correlation analysis of non probabilistic convex model and corresponding structural reliability technique compared with the probability approach the non probabilistic convex model only requires a small amount of samples to obtain the variation bounds of the imprecise parameters and whereby makes the reliability analysis very convenient and economical in this paper we attempt to propose and create a correlation analysis technique mathematically for the non probabilistic convex model and based on it develop an effective method to construct the multidimensional ellipsoids on the uncertainty a marginal convex model is defined to describe the variation range of each uncertain parameter and a covariance is defined to represent the correlation degree of two uncertain parameters for a multidimensional problem the covariance matrix and correlation matrix can be created through all marginal convex models and covariances based on which the required ellipsoid on the uncertainty can be conveniently achieved by combining the correlation analysis technique and the reliability index approach a non probabilistic reliability analysis method is also developed for uncertain structures six numerical examples are presented to demonstrate the effectiveness of the present method

finite range decomposition for families of gradient gaussian measures abstract let a family of gradient gaussian vector fields on z d be given we show the existence of a uniform finite range decomposition of the corresponding covariance operators that is the covariance operator can be written as a sum of covariance operators whose kernels are supported within cubes of diameters l k in addition we prove natural regularity for the subcovariance operators and we obtain regularity bounds as we vary within the given family of gradient gaussian measures

higher order failure probability calculation using nonlinear approximations use of approximations for calculating the structural failure probability under uncertainties is a necessary aspect but efficient use of suitable approximations at different stages of reliability prediction and design makes this valuable tool practical for many large scale structures the objective of this paper is to address the modern approximation concepts and their utility in structural reliability estimation emphasis is placed on building the higher order approximations using intervening variables to closely represent the nonlinear limit state functions the most probable failure point mpp search can be carried either in x space or u space for a wide variety of random variable distributions several test problems with highly nonlinear performance functions are used to demonstrate the accuracy of the proposed method

markov processes as a tool in field theory abstract in a recent paper brydges frohlich and spencer have successfully applied markov chains to classical spin systems using probabilistic methods in a more systematic way we improve some of their results the methods can be extended to fields with continuous parameter such as p fields

finite range decompositions of positive definite functions abstract we give sufficient conditions for a positive definite function to admit decomposition into a sum of positive definite functions which are compactly supported within disks of increasing diameters l n more generally we consider positive definite bilinear forms f v f f defined on c 0 we say v has a finite range decomposition if v can be written as a sum v g n of positive definite bilinear forms g n such that g n f g 0 when the supports of the test functions f g are separated by a distance greater or equal to l n we prove that such decompositions exist when v is dual to a bilinear form b 2 where b is a vector valued partial differential operator satisfying some regularity conditions

gaussian and non gaussian random fields associated with markov processes to every markov process with a symmetric transition density there correspond two random fields over the state space a gaussian field the free field and the occupation field t which describes amount of time the particle spends at each state a relation between these two random fields is established which is useful both for the field theory and theory of markov processes

a renormalisation group method v a single renormalisation group step this paper is the fifth in a series devoted to the development of a rigorous renormalisation group method applicable to lattice field theories containing boson and or fermion fields and comprises the core of the method in the renormalisation group method increasingly large scales are studied in a progressive manner with an interaction parametrised by a field polynomial which evolves with the scale under the renormalisation group map in our context the progressive analysis is performed via a finite range covariance decomposition perturbative calculations are used to track the flow of the coupling constants of the evolving polynomial but on their own perturbative calculations are insufficient to control error terms and to obtain mathematically rigorous results in this paper we define an additional non perturbative coordinate which together with the flow of coupling constants defines the complete evolution of the renormalisation group map we specify conditions under which the non perturbative coordinate is contractive under a single renormalisation group step our framework is essentially combinatorial but its implementation relies on analytic results developed earlier in the series of papers the results of this paper are applied elsewhere to analyse the critical behaviour of the 4 dimensional continuous time weakly self avoiding walk and of the 4 dimensional n component varphi 4 model in particular the existence of a logarithmic correction to mean field scaling for the susceptibility can be proved for both models together with other facts about critical exponents and critical behaviour

on a finite range decomposition of the resolvent of a fractional power of the laplacian ii the torus in previous papers m1 m2 m3 we proved the existence as well as regularity of a finite range decomposition for the resolvent g alpha x y m 2 delta alpha over 2 m 2 1 x y for 0 alpha 2 and all real m in the lattice bf z d for dimension d ge 2 in this paper which is a continuation of the previous one we extend those results by proving the existence as well as regularity of a finite range decomposition for the same resolvent but now on the lattice torus bf z d l n 1 bf z d for d ge 2 provided m neq 0 and 0 alpha 2 we also prove differentiability and uniform continuity properties with respect to the resolvent parameter m 2 here l is any odd positive integer and n ge 2 is any positive integer

towards h p adaptive generalized anova abstract this paper presents a novel h p adaptive generalized analysis of variance decomposition g anova for high dimensional stochastic computations firstly an iterative scheme based on variance based global sensitivity analysis is proposed to determine the optimal polynomial order and important component functions p adaptivity of g anova then a homotopy algorithm is employed to determine the unknown expansion coefficients moreover a novel distribution adaptive sequential experimental design h adaptivity is utilized at each step and the accuracy of the g anova based surrogate model is computed using leave one out statistical test it is observed that significantly less number of component functions are retained as a consequence the proposed approach is highly efficient and is applicable for solving problems involving large number of stochastic dimensions implementation of the proposed approach has been illustrated with four high dimensional applied mechanics problems the proposed approach is found to yield accurate and efficient results finally the proposed approach has been applied for structural reliability analysis of a large scale real life problem

cfd shape optimization using an incomplete gradient adjoint formulation a design methodology based on the adjoint approach for flow problems governed by the incompressible euler equations is presented the main feature of the algorithm is that it avoids solving the adjoint equations which saves an important amount of cpu time furthermore the methodology is general in the sense it does not depend on the geometry representation all the grid points on the surface to be optimized can be chosen as design parameters in addition the methodology can be applied to any type of mesh the partial derivatives of the flow equations with respect to the design parameters are computed by finite differences in this way this computation is independent of the numerical scheme employed to obtain the flow solution once the design parameters have been updated the new solid surface is obtained with a pseudo shell approach in such a way that local singularities which can degrade or inhibit the convergence to the optimal solution are avoided some 2d and 3d numerical examples are shown to demonstrate the proposed methodology copyright 2001 john wiley sons ltd

riesz transform gaussian bounds and the method of wave equation for an abstract self adjoint operator l and a local operator a we study the boundedness of the riesz transform al on lp for some 0 a very simple proof of the obtained result is based on the finite speed propagation property for the solution of the corresponding wave equation we also discuss the relation between the gaussian bounds and the finite speed propagation property using the wave equation methods we obtain a new natural form of the gaussian bounds for the heat kernels for a large class of the generating operators we describe a surprisingly elementary proof of the finite speed propagation property in a more general setting than it is usually considered in the literature

a factorized high dimensional model representation on the nodes of a finite hyperprismatic regular grid when the values of a multivariate function f x 1 x n having n independent variables like x 1 x n are given at the nodes of a cartesian product set in the space of the independent variables and an interpolation problem is defined to find out the analytical structure of this function some difficulties arise in the standard methods due to the multidimensionality of the problem here the main purpose is to partition this multivariate data into low variate data and to obtain the analytical structure of the multivariate function by using this partitioned data high dimensional model representation hdmr is used for these types of problems however if hdmr requires all components which means 2 n number of components to get a desired accuracy then factorized high dimensional model representation fhdmr can be used this method uses the components of hdmr this representation is needed when the sought multivariate function has a multiplicative nature in this work we introduce how to utilize fhdmr for these problems and present illustrative examples

a semi analytical framework for structural reliability analysis abstract estimation of the probability of failure of structural systems can often be computationally intensive and time consuming hence alternative techniques for efficient computation of the probability of failure of complex structures while retaining the accuracy is of paramount importance to structural engineering community this paper presents a semi analytical approach for computing failure probability of structure the proposed approach utilizes i polynomial correlated function expansion a novel surrogate modelling technique to determine the moments and ii maximum entropy method to determine an analytical expression for the response probability density function pdf once the analytical expression for the pdf is obtained failure probability is calculated by numerically integrating the pdf in the failure domain as a special case when only the first two moments of response are considered an analytical formula for the probability of failure is also proposed four structural engineering problems are shown to illustrate the performance of the proposed approach it is observed that the proposed approach outperforms other existing approaches

a simple method for finite range decomposition of quadratic forms and gaussian fields we present a simple method to decompose the green forms corresponding to a large class of interesting symmetric dirichlet forms into integrals over symmetric positive semi definite and finite range properly supported forms that are smoother than the original green form this result gives rise to multiscale decompositions of the associated gaussian free fields into sums of independent smoother gaussian fields with spatially localized correlations our method makes use of the finite propagation speed of the wave equation and chebyshev polynomials it improves several existing results and also gives simpler proofs

the encounter of interval and probabilistic approaches to structural reliability at the design point abstract the consideration of uncertainties in structural analysis has traditionally been made in the field of structural reliability within the framework of classical probability theory in many cases however there is no sufficient probabilistic information to substantiate such an approach in recent years several non probabilistic methods have been proposed as alternatives one of them is the interval analysis oriented to estimate the range of variation of output structural variables given the ranges of the input ones the method however requires the use of special arithmetics which renders its application unpractical for structural analysis based on finite elements or other numerical techniques in this paper a monte carlo method that can be applied to probabilistic as well as to interval approaches to reliability analysis is proposed it is based on the optimal properties of the classical design point of probabilistic structural reliability in particular it is shown that the order statistics of the output variable is concealed in a plot defined by the design point vector this facilitates the selection of the relevant samples for either interval or probability based analysis the examples show that the desired interval of the response or the failure probability can be accurately estimated on such a basis the paper also discusses the use of monte carlo methods for both reliability and interval analysis from the point of view of the proposed representation

polynomials of the occupation field and related random fields abstract to every symmetric markov process there correspond two random fields over the state space a gaussian free field x and the occupation field t x which describes the amount of time spent by a particle at each state for the brownian motion in d 2 dimensions both fields are generalized using a relation between t x and the field x x 2 2 established in a previous publication polynomials of the fields t and are investigated in particular polynomials of t characterize self intersections of the process

finite range decomposition for gaussian measures with improved regularity abstract we consider a family of gradient gaussian vector fields on the torus z l n z d adams koteck muller and independently bauerschmidt established the existence of a uniform finite range decomposition of the corresponding covariance operators i e the covariance can be written as a sum of covariance operators supported on increasing cubes with diameter l k we improve this result and show that the decay behaviour of the kernels in fourier space can be controlled then we show the regularity of the integration map that convolves functionals with the partial measures of the finite range decomposition in particular the new finite range decomposition avoids the loss of regularity which arises in the renormalisation group approach to anisotropic problems in statistical mechanics

four dimensional weakly self avoiding walk with contact self attraction we consider the critical behaviour of the continuous time weakly self avoiding walk with contact self attraction on mathbb z 4 for sufficiently small attraction we prove that the susceptibility and correlation length of order p for any p 0 have logarithmic corrections to mean field scaling and that the critical two point function is asymptotic to a multiple of x 2 this shows that small contact self attraction results in the same critical behaviour as no contact self attraction a collapse transition is predicted for larger self attraction the proof uses a supersymmetric representation of the two point function and is based on a rigorous renormalisation group method that has been used to prove the same results for the weakly self avoiding walk without self attraction

finite range decomposition of gaussian processes let be the finite difference laplacian associated to the lattice z d for dimension d 3 a 0 and l a sufficiently large positive dyadic integer we prove that the integral kernel of the resolvent g a a 1 can be decomposed as an infinite sum of positive semi definite functions v n of finite range v n x y 0 for x y o l n equivalently the gaussian process on the lattice with covariance g a admits a decomposition into independent gaussian processes with finite range covariances for a 0 v n has a limiting scaling form l n left d 2 right gamma c left tfrac x y l n right as n as a corollary such decompositions also exist for fractional powers 2 0 2 the results of this paper give an alternative to the block spin renormalization group on the lattice

erratum to on a finite range decomposition of the resolvent of a fractional power of the laplacian erratum to on a finite range decomposition of the resolvent of a fractional power of the laplacian

gaussian estimates for fundamental solutions of second order parabolic systems with time independent coefficients auscher mcintosh and tchamitchian studied the heat kernels of second order elliptic operators in divergence form with complex bounded measurable coefficients on r n in particular in the case when n 2 they obtained gaussian upper bound estimates for the heat kernel without imposing further assumption on the coefficients we study the fundamental solutions of the systems of second order parabolic equations in the divergence form with bounded measurable time independent coefficients and extend their results to the systems of parabolic equations

hybrid high dimensional model representation for reliability analysis abstract this paper presents a new probabilistic method based on hybrid high dimensional model representation hhdmr for predicting the failure probability of structural and mechanical systems subject to random loads material properties and geometry high dimensional model representation hdmr is a general set of quantitative model assessment and analysis tools for capturing the high dimensional relationships between sets of input and output model variables it is a very efficient formulation of the system response if higher order variable cooperative effects are weak and if the response function is dominantly of additive nature allowing the physical model to be captured by the first few lower order terms but if multiplicative nature of the response function is dominant then all right hand side components of hdmr must be used to be able to obtain the best result however if hdmr requires all components which means 2 n number of components to get a desired accuracy making the method very expensive in practice then factorized hdmr fhdmr can be used but in most cases the limit state performance function has neither additive nor multiplicative nature rather it has an intermediate nature this paper presents a new hhdmr based approximation for the limit state performance functions having an intermediate nature the proposed approximation of an implicit limit state performance function includes both hdmr and fhdmr expansions through a hybridity parameter as an alternative to the conventional methods for reliability analysis which are very computationally demanding when applied in conjunction with complex finite element models this study aims to assess how accurately and efficiently hhdmr technique can capture complex model output uncertainty once the approximate form of the original implicit limit state performance function is defined the failure probability can be obtained by statistical simulation results of six numerical examples involving mathematical functions and structural solid mechanics problems indicate that the failure probability obtained using hhdmr based approximation of an implicit limit state performance function provides significant accuracy when compared with the conventional monte carlo method while requiring fewer original model simulations

the random walk representation of classical spin systems and correlation inequalities ferromagnetic lattice spin systems can be expressed as gases of random walks interacting via a soft core repulsion by using a mixed spinrandom walk representation we present a unified approach to many recently established correlation inequalities as an application of these inequalities we obtain a simple proof of the mass gap for the 4 2 quantum field model we also establish new upper bounds on critical temperatures

a response surface approach for the static analysis of stochastic structures with geometrical nonlinearities a response surface approach for the finite element analysis of uncertain structures undergoing large displacements is presented this method is based on the use of ad hoc response surface functions built up by ratios of polynomials as opposite to commonly used linear or quadratic polynomials such functions are insensitive to the sampling point positions once the response surface form is defined response statistics can be approximated by analytical relationships or statistical simulation taking full advantage of sensitivity analysis numerical investigations demonstrated that a remarkable accuracy is achieved in the evaluation of both statistical moments and probability density functions of the response

on the convergence to the continuum of finite range lattice covariances in j stat phys 115 415 449 2004 brydges guadagni and mitter proved the existence of multiscale expansions of a class of lattice green s functions as sums of positive definite finite range functions called fluctuation covariances the lattice green s functions in the class considered are integral kernels of inverses of second order positive self adjoint elliptic operators with constant coefficients and fractional powers thereof the rescaled fluctuation covariance in the nth term of the expansion lives on a lattice with spacing l n and satisfies uniform bounds our main result in this note is that the sequence of these terms converges in appropriate norms at a rate l n 2 to a smooth positive definite finite range continuum function

hybrid high dimensional model representation hhdmr on the partitioned data a multivariate interpolation problem is generally constructed for appropriate determination of a multivariate function whose values are given at a finite number of nodes of a multivariate grid one way to construct the solution of this problem is to partition the given multivariate data into low variate data high dimensional model representation hdmr and generalized high dimensional model representation ghdmr methods are used to make this partitioning using the components of the hdmr or the ghdmr expansions the multivariate data can be partitioned when a cartesian product set in the space of the independent variables is given the hdmr expansion is used on the other hand if the nodes are the elements of a random discrete data the ghdmr expansion is used instead of hdmr these two expansions work well for the multivariate data that have the additive nature if the data have multiplicative nature then factorized high dimensional model representation fhdmr is used but in most cases the nature of the given multivariate data and the sought multivariate function have neither additive nor multiplicative nature they have a hybrid nature so a new method is developed to obtain better results and it is called hybrid high dimensional model representation hhdmr this new expansion includes both the hdmr or ghdmr and the fhdmr expansions through a hybridity parameter in this work the general structure of this hybrid expansion is given it has tried to obtain the best value for the hybridity parameter according to this value the analytical structure of the sought multivariate function can be determined via hhdmr

next order asymptotic expansion for n marginal optimal transport with coulomb and riesz costs abstract motivated by a problem arising from density functional theory we provide the sharp next order asymptotics for a class of multimarginal optimal transport problems with cost given by singular long range pairwise interaction potentials more precisely we consider an n marginal optimal transport problem with n equal marginals supported on r d and with cost of the form i j x i x j s in this setting we determine the second order term in the n asymptotic expansion of the minimum energy for the long range interactions corresponding to all exponents 0 s d we also prove a small oscillations property for this second order energy term our results can be extended to a larger class of models than power law type radial costs such as non rotationally invariant costs the key ingredient and main novelty in our proofs is a robust extension and simplification of the fefferman gregg decomposition 20 26 extended here to our class of kernels and which provides a unified method valid across our full range of exponents our first result generalizes a recent work of lewin lieb and seiringer 36 who dealt with the second order term for the coulomb case s 1 d 3

logarithmic correction for the susceptibility of the 4 dimensional weakly self avoiding walk a renormalisation group analysis we prove that the susceptibility of the continuous time weakly self avoiding walk on mathbb z d in the critical dimension d 4 has a logarithmic correction to mean field scaling behaviour as the critical point is approached with exponent frac 1 4 for the logarithm the susceptibility has been well understood previously for dimensions d 5 using the lace expansion but the lace expansion does not apply when d 4 the proof begins by rewriting the walk two point function as the two point function of a supersymmetric field theory the field theory is then analysed via a rigorous renormalisation group method developed in a companion series of papers by providing a setting where the methods of the companion papers are applied together the proof also serves as an example of how to assemble the various ingredients of the general renormalisation group method in a coordinated manner