high breakdown point and high efficiency robust estimates for regression a class of robust estimates for the linear model is introduced these estimates called mm estimates have simultaneously the following properties i they are highly efficient when the errors have a normal distribution and ii their breakdown point is 0 5 the mm estimates are defined by a three stage procedure in the first stage an initial regression estimate is computed which is consistent robust and with high breakdown point but not necessarily efficient in the second stage an m estimate of the errors scale is computed using residuals based on the initial estimate finally in the third stage an m estimate of the regression parameters based on a proper redescending psi function is computed consistency and asymptotical normality of the mm estimates assuming random carriers are proved a convergent iterative numerical algorithm is given finally the asymptotic biases under contamination of optimal bounded influence estimates and mm estimates are compared

multi element generalized polynomial chaos for arbitrary probability measures we develop a multi element generalized polynomial chaos me gpc method for arbitrary probability measures and apply it to solve ordinary and partial differential equations with stochastic inputs given a stochastic input with an arbitrary probability measure its random space is decomposed into smaller elements subsequently in each element a new random variable with respect to a conditional probability density function pdf is defined and a set of orthogonal polynomials in terms of this random variable is constructed numerically then the generalized polynomial chaos gpc method is implemented element by element numerical experiments show that the cost for the construction of orthogonal polynomials is negligible compared to the total time cost efficiency and convergence of me gpc are studied numerically by considering some commonly used random variables me gpc provides an efficient and flexible approach to solving differential equations with random inputs especially for problems related to long term integration large perturbation and stochastic discontinuities

on the maximum bias functions of mm estimates and constrained m estimates of regression we derive the maximum bias functions of the mm estimates and the constrained m estimates or cm estimates of regression and compare them to the maximum bias functions of the s estimates and the t estimates of regression in these comparisons the cm estimates tend to exhibit the most favorable bias robustness properties also under the gaussian model it is shown how one can construct a cm estimate which has a smaller maximum bias function than a given s estimate that is the resulting cm estimate dominates the s estimate in terms of maxbias and at the same time is considerably more efficient

intrusive galerkin methods with upwinding for uncertain nonlinear hyperbolic systems this paper deals with stochastic spectral methods for uncertainty propagation and quantification in nonlinear hyperbolic systems of conservation laws we consider problems with parametric uncertainty in initial conditions and model coefficients whose solutions exhibit discontinuities in the spatial as well as in the stochastic variables the stochastic spectral method relies on multi resolution schemes where the stochastic domain is discretized using tensor product stochastic elements supporting local polynomial bases a galerkin projection is used to derive a system of deterministic equations for the stochastic modes of the solution hyperbolicity of the resulting galerkin system is analyzed a finite volume scheme with a roe type solver is used for discretization of the spatial and time variables an original technique is introduced for the fast evaluation of approximate upwind matrices which is particularly well adapted to local polynomial bases efficiency and robustness of the overall method are assessed on the burgers and euler equations with shocks

a general result on the stabilization of linear systems using bounded controls we present two constructions of controllers that globally stabilize linear systems subject to control saturation we allow essentially arbitrary saturation functions the only conditions imposed on the system are the obvious necessary ones namely that no eigenvalues of the uncontrolled system have positive real part and that the standard stabilizability rank condition hold one of the constructions is in terms of a neural network type one hidden layer architecture while the other one is in terms of cascades of linear maps and saturations

sparse tensor discretization of elliptic spdes we propose and analyze sparse deterministic stochastic tensor galerkin finite element methods sparse sgfems for the numerical solution of elliptic partial differential equations pdes with random coefficients in a physical domain d subset mathbb r d in tensor product sgfems the variational solution to the boundary value problem is approximated in tensor product finite element spaces v gamma otimes v d where v gamma and v d denote suitable finite dimensional subspaces of the stochastic and deterministic function spaces respectively these approaches lead to sgfem algorithms of complexity o n gamma n d where n gamma dim v gamma and n d dim v d in this work we use hierarchic sequences v gamma 1 subset v gamma 2 subset ldots and v d 1 subset v d 2 subset ldots of finite dimensional spaces to approximate the law of the random solution the hierarchies of approximation spaces allow us to define sparse tensor product spaces v gamma ell hat otimes v d ell ell 1 2 dots yielding algorithms of o n gamma log n d n d log n gamma work and memory we estimate the convergence rate of sgfem for algebraic decay of the input random field karhunen loeve coefficients we give an algorithm for an input adapted a priori selection of deterministic and stochastic discretization spaces the convergence rate in terms of the total number of degrees of freedom of the proposed method is superior to monte carlo approximations numerical examples illustrate the theoretical results and demonstrate superiority of the sparse tensor product discretization proposed here versus the full tensor product approach

stabilization of partially observed stochastic evolution systems abstract the stability of an evolution system with state dependent noise has been introduced by haussmann 7 and ichikawa 9 10 using a semigroup formulation of the model following a similar approach we study here the corresponding stabilization problem assuming partial observations under a convenient hypothesis we show the existence of a stabilizing control which is a linear feedback on the observation this result is established by extending the compensator concept to the stochastic case the theory is illustrated by a significant example

constrained predictive control of linear distributed parameter systems the model predictive control problem with guaranteed stability for a class of linear distributed parameter systems with simultaneous amplitude and rate control constraints is investigated the approach is developed using a finite dimensional input output modelling technique for distributed parameter systems based on a distributed output projection onto a finite basis set the paper employs a dual mode prediction strategy in conjunction with the projection based state space system description and proposes algorithms for the design of polyhedral positive invariant and feasible sets which serve as terminal target sets for predicted states an extension to predictive control with integral action is described the proposed algorithms are shown to work well by means of both simulations and experimental results

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

an anisotropic sparse grid stochastic collocation method for partial differential equations with random input data this work proposes and analyzes an anisotropic sparse grid stochastic collocation method for solving partial differential equations with random coefficients and forcing terms input data of the model the method consists of a galerkin approximation in the space variables and a collocation in probability space on sparse tensor product grids utilizing either clenshaw curtis or gaussian knots even in the presence of nonlinearities the collocation approach leads to the solution of uncoupled deterministic problems just as in the monte carlo method this work includes a priori and a posteriori procedures to adapt the anisotropy of the sparse grids to each given problem these procedures seem to be very effective for the problems under study the proposed method combines the advantages of isotropic sparse collocation with those of anisotropic full tensor product collocation the first approach is effective for problems depending on random variables which weigh approximately equally in the solution while the benefits of the latter approach become apparent when solving highly anisotropic problems depending on a relatively small number of random variables as in the case where input random variables are karhunen loeve truncations of smooth random fields this work also provides a rigorous convergence analysis of the fully discrete problem and demonstrates sub exponential convergence in the asymptotic regime and algebraic convergence in the preasymptotic regime with respect to the total number of collocation points it also shows that the anisotropic approximation breaks the curse of dimensionality for a wide set of problems numerical examples illustrate the theoretical results and are used to compare this approach with several others including the standard monte carlo in particular for moderately large dimensional problems the sparse grid approach with a properly chosen anisotropy seems to be very efficient and superior to all examined methods

a minimax bias property of the least alpha quantile estimates a natural measure of the degree of robustness of an estimate mathbf t is the maximum asymptotic bias b mathbf t varepsilon over an varepsilon contamination neighborhood martin yohai and zamar have shown that the class of least alpha quantile regression estimates is minimax bias in the class of m estimates that is they minimize b mathbf t varepsilon with alpha depending on varepsilon in this paper we generalize this result proving that the least alpha quantile estimates are minimax bias in a much broader class of estimates which we call residual admissible and which includes most of the known robust estimates defined as a function of the regression residuals e g least median of squares least trimmed of squares s estimates tau estimates m estimates signed r estimates etc the minimax results obtained here likewise the results obtained by martin yohai and zamar require that the carriers have elliptical distribution under the central model

lower bounds for contamination bias globally minimax versus locally linear estimation we study how robust estimators can be in parametric families obtaining a lower bound on the contamination bias of an estimator that holds for a wide class of parametric families this lower bound includes as a special case the bound used to establish that the median is bias minimax among location equivariant estimators and it is tight or nearly tight in a variety of other settings such as scale estimation discrete exponential families and multiple linear regression the minimum variation distance estimator has contamination bias within a dimension free factor of this bound a second lower bound applies to locally linear estimates and implies that such estimates cannot be bias minimax among all fisher consistent estimates in higher dimensions in linear regression this class of estimates includes the familiar m estimates gm estimates and s estimates in discrete exponential families yet another lower bound implies that the proportion of zeros estimate has minimax bias if the median of the distribution is zero a common situation in some fields this bound also implies that the information standardized sensitivity of every fisher consistent estimate of the poisson mean and of the binomial proportion is unbounded

regulator problem for linear continuous time systems with nonsymmetrical constrained control the regulator problem is studied for linear continuous time systems with nonsymmetrical constrained control necessary and sufficient conditions allowing the largest nonsymmetrical polyhedral positively invariant domain w r t the system in the closed loop to be obtained are given the case of symmetrical constrained control is obtained as a particular case

solution verification goal oriented adaptive methods for stochastic advection diffusion problems abstract a goal oriented analysis of linear stochastic advection diffusion models is presented which provides both a method for solution verification as well as a basis for improving results through adaptation of both the mesh and the way random variables are approximated a class of model problems with random coefficients and source terms is cast in a variational setting specific quantities of interest are specified which are also random variables a stochastic adjoint problem associated with the quantities of interest is formulated and a posteriori error estimates are derived these are used to guide an adaptive algorithm which adjusts the sparse probabilistic grid so as to control the approximation error numerical examples are given to demonstrate the methodology for a specific model problem

on the positive invariance of polyhedral sets for discrete time systems abstract in this note necessary and sufficient conditions for a polyhedral set to be a positively invariant set of a linear discrete time system are established

stability and stabilizability of infinite dimensional systems in this paper we are concerned with stability problems for infinite dimensional systems first we review the theory for linear systems where the dynamics are governed by strongly continuous semigro

least median of squares regression abstract classical least squares regression consists of minimizing the sum of the squared residuals many authors have produced more robust versions of this estimator by replacing the square by something else such as the absolute value in this article a different approach is introduced in which the sum is replaced by the median of the squared residuals the resulting estimator can resist the effect of nearly 50 of contamination in the data in the special case of simple regression it corresponds to finding the narrowest strip covering half of the observations generalizations are possible to multivariate location orthogonal regression and hypothesis testing in linear models

high order methods as an alternative to using sparse tensor products for stochastic galerkin fem solutions of random elliptic boundary value problems admit efficient approximations by polynomials on the parameter domain each coefficient in such an expansion is a spatially dependent function and can be approximated within a hierarchy of finite element spaces if the finite elements are of sufficiently high order using just a single spatial mesh is predicted to achieve the same convergence rate with respect to the total number of degrees of freedom as sparse tensor product constructions and other multilevel stochastic galerkin approximations numerical computations for an elliptic two point boundary value problem confirm this and indicate no loss of accuracy for a single level method compared to using a sparse tensor product with the same total number of degrees of freedom

min max bias robust regression abstract this paper considers the problem of minimizing the maximum asymptotic bias of regression estimates over epsilon contamination neighborhoods for the joint distribution of the response and carriers two classes of estimates are treated 1 m estimates with bounded function rho applied to the scaled residuals using a very general class of scale estimates and 2 bounded influence function type generalized m estimates estimates in the first class are obtained as the solution of a minimization problem while estimates in the second class are specified by an estimating equation the first class of m estimates is sufficiently general to include both huber proposal 2 simultaneous estimates of regression coefficients and residuals scale and rousseeuw yohai s estimates of regression it is shown than an s estimate based on a jump function type rho solves the min max bias problem for the class of m estimates with very general scale this estimate is obtained by the minimization of the alpha quantile of the squared residuals where alpha epsilon depends on the fraction of contamination epsilon when epsilon approaches limit of 5 alpha epsilon approaches limit of 5 and the min max estimator approaches the least median of squared residuals estimator introduced by rousseeuw for the bounded influence class of gm estimates it is shown the a sign type nonlinearity yields the min max estimate this estimate coincides with the minimum gross error sensitivity gm estimate

efficient robust regression via two stage generalized empirical likelihood large and finite sample efficiency and resistance to outliers are the key goals of robust statistics although often not simultaneously attainable we develop and study a linear regression estimator that comes close efficiency is obtained from the estimator s close connection to generalized empirical likelihood and its favorable robustness properties are obtained by constraining the associated sum of weighted squared residuals we prove maximum attainable finite sample replacement breakdown point and full asymptotic efficiency for normal errors simulation evidence shows that compared to existing robust regression estimators the new estimator has relatively high efficiency for small sample sizes and comparable outlier resistance the estimator is further illustrated and compared to existing methods via application to a real dataset with purported outliers

maximum bias curves for robust regression with non elliptical regressors maximum bias curves for some regression estimates were previously derived assuming that i the intercept term is known and or ii the regressors have an elliptical distribution we present a single method to obtain the maximum bias curves for a large class of regression estimates our results are derived under very mild conditions and in particular do not require the restrictive assumptions i and ii above using these results it is shown that the maximum bias curves heavily depend on the shape of the regressors distribution which we call the x configuration despite this big effect the relative performance of different estimates remains unchanged under different x configurations we also explore the links between maxbias curves and bias bounds finally we compare the robustness properties of some estimates for the intercept parameter

a stochastic collocation method for elliptic partial differential equations with random input data in this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms input data of the model the input data are assumed to depend on a finite number of random variables the method consists in a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it can be seen as a generalization of the stochastic galerkin method proposed in i babuscka r tempone and g e zouraris siam j numer anal 42 2004 pp 800 825 and allows one to treat easily a wider range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction in the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method

feedback stabilization of a class of distributed parameter systems with control constraints abstract the feedback stabilization problem for a class of infinite dimensional linear systems with control constraints is investigated the approach is developed using a state space system framework which is based on a semigroup formulation in contrast to the previous works in this direction it is not assumed that the c 0 semigroup we deal with is a semigroup of contractions the main result links the problem with the positive invariance of appropriate polyhedral sets verifiable necessary and sufficient conditions for this latter property are stated

constrained regulation of linear continuous time dynamical systems abstract in this paper the linear constrained regulation problem lcrp for continuous time dynamical systems is studied the first part of the paper deals with the existence of linear state feedback control laws that transfer asymptotically to the origin all initial states belonging to a given polyhedral subset of the state space while respecting linear constraints on both state and control vectors then the lcrp is formulated as an optimization problem which can be solved by applying linear programming techniques

bounded feedback stabilization and global separation principle of distributed parameter systems we show that the infinite dimensional system spl sigma x spl dot t ax t bu t x sub 0 spl epsiv h is globally strongly asymptotically stabilizable by an arbitrarily small smooth feedback here the operator a is the infinitesimal generator of a c sub 0 semigroup of contractions e sup ta on real hilbert space h and b is a bounded linear operator mapping a hilbert space of controls u into h an explicit smooth feedback control law is given further we identify the class of perturbations for which the system is still stabilizable by the same feedback law as for the nominal system based on these results and some differential lyapunov operator equations we then establish a global separation principle for the system spl sigma with a kalman like observer finally these results are illustrated via an example dealing with the wave equation

high breakdown point estimates of regression by means of the minimization of an efficient scale abstract a new class of robust estimates estimates is introduced the estimates have simultaneously the following properties a they are qualitatively robust b their breakdown point is 5 and c they are highly efficient for regression models with normal errors they are defined by minimizing a new scale estimate applied to the residuals asymptotically a estimate is equivalent to an m estimate with a function given by a weighted average of two functions one corresponding to a very robust estimate and the other to a highly efficient estimate the weights are adaptive and depend on the underlying error distribution we prove consistency and asymptotic normality and give a convergent iterative computing algorithm finally we compare the biases produced by gross error contamination in the estimates and optimal bounded influence estimates