efficient global reliability analysis for nonlinear implicit performance functions many engineering applications are characterized by implicit response functions that are expensive to evaluate and sometimes nonlinear in their behavior making reliability analysis difficult this paper develops an efficient reliability analysis method that accurately characterizes the limit state throughout the random variable space the method begins with a gaussian process model built from a very small number of samples and then adaptively chooses where to generate subsequent samples to ensure that the model is accurate in the vicinity of the limit state the resulting gaussian process model is then sampled using multimodal adaptive importance sampling to calculate the probability of exceeding or failing to exceed the response level of interest by locating multiple points on or near the limit state more complex and nonlinear limit states can be modeled leading to more accurate probability integration by concentrating the samples in the area where accuracy is important i e in the vicinity of the limit state only a small number of true function evaluations are required to build a quality surrogate model the resulting method is both accurate for any arbitrarily shaped limit state and computationally efficient even for expensive response functions this new method is applied to a collection of example problems including one that analyzes the reliability of a microelectromechanical system device that current available methods have difficulty solving either accurately or efficiently

the homological degree of a module a homological degree of a graded module m is an extension of the usual notion of multiplicity tailored to provide a numerical signature for the module even when m is not cohen macaulay we construct a degree hdeg m that behaves well under hyperplane sections and the modding out of elements of finite support when carried out in a local algebra this degree gives a simulacrum of complexity a la castelnuovo mumford s regularity several applications for estimating reduction numbers of ideals and predictions on the outcome of noether normalizations are given

string structures on loop bundles differential geometry and topology of principal loop bundles bundles of loop groups over loop spaces are investigated string structures defined as bundle extensions corresponding to the central extension of the structure group do not always exist various methods for deriving the obstruction to the existence of string structures are discussed

effective two point function approximation for design optimization anew two pointapproximationofafunctionthatisverystraightforwardtobuildandthatprovidesaccurateand stable approximation is presented earlier developments of the two point approximations had either incomplete matches at two data points or needed the solution of additional equations to get all of the parameters the present two point approximation is an incomplete second order taylor series expansion in terms of intervening variables thehessianmatrixhasonlydiagonalelements anditdependsondesignvariables theexponentofeachintervening design variable and the unknown constant of the second order terms are evaluated by matching the derivatives and the value of the approximation with the previous data point gradients and the value of the original function respectively all of theunknownsareidentie ed in aclosedform both thefunctionandthegradientof thetwo point approximation are equal to those of the original function at two data points several examples are given to show the accuracy and efe ciency of this method

twistor spaces and spinors over loop spaces in this paper two natural twistor spaces over the loop space of a riemannian manifold are constructed and their equivalence is shown in the kahlerian case this relies on a detailed study of frame bundles of loop spaces on the one hand and on the other hand on an explicit local trivialization of the atiyah operator family defined in atiyah smf 131 43 59 1985 associated to a loop space we relate these constructions to the dixmier douady obstruction class against the existence of a string structure as well as to pseudo line bundle gerbes in the sense of brylinski loop spaces characteristic classes and geometric quantization birkhauser basel 1993

on some conjectures about the chern numbers of filtrations abstract let i be an m primary ideal of a noetherian local ring r m of positive dimension the coefficient e 1 a of the hilbert polynomial of an i admissible filtration a is called the chern number of a the positivity conjecture of vasconcelos for the chern number of the integral closure filtration i n is proved for a 2 dimensional complete local domain and more generally for any analytically unramified local ring r whose integral closure in its total ring of fractions is cohen macaulay as an r module it is proved that if i is a parameter ideal then the chern number of the i adic filtration is non negative several other results on the chern number of the integral closure filtration are established especially in the case when r is not necessarily cohen macaulay

nested polynomial trends for the improvement of gaussian process based predictors the role of simulation has kept increasing for the sensitivity analysis and the uncertainty quantification of complex systems such numerical procedures are generally based on the processing of a huge amount of code evaluations when the computational cost associated with one particular evaluation of the code is high such direct approaches based on the computer code only can be not affordable surrogate models have therefore to be introduced to interpolate the information given by a fixed set of code evaluations to the whole input space when confronted to deterministic mappings the gaussian process based regression gpr or kriging presents a good compromise between complexity efficiency and error control such a method considers the quantity of interest of the system as a particular realization of a gaussian stochastic process which mean and covariance functions have to be identified from the available code evaluations in this context this work proposes an innovative parameterization of this mean function which is based on the composition of two polynomials this approach is particularly relevant for the approximation of strongly non linear quantities of interest from very little information after presenting the theoretical basis of this method this work compares its efficiency to alternative approaches on a series of examples

an efficient method combining active learning kriging and monte carlo simulation for profust failure probability abstract for more and more complicated engineering structures it is a challenge to efficiently estimate the profust failure probability based on the probability inputs and fuzzy state assumption by combining active learning kriging with monte carlo simulation ak mcs an efficient method is proposed to estimate the profust failure probability firstly the profust failure probability is transformed into an integral of the classical failure probability by introducing a variable related to the fuzzy state assumption this integral is further reorganized as a weighted sum of a series of classical failure probabilities by gaussian quadrature and the series of the classical failure probabilities have the similar limit state function constructions constrained by different thresholds secondly mcs is used according to the probability input distribution to generate the sample pool in which the active learning kriging is used to establish the surrogates of the series of similar limit state functions with different thresholds an improved learning function is proposed by minimizing the u learning function minima corresponding to all limit state functions so that the candidate with the largest effect on the surrogating quality of all limit states can be selected as a training point to update the kriging model once the updating process of the kriging model converges all limit state functions can be identified by the kriging model and the profust failure probability can be estimated by using the kriging model without any extra model evaluation several examples are used to demonstrate the feasibility of the proposed strategy for estimating the profust failure probability

estimation of a density from an imperfect simulation model uncertainty quantification of a technical system can be done using density estimation we usually start with a stochastic model which is fitted to the technical system and the density estimation is done using data from this stochastic model however in any application such a stochastic model will not be perfect and the estimation of the density should take into account the inadequacy of the stochastic model in this paper we show how observed data of the real system together with an imperfect simulation model can be used to derive confidence bands for the density of the technical system our main result is that the newly introduced confidence bands allow to derive lower and upper bounds on the probability of intervals in the technical system furthermore we present an upper bound on the area of the confidence band in case of a smooth density the results are illustrated by applying the estimates to simulated and real data

string structures and the path fibration of a group we use the realisation of the universal bundle for the loop group as the path fibration of the group to investigate the string class that is the obstruction to a loop group bundle lifting to a kac moody group bundle in the case that the loop group bundle is constructed by taking loops into a principal bundle we show that the classifying map is the holonomy around loops and give an explicit formula for the string class relating it to the pontrjangin class of the principal bunble

sectional genera of parameter ideals let m be a finitely generated module over a noetherian local ring this paper reports for a given parameter ideal q for m a criterion for the equality gs q m hdegq m eq0 m tq1 m where gs q m hdegq m eq0 m and tq1 m respectively denote the sectional genus the homological degree the multiplicity and the homological torsion of m with respect to q

the homology of parameter ideals abstract for a noetherian local ring we analyze conjectural relationships between the first hilbert coefficient of a parameter ideal and the first partial euler characteristic of its koszul complex given their similar role as predictors of the cohen macaulay property we consider a direct comparison between them for parameter ideals generated by d sequences these numbers are related in an explicit formula we then turn to study of families of parameter ideals that have the same hilbert function

adaptive estimation of quantiles in a simulation model let x be an mathbb r d valued random variable let m mathbb r d rightarrow mathbb r be a measurable function and set y m x given a sample of x y of size n we consider the problem of estimating the quantile of y of a given level alpha in 0 1 a method for choosing the parameter of a surrogate model of m is introduced and it is shown that the corresponding surrogate quantile estimate achieves the rate of convergence bounded by the sum of the minimal rate of convergence of the quantile estimates corresponding to the given surrogate estimates and a term of order log n n the finite sample size behavior of this quantile estimate is illustrated by applying it to simulated data and to a quantile estimation problem in mechanical engineering

the dirac ramond operator on loops in flat space in this paper a rigorous construction of the s1 equivariant dirac operator i e dirac ramond operator on the space of mean zero loops in rd is given and its equivariant l2 index computed essential use is made of infinite tensor product representations of the canonical anticommutation relations algebra

gaussian process surrogates for failure detection an important task of uncertainty quantification is to identify the probability of undesired events in particular system failures caused by various sources of uncertainties in this work we consider the construction of gaussian process surrogates for failure detection and failure probability estimation in particular we consider the situation that the underlying computer models are extremely expensive and in this setting determining the sampling points in the state space is of essential importance we formulate the problem as an optimal experimental design for bayesian inferences of the limit state i e the failure boundary and propose an efficient numerical scheme to solve the resulting optimization problem in particular the proposed limit state inference method is capable of determining multiple sampling points at a time and thus it is well suited for problems where multiple computer simulations can be performed in parallel the accuracy and performance of the proposed method is demonstrated by both academic and practical examples

fast parallel kriging based stepwise uncertainty reduction with application to the identification of an excursion set stepwise uncertainty reduction sur strategies aim at constructing a sequence of points for evaluating a function f in such a way that the residual uncertainty about a quantity of interest progressively decreases to zero using such strategies in the framework of gaussian process modeling has been shown to be efficient for estimating the volume of excursion of f above a fixed threshold however sur strategies remain cumbersome to use in practice because of their high computational complexity and the fact that they deliver a single point at each iteration in this article we introduce several multipoint sampling criteria allowing the selection of batches of points at which f can be evaluated in parallel such criteria are of particular interest when f is costly to evaluate and several cpus are simultaneously available we also manage to drastically reduce the computational cost of these strategies through the use of closed form formulas we illustrate their performances in various numerical experiment

calibration and uncertainty analysis for computer simulations with multivariate output model calibration analysis is concerned with the estimation of unobservable modeling parameters using observations of system response when the model being calibrated is an expensive computer simulation special techniques such as surrogate modeling and bayesian inference are often fruitful in this paper we show how the flexibility of the bayesian calibration approach can be exploited to account for a wide variety of uncertainty sources in the calibration process we propose a straightforward approach for simultaneously handling gaussian and non gaussian errors as well as a framework for studying the effects of prescribed uncertainty distributions for model inputs that are not treated as calibration parameters further we discuss how gaussian process surrogate models can be used effectively when simulator response may be a function of time and or space multivariate output the proposed methods are illustrated through the calibration of a simulation of thermally decomposing foam

a density matching approach for optimization under uncertainty abstract modern computers enable methods for design optimization that account for uncertainty in the system so called optimization under uncertainty ouu we propose a metric for ouu that measures the distance between a designer specified probability density function of the system response the target and the system response s density function at a given design we study an ouu formulation that minimizes this distance metric over all designs we discretize the objective function with numerical quadrature and we approximate the response density function with a gaussian kernel density estimate we offer heuristics for addressing issues that arise in this formulation and we apply the approach to a cfd based airfoil shape optimization problem we qualitatively compare the density matching approach to a multi objective robust design optimization to gain insight into the method