that the nodes do concentrate sharply and move with the shocks as desired the conclusiveness of these preliminary numerical trials is marred somewhat by the fact that we never successfully debugged a newton s method for our implicit stiff ode solver and were thus limited to ver

distributionally robust optimization with matrix moment constraints lagrange duality and cutting plane methods a key step in solving minimax distributionally robust optimization dro problems is to reformulate the inner maximization w r t probability measure as a semiinfinite programming problem through lagrange dual slater type conditions have been widely used for strong duality zero dual gap when the ambiguity set is defined through moments in this paper we investigate effective ways for verifying the slater type conditions and introduce other conditions which are based on lower semicontinuity of the optimal value function of the inner maximization problem moreover we propose two discretization schemes for solving the dro with one for the dualized dro and the other directly through the ambiguity set of the dro in the absence of strong duality the discretization scheme via lagrange duality may provide an upper bound for the optimal value of the dro whereas the direct discretization approach provides a lower bound two cutting plane schemes are consequently proposed one for the discretized dualized dro and the other for the minimax dro with discretized ambiguity set convergence analysis is presented for the approximation schemes in terms of the optimal value optimal solutions and stationary points comparative numerical results are reported for the resulting algorithms

theory of singular vortex solutions of the nonlinear schrodinger equation we present a systematic study of singular vortex solutions of the critical and supercritical two dimensional nonlinear schrodinger equation in particular we study the critical power for collapse and the asymptotic blowup profile of singular vortices

adaptive local overlapping grid methods for parabolic systems in two space dimensions abstract adaptive mesh refinement techniques are described for two dimensional systems of parabolic partial differential equations solutions are calculated using galerkin s method with a piecewise bilinear basis in space and backward euler integration in time a posteriori estimates of the local discretization error of piecewise bilinear finite element solutions are obtained by a p refinement technique these error estimates are used to control a local h refinement strategy where finer grids are recursively introduced in regions where a prescribed tolerance is exceeded fine grids at a given level of refinement may overlap each other and independent solutions are generated on each of them a version of the schwarz alternating principle is used to coordinate solutions between overlapping fine grids computational results demonstrating the performance of the adaptive procedure on linear and nonlinear problems and apparent convergence of the error estimate for linear heat conduction problem and uniform global refinement is presented

ambiguous chance constrained problems and robust optimization in this paper we study ambiguous chance constrained problems where the distributions of the random parameters in the problem are themselves uncertain we focus primarily on the special case where the uncertainty set of the distributions is of the form where p denotes the prohorov metric the ambiguous chance constrained problem is approximated by a robust sampled problem where each constraint is a robust constraint centered at a sample drawn according to the central measure the main contribution of this paper is to show that the robust sampled problem is a good approximation for the ambiguous chance constrained problem with a high probability this result is established using the strassen dudley representation theorem that states that when the distributions of two random variables are close in the prohorov metric one can construct a coupling of the random variables such that the samples are close with a high probability we also show that the robust sampled problem can be solved efficiently both in theory and in practice

sample average approximation method for compound stochastic optimization problems the paper studies stochastic optimization programming problems with compound functions containing expectations and extreme values of other random functions as arguments compound functions arise in various applications a typical example is a variance function of nonlinear outcomes other examples include stochastic minimax problems econometric models with latent variables and multilevel and multicriteria stochastic optimization problems as a solution technique a sample average approximation saa method also known as statistical or empirical sample mean method is used the method consists in approximation of all expectation functions by their empirical means and solving the resulting approximate deterministic optimization problems in stochastic optimization this method is widely used for optimization of standard expectation functions under constraints in this paper saa method is extended to general compound stochastic optimization problems the conditions for convergence in mean almost sure

asymptotic theory for solutions in statistical estimation and stochastic programming new techniques of local sensitivity analysis for nonsmooth generalized equations are applied to the study of sequences of statistical estimates and empirical approximations to solutions of stochastic programs consistency is shown to follow from a certain local invertibility property and asymptotic distributions are derived from a generalized implicit function theorem that characterizes asymptotic behavior in situations where estimates are subjected to constraints and estimation functionals are nonsmooth

asymptotic behavior of statistical estimators and of optimal solutions of stochastic optimization problems we study the asymptotic behavior of the statistical estimators that maximize a not necessarily dieren tiable criterion function possibly subject to side constraints equalities and inequalities the consistency results generalize those of wald and huber conditions are also given under which one is still able to obtain asymptotic normality the analysis brings to the fore the relationship between the problem of nding statistical estimators and that of nding the optimal solutions of stochastic optimization problems with partial information the last section is devoted to the properties of the saddle points of the associated lagrangians

an adaptive grid method for two dimensional viscous flows this paper extends the gas kinetic bgk ns scheme to an adaptive grid for the viscous flow simulations the grid movement and adaptation is controlled by a monitor function which may depend on velocity gradient or other flow variables such as density or pressure for the viscous flow computation the use of adaptive mesh much improves the efficiency and accuracy of the method in comparison with the methods with static mesh points the current method is an accurate and efficient method for the viscous flow computation where the grid points can be easily moved and concentrated on the regions with large velocity and density gradients such as the boundary layer and multi material interface many numerical examples validate the current approach for the viscous flow simulations

a simple adaptive grid method in two dimensions this paper gives an interpretation of the concept of equidistribution in the context of adaptive grid generation for multidimensional problems it is shown that the equidistribution principle cannot be satisfied throughout the domain of the problem and based on this recognition a local equidistribution principle is developed a discrete formulation is described for grid generation in two space dimensions and a smoothing mechanism is presented for improving mesh quality the adaptive grid method that is constructed contains three grid quality parameters numerical examples illustrate adaptive grid generation using a prescribed monitor function and grid generation for numerical solution of partial differential equations results show that the method produces high quality grids and that it is fairly insensitive to the choice of parameters

singular ring solutions of critical and supercritical nonlinear schrodinger equations abstract we present new singular solutions of the nonlinear schrodinger equation nls i t t r r r d 1 r r 2 0 1 d 2 d 2 these solutions collapse with a quasi self similar ring profile q i e q where q 1 l 1 t q r r m t l exp i 0 t d s l 2 s i l t 4 l r 2 1 r r m t 2 l t is the ring width that vanishes at the singularity r m t r 0 l t is the ring radius and 2 d 1 the blowup rate of these solutions is 1 1 for 2 d 2 and 1 d 0 1 and a square root with a loglog correction the loglog law when 2 and 1 d 0 therefore the nls has solutions that collapse with any blowup rate p for 1 2 p 1 this study extends the results of g fibich n gavish x wang new singular solutions of the nonlinear schrodinger equation physica d 211 2005 193 220 for 1 and d 2 and of p raphael existence and stability of a solution blowing up on a sphere for a l 2 super critical non linear schrodinger equation duke math j 134 2 2006 199 258 for 2 and d 2 to all 2 d 2 and 1 d

convergence of stationary points of sample average two stage stochastic programs a generalized equation approach this paper presents an asymptotic analysis of a monte carlo method variously known as sample average approximation saa or sample path optimization spo for a general two stage stochastic minimization problem we study the case when the second stage problem may have multiple local optima or stationary points that are not global solutions and saa is implemented using a general nonlinear programming solver that is only guaranteed to find stationary points new optimality conditions are developed for both the true problem and its saa problem to accommodate karush kuhn tucker points because the optimality conditions are essentially stochastic generalized equations the asymptotic analysis is carried out for the generalized equations first and then applied to optimality conditions for this purpose we analyze piecewise continuous pc0 stochastic mappings to understand when their expectations are piecewise continuous and thereby derive exponential convergence of saa it is shown under moderate conditions that with probability one an accumulation point of the saa stationary points satisfies a relaxed stationary condition for the true problem and further that with probability approaching one exponentially fast with increasing sample size a stationary point of saa converges to the set of relaxed stationary points these results strengthen or complement existing results where the second stage problem is often assumed to have a unique solution and the exponential convergence is focused on how fast a solution of the true problem becomes an approximate solution of an saa problem rather than the other way around

on rates of convergence for stochastic optimization problems under non independent and identically distributed sampling in this paper we discuss the issue of solving stochastic optimization problems by means of sample average approximations our focus is on rates of convergence of estimators of optimal solutions and optimal values with respect to the sample size this is a well studied problem in case the samples are independent and identically distributed i e when standard monte carlo simulation is used here we study the case where that assumption is dropped broadly speaking our results show that under appropriate assumptions the rates of convergence for pointwise estimators under a sampling scheme carry over to the optimization case in the sense that convergence of approximating optimal solutions and optimal values to their true counterparts has the same rates as in pointwise estimation we apply our results to two well established sampling schemes namely latin hypercube sampling and randomized quasi monte carlo qmc the novelty of our work arises from the fact that while there has been some work on the use of variance reduction techniques and qmc methods in stochastic optimization none of the existing work to the best of our knowledge has provided a theoretical study on the effect of these techniques on rates of convergence for the optimization problem we present numerical results for some two stage stochastic programs from the literature to illustrate the discussed ideas

an incremental method for solving convex finite min max problems we introduce a new approach to minimizing a function defined as the pointwise maximum over finitely many convex real functions next referred to as the component functions with the aim of working on the basis of incomplete knowledge of the objective function a descent algorithm is proposed which need not require at the current point the evaluation of the actual value of the objective function namely of all the component functions thus extending to min max problems the philosophy of the incremental approaches widely adopted in the nonlinear least squares literature given the nonsmooth nature of the problem we resort to the well established machinery of bundle methods we provide global convergence analysis of our method and in addition we study a subgradient aggregation scheme aimed at simplifying the problem of finding a tentative step this paper is completed by the numerical results obtained on a set of standard test problems

models and algorithms for distributionally robust least squares problems we present three different robust frameworks using probabilistic ambiguity descriptions of the data in least squares problems these probability ambiguity descriptions are given by 1 confidence region over the first two moments 2 bounds on the probability measure with moments constraints 3 the kantorovich probability distance from a given measure for the first case we give an equivalent formulation and show that the optimization problem can be solved using a semidefinite optimization reformulation or polynomial time algorithms for the second case we derive the equivalent lagrangian problem and show that it is a convex stochastic programming problem we further analyze three special subcases i finite support ii measure bounds by a reference probability measure iii measure bounds by two reference probability measures with known density functions we show that case i has an equivalent semidefinite programming reformulation and the sample average approximations of case ii and iii have equivalent semidefinite programming reformulations for ambiguity description 3 we show that the finite support case can be solved by using an equivalent second order cone programming reformulation

a grid redistribution method for singular problems many physical phenomena develop singular or nearly singular behavior in localized regions e g boundary layers or blowup solutions using uniform grids for such problems becomes computationally prohibitive as the solution approaches singularity ren and wang developed a semi static adaptive grid method w ren x p wang an iterative grid redistribution method for singular problems in multiple dimensions j comput phys 159 2000 246 273 for the solution of these problems known as the iterative grid redistribution igr method in this study we develop a theoretical basis for semi static adaptive grid method for singular problems based on this theory we obtain the key result of this study a methodology for designing robust weight functionals which ensures grid resolution in the singular region as well as control of the maximal grid spacing in the outer region using this methodology we introduce a semi static adaptive grid method which does not involve an iterative procedure for grid redistribution as in the igr method we demonstrate the efficacy of this method with numerical examples of solutions which localize by more than nine orders of magnitude

an iterative grid redistribution method for singular problems in multiple dimensions we introduce an iterative grid redistribution method based on the variational approach the iterative procedure enables us to gain more precise control of the grid distribution near the regions of large solution variations the method is particularly effective for solving partial differential equations with singular solutions e g blowup solutions our method requires little prior information of the singular solutions and can handle multiple singularities the method is successfully applied to the nonlinear schrodinger equation and the keller segal equations where solutions with multiple blowup points can be solved up to times very close to the blowup time

hybrid methods for solving the educational testing problem abstract methods for solving the educational testing problem are considered one approach glunt 1995 is to formulate the problem as a linear convex programming problem in which the constraint is the intersection of three convex sets this method is globally convergent but the rate of convergence is slow however the method does have the capability of determining the correct rank of the solution matrix and this can be done in relatively few iterations if the correct rank of the solution matrix is known it is shown how to formulate the problem as a smooth nonlinear minimization problem for which a rapid convergence can be obtained by l1sqp method 6 this paper studies hybrid methods that attempt to combine the best features of both types of method an important feature concerns the interfacing of the component methods thus it has to be decided which method to use first and when to switch between methods difficulties such as these are addressed in the paper comparative numerical results are also reported

on a class of minimax stochastic programs for a particular class of minimax stochastic programming models we show that the problem can be equivalently reformulated into a standard stochastic programming problem this permits the direct use of standard decomposition and sampling methods developed for stochastic programming we also show that this class of minimax stochastic programs is closely related to a large family of mean risk stochastic programs where risk is measured in terms of deviations from a quantile

numerical study of blow up to the purely elliptic generalized daveystewartson system blow up solutions for the purely elliptic generalized daveystewartson system are studied by using a relaxation numerical method the numerical method is based on an implicit finite difference scheme with a second order accuracy in both time and space the stability of the numerical method is analyzed by investigating the linear stability of plane wave solutions to evaluate the ability of the relaxation method to detect blow up numerical simulations are conducted for several test problems a particular attention is paid to the gap interval neither a global existence nor a blow up result is established the monotonicity properties of blow up time on the coupling parameter are also investigated numerically

robust decision making using a general utility set abstract we address the problem of ambiguity and inconsistency in a decision maker s dm assessed utility function by using a maxmin framework in this framework dm s utility function belongs to a set of functions the member functions of the set are nondecreasing and satisfy additional boundary and auxiliary conditions the maxmin framework provides robustness in decision making alternatively it allows us to perform sensitivity or parametric analysis for the optimal decision when a reference utility function is known for this purpose we use a cost of ambiguity concept and show that this cost is increasing and concave when extent of ambiguity is parametrically increased next we develop a lagrangian based solution approach for the decision problem we show that under suitable conditions a sample average approximation saa of the lagrangian model can be solved using a mixed integer linear program milp we also show that the set of optimal solutions of the saa converges to that of its true counterpart and the optimum objective value of the saa converges to the true objective value at an exponential rate we use this convergence property to develop a heuristic for identifying a solution of the saa milp with increasing sample size we illustrate the properties of the maxmin model using two examples of portfolio investment and streaming bandwidth we provide a discussion on the performance of a commercial solver used to solve saa milps and the quality of solutions generated by the proposed heuristic

an efficient dynamically adaptive mesh for potentially singular solutions abstract we develop an efficient dynamically adaptive mesh generator for time dependent problems in two or more dimensions the mesh generator is motivated by the variational approach and is based on solving a new set of nonlinear elliptic pdes for the mesh map when coupled to a physical problem the mesh map evolves with the underlying solution and maintains high adaptivity as the solution develops complicated structures and even singular behavior the overall mesh strategy is simple to implement avoids interpolation and can be easily incorporated into a broad range of applications the efficacy of the mesh is first demonstrated by two examples of blowing up solutions to the 2 d semilinear heat equation these examples show that the mesh can follow with high adaptivity a finite time singularity process the focus of applications presented here is however the baroclinic generation of vorticity in a strongly layered 2 d boussinesq fluid a challenging problem the moving mesh follows effectively the flow resolving both its global features and the almost singular shear layers developed dynamically the numerical results show the fast collapse to small scales and an exponential vorticity growth

grid evolution in time asymptotic problems the selection of the proper coordinate system in solving any fluid flow or heat transfer problem is a very important consideration a new technique of moving mesh points in physical space is introduced so as to reduce the error in a computed asymptotic solution relative to that obtained using a fixed mesh the technique has been used to solve the simple viscous burgers equation in one and two dimensions substantial error reductions are demonstrated the treatment of boundary points and the effect of using different error criteria in generating grids are discussed

projective and coarse projective integration for problems with continuous symmetries temporal integration of equations possessing continuous symmetries e g systems with translational invariance associated with traveling solutions and scale invariance associated with self similar solutions in a co evolving frame i e a frame which is co traveling co collapsing or co exploding with the evolving solution leads to improved accuracy because of the smaller time derivative in the new spatial frame the slower time behavior permits the use of projective and coarse projective integration with longer projective steps in the computation of the time evolution of partial differential equations and multiscale systems respectively these methods are also demonstrated to be effective for systems which only approximately or asymptotically possess continuous symmetries the ideas of projective integration in a co evolving frame are illustrated on the one dimensional translationally invariant nagumo partial differential equation pde a corresponding kinetic monte carlo model motivated from the nagumo kinetics is used to illustrate the coarse grained method a simple one dimensional diffusion problem is used to illustrate the scale invariant case the efficiency of projective integration in the co evolving frame for both the macroscopic diffusion pde and for a random walker particle based model is again demonstrated

variational theory for optimization under stochastic ambiguity stochastic ambiguity provides a rich class of uncertainty models that includes those in stochastic robust risk based and semi infinite optimization and that accounts for uncertainty about parameter values as well as incompleteness of the description of uncertainty we provide a novel unifying perspective on optimization under stochastic ambiguity that rests on two pillars first ambiguity is formulated in terms of the cumulative probability distribution associated with the random elements more specifically ambiguity is expressed by letting this distribution belong to a subfamily of distributions that might or might not depend on the decision variable we derive a series of estimates by introducing a metric for the space of distribution functions based on the hypo distance between upper semicontinuous functions in the process we show that this metric is consistent with convergence in distribution weak star convergence of the associated probability measures second we rely on the theory

dual stochastic dominance and related mean risk models we consider the problem of constructing mean risk models which are consistent with the second degree stochastic dominance relation by exploiting duality relations of convex analysis we develop the quantile model of stochastic dominance for general distributions this allows us to show that several models using quantiles and tail characteristics of the distribution are in harmony with the stochastic dominance relation we also provide stochastic linear programming formulations of these models

robustifying convex risk measures for linear portfolios a nonparametric approach this paper introduces a framework for robustifying convex law invariant risk measures the robustified risk measures are defined as the worst case portfolio risk over neighborhoods of a reference probability measure which represent the investors beliefs about the distribution of future asset losses it is shown that under mild conditions the infinite dimensional optimization problem of finding the worst case risk can be solved analytically and closed form expressions for the robust risk measures are obtained using these results robust versions of several risk measures including the standard deviation the conditional value at risk and the general class of distortion functionals are derived the resulting robust risk measures are convex and can be easily incorporated into portfolio optimization problems and a numerical study shows that in most cases they perform significantly better out of sample than their nonrobust variants in terms of risk expected losses and turnover

price of correlations in stochastic optimization when decisions are made in the presence of high dimensional stochastic data handling joint distribution of correlated random variables can present a formidable task both in terms of sampling and estimation as well as algorithmic complexity a common heuristic is to estimate only marginal distributions and substitute joint distribution by independent product distribution in this paper we study possible loss incurred on ignoring correlations through a distributionally robust stochastic programming model and we quantify that loss as price of correlations poc using techniques of cost sharing from game theory we identify a wide class of problems for which poc has a small upper bound to our interest this class will include many stochastic convex programs uncapacitated facility location steiner tree and submodular functions suggesting that the intuitive approach of assuming independent distribution acceptably approximates the robust model for these stochastic optimization problems additionally we demonstrate hardness of bounding poc via examples of subadditive and supermodular functions that have large poc we find that our results are also useful for solving many deterministic optimization problems like welfare maximization k dimensional matching and transportation problems under certain conditions

decomposition algorithms for two stage distributionally robust mixed binary programs in this paper we introduce and study a two stage distributionally robust mixed binary problem tsdr mbp in which the random parameters follow the worst case distribution belonging to an uncertain

singular solutions of the biharmonic nonlinear schrodinger equation we consider singular solutions of the l 2 critical biharmonic nonlinear schrodinger equation we prove that the blowup rate is bounded by a quartic root the solution approaches a quasi self similar profile and a finite amount of l 2 norm which is no less than the critical power concentrates into the singularity we also prove the existence of a ground state solution we use asymptotic analysis to show that the blowup rate of peak type singular solutions is slightly faster than that of a quartic root and the self similar profile is given by the ground state standing wave these findings are verified numerically up to focusing levels of 10 8 using an adaptive grid method we also use the spectral renormalization method to compute the ground state of the standing wave equation and the critical power for collapse in one two and three dimensions

optimal anisotropic meshes for minimizing interpolation errors in l p norm in this paper we present a new optimal interpolation error estimate in l p norm 1 p for finite element simplicial meshes in any spatial dimension a sufficient condition for a mesh to be nearly optimal is that it is quasi uniform under a new metric defined by a modified hessian matrix of the function to be interpolated we also give new functionals for the global moving mesh method and obtain optimal monitor functions from the viewpoint of minimizing interpolation error in the l p norm some numerical examples are also given to support the theoretical estimates

explicit reformulations for robust optimization problems with general uncertainty sets we consider a rather general class of mathematical programming problems with data uncertainty where the uncertainty set is represented by a system of convex inequalities we prove that the robust counterparts of this class of problems can be reformulated equivalently as finite and explicit optimization problems moreover we develop simplified reformulations for problems with uncertainty sets defined by convex homogeneous functions our results provide a unified treatment of many situations that have been investigated in the literature and are applicable to a wider range of problems and more complicated uncertainty sets than those considered before the analysis in this paper makes it possible to use existing continuous optimization algorithms to solve more complicated robust optimization problems the analysis also shows how the structure of the resulting reformulation of the robust counterpart depends both on the structure of the original nominal optimization problem and on the structure of the uncertainty set

a three dimensional adaptive method based on the iterative grid redistribution in this paper we develop a three dimensional adaptive method based on iterative grid redistribution technique introduced in j comput phys 159 2000 246 the key step for the successful implementation is a fast algorithm for grid generation which is composed of solving the linear grid equation systems and an efficient method for inverting a map by computing iso surface intersections to carry out the three dimensional calculations the whole procedure is also parallelized on a pc cluster the improved and extended three dimensional adaptive method is applied to solve pdes with singular solutions that have three dimensional structures numerical experiments have demonstrated the method s effectiveness

a note on uniform exponential convergence of sample average approximation of random functions abstract shapiro and xu 2008 17 investigated uniform large deviation of a class of holder continuous random functions it is shown under some standard moment conditions that with probability approaching one at exponential rate with the increase of sample size the sample average approximation of the random function converges to its expected value uniformly over a compact set this note extends the result to a class of discontinuous functions whose expected values are continuous and the holder continuity may be violated for some negligible random realizations the extension entails the application of the exponential convergence result to a substantially larger class of practically interesting functions in stochastic optimization

an approximation framework for two stage ambiguous stochastic integer programs under mean mad information abstract we consider two stage recourse models in which only limited information is available on the probability distributions of the random parameters in the model if all decision variables are continuous then we are able to derive the worst case and best case probability distributions under the assumption that only the means and mean absolute deviations of the random parameters are known contrary to most existing results in the literature these probability distributions are the same for every first stage decision the ambiguity set that we use in this paper also turns out to be particularly suitable for ambiguous recourse models involving integer decisions variables for such problems we develop a general approximation framework and derive error bounds for using these approximatons we apply this approximation framework to mixed ambiguous mixed integer recourse models in which some of the probability distributions of the random parameters are known and others are ambiguous to illustrate these results we carry out numerical experiments on a surgery block allocation problem

blow up properties for a degenerate parabolic system with nonlinear localized sources in this paper we investigate the initial boundary problem of a degenerate parabolic system with nonlinear localized sources we classify the blow up solutions into global blow up cases and single point blow up cases according to the values of m n p i q i furthermore we obtain the uniform blow up profiles of solutions for the global blow up case finally we give some numerical examples to verify the results these extend and generalize a recent work of one of the authors l du blow up for a degenerate reaction diffusion systems with nonlinear localized sources j math anal appl 324 2006 304 320 which only considered uniform blow up profiles under the special case p 1 p 2 0

discontinuous piecewise linear optimization a theoretical framework and a practical algorithm are presented to solve discontinuous piecewise linear optimization problems dealing with functions for which theridges are known a penalty approach allows one to consider such problems subject to a wide range of constraints involving piecewise linear functions although the theory is expounded in detail in the special case of discontinuous piecewiselinear functions it is straightforwardly extendable using standard nonlinear programming techniques tononlinear discontinuous piecewise differentiable functions the descent algorithm which is elaborated uses active set and projected gradient approaches it is a generalization of the ideas used by conn to deal with nonsmoothness in thel1 exact penalty function and it is based on the notion ofdecomposition of a function into a smooth and a nonsmooth part the constrained case is reduced to the unconstrained minimization of a piecewise linear l1 exact penalty function we also discuss how the algorithm is modified when it encounters degenerate points preliminary numerical results are presented the algorithm is applied to discontinuous optimization problems from models in industrial engineering 1998 the mathematical programming society inc published by elsevier science b v

convergence analysis for distributionally robust optimization and equilibrium problems in this paper we study distributionally robust optimization approaches for a one stage stochastic minimization problem where the true distribution of the underlying random variables is unknown but it is possible to construct a set of probability distributions which contains the true distribution and optimal decision is taken on the basis of the worst possible distribution from that set we consider the case when the distributional set which is also known as the ambiguity set varies and its impact on the optimal value and the optimal solutions a typical example is when the ambiguity set is constructed through samples and we need to look into the impact of increasing the sample size the analysis provides a unified framework for convergence of some problems where the ambiguity set is approximated in a process with increasing information on uncertainty and extends the classical convergence analysis in stochastic programming the discussion is extended briefly to a stochastic nash equilibrium problem wh