parallel sorting with constant time for comparisons we prove that there exist graphs with n vertices and at most 2n 5 3 log n edges for which every acyclic orientation has in its transitive closure at least begin pmatrix n 2 end pmatrix 10n 5 3 arcs we conclude that with 2n 5 3 log n parallel processors n items may be sorted with all comparisons arranged in two time intervals we also show that frac 1 9 n 3 2 processors are not sufficient to achieve the same end these results are extended to parallel sorting in k time intervals and related to other work on parallel sorting the existence of sorting algorithms achieving the bounds is proved by nonconstructive methods the constants quoted in the abstract are somewhat improved in the paper

maximum entropy and the nearly black object summary maximum entropy me inversion is a non linear inversion technique for inverse problems where the object to be recovered is known to be positive it has been applied in areas ranging from radio astronomy to various forms of spectroscopy sometimes with dramatic success in some cases me has attained an order of magnitude finer resolution and or an order of magnitude smaller noise level than that obtainable by standard linear methods the dramatic successes all seem to occur in cases where the object to be recovered is nearly black essentially zero in the vast majority of samples we show that near blackness is required both for signal to noise enhancements and for superresolution however other methods in particular minimum 1 norm reconstruction may exploit near blackness to an even greater extent

numerical schemes for hamilton jacobi equations on unstructured meshes in this paper a numerical scheme is presented by applying the finite element method to the viscosity equations of the hamilton jacobi equations on unstructured meshes by improving the finite element scheme another numerical scheme is constructed under certain limitations the numerical solutions of the two schemes converge to the viscosity solutions of the hamilton jacobi equations the latter numerical scheme has weaker restrictions than the former scheme for convergence numerical examples are provided to test the stability convergence and sensitivity to different meshes

entropic approach to interior point solution of linear programs we present a maxentropic approach for obtaining interior suboptimal solutions to a linear programming problem not necessarily in karmarkar s canonical form each suboptimal solution lies in the relative interior set of feasible solutions we present an application to the problem of obtaining reconstructions of minimal weighted l 1 norm of a function from the knowledge of a few of its fourier coefficients

finding an approximate maximum suppose that there are n elements from a totally ordered domain the objective is to find in a minimum possible number of rounds an element that belongs to the biggest n 2 where in each round one is allowed to ask n binary comparisons it is shown that log n theta 1 rounds are both necessary and sufficient in the best algorithm for this problem

approximation schemes for viscosity solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of applications including the calculus of variations control theory and differential games recently m g crandall and p l lions established the correct notion of generalized solutions for these equations this article discusses the convergence of general approximation schemes to this solution and gives under certain hypotheses explicit error estimates these results are then applied to obtain various representations as limits of solutions of general explicit and implicit finite difference schemes with error estimates

explicit solution of a general consumption investment problem this paper solves a general consumption and investment decision problem in closed form an investor seeks to maximize total expected discounted utility of consumption there are n distinct risky investments modelled by dependent geometric brownian motion processes and one riskless deterministic investment the analysis allows for a general utility function and general rates of return the model and analysis take into consideration the inherent nonnegativity of consumption and consider bankruptcy so this paper generalizes many of the results of lehoczky sethi and shreve lehoczky j s sethi s shreva 1983 optimal consumption and investment policies allowing consumption constraints and bankruptcy math oper res 8 613 636 the value function is determined explicitly as are the optimal consumption and investment policies the analysis is extended to consider more general risky investments under certain conditions the value functions derived for geometric brownian motion are shown to provide upper and lower bounds on the value functions in the more general context

bayesian methods and maximum entropy for ill posed inverse problems in this paper we study linear inverse problems where some generalized moments of an unknown positive measure are observed we introduce a new construction called the maximum entropy on the mean method mem which relies on a suitable sequence of finite dimensional discretized inverse problems its advantage is threefold it allows us to interpret all usual deterministic methods as bayesian methods it gives a very convenient way of taking into account prior information it also leads to new criteria for the existence question concerning the linear inverse problem which will be a starting point for the investigation of superresolution phenomena the key tool in this work is the large deviations property of some discrete random measure connected with the reconstruction procedure

partially finite programming in l_1 and the existence of maximum entropy estimates best entropy estimation is a technique that has been widely applied in many areas of science it consists of estimating an unknown density from some of its moments by maximizing some measure of the entropy of the estimate this problem can be modelled as a partially finite convex program with an integrable function as the variable a complete duality and existence theory is developed for this problem and for an associated extended problem which allows singular measure theoretic solutions this theory explains the appearance of singular components observed in the literature when the burg entropy is used it also provides a unified treatment of existence conditions when the burg boltzmann shannon or some other entropy is used as the objective some examples are discussed

blind deconvolution of discrete linear systems we study the blind deconvolution problem in the case where the input noise has a finite discrete support and the transfer linear system is not necessarily minimum phase we propose a new family of estimators built using algebraic considerations the estimates are consistent under very wide assumptions the input signal need not be independently distributed the cardinality of the finite support may be estimated simultaneously we consider in particular ar systems in this case we prove that the estimator of the parameters is perfect a s with a finite number of observations

parallel comparison merging of many ordered lists we consider the problem of merging m disjoint ordered lists each of size n m we determine up to a constant factor the worst case and average case deterministic and randomized parallel comparison complexity of the problem for all the range of n m and p where p is the number of processors used the worst case deterministic time complexity is log mlog 1 p n loglog nlog 2 p n that means n log mp log log n for p 2n and log mlog p n loglog nlog p n for p 2n clearly merging two equal lists and sorting are special cases of this problem for m 2 and m n respectively we also prove that these bounds hold for randomized algorithms and even for the average case of deterministic or randomized ones therefore the average case of the best deterministic or randomized algorithm for this problem is not faster than the worst case of the best deterministic one by more than a constant factor

random time changes and convergence in distribution under the meyer zheng conditions an analog of conditions of meyer and zheng for the relative compactness in the sense of convergence in distribution of a sequence of stochastic processes is formulated for general separable metric spaces and the corresponding notion of convergence is characterized in terms of the convergence in the skorohod topology of time changes of the original processes in addition convergence in distribution under the topology of convergence in measure is discussed and results of jacod memin and metivier on convergence under the skorohod topology are extended

numerical methods for optimal dividend payment and investment strategies of regime switching jump diffusion models with capital injections this work focuses on numerical methods for finding optimal investment dividend payment and capital injection policies to maximize the present value of the difference between the cumulative dividend payment and the possible capital injections the surplus is modeled by a regime switching jump diffusion process subject to both regular and singular controls using the dynamic programming principle the value function is a solution of the coupled system of nonlinear integro differential quasi variational inequalities in this paper the state constraint of the impulsive control gives rise to a capital injection region with free boundary which makes the problem even more difficult to analyze together with the regular control and regime switching the closed form solutions are virtually impossible to obtain we use markov chain approximation techniques to construct a discrete time controlled markov chain to approximate the value function and optimal controls convergence of the approximation algorithms is proved examples are presented to illustrate the applicability of the numerical methods

two approximations of solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of application including the calculus of variations control theory and differential games the associated initial value problems almost never have global time classical solutions and one must deal with suitable generalized solutions the correct class of generalized solutions has only recently been established by the authors this article establishes the convergence of a class of difference approximations to these solutions by obtaining explicit error estimates analogous results are proved by similar means for the method of vanishing viscosity author

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

a new look at entropy for solving linear inverse problems entropy based methods are widely used for solving inverse problems particularly when the solution is known to be positive here we address linear ill posed and noisy inverse problems of the form z ax n with a general convex constraint x spl isin x where x is a convex set although projective methods are well adapted to this context we study alternative methods which rely highly on some information based criteria our goal is to clarify the role played by entropy in this field and to present a new point of view on entropy using general tools and results coming from convex analysis we present then a new and broad scheme for entropic based inversion of linear noisy inverse problems this scheme was introduced by navaza in 1985 in connection with a physical modeling for crystallographic applications and further studied by dacunha castelle and gamboa 1990 important features of this paper are i a unified presentation of many well known reconstruction criteria ii proposal of new criteria for reconstruction under various prior knowledge and with various noise statistics iii a description of practical inversion of data using the aforementioned criteria and iv a presentation of some reconstruction results

convergent numerical scheme for singular stochastic control with state constraints in a portfolio selection problem we consider a singular stochastic control problem with state constraints that arises in problems of optimal consumption and investment under transaction costs numerical approximations for the value function using the markov chain approximation method of kushner and dupuis are studied the main result of the paper shows that the value function of the markov decision problem mdp corresponding to the approximating controlled markov chain converges to that of the original stochastic control problem as various parameters in the approximation approach suitable limits all our convergence arguments are probabilistic the main assumption that we make is that the value function be finite and continuous in particular uniqueness of the solutions of the associated hjb equations is neither needed nor available in the generality under which the problem is considered specific features of the problem that make the convergence analysis nontrivial include unboundedness of the state and control space and the cost function degeneracies in the dynamics mixed boundary dirichlet neumann conditions and presence of both singular and absolutely continuous controls in the dynamics finally schemes for computing the value function and optimal control policies for the mdp are presented and illustrated with a numerical study

additive control of stochastic linear systems with finite horizon we consider a dynamic system whose state is governed by a linear stochastic differential equation with time dependent coefficients the control acts additively on the state of the system our objective is to minimize an integral cost which depends upon the evolution of the state and the total variation of the control process it is proved that the optimal cost is the unique solution of an appropriate free boundary problem in a space time domain by using some decomposition arguments the problems of a two sided control i e optimal corrections and the case with constraints on the resources i e finite fuel can be reduced to a simpler case of only one sided control i e a monotone follower these results are applied to solving some examples by the so called method of similarity solutions

heavy traffic convergence of a controlled multiclass queueing system this paper provides a rigorous proof of the connection between the optimal sequencing problem for a two station two customer class queueing network and the problem of control of a multidimensional diffusion process obtained as a heavy traffic limit of the queueing problem in particular the diffusion problem which is one of singular control of a brownian motion is used to develop policies which are shown to be asymptotically nearly optimal as the traffic intensity approaches one in the queueing network the results are proved by a viscosity solution analysis of the related hamilton jacobi bellman equations

numerical solutions of optimal risk control and dividend optimization policies under a generalized singular control formulation this paper develops numerical methods for finding optimal dividend pay out and reinsurance policies a generalized singular control formulation of surplus and discounted payoff function is introduced where the surplus is modeled by a regime switching process subject to both regular and singular controls to approximate the value function and optimal controls markov chain approximation techniques are used to construct a discrete time controlled markov chain the proofs of the convergence of the approximation sequence to the surplus process and the value function are given examples of proportional and excess of loss reinsurance are presented to illustrate the applicability of numerical methods

existence of viscosity solutions of hamilton jacobi equations abstract equations of hamilton jacobi type arise in many areas of application including the calculus of variations control theory and differential games recently m g crandall and p l lions trans amer math soc 277 1983 1 42 introduced the class of viscosity solutions of these equations and proved uniqueness within this class this paper discusses the existence of these solutions under assumptions closely related to the ones which guarantee the uniqueness

routing and singular control for queueing networks in heavy traffic the problem of routing control in an open queueing network under conditions of heavy traffic and finite scaled buffers is dealt with the operating statistics can be state dependent the sequence of scaled controlled state processes converges to a singularly controlled reflected diffusion with the associated costs under broad conditions due to the nature of the controls a scaling method is introduced to obtain the convergence since the actual sequence of processes does not necessarily converge in the skorokhod topology owing to finite buffers an extension of the reflection mapping needs to be obtained the optimal value functions for the physical processes converge to the optimal value function of the limit process under broad conditions approximations to the optimal control for the limit process are obtained as well as properties of the sequence of physical processes the optimal or controlled but not necessarily optimal limit process can be used to approximate a large variety of functio

numerical methods for an optimal investment consumption model this paper examines some numerical techniques for an investment consumption problem considered by fleming and zariphopoulou the value function v x satisfies the differential equation of dynamic programming for x 0 special monotonicity and concavity features of the problem allow us to prove convergence not only of discrete approximations to v x but of the corresponding discrete approximations to optimal investment and consumption policies

the maximum entropy method on the mean applications to linear programming and superresolution in this paper we give two different results we propose new methods to solve classical optimization problems in linear programming we also obtain precise quantitative results for the superresolution phenomenon as observed earlier by practical searchers on specific algorithms the common background of our work is the generalized moment problem which is known to be connected with linear programming and superresolution we describe the maximum entropy method on the mean that provides solution to the problem and leads to computational criteria to decide the existence of solutions or not

reflected brownian motion on an orthant we consider a k dimensional diffusion process z whose state space is the nonnegative orthant on the interior of the orthant z behaves like a kdimensional brownian motion with arbitrary covariance matrix and drift vector at each of the k 1 dimensional hyperplanes that form the boundary of the orthant z reflects instantaneously in a direction that is constant over that hyperplane there is no extant theory of multidimensional diffusion that applies to this process because the boundary of its state space is not smooth we adopt an approach that requires a restriction on the directions of reflection but reiman has shown that this restriction is met by all diffusions z arising as heavy traffic limits in open k station queuing networks our process z is defined as a path to path mapping of k dimensional brownian motion from this construction it follows that z is a continuous markov process and a semimartingale using the latter property we obtain a change of variable formula from which one can develop a complete analytical theory for the process z

numerical methods for stochastic singular control problems the paper develops a powerful class of numerical methods for stochastic singular control problems the basic models used are diffusion or reflected diffusions but the method is of general applicability the central idea is that of the markov chain approximation method where an approximation to the control problem is found for which an optimal solution is computable and which is an arbitrarily good approximation to the original problem and its optimal value function the methods are convenient to program and use and they have been used with success and they cover a wide variety of problems in fact for the singular problem they seem to be the only ones currently available owing to problems in proving tightness of certain processes that occur in the convergence proofs the methods of proof used for the nonsingular problems need modifications examples of useful approximations the algorithms and the convergence proofs are given to illustrate the power of the methods two classes of problems are d

numerical methods for optimal harvesting strategies in random environments under partial observations this work is concerned with optimal harvesting problems in random environments in contrast to the existing literature the markov chain is hidden and can only be observed in a gaussian white noise in our work we first use the wonham filter to estimate the markov chain from the observable evolution of the given process so as to convert the original problem to a completely observable one then we treat the resulting optimal control problem because the problem is virtually impossible to solve in closed form our main effort is devoted to developing numerical approximation algorithms to approximate the value function and optimal strategies markov chain approximation methods are used to construct a discrete time controlled markov chain convergence of the algorithm is proved by weak convergence method and suitable scaling a numerical example is provided to demonstrate the results

duality relationships for entropy like minimization problems this paper considers the minimization of a convex integral functional over the positive cone of an l p space subject to a finite number of linear equality constraints such problems arise in spectral estimation where the bjective function is often entropy like and in constrained approximation the lagrangian dual problem is finite dimensional and unconstrained under a quasi interior constraint qualification the primal and dual values are equal with dual attainment examples show the primal value may not be attained conditions are given that ensure that the primal optimal solution can be calculated directly from a dual optimum these conditions are satisfied in many examples

an implementation of a discretization method for semi infinite programming for linear semi infinite programming problems a discretization method is presented a first coarse grid is successively refined in such a way that the solution on the foregoing grids can be used on the one hand as starting points for the subsequent grids and on the other hand to considerably reduce the number of constraints which have to be considered in the subsequent problems this enables an efficient treatment of large problems with moderate storage requirements a numerically stable simplex algorithm is used to solve the lp subproblems numerical examples from bivariate chebyshev approximation are presented

iterative methods for the solution of a singular control formulation of a gmwb pricing problem discretized singular control problems in finance result in highly nonlinear algebraic equations which must be solved at each timestep we consider a singular stochastic control problem arising in pricing a guaranteed minimum withdrawal benefit gmwb where the underlying asset is assumed to follow a jump diffusion process we use a scaled direct control formulation of the singular control problem and examine the conditions required to ensure that a fast fixed point policy iteration scheme converges our methods take advantage of the special structure of the gmwb problem in order to obtain a rapidly convergent iteration the direct control method has a scaling parameter which must be set by the user we give estimates for bounds on the scaling parameter so that convergence can be expected in the presence of round off error example computations verify that these estimates are of the correct order finally we compare the scaled direct control formulation to a formulation based on a block version of the penalty method huang and forsyth in ima j numer anal 32 320 351 2012 we show that the scaled direct control method has some advantages over the penalty method

application of the pade method to solving the noisy trigonometric moment problem some initial results the noisy trigonometric moment problem for a finite linear combination of box functions is considered and a research program possibly leading to a superresolving method is outlined and some initial steps are performed the method is based on the remark that the poles of the pade approximant to the z transform of the noiseless moments show asymptotically a regular pattern in the complex plane the pattern can be described by a set of arcs connecting points on the unit circle and a pole density function defined on the arcs when a moderate noise affects the moments more arcs are needed to describe the pole pattern but the noiseless pattern is slightly deformed still allowing its identification when this identification is possible a very effective noise filter and moment extrapolator should be easily constructed in this paper only some preliminary steps of the above research program are performed specifically the case of one box function is considered a method for computing the pole patterns

on some cheap control problems for diffusion processes on considere plusieurs cas de problemes de commande de processus de diffusion dans lesquels la fonctionnelle de cout ne depend pas explicitement de la commande on demontre la continuite de la fonction cout optimale et on en donne une caracterisation a l aide d une inegalite quasivariationnelle interpretant le probleme comme la limite d un probleme de commande impulsionnelle dont le cout tend vers zero

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

numerical methods for stochastic singular control problems with state dependent control the markov chain approximation method is a widely used numerical approach to computing optimal controls and value functions for general nonlinear jump diffusions with a possible reflecting boundary we extend the method to models with singular controls where the control increment has the form g x t dh t which we call state dependent owing to the multiplier g x for the most part past work concerned the case where g cdot is constant there are major differences in the properties and treatments of the two cases owing to the possibility of multiple simultaneous impulses h cdot must be interpreted in a generalized sense and the analysis must be done in a stretched out time scale analogously to the approach previously used by the author and colleagues

portfolio selection with transaction costs in this paper optimal consumption and investment decisions are studied for an investor who has available a bank account paying a fixed rate of interest and a stock whose price is a log normal diffusion this problem was solved by merton and others when transactions between bank and stock are costless here we suppose that there are charges on all transactions equal to a fixed percentage of the amount transacted it is shown that the optimal buying and selling policies are the local times of the two dimensional process of bank and stock holdings at the boundaries of a wedge shaped region which is determined by the solution of a nonlinear free boundary problem an algorithm for solving the free boundary problem is given