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

adaptive spectral viscosity for hyperbolic conservation laws spectral approximations to nonlinear hyperbolic conservation laws require dissipative regularization for stability the dissipative mechanism must on the other hand be small enough in order to retain the spectral accuracy in regions where the solution is smooth we introduce a new form of viscous regularization which is activated only in the local neighborhood of shock discontinuities the basic idea is to employ a spectral edge detection algorithm as a dynamical indicator of where in physical space to apply numerical viscosity the resulting spatially local viscosity is successfully combined with spectral viscosity where a much higher than usual cut off frequency can be used numerical results show that the new adaptive spectral viscosity scheme significantly improves the accuracy of the standard spectral viscosity scheme in particular results are improved near the shocks and at low resolutions examples include numerical simulations of burgers equation shallow water with bottom topography and the isothermal euler equations we also test the schemes on a nonconvex scalar problem finding that the new scheme approximates the entropy solution more reliably than the standard spectral viscosity scheme

adjustable robust solutions of uncertain linear programs we consider linear programs with uncertain parameters lying in some prescribed uncertainty set where part of the variables must be determined before the realization of the uncertain parameters non adjustable variables while the other part are variables that can be chosen after the realization adjustable variables we extend the robust optimization methodology 1 3 6 9 13 14 to this situation by introducing the adjustable robust counterpart arc associated with an lp of the above structure often the arc is significantly less conservative than the usual robust counterpart rc however in most cases the arc is computationally intractable np hard this difficulty is addressed by restricting the adjustable variables to be affine functions of the uncertain data the ensuing affinely adjustable robust counterpart aarc problem is then shown to be in certain important cases equivalent to a tractable optimization problem typically an lp or a semidefinite problem and in other cases having a tight approximation which is tractable the aarc approach is illustrated by applying it to a multi stage inventory management problem

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

chaos revealing multiplicative representation of quantum eigenstates the quantisation of the two dimensional toric and spherical phase spaces is considered in analytic coherent state representations every pure quantum state admits therein a finite multiplicative parametrisation by the zeros of its husimi function for eigenstates of quantised systems this description explicitly reflects the nature of the underlying classical dynamics in the semiclassical regime the distribution of the zeros in the phase space becomes one dimensional for integrable systems and highly spread out conceivably uniform for chaotic systems this multiplicative representation thereby acquires a special relevance for semiclassical analysis in chaotic systems

on the zeros of the husimi distribution the basic features of the zeros of the husimi phase space density of a quantum eigenstate are discussed it is demonstrated that some special properties of the harmonic oscillator related to the number of zeros and their location within the classical energy shell are not valid for anharmonic potentials as illustrated by numerical examples for the morse oscillator also discussed is the distribution of zeros in quantum poincare sections for systems with two degrees of freedom

introduction to convex optimization in financial markets convexity arises naturally in financial risk management in risk preferences concerning random cash flows convexity corresponds to the fundamental diversification principle convexity is a basic property also of budget constraints both in classical linear models as well as in more realistic models with transaction costs and constraints moreover modern securities markets are based on trading protocols that result in convex trading costs the first part of this paper gives an introduction to certain basic concepts and principles of financial risk management in simple optimization terms the second part reviews some convex optimization techniques used in mathematical and numerical analysis of financial optimization problems

the rate of convergence of spectral viscosity methods for periodic scalar conservation laws a rate of convergence is proven for spectral viscosity methods for periodic scalar conservation laws this rate is obtained by showing the discretization error to be small enough that the difference between the solutions of the spectral viscosity method and the ordinary viscosity method tends to zero in l 1 as the number of discrete modes tends to infinity and the viscosity simultaneously tends to zero at the appropriate rate the method is also used to obtain an l infty bound for the spectral viscosity approximations to the elasticity equations convergence although without a rate then follows from compensated compactness theory

phase space approach to quantum dynamics replaces the schrodinger equation for the time propagation of states of a quantized 2d spherical phase space by the dynamics of a system of n particles lying in phase space this is done through factorization formulae of an analytic function theory arising in a coherent state representation the particles being the zeroes of the quantum state for linear hamiltonians like a spin in a uniform magnetic field the motion of the particles is classical however nonlinear terms induce interactions between the particles their time propagation is studied and it is shown that contrary to integrable systems for chaotic maps they tend to fill as their classical counterpart the whole phase space in a uniform way

detection of edges in spectral data ii nonlinear enhancement we discuss a general framework for recovering edges in piecewise smooth functions with finitely many jump discontinuities where f x f x f x neq 0 our approach is based on two main aspects localization using appropriate concentration kernels and separation of scales by nonlinear enhancement to detect such edges one employs concentration kernels k epsilon cdot depending on the small scale epsilon it is shown that odd kernels properly scaled and admissible in the sense of having small w 1 infty moments of order cal o epsilon satisfy k epsilon f x f x cal o epsilon thus recovering both the location and amplitudes of all edges as an example we consider general concentration kernels of the form k sigma n t sum sigma k n sin kt to detect edges from the first 1 epsilon n spectral modes of piecewise smooth f s here we improve in generality and simplicity over our previous study in a gelb and e tadmor appl comput harmon anal 7 1999 pp 101 135 both periodic and nonperiodic spectral projections are considered we identify in particular a new family of exponential factors sigma exp cdot with superior localization properties the other aspect of our edge detection involves a nonlinear enhancement procedure which is based on separation of scales between the edges where k epsilon f x sim f x neq 0 and the smooth regions where k epsilon f cal o epsilon sim 0 numerical examples demonstrate that by coupling concentration kernels with nonlinear enhancement one arrives at effective edge detectors

duality and martingales a stochastic programming perspective on contingent claims the hedging of contingent claims in the discrete time discrete state case is analyzed from the perspective of modeling the hedging problem as a stochastic program application of conjugate duality leads to the arbitrage pricing theorems of financial mathematics namely the equivalence of absence of arbitrage and the existence of a probability measure that makes the price process into a martingale the model easily extends to the analysis of options pricing when modeling risk management concerns and the impact of spreads and margin requirements for writers of contingent claims however we find that arbitrage pricing in incomplete markets fails to model incentives to buy or sell options an extension of the model to incorporate pre existing liabilities and endowments reveals the reasons why buyers and sellers trade in options the model also indicates the importance of financial equilibrium analysis for the understanding of options prices in incomplete markets

legendre pseudospectral viscosity method for nonlinear conservation laws in this paper the legendre spectral viscosity sv method for the approximate solution of initial boundary value problems associated with nonlinear conservation laws is studied the authors prove that by adding a small amount of sv bounded solutions of the legendre sv method converge to the exact scalar entropy solution the convergence proof is based on compensated compactness arguments and therefore applies to certain 2 times 2 systems finally numerical experiments for scalar as well as the one dimensional system of gas dynamics equations are presented which confirm the convergence of the legendre sv method moreover these numerical experiments indicate that by post processing the sv approximation one can recover the entropy solution within spectral accuracy

epi convergent discretizations of multistage stochastic programs in many dynamic stochastic optimization problems in practice the uncertain factors are best modeled as random variables with an infinite support this results in infinite dimensional optimization problems that can rarely be solved directly therefore the random variables stochastic processes are often approximated by finitely supported ones scenario trees which result in finite dimensional optimization problems that are more likely to be solvable by available optimization tools this paper presents conditions under which such finite dimensional optimization problems can be shown to epi converge to the original infinite dimensional problem epi convergence implies the convergence of optimal values and solutions as the discretizations are made finer our convergence result applies to a general class of convex problems where neither linearity nor complete recourse are assumed

sparse low energy decomposition for viscous conservation laws for viscous conservation laws solutions contain smooth but high contrast features which require the use of fine grids to properly resolve on coarse grids these high contrast jumps resemble shocks rather than their true viscous profiles which could lead to issues in the numerical approximation of their underlying dynamics in many cases the equations of motion emit traveling wave solutions which can be used to represent the viscous profiles analytically the traveling wave solutions can be thought of as a lower dimensional representation of the motion since they contain information from the evolution equation but are constant along certain time space curves using a parameterized basis involving the traveling waves along with the sparse low energy decompositions found in imaging sciences we propose an approximation to viscous conservation laws which separates the coarse smooth component from the sharp fine one our method provides an appropriate approximation to the solution on a coarse grid thereby accurately under resolving the viscous profile this is similar to the philosophy of shock capturing methods in the sense that we want to capture the viscous front without needing to resolve the profile theoretical results on the consistency of our method are shown in general we provide several computational examples for convex and non convex fluxes

epi convergent discretizations of multistage stochastic programs via integration quadratures this paper presents procedures for constructing numerically solvable discretizations of multistage stochastic programs that epi converge to the original problem as the discretizations are made finer epi convergence implies in particular that the cluster points of the first stage solutions of the discretized problems are optimal first stage solutions of the original problem the discretization procedures apply to a general class of nonlinear stochastic programs where the uncertain factors are driven by time series models using existing routines for numerical integration allows for an easy and efficient implementation of the procedures

renyi entropies characterizing the shape and the extension of the phase space representation of quantum wave functions in disordered systems we discuss some properties of the generalized entropies called renyi entropies and their application to the case of continuous distributions in particular it is shown that these measures of complexity can be divergent however their differences are free from these divergences thus enabling them to be good candidates for the description of the extension and the shape of continuous distributions we apply this formalism to the projection of wave functions onto the coherent state basis i e to the husimi representation we also show how the localization properties of the husimi distribution on average can be reconstructed from its marginal distributions that are calculated in position and momentum space in the case when the phase space has no structure i e no classical limit can be defined numerical simulations on a one dimensional disordered system corroborate our expectations

log optimal and rapid paths in von neumann gale dynamical systems abstract von neumann gale dynamical systems are defined in terms of multivalued operators in spaces of random vectors possessing certain properties of convexity and homogeneity a central role in the theory of such systems is played by a special class of paths trajectories called rapid they grow over each time period t 1 t in a sense faster than others the paper establishes existence and characterization theorems for such paths showing in particular that any trajectory maximizing a logarithmic functional over a finite time horizon is rapid the proof of this result is based on the methods of convex analysis in spaces of measurable functions the study is motivated by the applications of the theory of von neumann gale dynamical systems to the modeling of capital growth in financial markets with friction

total variation and error estimates for spectral viscosity approximations we study the behavior of spectral viscosity approximations to nonlinear scalar conservation laws we show how the spectral viscosity method compromises between the total variation bounded viscosity approximations which are restricted to first order accuracy and the spectrally accurate yet unstable fourier method in particular we prove that the spectral viscosity method is l sup 1 stable and hence total variation bounded moreover the spectral viscosity solutions are shown to be lip sup stable in agreement with oleinik s e entropy condition this essentially nonoscillatory behavior of the spectral viscosity method implies convergence to the exact entropy solution and we provide convergence rate estimates of both global and local types 16 refs

convex duality in stochastic optimization and mathematical finance this paper proposes a general duality framework for the problem of minimizing a convex integral functional over a space of stochastic processes adapted to a given filtration the framework unifies many well known duality frameworks from operations research and mathematical finance the unification allows the extension of some useful techniques from these two fields to a much wider class of problems in particular combining certain finite dimensional techniques from convex analysis with measure theoretic techniques from mathematical finance we are able to close the duality gap in some situations where traditional topological arguments fail

stochastic programs without duality gaps this paper studies dynamic stochastic optimization problems parameterized by a random variable such problems arise in many applications in operations research and mathematical finance we give sufficient conditions for the existence of solutions and the absence of a duality gap our proof uses extended dynamic programming equations whose validity is established under new relaxed conditions that generalize certain no arbitrage conditions from mathematical finance

analysis of the spectral vanishing viscosity method for periodic conservation laws the convergence of the spectral vanishing method for both the spectral and pseudospectral discretizations of the inviscid burgers equation is analyzed it is proved that this kind of vanishing viscosity is responsible for spectral decay of those fourier coefficients located toward the end of the computed spectrum consequently the discretization error is shown to be spectrally small independently of whether or not the underlying solution is smooth this in turn implies that the numerical solution remains uniformly bounded and convergence follows by compensated compactness arguments

non classical shocks and kinetic relations scalar conservation laws this paper analyzes the non classical shock waves which arise as limits of certain diffusive dispersive approximations to hyperbolic conservation laws such shocks occur for non convex fluxes and connect regions of different convexity they have negative entropy dissipation for a single convex entropy function but not all convex entropies and do not obey the classical oleinik entropy criterion we derive necessary conditions for the existence of non classical shock waves and construct them as limits of traveling wave solutions for several diffusive dispersive approximations

expected utility penalty functions and duality in stochastic nonlinear programming we consider nonlinear programming problem p with stochastic constraints the lagrangean corresponding to such problems has a stochastic part which in this work is replaced by its certainty equivalent in the sense of expected utility theory it is shown that the deterministic surrogate problem ce p thus obtained contains a penalty function which penalizes violation of the constraints in the mean the approach is related to several known methods in stochastic programming such as chance constraints stochastic goal programming reliability programming and mean variance analysis the dual problem of ce p is studied for problems with stochastic righthand sides in the constraints and a comprehensive duality theory is developed by introducing a new certainty equivalent nce concept motivation for the nce and its potential role in decision theory are discussed as well as mean variance approximations

spectral vanishing viscosity method for nonlinear conservation laws we propose a new spectral viscosity sv scheme for the accurate solution of nonlinear conservation laws it is proved that the sv solution converges to the unique entropy solution under appropriate reasonable conditions the proposed sv scheme is implemented directly on high modes of the computed solution this should be compared with the original nonperiodic sv scheme introduced by maday ould kaber and tadmor in siam j numer anal 30 1993 321 342 where sv is activated on the derivative of the sv solution the new proposed sv method could be viewed as a correction of the former and it offers an improvement which is confirmed by our numerical experiments a postprocessing method is implemented to greatly enhance the accuracy of the computed sv solution the numerical results show the efficiency of the new method

phase retrieval problem in quantum chaos and its relation to the origin of irreversibility i abstract the purpose of this paper are i to study characteristics features of phase in quantum dynamics corresponding to chaos and ii to discuss implication of these features for the origin of irreversibility the first is done by focusing attention on topological defects of phase in the husimi representation we find that the process of generating topological defects corresponds to the folding of morse smale horseshoe dynamics moreover the distribution of topological defects changes from regular to random under the time evolution this change is a distinctive feature of quantum dynamics corresponding to chaos the second is done in terms of phase retrieval problem we present numerical results which strongly suggest the following the randomness in the distribution of topological defects is the cause of difficulty in retrieving phase based on these results we present the following hypothesis the loss of phase information in statistical mechanics can be understood in terms of the randomness of phase in quantum dynamics corresponding to chaos

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

intrinsic ordering of quasienergy states for mixed regular chaotic quantum systems zeros of the husimi distribution a method is proposed which allows an intrinsic ordering of the quasienergy states for a time periodic quantum system according to the number of essential zeros of the husimi distribution in phase space the corresponding classical system shows a typical mixed dynamics with coexisting regular and chaotic regions in phase space the ordering of states allows a straightforward comparison with semiclassical quantities moreover a direct connection of the organization of zeros with the underlying classical phase space dynamics is demonstrated

convergence of spectral methods for nonlinear conservation laws we discuss the convergence of fourier methods for scalar nonlinear conservation laws that exhibit spontaneous shock discontinuities numerical tests indicate that the convergence may and in fact in some cases we prove it must fail with or without post processing of the numerical solution instead we introduce here a new kind of spectrally accurate vanishing viscosity to augment the fourier approximation of such nonlinear conservation laws using compensated compactness arguments augmented by the assumption of l infty stability we show for the inviscid burgers model equation that this spectral viscosity method prevents oscillations and convergence to the unique entropy solution follows