a new algorithm for high dimensional uncertainty quantification based on dimension adaptive sparse grid approximation and reduced basis methods in this work we develop an adaptive and reduced computational algorithm based on dimension adaptive sparse grid approximation and reduced basis methods for solving high dimensional uncertainty quantification uq problems in order to tackle the computational challenge of curse of dimensionality commonly faced by these problems we employ a dimension adaptive tensor product algorithm 16 and propose a verified version to enable effective removal of the stagnation phenomenon besides automatically detecting the importance and interaction of different dimensions to reduce the heavy computational cost of uq problems modelled by partial differential equations pde we adopt a weighted reduced basis method 7 and develop an adaptive greedy algorithm in combination with the previous verified algorithm for efficient construction of an accurate reduced basis approximation the efficiency and accuracy of the proposed algorithm are demonstrated by several numerical experiments

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

an optimal method for stochastic composite optimization this paper considers an important class of convex programming cp problems namely the stochastic composite optimization sco whose objective function is given by the summation of general nonsmooth and smooth stochastic components since sco covers non smooth smooth and stochastic cp as certain special cases a valid lower bound on the rate of convergence for solving these problems is known from the classic complexity theory of convex programming note however that the optimization algorithms that can achieve this lower bound had never been developed in this paper we show that the simple mirror descent stochastic approximation method exhibits the best known rate of convergence for solving these problems our major contribution is to introduce the accelerated stochastic approximation ac sa algorithm based on nesterov s optimal method for smooth cp nesterov in doklady an sssr 269 543 547 1983 nesterov in math program 103 127 152 2005 and show that the ac sa algorithm can achieve the aforementioned lower bound on the rate of convergence for sco to the best of our knowledge it is also the first universally optimal algorithm in the literature for solving non smooth smooth and stochastic cp problems we illustrate the significant advantages of the ac sa algorithm over existing methods in the context of solving a special but broad class of stochastic programming problems

distributionally robust optimization under moment uncertainty with application to data driven problems stochastic programming can effectively describe many decision making problems in uncertain environments unfortunately such programs are often computationally demanding to solve in addition their solution can be misleading when there is ambiguity in the choice of a distribution for the random parameters in this paper we propose a model that describes uncertainty in both the distribution form discrete gaussian exponential etc and moments mean and covariance matrix we demonstrate that for a wide range of cost functions the associated distributionally robust or min max stochastic program can be solved efficiently furthermore by deriving a new confidence region for the mean and the covariance matrix of a random vector we provide probabilistic arguments for using our model in problems that rely heavily on historical data these arguments are confirmed in a practical example of portfolio selection where our framework leads to better performing policies on the true distribution underlying the daily returns of financial assets

reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation we present a new approach to treating nonlinear operators in reduced basis approximations of parametrized evolution equations our approach is based on empirical interpolation of nonlinear differential operators and their frechet derivatives efficient offline online decomposition is obtained for discrete operators that allow an efficient evaluation for a certain set of interpolation functionals an a posteriori error estimate for the resulting reduced basis method is derived and analyzed numerically we introduce a new algorithm the podei greedy algorithm which constructs the reduced basis spaces for the empirical interpolation and for the numerical scheme in a synchronized way the approach is applied to nonlinear parabolic and hyperbolic equations based on explicit or implicit finite volume discretizations we show that the resulting reduced scheme is able to capture the evolution of both smooth and discontinuous solutions in case of symmetries of the problem the approach realizes an automatic and intuitive space compression or even space dimensionality reduction we perform empirical investigations of the error convergence and run times in all cases we obtain a good run time acceleration

a reduced order discontinuous galerkin method based on pod for electromagnetic simulation this paper is concerned with the design of a reduced order discontinuous galerkin dg method based on the proper orthogonal decomposition pod method for electromagnetic simulation a centered flux approximation for surface integral and a second order leap frog scheme for advancing in time are applied in the classical dg method the pod basis is created by the eigensystem of the correlation matrix which is generated by the snapshot matrix whose columns are the snapshot vectors extracted from the high fidelity dg simulation the pod discontinuous galerkin time domain formulation with lower dimension and sufficiently high accuracy is established by applying a galerkin projection for the semidiscrete dg scheme the overall goal is to reduce the computational cost while maintaining an acceptable level of accuracy numerical experiments for electromagnetic problems illustrate the performance of the proposed reduction method

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

robust convex optimization we study convex optimization problems for which the data is not specified exactly and it is only known to belong to a given uncertainty set u yet the constraints must hold for all possible values of the data from u the ensuing optimization problem is called robust optimization in this paper we lay the foundation of robust convex optimization in the main part of the paper we show that if u is an ellipsoidal uncertainty set then for some of the most important generic convex optimization problems linear programming quadratically constrained programming semidefinite programming and others the corresponding robust convex program is either exactly or approximately a tractable problem which lends itself to efficientalgorithms such as polynomial time interior point methods

an arima model based approach to seasonal adjustment abstract this article proposes a model based procedure to decompose a time series uniquely into mutually independent additive seasonal trend and irregular noise components the series is assumed to follow the gaussian arima model properties of the procedure are discussed and an actual example is given

sample average approximation of stochastic dominance constrained programs in this paper we study optimization problems with second order stochastic dominance constraints this class of problems allows for the modeling of optimization problems where a risk averse decision maker wants to ensure that the solution produced by the model dominates certain benchmarks here we deal with the case of multi variate stochastic dominance under general distributions and nonlinear functions we introduce the concept of mathcal c dominance which generalizes some notions of multi variate dominance found in the literature we apply the sample average approximation saa method to this problem which results in a semi infinite program and study asymptotic convergence of optimal values and optimal solutions as well as the rate of convergence of the feasibility set of the resulting semi infinite program as the sample size goes to infinity we develop a finitely convergent method to find an epsilon optimal solution of the saa problem an important aspect of our contribution is the construction of practical statistical lower and upper bounds for the true optimal objective value we also show that the bounds are asymptotically tight as the sample size goes to infinity

icp registration using invariant features investigates the use of euclidean invariant features in a generalization of iterative closest point icp registration of range images pointwise correspondences are chosen as the closest point with respect to a weighted linear combination of positional and feature distances it is shown that under ideal noise free conditions correspondences formed using this distance function are correct more often than correspondences formed using the positional distance alone in addition monotonic convergence to at least a local minimum is shown to hold for this method when noise is present a method that automatically sets the optimal relative contribution of features and positions is described this method trades off the error in feature values due to noise against the error in positions due to misalignment experimental results suggest that using invariant features decreases the probability of being trapped in a local minimum and may be an effective solution for difficult range image registration problems where the scene is very small compared to the model

pod galerkin reduced order modeling with adaptive finite element snapshots we consider model order reduction by proper orthogonal decomposition pod for parametrized partial differential equations where the underlying snapshots are computed with adaptive finite elements we address computational and theoretical issues arising from the fact that the snapshots are members of different finite element spaces we propose a method to create a pod galerkin model without interpolating the snapshots onto their common finite element mesh the error of the reduced order solution is not necessarily galerkin orthogonal to the reduced space created from space adapted snapshot we analyze how this influences the error assessment for pod galerkin models of linear elliptic boundary value problems as a numerical example we consider a two dimensional convection diffusion equation with a parametrized convective direction to illustrate the applicability of our techniques to non linear time dependent problems we present a test case of a two dimensional viscous burgers equation with parametrized initial data

a class of utility functions containing all the common utility functions this paper presents a class of utility functions class a that contains all of the utility functions commonly used for mathematical modeling the class consisting of those utility functions whose derivatives alternate in sign a simple representation via mixtures of exponential utilities is provided for this class which is both mathematically convenient and conducive to functional operations a connection with laplace transforms and the resultant implications for preference relations aggregation and utility assessment are discussed

optimization with stochastic dominance constraints we introduce stochastic optimization problems involving stochastic dominance constraints we develop necessary and sufficient conditions of optimality and duality theory for these models and show that the lagrange multipliers corresponding to dominance constraints are concave nondecreasing utility functions the models and results are illustrated on a portfolio optimization problem

convergence analysis of stationary points in sample average approximation of stochastic programs with second order stochastic dominance constraints sample average approximation saa method has recently been applied to solve stochastic programs with second order stochastic dominance ssd constraints in particular hu et al math program 133 171 201 2012 presented a detailed convergence analysis of epsilon optimal values and epsilon optimal solutions of sample average approximated stochastic programs with polyhedral ssd constraints in this paper we complement the existing research by presenting convergence analysis of stationary points when saa is applied to a class of stochastic minimization problems with ssd constraints specifically under some moderate conditions we prove that optimal solutions and stationary points obtained from solving sample average approximated problems converge with probability one to their true counterparts moreover by exploiting some recent results on large deviation of random functions and sensitivity analysis of generalized equations we derive exponential rate of convergence of stationary points

statistical robustness in utility preference robust optimization models utility preference robust optimization pro concerns decision making problems where information on decision maker s utility preference is incomplete and has to be elicited through partial information and the optimal decision is based on the worst case utility function elicited a key assumption in the pro models is that the true probability distribution is either known or can be recovered by real data generated by the true distribution in data driven optimization this assumption may not be satisfied when perceived data differ from real data and consequently it raises a question as to whether statistical estimators of the pro models based on perceived data are reliable in this paper we investigate the issue which is also known as qualitative robustness in the literature of statistics huber in robust statistics 3rd edn wiley new york 1981 and risk management kratschmer et al in finance stoch 18 271 295 2014 by utilizing the framework proposed by kratschmer et al 2014 we derive moderate sufficient conditions under which the optimal value and optimal solution of the pro models are robust against perturbation of the exogenous uncertainty data and examine how the tail behaviour of utility functions affects the robustness moreover under some additional conditions on the lipschitz continuity of the underlying functions with respect to random data we establish quantitative robustness of the statistical estimators under the kantorovich metric finally we investigate uniform consistency of the optimal value and optimal solution of the pro models the results cover utility selection problems and stochastic optimization problems as special cases

quasi optimal convergence rate for an adaptive finite element method we analyze the simplest and most standard adaptive finite element method afem with any polynomial degree for general second order linear symmetric elliptic operators as is customary in practice the afem marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property we prove that the afem is a contraction for the sum of the energy error and the scaled error estimator between two consecutive adaptive loops this geometric decay is instrumental to derive the optimal cardinality of the afem we show that the afem yields a decay rate of the energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity

reduced basis method for parametrized elliptic optimal control problems we propose a suitable model reduction paradigm the certied reduced basis method rb for the rapid and reliable solution of parametrized optimal control problems governed by partial dierential equations pdes in particular we develop the methodology for parametrized quadratic optimization problems with elliptic equations as constraint and infinite dimensional control variable firstly we recast the optimal control problem in the framework of saddle point problems in order to take advantage of the already developed rb theory for stokes type problems then the usual ingredients of the rb methodology are called into play a galerkin projection onto a low dimensional space of basis functions properly selected by an adaptive procedure an affine parametric dependence enabling to perform competitive offline online splitting in the computational procedure an efficient and rigorous a posteriori error estimate on the state control and adjoint variables as well as on the cost functional finally we address some numerical tests that confirm our theoretical results and show the efficiency of the proposed technique

dimension reduction in time series and the dynamic factor model this note shows that the dimension reduction method proposed by li shedden 2002 is equivalent to the dynamic factor model introduced by pena box 1987

galerkin proper orthogonal decomposition methods for parabolic problems in this work error estimates for galerkin proper orthogonal decomposition pod methods for linear and certain non linear parabolic systems are proved the resulting error bounds depend on the number of pod basis functions and on the time discretization numerical examples are included

identification of shared components in large ensembles of time series using dimension reduction in this article we present a framework for parsimonious modeling of large ensembles of time series the idea is to identify a small number of stochastic time series components such that each series in the ensemble is a weighted sum of series specific realizations of the components we present an estimation procedure that is computationally tractable for large datasets through simulations we argue that the estimators perform well we illustrate the method using two datasets monthly u s unemployment rates from 2069 counties over 134 months and a functional magnetic resonance imaging series covering 10 740 voxels over 108 time points our framework is conceptually different from the principal component analysis pca commonly used in multivariate data analysis we argue that applying pca directly to an ensemble of time series captures a shared deterministic structure rather than the series specific stochastic components that we seek

computational geometry an introduction from the reviews this book offers a coherent treatment at the graduate textbook level of the field that has come to be known in the last decade or so as computational geometry the book is well organized and lucidly written a timely contribution by two founders of the field it clearly demonstrates that computational geometry in the plane is now a fairly well understood branch of computer science and mathematics it also points the way to the solution of the more challenging problems in dimensions higher than two

stochastically weighted stochastic dominance concepts with an application in capital budgeting abstract the problem of comparing random vectors arises in many applications we propose three new concepts of stochastically weighted dominance for comparing random vectors x and y the main idea is to use a random vector v to scalarize x and y as v t x and v t y and subsequently use available concepts from stochastic dominance and stochastic optimization for comparison for the case where the distributions of x y and v have finite support we give mixed integer linear inequalities that can be used for random vector comparison as well as for modeling of optimization problems where one of the random vectors depends on decisions to be optimized some advantages of the proposed new concepts are illustrated with the help of a capital budgeting example

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

portfolio construction based on stochastic dominance and target return distributions mean risk models have been widely used in portfolio optimization however such models may produce portfolios that are dominated with respect to second order stochastic dominance and therefore not optimal for rational and risk averse investors this paper considers the problem of constructing a portfolio which is non dominated with respect to second order stochastic dominance and whose return distribution has specified desirable properties the problem is multi objective and is transformed into a single objective problem by using the reference point method in which target levels known as aspiration points are specified for the objective functions a model is proposed in which the aspiration points relate to ordered outcomes for the portfolio return this concept is extended by additionally specifying reservation points which act pre emptively in the optimization model the theoretical properties of the models are studied the performance of the models on real data drawn from the hang seng index is also investigated

robust spectral risk optimization when information on risk spectrum is incomplete a spectral risk measure srm is a weighted average of value at risk where the weighting function also known as risk spectrum or distortion function characterizes a decision maker s risk attitude

optimality and duality theory for stochastic optimization problems with nonlinear dominance constraints we consider a new class of optimization problems involving stochastic dominance constraints of second order we develop a new splitting approach to these models optimality conditions and duality theory these results are used to construct special decomposition methods

conditioning of convex piecewise linear stochastic programs in this paper we consider stochastic programming problems where the objective function is given as an expected value of a convex piecewise linear random function with an optimal solution of such a problem we associate a condition number which characterizes well or ill conditioning of the problem using theory of large deviations we show that the sample size needed to calculate the optimal solution of such problem with a given probability is approximately proportional to the condition number

element boundary terms in reduced order models for flow problems domain decomposition and adaptive coarse mesh hyper reduction abstract in this paper we present a finite element based reduced order model and in particular we consider two aspects related to the introduction of inter element boundary terms in the formulation the first is a domain decomposition strategy in which the transmission conditions involve boundary terms to account for non matching meshes and discontinuous physical properties the second is a coarse mesh hyper reduction for which we propose an adaptive refinement driven by an a posteriori error estimator that contains element boundary terms as the finite element full order model the reduced order model is based on the variational multi scale framework with sub grid scales defined not only in the element interiors but also on the inter element boundaries we present some examples of application using the incompressible navier stokes equations and the boussinesq approximation

an information maximization approach to blind separation and blind deconvolution we derive a new self organizing learning algorithm that maximizes the information transferred in a network of nonlinear units the algorithm does not assume any knowledge of the input distributions and is defined here for the zero noise limit under these conditions information maximization has extra properties not found in the linear case linsker 1989 the nonlinearities in the transfer function are able to pick up higher order moments of the input distributions and perform something akin to true redundancy reduction between units in the output representation this enables the network to separate statistically independent components in the inputs a higher order generalization of principal components analysis we apply the network to the source separation or cocktail party problem successfully separating unknown mixtures of up to 10 speakers we also show that a variant on the network architecture is able to perform blind deconvolution cancellation of unknown echoes and reverberation in a speech signal finally we derive dependencies of information transfer on time delays we suggest that information maximization provides a unifying framework for problems in blind signal processing