the riemann hilbert approach to strong asymptotics for orthogonal polynomials on 1 1 abstract we consider polynomials that are orthogonal on 1 1 with respect to a modified jacobi weight 1 x 1 x h x with 1 and h real analytic and strictly positive on 1 1 we obtain full asymptotic expansions for the monic and orthonormal polynomials outside the interval 1 1 for the recurrence coefficients and for the leading coefficients of the orthonormal polynomials we also deduce asymptotic behavior for the hankel determinants and for the monic orthogonal polynomials on the interval 1 1 for the asymptotic analysis we use the steepest descent technique for riemann hilbert problems developed by deift and zhou and applied to orthogonal polynomials on the real line by deift kriecherbauer mclaughlin venakides and zhou in the steepest descent method we will use the szeg function associated with the weight and for the local analysis around the endpoints 1 we use bessel functions of appropriate order whereas deift et al use airy functions

efficient low rank approximation of the stochastic galerkin matrix in tensor formats in this article we describe an efficient approximation of the stochastic galerkin matrix which stems from a stationary diffusion equation the uncertain permeability coefficient is assumed to be a log normal random field with given covariance and mean functions the approximation is done in the canonical tensor format and then compared numerically with the tensor train and hierarchical tensor formats it will be shown that under additional assumptions the approximation error depends only on the smoothness of the covariance function and does not depend either on the number of random variables nor the degree of the multivariate hermite polynomials

level spacing distributions and the bessel kernel scaling models of randomn n hermitian matrices and passing to the limitn leads to integral operators whose fredholm determinants describe the statistics of the spacing of the eigenvalues of hermitian matrices of large order for the gaussian unitary ensemble and for many others as well the kernel one obtains by scaling in the bulk of the spectrum is the sine kernel frac sin pi x y pi x y rescaling the gue at the edge of the spectrum leads to the kernel frac ai x ai y ai x ai y x y where ai is the airy function in previous work we found several analogies between properties of this airy kernel and known properties of the sine kernel a system of partial differential equations associated with the logarithmic differential of the fredholm determinant when the underlying domain is a union of intervals a representation of the fredholm determinant in terms of a painleve transcendent in the case of a single interval and also in this case asymptotic expansions for these determinants and related quantities achieved with the help of a differential operator which commutes with the integral operator in this paper we show that there are completely analogous properties for a class of kernels which arise when one rescales the laguerre or jacobi ensembles at the edge of the spectrum namely frac j alpha sqrt x sqrt y j alpha sqrt y sqrt x j alpha sqrt x j alpha sqrt y 2 x y wherej z is the bessel function of order in the cases 1 2 these become after a variable change the kernels which arise when taking scaling limits in the bulk of the spectrum for the gaussian orthogonal and symplectic ensembles in particular an asymptotic expansion we derive will generalize ones found by dyson for the fredholm determinants of these kernels

the asymptotics of a bessel kernel determinant which arises in random matrix theory in random matrix theory the local correlations of the laguerre and jacobi unitary ensemble in the hard edge scaling limit can be described in terms of the bessel kernel b x y xyj x yj y j y xj x x2 y2 x y 0 1 in particular the so called hard edge gap probabilities p r can be expressed as the fredholm determinants of the corresponding integral operator b restricted to the finite interval 0 r using operator theoretic methods we are going to compute their asymptotics as r i e we show that p r det i b l2 0 r exp r24 r 22logr g 1 2 2 where g stands for the barnes g function in fact this asymptotic formula will be proved for all complex parameters satisfying re 1

analytic regularity and collocation approximation for elliptic pdes with random domain deformations in this work we consider the problem of approximating the statistics of a given quantity of interest qoi that depends on the solution of a linear elliptic pde defined over a random domain parameterized by n random variables the elliptic problem is remapped onto a corresponding pde with a fixed deterministic domain we show that the solution can be analytically extended to a well defined region in c n with respect to the random variables a sparse grid stochastic collocation method is then used to compute the mean and variance of the qoi finally convergence rates for the mean and variance of the qoi are derived and compared to those obtained in numerical experiments

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

analysis and implementation issues for the numerical approximation of parabolic equations with random coefficients we consider the problem of numerically approximating statistical moments of the solution of a time dependent linear parabolic partial differential equation pde whose coefficients and or forcing terms are spatially correlated random fields the stochastic coefficients of the pde are approximated by truncated karhunen loeve expansions driven by a finite number of uncorrelated random variables after approximating the stochastic coefficients the original stochastic pde turns into a new deterministic parametric pde of the same type the dimension of the parameter set being equal to the number of random variables introduced after proving that the solution of the parametric pde problem is analytic with respect to the parameters we consider global polynomial approximations based on tensor product total degree or sparse polynomial spaces and constructed by either a stochastic galerkin or a stochastic collocation approach we derive convergence rates for the different cases and present numerical results that show how these approaches are a valid alternative to the more traditional monte carlo method for this class of problems copyright 2009 john wiley sons ltd

is gauss quadrature better than clenshaw curtis we compare the convergence behavior of gauss quadrature with that of its younger brother clenshaw curtis seven line matlab codes are presented that implement both methods and experiments show that the supposed factor of 2 advantage of gauss quadrature is rarely realized theorems are given to explain this effect first following o hara and smith in the 1960s the phenomenon is explained as a consequence of aliasing of coefficients in chebyshev expansions then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of log z 1 z 1 in the complex plane gauss quadrature corresponds to pade approximation at z infty clenshaw curtis quadrature corresponds to an approximation whose order of accuracy at z infty is only half as high but which is nevertheless equally accurate near 1 1

risk averse pde constrained optimization using the conditional value at risk uncertainty is inevitable when solving science and engineering application problems in the face of uncertainty it is essential to determine robust and risk averse solutions in this work we consider a class of pde constrained optimization problems in which the pde coefficients and inputs may be uncertain we introduce two approximations for minimizing the conditional value at risk cvar for such pde constrained optimization problems these approximations are based on the primal and dual formulations of cvar for the primal problem we introduce a smooth approximation of cvar in order to utilize derivative based optimization algorithms and to take advantage of the convergence properties of quadrature based discretizations for this smoothed cvar we prove differentiability as well as consistency of our approximation for the dual problem we regularize the inner maximization problem rigorously derive optimality conditions and demonstrate the consistency of our approximation furthermore we propose a

strong and weak error estimates for elliptic partial differential equations with random coefficients we consider the problem of numerically approximating the solution of an elliptic partial differential equation with random coefficients and homogeneous dirichlet boundary conditions we focus on the case of a lognormal coefficient and deal with the lack of uniform coercivity and uniform boundedness with respect to the randomness this model is frequently used in hydrogeology we approximate this coefficient by a finite dimensional noise using a truncated karhunen loeve expansion we give estimates of the corresponding error on the solution both a strong error estimate and a weak error estimate that is an estimate of the error commited on the law of the solution we obtain a weak rate of convergence which is twice the strong one in addition we give a complete error estimate for the stochastic collocation method in this case where neither coercivity nor boundedness is stochastically uniform to conclude we apply these results of strong and weak convergence to two classical cases of covariance kernel choices the case of an exponential covariance kernel on a box and the case of an analytic covariance kernel yielding explicit weak and strong convergence rates

error analysis of a stochastic collocation method for parabolic partial differential equations with random input data a stochastic collocation method for solving linear parabolic partial differential equations with random coefficients forcing terms and initial conditions is analyzed the input data are assumed to depend on a finite number of random variables unlike previous analyses a wider range of situations are considered including input data that depend nonlinearly on the random variables and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate the exponential decay of the interpolation error in the probability space for both finite element semidiscrete spatial discretizations and for finite element crank nicolson fully discrete space time discretizations ingredients in the convergence analysis include the proof of the analyticity with respect to the probabilistic parameters of the semidiscrete and fully discrete approximate solutions a numerical example is provided to illustrate the analyses

modeling uncertainty in steady state diffusion problems via generalized polynomial chaos abstract we present a generalized polynomial chaos algorithm for the solution of stochastic elliptic partial differential equations subject to uncertain inputs in particular we focus on the solution of the poisson equation with random diffusivity forcing and boundary conditions the stochastic input and solution are represented spectrally by employing the orthogonal polynomial functionals from the askey scheme as a generalization of the original polynomial chaos idea of wiener amer j math 60 1938 897 a galerkin projection in random space is applied to derive the equations in the weak form the resulting set of deterministic equations for each random mode is solved iteratively by a block gauss seidel iteration technique both discrete and continuous random distributions are considered and convergence is verified in model problems and against monte carlo simulations

solving elliptic boundary value problems with uncertain coefficients by the finite element method the stochastic formulation this work studies a linear elliptic problem with uncertainty the introduction gives a survey of different formulations of the uncertainty and resulting numerical approximations the major emphasis

uncertainty propagation using wiener haar expansions an uncertainty quantification scheme is constructed based on generalized polynomial chaos pc representations two such representations are considered based on the orthogonal projection of uncertain data and solution variables using either a haar or a legendre basis governing equations for the unknown coefficients in the resulting representations are derived using a galerkin procedure and then integrated in order to determine the behavior of the stochastic process the schemes are applied to a model problem involving a simplified dynamical system and to the classical problem of rayleigh benard instability for situations involving random parameters close to a critical point the computational implementations show that the wiener haar wha representation provides more robust predictions that those based on a wiener legendre wle decomposition however when the solution depends smoothly on the random data the wle scheme exhibits superior convergence suggestions regarding future extensions are finally drawn based on these experiences

a multiscale preconditioner for stochastic mortar mixed finite elements the aim of this paper is to introduce a new approach to efficiently solve sequences of problems that typically arise when modeling flow in stochastic porous media the governing equations are based on darcy s law with a stochastic permeability field starting from a specified covariance relationship the log permeability is decomposed using a truncated karhunen loeve expansion multiscale mortar mixed finite element approximations are used in the spatial domain and a nonintrusive sampling method is used in the stochastic dimensions a multiscale mortar flux basis is computed for a single permeability called a training permeability that captures the main characteristics of the porous media and is used as a preconditioner for each stochastic realization we prove that the condition number of the preconditioned interface operator is independent of the subdomain mesh size and the mortar mesh size computational results confirm that our approach provides an efficient means to quantify the uncertainty for stochastic flow in porous media

an adaptive high dimensional stochastic model representation technique for the solution of stochastic partial differential equations a computational methodology is developed to address the solution of high dimensional stochastic problems it utilizes high dimensional model representation hdmr technique in the stochastic space to represent the model output as a finite hierarchical correlated function expansion in terms of the stochastic inputs starting from lower order to higher order component functions hdmr is efficient at capturing the high dimensional input output relationship such that the behavior for many physical systems can be modeled to good accuracy only by the first few lower order terms an adaptive version of hdmr is also developed to automatically detect the important dimensions and construct higher order terms using only the important dimensions the newly developed adaptive sparse grid collocation asgc method is incorporated into hdmr to solve the resulting sub problems by integrating hdmr and asgc it is computationally possible to construct a low dimensional stochastic reduced order model of the high dimensional stochastic problem and easily perform various statistic analysis on the output several numerical examples involving elementary mathematical functions and fluid mechanics problems are considered to illustrate the proposed method the cases examined show that the method provides accurate results for stochastic dimensionality as high as 500 even with large input variability the efficiency of the proposed method is examined by comparing with monte carlo mc simulation

the isomonodromy approach to matric models in 2d quantum gravity we consider the double scaling limit in the hermitian matrix model for 2d quantum gravity associated with the measure exp sum limits j 1 n t j z 2j n geqq 3 we show that after the appropriate modification of the contour of integration the cross migdal douglas shenker limit to the painleve i equation in the generic case of the pure gravity is valid and calculate the nonperturbative parameters of the corresponding painleve function our approach is based on the wkb analysis of the l a pair corresponding to the discrete string equation in the framework of the inverse monodromy method here we extend our results which were obtained before for the particular casesn 2 3 our analysis complements the isomonodromy approach proposed by g moore to the general string equations that come from the matrix model in the continuous limit and differ in that we apply the isomonodromy technique to investigate the double scaling limit itself

polynomial chaos expansions for dependent random variables abstract polynomial chaos expansions pce are well suited to quantifying uncertainty in models parameterized by independent random variables the assumption of independence leads to simple strategies for building multivariate orthonormal bases and for sampling strategies to evaluate pce coefficients in contrast the application of pce to models of dependent variables is much more challenging three approaches can be used to construct pce of models of dependent variables the first approach uses mapping methods where measure transformations such as the nataf and rosenblatt transformation can be used to map dependent random variables to independent ones however we show that this can significantly degrade performance since the jacobian of the map must be approximated a second strategy is the class of dominating support methods in these approaches a pce is built using independent random variables whose distributional support dominates the support of the true dependent joint density we provide evidence that this approach appears to produce approximations with suboptimal accuracy a third approach the novel method proposed here uses gram schmidt orthogonalization gso to numerically compute orthonormal polynomials for the dependent random variables this approach has been used successfully when solving differential equations using the intrusive stochastic galerkin method and in this paper we use gso to build pce using a non intrusive stochastic collocation method the stochastic collocation method treats the model as a black box and builds approximations of the input output map from a set of samples building pce from samples can introduce ill conditioning which does not plague stochastic galerkin methods to mitigate this ill conditioning we generate weighted leja sequences which are nested sample sets to build accurate polynomial interpolants we show that our proposed approach gso with weighted leja sequences produces pce which are orders of magnitude more accurate than pce constructed using mapping or dominating support methods

high order collocation methods for differential equations with random inputs recently there has been a growing interest in designing efficient methods for the solution of ordinary partial differential equations with random inputs to this end stochastic galerkin methods appear to be superior to other nonsampling methods and in many cases to several sampling methods however when the governing equations take complicated forms numerical implementations of stochastic galerkin methods can become nontrivial and care is needed to design robust and efficient solvers for the resulting equations on the other hand the traditional sampling methods e g monte carlo methods are straightforward to implement but they do not offer convergence as fast as stochastic galerkin methods in this paper a high order stochastic collocation approach is proposed similar to stochastic galerkin methods the collocation methods take advantage of an assumption of smoothness of the solution in random space to achieve fast convergence however the numerical implementation of stochastic collocation is trivial as it requires only repetitive runs of an existing deterministic solver similar to monte carlo methods the computational cost of the collocation methods depends on the choice of the collocation points and we present several feasible constructions one particular choice based on sparse grids depends weakly on the dimensionality of the random space and is more suitable for highly accurate computations of practical applications with large dimensional random inputs numerical examples are presented to demonstrate the accuracy and efficiency of the stochastic collocation methods

application of the τ function theory of painleve equations to random matrices piv pii and the gue tracy and widom have evaluated the cumulative distribution of the largest eigenvalue for the finite and scaled infinite gue in terms of a piv and pii transcendent respectively we generalise these results to the evaluation of where for and otherwise and the average is with respect to the joint eigenvalue distribution of the gue as well as to the evaluation of of particular interest are and f n 2 and their scaled limits which give the distribution of the largest eigenvalue and the density respectively our results are obtained by applying the okamoto function theory of piv and pii for which we give a self contained presentation based on the recent work of noumi and yamada we point out that the same approach can be used to study the quantities and f n a for the other classical matrix ensembles

efficient adaptive stochastic galerkin methods for parametric operator equations this paper is concerned with the design and implementation of efficient solution algorithms for elliptic pde problems with correlated random data the energy orthogonality that is built into stochastic galerkin approximations is cleverly exploited to give an innovative energy error estimation strategy that utilizes the tensor product structure of the approximation space an associated error estimator is constructed and shown theoretically and numerically to be an effective mechanism for driving an adaptive refinement process the codes used in the numerical studies are available online

hankel determinant and orthogonal polynomials for a gaussian weight with a discontinuity at the edge we compute asymptotics for hankel determinants and orthogonal polynomials with respect to a discontinuous gaussian weight in a critical regime where the discontinuity is close to the edge of the associated equilibrium measure support their behavior is described in terms of the ablowitz segur family of solutions to the painleve ii equation our results complement the ones in 33 as consequences of our results we conjecture asymptotics for an airy kernel fredholm determinant and total integral identities for painleve ii transcendents and we also prove a new result on the poles of the ablowitz segur solutions to the painleve ii equation we also highlight applications of our results in random matrix theory

asymptotics of toeplitz hankel and toeplitz hankel determinants with fisher hartwig singularities we study the asymptotics in n for n dimensional toeplitz determinants whose symbols possess fisher hartwig singularities on a smooth background we prove the general nondegenerate asymptotic behavior as conjectured by basor and tracy we also obtain asymptotics of hankel determinants on a nite interval as well as determinants of toeplitz hankel type our analysis is based on a study of the related system of orthogonal polynomials on the unit circle using the riemann hilbert approach

a stochastic collocation method for the second order wave equation with a discontinuous random speed in this paper we propose and analyze a stochastic collocation method for solving the second order wave equation with a random wave speed and subjected to deterministic boundary and initial conditions the speed is piecewise smooth in the physical space and depends on a finite number of random variables the numerical scheme consists of a finite difference or finite element method in the physical space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space this approach leads to the solution of uncoupled deterministic problems as in the monte carlo method we consider both full and sparse tensor product spaces of orthogonal polynomials we provide a rigorous convergence analysis and demonstrate different types of convergence of the probability error with respect to the number of collocation points for full and sparse tensor product spaces and under some regularity assumptions on the data in particular we show that unlike in elliptic and parabolic problems the solution to hyperbolic problems is not in general analytic with respect to the random variables therefore the rate of convergence may only be algebraic an exponential fast rate of convergence is still possible for some quantities of interest and for the wave solution with particular types of data we present numerical examples which confirm the analysis and show that the collocation method is a valid alternative to the more traditional monte carlo method for this class of problems

stochastic collocation and mixed finite elements for flow in porous media the aim of this paper is to quantify uncertainty of flow in porous media through stochastic modeling and computation of statistical moments the governing equations are based on darcy s law with stochastic permeability starting from a specified covariance relationship the log permeability is decomposed using a truncated karhunen loeve expansion mixed finite element approximations are used in the spatial domain and collocation at the zeros of tensor product hermite polynomials is used in the stochastic dimensions error analysis is performed and experimentally verified with numerical simulations computational results include incompressible and slightly compressible single and two phase flow

explicit cost bounds of stochastic galerkin approximations for parameterized pdes with random coefficients this work analyzes the overall computational complexity of the stochastic galerkin finite element method sgfem for approximating the solution of parameterized elliptic partial differential equations with both affine and non affine random coefficients to compute the fully discrete solution such approaches employ a galerkin projection in both the deterministic and stochastic domains produced here by a combination of finite elements and a global orthogonal basis defined on an isotopic total degree index set respectively to account for the sparsity of the resulting system we present a rigorous cost analysis that considers the total number of coupled finite element systems that must be simultaneously solved in the sgfem however to maintain sparsity as the coefficient becomes increasingly nonlinear in the parameterization it is necessary to also approximate the coefficient by an additional orthogonal expansion in this case we prove a rigorous complexity estimate for the number of floating point operations flops required per matrix vector multiplication of the coupled system based on such complexity estimates we also develop explicit cost bounds in terms of flops to solve the stochastic galerkin sg systems to a prescribed tolerance which are used to compare with the minimal complexity estimates of a stochastic collocation finite element method scfem shown in our previous work galindo et al 2015 finally computational evidence complements the theoretical estimates and supports our conclusion that in the case that the coefficient is affine the coupled sg system can be solved more efficiently than the decoupled sc systems however as the coefficient becomes more nonlinear it becomes prohibitively expensive to obtain an approximation with the sgfem

asymptotics of the airy kernel determinant the authors use riemann hilbert methods to compute the constant that arises in the asymptotic behavior of the airy kernel determinant of random matrix theory

error analysis of the dynamically orthogonal approximation of time dependent random pdes in this work we discuss the dynamically orthogonal do approximation of time dependent partial differential equations with random data the approximate solution is expanded at each time instant on a time dependent orthonormal basis in the physical domain with a fixed and small number of terms dynamic equations are written for the evolution of the basis as well as the evolution of the stochastic coefficients of the expansion we analyze the case of a linear parabolic equation with random data and derive a theoretical bound for the approximation error of the s terms do solution by the corresponding s terms best approximation i e the truncated s terms karhunen loeve expansion at each time instant the bound is applicable on the largest time interval in which the best s terms approximation is continuously time differentiable properties of the do approximations are analyzed on simple cases of deterministic equations with random initial data numerical tests are presented that confirm the theoret

dyson s constant in the asymptotics of the fredholm determinant of the sine kernel we prove that the asymptotics of the fredholm determinant of i k where k is the integral operator with the sine kernel open image in new window on the interval 0 are given by

ingredients for a general purpose stochastic finite elements implementation abstract the stochastic finite element method has seen a number of incarnations over the past few years ranging from perturbation based methods and neumann expansion based methods to methods for evaluating bounds on the response variability and more recent functional expansions combined with error minimization techniques of these only the last class of methods provides an explicit expression to the solution of the problem as opposed to evaluating a global norm of this solution and can thus be viewed as a rational extension of concepts developed for the deterministic finite element method the main obstacle in developing a substantial user community of various stochastic finite element methods has been the lack of a general purpose formalism for addressing issues of general engineering interest this paper presents the main ingredients for developing a general purpose version of the spectral stochastic finite element method

robust shape optimization of continuous structures via the level set method abstract this work proposes a stochastic shape optimization method for continuous structures using the level set method such a method aims to minimize the expected compliance and its variance as measures of the structural robustness the behavior of continuous structures is modeled by linear elasticity equations with uncertain loading and material this uncertainty can be modeled using random variables with different probability distributions as well as random fields the proper problem formulation is ensured by the proof of the existence colorrev of solution under certain geometrical constraints on the set of admissible shapes the proposed method addresses the stochastic linear elasticity problem in its weak form obtaining the explicit expressions for the continuous shape derivatives some numerical examples are presented to show the effectiveness of the proposed approach

adaptive stochastic galerkin fem for lognormal coefficients in hierarchical tensor representations stochastic galerkin methods for non affine coefficient representations are known to cause major difficulties from theoretical and numerical points of view in this work an adaptive galerkin fe method for linear parametric pdes with lognormal coefficients discretized in hermite chaos polynomials is derived it employs problem adapted function spaces to ensure solvability of the variational formulation the inherently high computational complexity of the parametric operator is made tractable by using hierarchical tensor representations for this a new tensor train format of the lognormal coefficient is derived and verified numerically the central novelty is the derivation of a reliable residual based a posteriori error estimator this can be regarded as a unique feature of stochastic galerkin methods it allows for an adaptive algorithm to steer the refinements of the physical mesh and the anisotropic wiener chaos polynomial degrees for the evaluation of the error estimator to become feasible a numerically efficient tensor format discretization is developed benchmark examples with unbounded lognormal coefficient fields illustrate the performance of the proposed galerkin discretization and the fully adaptive algorithm

multi critical unitary random matrix ensembles and the general painleve ii equation we study unitary random matrix ensembles of the form z n detm 2 e ntrvwdm where a 1 2 and v is such that the limiting mean eigenvalue density for n n oc and n n 1 vanishes quadratically at the origin in order to compute the double scaling limits of the eigenvalue correlation kernel near the origin we use the deift zhou steepest descent method applied to the riemann hilbert problem for orthogonal polynomials on the real line with respect to the weight x 2ae nv x here the main focus is on the construction of a local parametrix near the origin with functions associated with a special solution qa of the painleve ii equation q sq 2q3 a we show that qa has no real poles for a 1 2 by proving the solvability of the corresponding riemann hilbert problem we also show that the asymptotics of the recurrence coefficients of the orthogonal polynomials can be expressed in terms of qa in the double scaling limit

asymptotics of tracy widom distributions and the total integral of a painleve ii function the tracy widom distribution functions involve integrals of a painleve ii function starting from positive infinity in this paper we express the tracy widom distribution functions in terms of integrals starting from minus infinity there are two consequences of these new representations the first is the evaluation of the total integral of the hastings mcleod solution of the painleve ii equation the second is the evaluation of the constant term of the asymptotic expansions of the tracy widom distribution functions as the distribution parameter approaches minus infinity for the gue tracy widom distribution function this gives an alternative proof of the recent work of deift its and krasovsky the constant terms for the goe and gse tracy widom distribution functions are new

adaptive stochastic galerkin fem abstract a framework for residual based a posteriori error estimation and adaptive mesh refinement and polynomial chaos expansion for general second order linear elliptic pdes with random coefficients is presented a parametric deterministic elliptic boundary value problem on an infinite dimensional parameter space is discretized by means of a galerkin projection onto finite generalized polynomial chaos gpc expansions and by discretizing each gpc coefficient by a fem in the physical domain an anisotropic residual based a posteriori error estimator is developed it contains bounds for both contributions to the overall error the error due to gpc discretization and the error due to finite element discretization of the gpc coefficients in the expansion the reliability of the residual estimator is established based on the explicit form of the residual estimator an adaptive refinement strategy is presented which allows to steer the polynomial degree adaptation and the dimension adaptation in the stochastic galerkin discretization and embedded in the gpc adaptation loop also the finite element mesh refinement of the gpc coefficients in the physical domain asynchronous mesh adaptation for different gpc coefficients is permitted subject to a minimal compatibility requirement on meshes used for different gpc coefficients details on the implementation with the open source software framework alea are presented it is generic and is based on available stiffness and mass matrices of a fem for the deterministic nonparametric nominal problem evaluated in the fenics environment preconditioning of the resulting matrix equation and iterative solution are discussed numerical experiments in two spatial dimensions for membrane and plane stress boundary value problems on polygons are presented they indicate substantial savings in total computational complexity due to fe mesh coarsening in high gpc coefficients

deformations of the tracy widom distribution in random matrix theory the tracy widom tw distribution describes the behavior of the largest eigenvalue we consider here two models in which tw undergoes transformations in the first one disorder is introduced in the gaussian ensembles by superimposing an external source of randomness a competition between tw and a normal gaussian distribution results depending on the spreading of the disorder the second model consists of removing at random a fraction of correlated eigenvalues of a random matrix the usual formalism of fredholm determinants extends naturally a continuous transition from tw to the weilbull distribution characteristic of extreme values of an uncorrelated sequence is obtained

inverse regression based uncertainty quantification algorithms for high dimensional models many uncertainty quantification uq approaches suffer from the curse of dimensionality that is their computational costs become intractable for problems involving a large number of uncertainty parameters in these situations the classic monte carlo often remains the preferred method of choice because its convergence rate o n 1 2 where n is the required number of model simulations does not depend on the dimension of the problem however many high dimensional uq problems are intrinsically low dimensional because the variation of the quantity of interest qoi is often caused by only a few latent parameters varying within a low dimensional subspace known as the sufficient dimension reduction sdr subspace in the statistics literature motivated by this observation we propose two inverse regression based uq algorithms iruq for high dimensional problems both algorithms use inverse regression to convert the original high dimensional problem to a low dimensional one which is then efficiently solved by building a response surface for the reduced model for example via the polynomial chaos expansion the first algorithm which is for the situations where an exact sdr subspace exists is proved to converge at rate o n 1 hence much faster than mc the second algorithm which doesn t require an exact sdr employs the reduced model as a control variate to reduce the error of the mc estimate the accuracy gain could still be significant depending on how well the reduced model approximates the original high dimensional one iruq also provides several additional practical advantages it is non intrusive it does not require computing the high dimensional gradient of the qoi and it reports an error bar so the user knows how reliable the result is

a method for solving stochastic equations by reduced order models and local approximations a method is proposed for solving equations with random entries referred to as stochastic equations ses the method is based on two recent developments the first approximates the response surface giving the solution of a stochastic equation as a function of its random parameters by a finite set of hyperplanes tangent to it at expansion points selected by geometrical arguments the second approximates the vector of random parameters in the definition of a stochastic equation by a simple random vector referred to as stochastic reduced order model srom and uses it to construct a srom for the solution of this equation the proposed method is a direct extension of these two methods it uses sroms to select expansion points rather than selecting these points by geometrical considerations and represents the solution by linear and or higher order local approximations the implementation and the performance of the method are illustrated by numerical examples involving random eigenvalue problems and stochastic algebraic differential equations the method is conceptually simple non intrusive efficient relative to classical monte carlo simulation accurate and guaranteed to converge to the exact solution

approximating infinity dimensional stochastic darcy s equations without uniform ellipticity we consider a stochastic darcy s pressure equation whose coefficient is generated by a white noise process on a hilbert space employing the ordinary rather than the wick product a weak form of this equation involves different spaces for the solution and test functions and we establish a continuous inf sup condition and well posedness of the problem we generalize the numerical approximations proposed in benth and theting stochastic anal appl 20 2002 pp 1191 1223 for wick stochastic partial differential equations to the ordinary product stochastic pressure equation we establish discrete inf sup conditions and provide a priori error estimates for a wide class of norms the proposed numerical approximation is based on wiener chaos finite element methods and yields a positive definite symmetric linear system we also improve and generalize the approximation results of benth and gjerde stochastics stochastics rep 63 1998 pp 313 326 and cao stochastics 78 2006 pp 179 187 when a generalized process is truncated by a finite wiener chaos expansion finally we present numerical experiments to validate the results

quasi monte carlo methods for elliptic pdes with random coefficients and applications we devise and implement quasi monte carlo methods for computing the expectations of nonlinear functionals of solutions of a class of elliptic partial differential equations with random coefficients our motivation comes from fluid flow in random porous media where relevant functionals include the fluid pressure velocity at any point in space or the breakthrough time of a pollution plume being transported by the velocity field our emphasis is on situations where a very large number of random variables is needed to model the coefficient field as an alternative to classical monte carlo we here employ quasi monte carlo methods which use deterministically chosen sample points in an appropriate usually high dimensional parameter space each realization of the pde solution requires a finite element fe approximation in space and this is done using a realization of the coefficient field restricted to a suitable regular spatial grid not necessarily the same as the fe grid in the statistically homogeneous case the corresponding covariance matrix can be diagonalized and the required coefficient realizations can be computed efficiently using fft in this way we avoid the use of a truncated karhunen loeve expansion but introduce high nominal dimension in parameter space numerical experiments with 2 dimensional rough random fields high variance and small length scale are reported showing that the quasi monte carlo method consistently outperforms the monte carlo method with a smaller error and a noticeably better than o n 1 2 convergence rate where n is the number of samples moreover the rate of convergence of the quasi monte carlo method does not appear to degrade as the nominal dimension increases examples with dimension as high as 10 6 are reported

adaptive stochastic galerkin fem with hierarchical tensor representations the solution of pde with stochastic data commonly leads to very high dimensional algebraic problems e g when multiplicative noise is present the stochastic galerkin fem considered in this paper then suffers from the curse of dimensionality this is directly related to the number of random variables required for an adequate representation of the random fields included in the pde with the presented new approach we circumvent this major complexity obstacle by combining two highly efficient model reduction strategies namely a modern low rank tensor representation in the tensor train format of the problem and a refinement algorithm on the basis of a posteriori error estimates to adaptively adjust the different employed discretizations the adaptive adjustment includes the refinement of the fe mesh based on a residual estimator the problem adapted stochastic discretization in anisotropic legendre wiener chaos and the successive increase of the tensor rank computable a posteriori error estimators are derived for all error terms emanating from the discretizations and the iterative solution with a preconditioned als scheme of the problem strikingly it is possible to exploit the tensor structure of the problem to evaluate all error terms very efficiently a set of benchmark problems illustrates the performance of the adaptive algorithm with higher order fe moreover the influence of the tensor rank on the approximation quality is investigated

breaking the curse of dimensionality or how to use svd in many dimensions for d dimensional tensors with possibly large d 3 an hierarchical data structure called the tree tucker format is presented as an alternative to the canonical decomposition it has asymptotically the same and often even smaller number of representation parameters and viable stability properties the approach involves a recursive construction described by a tree with the leafs corresponding to the tucker decompositions of three dimensional tensors and is based on a sequence of svds for the recursively obtained unfolding matrices and on the auxiliary dimensions added to the initial spatial dimensions it is shown how this format can be applied to the problem of multidimensional convolution convincing numerical examples are given

efficient iterative solvers for stochastic galerkin discretizations of log transformed random diffusion problems we consider the numerical solution of a steady state diffusion problem where the diffusion coefficient is the exponent of a random field the standard stochastic galerkin formulation of this problem is computationally demanding because of the nonlinear structure of the uncertain component of it we consider a reformulated version of this problem as a stochastic convection diffusion problem with random convective velocity that depends linearly on a fixed number of independent truncated gaussian random variables the associated galerkin matrix is nonsymmetric but sparse and allows for fast matrix vector multiplications with optimal complexity we construct and analyze two block diagonal preconditioners for this galerkin matrix for use with krylov subspace methods such as the generalized minimal residual method we test the efficiency of the proposed preconditioning approaches and compare the iterative solver performance for a model problem posed in both diffusion and convection diffusion formulations

convergence rates of multilevel and sparse tensor approximations for a random elliptic pde partial differential equations with random coefficients can be cast as parametric problems with a potentially infinite dimensional parameter domain for a class of elliptic equations we derive convergence rates of approximations based on a tensorized polynomial basis on the parameter domain and a sequence of spatial discretizations we prove that a sparse tensor product construction achieves essentially the same convergence rate as a multilevel approximation in which the spatial discretization level is chosen separately for each coefficient of the solution with respect to the polynomial basis in some cases the same rate is attained also by a single level approximation using just one spatial discretization for all coefficients we suggest an adaptive algorithm that reaches the optimal convergence rate with respect to the total number of degrees of freedom in a multilevel setting without prior knowledge of this rate numerical computations confirm theoretical results

finite elements for elliptic problems with stochastic coefficients abstract we describe a deterministic finite element fe solution algorithm for a stochastic elliptic boundary value problem sbvp whose coefficients are assumed to be random fields with finite second moments and known piecewise smooth two point spatial correlation function separation of random and deterministic variables parametrization of the uncertainty is achieved via a karhunen loeve kl expansion an o n log n algorithm for the computation of the kl eigenvalues is presented based on a kernel independent fast multipole method fmm truncation of the kl expansion gives an m 1 wiener polynomial chaos pc expansion of the stochastic coefficient and is shown to lead to a high dimensional deterministic boundary value problem dbvp analyticity of its solution in the stochastic variables with sharp bounds for the domain of analyticity are used to prescribe variable stochastic polynomial degree r r1 rm in an m r wiener pc expansion for the approximate solution pointwise error bounds for the fem approximations of kl eigenpairs the truncation of the kl expansion and the fe solution to the dbvp are given numerical examples show that m depends on the spatial correlation length of the random diffusion coefficient the variable polynomial degree r in pc stochastic galerkin fem allows to handle kl expansions with m up to 30 and r1 up to 10 in moderate time

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 sparse grid stochastic collocation method for partial differential equations with random input data this work proposes and analyzes a smolyak type sparse grid stochastic collocation method for the approximation of statistical quantities related to the solution of partial differential equations with random coefficients and forcing terms input data of the model to compute solution statistics the sparse grid stochastic collocation method uses approximate solutions produced here by finite elements corresponding to a deterministic set of points in the random input space this naturally requires solving uncoupled deterministic problems as in the monte carlo method if the number of random variables needed to describe the input data is moderately large full tensor product spaces are computationally expensive to use due to the curse of dimensionality in this case the sparse grid approach is still expected to be competitive with the classical monte carlo method therefore it is of major practical relevance to understand in which situations the sparse grid stochastic collocation method is more efficient than monte carlo this work provides error estimates for the fully discrete solution using l q norms and analyzes the computational efficiency of the proposed method in particular it demonstrates algebraic convergence with respect to the total number of collocation points and quantifies the effect of the dimension of the problem number of input random variables in the final estimates the derived estimates are then used to compare the method with monte carlo indicating for which problems the former is more efficient than the latter computational evidence complements the present theory and shows the effectiveness of the sparse grid stochastic collocation method compared to full tensor and monte carlo approaches

sparse grid collocation schemes for stochastic natural convection problems in recent years there has been an interest in analyzing and quantifying the effects of random inputs in the solution of partial differential equations that describe thermal and fluid flow problems spectral stochastic methods and monte carlo based sampling methods are two approaches that have been used to analyze these problems as the complexity of the problem or the number of random variables involved in describing the input uncertainties increases these approaches become highly impractical from implementation and convergence points of view this is especially true in the context of realistic thermal flow problems where uncertainties in the topology of the boundary domain boundary flux conditions and heterogeneous physical properties usually require high dimensional random descriptors the sparse grid collocation method based on the smolyak algorithm offers a viable alternate method for solving high dimensional stochastic partial differential equations an extension of the collocation approach to include adaptive refinement in important stochastic dimensions is utilized to further reduce the numerical effort necessary for simulation we show case the collocation based approach to efficiently solve natural convection problems involving large stochastic dimensions equilibrium jumps occurring due to surface roughness and heterogeneous porosity are captured comparison of the present method with the generalized polynomial chaos expansion and monte carlo methods are made

a two stage adaptive stochastic collocation method on nested sparse grids for multiphase flow in randomly heterogeneous porous media a new computational method is proposed for efficient uncertainty quantification of multiphase flow in porous media with stochastic permeability for pressure estimation it combines the dimension adaptive stochastic collocation method on smolyak sparse grids and the kronrod patterson hermite nested quadrature formulas for saturation estimation an additional stage is developed in which the pressure and velocity samples are first generated by the sparse grid interpolation and then substituted into the transport equation to solve for the saturation samples to address the low regularity problem of the saturation numerical examples are presented for multiphase flow with stochastic permeability fields to demonstrate accuracy and efficiency of the proposed two stage adaptive stochastic collocation method on nested sparse grids

galerkin methods for linear and nonlinear elliptic stochastic partial differential equations abstract stationary systems modelled by elliptic partial differential equations linear as well as nonlinear with stochastic coefficients random fields are considered the mathematical setting as a variational problem existence theorems and possible discretisations in particular with respect to the stochastic part are given and investigated with regard to stability different and increasingly sophisticated computational approaches involving both wiener s polynomial chaos as well as the karhunen loeve expansion are addressed in conjunction with stochastic galerkin procedures and stability within the galerkin framework is established new and effective algorithms to compute the mean and covariance of the solution are proposed the similarities and differences with better known monte carlo methods are exhibited as well as alternatives to integration in high dimensional spaces hints are given regarding the numerical implementation and parallelisation numerical examples serve as illustration

galerkin finite element approximations of stochastic elliptic partial differential equations we describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous dirichlet boundary conditions here the aim of the com putations is to approximate statistical moments of the solution and in particular we give a priori error estimates for the computation of the expected value of the solution the first method gener ates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard galerkin finite element variational formulation the monte carlo method then uses these approximations to compute corresponding sample averages the sec ond method is based on a finite dimensional approximation of the stochastic coefficients turning the original stochastic problem into a deterministic parametric elliptic problem a galerkin finite element method of either the h o rp version then approximates the corresponding deterministic solution yielding approximations of the desired statistics we present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy this comparison suggests intuitive conditions for an optimal selection of the numerical approximation

stochastic collocation method for secondary bifurcation of a nonlinear aeroelastic system abstract a stochastic collocation method is proposed to investigate the secondary bifurcation of a two dimensional aeroelastic system with structural nonlinearity represented by cubic restoring forces and uncertainties expressed by random parameters in the cubic stiffness coefficient and in the initial pitch angle the accuracy of the stochastic collocation method is improved by incorporating higher order schemes such as piecewise cubic interpolation and piecewise cubic spline interpolation instead of a piecewise linear interpolation formula for an aeroelastic problem with the uncertainty expressed by a time dependent combination of five random variables an efficient collocation method is developed using a sparse grid approach with a dimension adaptive strategy numerical simulations are carried out to demonstrate the effectiveness of the proposed method for long term computation and discontinuous problems

reconstruction of dynamic displacement and velocity from measured accelerations using the variational statement of an inverse problem this paper presents two types of finite impulse response fir filters to reconstruct dynamic displacement induced by structural vibration from measured acceleration the governing equation for the reconstruction is derived by taking the variation of a minimization problem which defines an inverse problem on displacement a regularization function for overcoming the ill posedness of the inverse problem is included in the minimization problem the governing equation of the inverse problem becomes the same type of differential equation as that of a beam on an elastic foundation the conventional fir cfir filter directly approximates the transfer function of the governing equation while the fem based fir ffir filter is formulated by the discretization of the minimization problem with the finite element method for the finite element discretization the hermitian shape function is utilized the proposed ffir filter is capable of reconstructing displacement and velocity simultaneously the fundamental characteristics of the proposed filters are investigated in the frequency domain using the transfer and accuracy functions it is shown that the proposed fir filters suppress low frequency noise components in measured accelerations effectively and reconstruct physically meaningful displacement accurately the validity of the proposed filters is demonstrated through a numerical simulation study a field experiment and an evaluation of flutter derivatives using measurements taken from a wind tunnel test

on manifolds of tensors of fixed tt rank recently the format of tt tensors hackbusch and kuhn in j fourier anal appl 15 706 722 2009 oseledets in siam j sci comput 2009 submitted oseledets and tyrtyshnikov in siam j sci comput 31 5 2009 oseledets and tyrtyshnikov in linear algebra appl 2009 submitted has turned out to be a promising new format for the approximation of solutions of high dimensional problems in this paper we prove some new results for the tt representation of a tensor u in mathbb r n 1 times cdots times n d and for the manifold of tensors of tt rank underline r as a first result we prove that the tt or compression ranks r i of a tensor u are unique and equal to the respective separation ranks of u if the components of the tt decomposition are required to fulfil a certain maximal rank condition we then show that the set mathbb t of tt tensors of fixed rank underline r locally forms an embedded manifold in mathbb r n 1 times cdots times n d therefore preserving the essential theoretical properties of the tucker format but often showing an improved scaling behaviour extending a similar approach for matrices conte and lubich in m2an 44 759 2010 we introduce certain gauge conditions to obtain a unique representation of the tangent space mathcal t u mathbb t of mathbb t and deduce a local parametrization of the tt manifold the parametrisation of mathcal t u mathbb t is often crucial for an algorithmic treatment of high dimensional time dependent pdes and minimisation problems lubich in from quantum to classical molecular dynamics reduced methods and numerical analysis 2008 we conclude with remarks on those applications and present some numerical examples

multi element probabilistic collocation method in high dimensions we combine multi element polynomial chaos with analysis of variance anova functional decomposition to enhance the convergence rate of polynomial chaos in high dimensions and in problems with low stochastic regularity specifically we employ the multi element probabilistic collocation method mepcm 1 and so we refer to the new method as mepcm a we investigate the dependence of the convergence of mepcm a on two decomposition parameters the polynomial order m and the effective dimension n with n n and n the nominal dimension numerical tests for multi dimensional integration and for stochastic elliptic problems suggest that n m for monotonic convergence of the method we also employ mepcm a to obtain error bars for the piezometric head at the hanford nuclear waste site under stochastic hydraulic conductivity conditions finally we compare the cost of mepcm a against monte carlo in several hundred dimensions and we find mepcm a to be more efficient for up to 600 dimensions for a specific multi dimensional integration problem involving a discontinuous function

efficient posterior exploration of a high dimensional groundwater model from two stage markov chain monte carlo simulation and polynomial chaos expansion 1 this study reports on two strategies for accelerating posterior inference of a highly parameterized and cpu demanding groundwater flow model our method builds on previous stochastic collocation approaches e g marzouk and xiu 2009 and marzouk and najm 2009 and uses generalized polynomial chaos gpc theory and dimensionality reduction to emulate the output of a large scale groundwater flow model the resulting surrogate model is cpu efficient and serves to explore the posterior distribution at a much lower computational cost using two stage mcmc simulation the case study reported in this paper demonstrates a two to five times speed up in sampling efficiency

a weighted reduced basis method for elliptic partial differential equations with random input data in this work we propose and analyze a weighted reduced basis method to solve elliptic partial differential equations pdes with random input data the pdes are first transformed into a weighted parametric elliptic problem depending on a finite number of parameters distinctive importance of the solution at different values of the parameters is taken into account by assigning different weights to the samples in the greedy sampling procedure a priori convergence analysis is carried out by constructive approximation of the exact solution with respect to the weighted parameters numerical examples are provided for the assessment of the advantages of the proposed method over the reduced basis method and the stochastic collocation method in both univariate and multivariate stochastic problems

a multiscale probabilistic collocation method for subsurface flow in heterogeneous media 1 owing to the spatial variability of the media properties uncertainty quantification for subsurface flow and solute transport usually requires high resolution simulations in this work a multiscale probabilistic collocation method mspcm is developed for solving such problems in a computationally efficient manner the subsurface flow problem is cast in a stochastic framework and probabilistic collocation strategy is used to represent the original stochastic differential equation the resulting equations are a set of decoupled deterministic equations with respect to collocation points a multiscale finite element method is utilized to solve these deterministic problems on a coarse mesh coarse scale basis functions are constructed on a field in which the conductivity varies spatially at each set of stochastic collocation points the coarse scale solution is then obtained by solving a modified coarse formulation that takes into account the fine scale heterogeneity the fine scale solution is reconstructed after the coarse scale solution is available since the pcm and multiscale finite element method are implemented at different levels the mspcm inherits their respective advantages in which a stochastic problem is decomposed by fewer realizations and is solved on a coarser grid the performance of the proposed method is demonstrated with numerical examples the capability of mspcm in reproducing the probability density functions pdfs of head and velocity is investigated the numerical results show that the mspcm with proper coarsening level is able to capture small scale heterogeneity with a coarse mesh and to generate satisfactory probability density functions of head and velocity

energy norm a posteriori error estimation for parametric operator equations stochastic galerkin approximation is an increasingly popular approach for the solution of elliptic pde problems with correlated random data a typical strategy is to combine conventional h finite element approximation on the spatial domain with spectral p approximation on a finite dimensional manifold in the stochastic parameter domain the issues involved in a posteriori error analysis of computed solutions are outlined in this paper using an abstract setting of parametric operator equations a novel energy error estimator that uses a parameter free part of the underlying differential operator is introduced which effectively exploits the tensor product structure of the approximation space we prove that our error estimator is reliable and efficient we also discuss different strategies for enriching the approximation space and prove two sided estimates of the error reduction for the corresponding enhanced approximations these give computable estimates of the error reduction that depend only on

multilevel and weighted reduced basis method for stochastic optimal control problems constrained by stokes equations in this paper we develop and analyze a multilevel weighted reduced basis method for solving stochastic optimal control problems constrained by stokes equations we prove the analytic regularity of the optimal solution in the probability space under certain assumptions on the random input data the finite element method and the stochastic collocation method are employed for the numerical approximation of the problem in the deterministic space and the probability space respectively resulting in many large scale optimality systems to solve in order to reduce the unaffordable computational effort we propose a reduced basis method using a multilevel greedy algorithm in combination with isotropic and anisotropic sparse grid techniques a weighted a posteriori error bound highlights the contribution stemming from each method numerical tests on stochastic dimensions ranging from 10 to 100 demonstrate that our method is very efficient especially for solving high dimensional and large scale optimization problems

tracy widom distributions in critical unitary random matrix ensembles and the coupled painleve ii system we study fredholm determinants of the painleve ii and painleve xxxiv kernels in certain critical unitary random matrix ensembles these determinants describe special gap probabilities of eigenvalues we obtain tracy widom formulas for the fredholm determinants which are explicitly given in terms of integrals involving a family of distinguished solutions to the coupled painleve ii system in dimension four moreover the large gap asymptotics for these fredholm determinants are derived where the constant terms are given explicitly in terms of the riemann zeta function

reduced basis methods for nonlocal diffusion problems with random input data abstract the construction analysis and application of reduced basis methods for uncertainty quantification problems involving nonlocal diffusion problems with random input data is the subject of this work because of the lack of sparsity of discretized nonlocal models relative to analogous local partial differential equation models the need for reduced order modeling is much more acute in the nonlocal setting in this effort we develop reduced basis approximations for nonlocal diffusion equations with affine random coefficients efficiency estimates of the proposed greedy reduced basis methods are provided numerical examples are used to illustrate the effect varying various model parameters have on the efficiency and accuracy of the reduced basis method relative to sparse grid interpolation using the full finite element method it is shown that the proposed reduced basis approach can indeed provide substantial savings over standard sparse grid methods

a rapid and efficient isogeometric design space exploration framework with application to structural mechanics abstract in this paper we present an isogeometric analysis framework for design space exploration while the methodology is presented in the setting of structural mechanics it is applicable to any system of parametric partial differential equations the design space exploration framework elucidates design parameter sensitivities used to inform initial and early stage design moreover this framework enables the visualization of a full system response including the displacement and stress fields throughout the domain by providing an approximation to the system solution vector this is accomplished through a collocation like approach where various geometries throughout the design space under consideration are sampled the sampling scheme follows a quadrature rule while the physical solutions to these sampled geometries are obtained through an isogeometric method a surrogate model to the design space solution manifold is constructed through either an interpolating polynomial or pseudospectral expansion examples of this framework are presented with applications to the scordelis lo roof a flat l bracket and an nrel 5 mw wind turbine blade

monodromy dependence and connection formulae for isomonodromic tau functions we discuss an extension of the jimbo miwa ueno differential 1 form to a form closed on the full space of extended monodromy data of systems of linear ordinary differential equations with rational coefficients this extension is based on the results of m bertola generalizing a previous construction by b malgrange we show how this 1 form can be used to solve a long standing problem of evaluation of the connection formulae for the isomonodromic tau functions which would include an explicit computation of the relevant constant factors we explain how this scheme works for fuchsian systems and in particular calculate the connection constant for the generic painleve vi tau function the result proves the conjectural formula for this constant proposed by iorgov lisovyy and tykhyy we also apply the method to non fuchsian systems and evaluate constant factors in the asymptotics of the painleve ii tau function

on integrals of the tronquee solutions and the associated hamiltonians for the painleve ii equation abstract we consider a family of tronquee solutions of the painleve ii equation q s 2 q s 3 s q s 2 1 2 1 2 which is characterized by the stokes multipliers s 1 e 2 i s 2 s 3 e 2 i with being a free parameter these solutions include the well known generalized hastings mcleod solution as a special case if 0 we derive asymptotics of integrals of the tronquee solutions and the associated hamiltonians over the real axis for 1 2 and 0 with the constant terms evaluated explicitly our results agree with those already known in the literature if the parameters and are chosen to be special values some applications of our results in random matrix theory are also discussed

analysis of the domain mapping method for elliptic diffusion problems on random domains in this article we provide a rigorous analysis of the solution to elliptic diffusion problems on random domains in particular based on the decay of the karhunen loeve expansion of the domain perturbation field we establish decay rates for the derivatives of the random solution that are independent of the stochastic dimension for the implementation of a related approximation scheme like quasi monte carlo quadrature stochastic collocation etc we propose parametric finite elements to compute the solution of the diffusion problem on each individual realization of the domain generated by the perturbation field this simplifies the implementation and yields a non intrusive approach having this machinery at hand we can easily transfer it to stochastic interface problems the theoretical findings are complemented by numerical examples for both stochastic interface problems and boundary value problems on random domains

accurate and efficient evaluation of failure probability for partial different equations with random input data several computational challenges arise when evaluating the failure probability of a given system in the context of risk prediction or reliability analysis when the dimension of the uncertainties becomes high well established direct numerical methods can not be employed because of the curse of dimensionality many surrogate models have been proposed with the aim of reducing computational effort however most of them fail in computing an accurate failure probability when the limit state surface defined by the failure event in the probability space lacks smoothness in addition for a stochastic system modeled by partial differential equations pdes with random input only a limited number of the underlying pdes order of a few tens are affordable to solve in practice due to the considerable computational effort therefore preventing the application of many numerical methods especially for high dimensional random inputs in this work we develop hybrid and goal oriented adaptive reduced basis methods to tackle these challenges by accurately and efficiently computing the failure probability of a stochastic pde the curse of dimensionality is significantly alleviated by reduced basis approximation whose bases are constructed by goal oriented adaptation moreover an accurate evaluation of the failure probability for pde system with solution of low regularity in probability space is guaranteed by the certified a posteriori error bound for the output approximation error at the end of this paper we suitably extend our proposed method to deal with more general pde models finally we perform several numerical experiments to illustrate its computational accuracy and efficiency c 2013 elsevier b v all rights reserved

asymptotics of a τ function arising in the two dimensional ising model the short distance assymptotics of the function associated to the 2 point function of the two dimensional ising model is computed as a function of the integration constant defined from the long distance behavior of the function the result is expressible in terms of the barnes double gamma function equivalently the barnesg function

on the design of linear phase fir integrators for midband frequencies the ideal frequency responses 1 j spl omega an integrator and 1 spl omega an integrating hilbert transformer are usually approximated by iir structures this paper suggests their fir approximations maximal flatness at spl omega spl pi 2 yields exact mathematical formulas for computing the weights the proposed designs are particularly suitable for the frequency range 0 1 spl pi spl les spl omega spl les 0 90 spl pi

uncertainty quantification of geochemical and mechanical compaction in layered sedimentary basins abstract in this work we propose an uncertainty quantification methodology for sedimentary basins evolution under mechanical and geochemical compaction processes which we model as a coupled time dependent non linear monodimensional depth only system of pdes with uncertain parameters while in previous works formaggia et al 2013 porta et al 2014 we assumed a simplified depositional history with only one material in this work we consider multi layered basins in which each layer is characterized by a different material and hence by different properties this setting requires several improvements with respect to our earlier works both concerning the deterministic solver and the stochastic discretization on the deterministic side we replace the previous fixed point iterative solver with a more efficient newton solver at each step of the time discretization on the stochastic side the multi layered structure gives rise to discontinuities in the dependence of the state variables on the uncertain parameters that need an appropriate treatment for surrogate modeling techniques such as sparse grids to be effective we propose an innovative methodology to this end which relies on a change of coordinate system to align the discontinuities of the target function within the random parameter space the reference coordinate system is built upon exploiting physical features of the problem at hand we employ the fact that the locations of the material interfaces display a smooth dependence on the random parameters and are therefore amenable to sparse grid polynomial approximations we showcase the capabilities of our numerical methodologies through two synthetic test cases in particular we show that our methodology reproduces with high accuracy multi modal probability density functions displayed by target state variables e g porosity

convergence of quasi optimal sparse grid approximation of hilbert space valued functions application to random elliptic pdes in this work we provide a convergence analysis for the quasi optimal version of the sparse grids stochastic collocation method we presented in a previous work on the optimal polynomial approximation of stochastic pdes by galerkin and collocation methods beck et al math models methods appl sci 22 09 2012 the construction of a sparse grid is recast into a knapsack problem a profit is assigned to each hierarchical surplus and only the most profitable ones are added to the sparse grid the convergence rate of the sparse grid approximation error with respect to the number of points in the grid is then shown to depend on weighted summability properties of the sequence of profits this is a very general argument that can be applied to sparse grids built with any uni variate family of points both nested and non nested as an example we apply such quasi optimal sparse grids to the solution of a particular elliptic pde with stochastic diffusion coefficients namely the inclusions problem we detail the convergence estimates obtained in this case using polynomial interpolation on either nested clenshaw curtis or non nested gauss legendre abscissas verify their sharpness numerically and compare the performance of the resulting quasi optimal grids with a few alternative sparse grid construction schemes recently proposed in the literature

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

on the analysis of incomplete spectra in random matrix theory through an extension of the jimbo miwa ueno differential abstract several distribution functions in the classical unitarily invariant matrix ensembles are prime examples of isomonodromic tau functions as introduced by jimbo miwa and ueno jmu in the early 1980s 45 recent advances in the theory of tau functions 41 based on earlier works of b malgrange and m bertola have allowed to extend the original jimbo miwa ueno differential form to a 1 form closed on the full space of extended monodromy data of the underlying lax pairs this in turn has yielded a novel approach for the asymptotic evaluation of isomonodromic tau functions including the exact computation of all relevant constant factors we use this method to efficiently compute the tail asymptotics of soft edge hard edge and bulk scaled distribution and gap functions in the complex wishart ensemble provided each eigenvalue particle has been removed independently with probability 1 0 1

design of an fir filter for the displacement reconstruction using measured acceleration in low frequency dominant structures this paper presents a new class of displacement reconstruction scheme using only acceleration measured from a structure for a given set of acceleration data the reconstruction problem is formulated as a boundary value problem in which the acceleration is approximated by the second order central finite difference of displacement the displacement is reconstructed by minimizing the least squared errors between measured and approximated acceleration within a finite time interval an overlapping time window is introduced to improve the accuracy of the reconstructed displacement the displacement reconstruction problem becomes ill posed because the boundary conditions at both ends of each time window are not known a priori furthermore random noise in measured acceleration causes physically inadmissible errors in the reconstructed displacement a tikhonov regularization scheme is adopted to alleviate the ill posedness it is shown that the proposed method is equivalent to an fir filter designed in the time domain the fundamental characteristics of the proposed method are presented in the frequency domain using the transfer function and the accuracy function the validity of the proposed method is demonstrated by a numerical example a laboratory experiment and a field test copyright 2009 john wiley sons ltd

large deformations of the tracy widom distribution i non oscillatory asymptotics we analyze the left tail asymptotics of deformed tracy widom distribution functions describing the fluctuations of the largest eigenvalue in invariant random matrix ensembles after removing each soft edge eigenvalue independently with probability 1 gamma in 0 1 as gamma varies a transition from tracy widom statistics gamma 1 to classical weibull statistics gamma 0 was observed in the physics literature by bohigas et al phys rev e 79 031117 2009 we provide a description of this transition by rigorously computing the leading order left tail asymptotics of the thinned goe gue and gse tracy widom distributions in this paper we obtain the asymptotic behavior in the non oscillatory region with gamma in 0 1 fixed for the goe gue and gse distributions and gamma uparrow 1 at a controlled rate for the gue distribution this is the first step in an ongoing program to completely describe the transition between tracy widom and weibull statistics as a corollary to our results we obtain a new total integral formula involving the ablowitz segur solution to the second painleve equation

reweighted l1 minimization method for stochastic elliptic differential equations abstract we consider elliptic stochastic partial differential equations spdes with random coefficients and solve them by expanding the solution using generalized polynomial chaos gpc under some mild conditions on the coefficients the solution is sparse in the random space i e only a small number of gpc basis makes considerable contribution to the solution to exploit this sparsity we employ reweighted l 1 minimization to recover the coefficients of the gpc expansion we also combine this method with random sampling points based on the chebyshev probability measure to further increase the accuracy of the recovery of the gpc coefficients we first present a one dimensional test to demonstrate the main idea and then we consider 14 and 40 dimensional elliptic spdes to demonstrate the significant improvement of this method over the standard l 1 minimization method for moderately high dimensional 10 problems the combination of chebyshev measure with reweighted l 1 minimization performs well while for higher dimensional problems reweighted l 1 only is sufficient the proposed approach is especially suitable for problems for which the deterministic solver is very expensive since it reuses the sampling results and exploits all the information available from limited sources