characterization of discontinuities in high dimensional stochastic problems on adaptive sparse grids in this paper we present a set of efficient algorithms for detection and identification of discontinuities in high dimensional space the method is based on extension of polynomial annihilation for discontinuity detection in low dimensions compared to the earlier work the present method poses significant improvements for high dimensional problems the core of the algorithms relies on adaptive refinement of sparse grids it is demonstrated that in the commonly encountered cases where a discontinuity resides on a small subset of the dimensions the present method becomes optimal in the sense that the total number of points required for function evaluations depends linearly on the dimensionality of the space the details of the algorithms will be presented and various numerical examples are utilized to demonstrate the efficacy of the method

stochastic inverse heat conduction using a spectral approach an adjoint based functional optimization technique in conjunction with the spectral stochastic finite element method is proposed for the solution of an inverse heat conduction problem in the presence of uncertainties in material data process conditions and measurement noise the ill posed stochastic inverse problem is restated as a conditionally well posed l 2 optimization problem the gradient of the objective function is obtained in a distributional sense by defining an appropriate stochastic adjoint field the l 2 optimization problem is solved using a conjugate gradient approach accuracy and effectiveness of the proposed approach is appraised with the solution of several stochastic inverse heat conduction problems

sparse finite elements for elliptic problems with stochastic loading we formulate elliptic boundary value problems with stochastic loading in a bounded domain d d we show well posedness of the problem in stochastic sobolev spaces and we derive a deterministic elliptic pde in d d for the spatial correlation of the random solution we show well posedness and regularity results for this pde in a scale of weighted sobolev spaces with mixed highest order derivatives discretization with sparse tensor products of any hierarchic finite element fe spaces in d yields optimal asymptotic rates of convergence for the spatial correlation even in the presence of singularities or for spatially completely uncorrelated data multilevel preconditioning in d d allows iterative solution of the discrete equation for the correlation kernel in essentially the same complexity as the solution of the mean field equation

building blocks for computer vision with stochastic partial differential equations we discuss the basic concepts of computer vision with stochastic partial differential equations spdes in typical approaches based on partial differential equations pdes the end result in the best case is usually one value per pixel the expected value error estimates or even full probability density functions pdfs are usually not available this paper provides a framework allowing one to derive such pdfs rendering computer vision approaches into measurements fulfilling scientific standards due to full error propagation we identify the image data with random fields in order to model images and image sequences which carry uncertainty in their gray values e g due to noise in the acquisition process the noisy behaviors of gray values is modeled as stochastic processes which are approximated with the method of generalized polynomial chaos wiener askey chaos the wiener askey polynomial chaos is combined with a standard spatial approximation based upon piecewise multi linear finite elements we present the basic building blocks needed for computer vision and image processing in this stochastic setting i e we discuss the computation of stochastic moments projections gradient magnitudes edge indicators structure tensors etc finally we show applications of our framework to derive stochastic analogs of well known pdes for de noising and optical flow extraction these models are discretized with the stochastic galerkin method our selection of spde models allows us to draw connections to the classical deterministic models as well as to stochastic image processing not based on pdes several examples guide the reader through the presentation and show the usefulness of the framework

maximum likelihood estimation of stochastic chaos representations from experimental data this paper deals with the identification of probabilistic models of the random coefficients in stochastic boundary value problems sbvp the data used in the identification correspond to measurements of the displacement field along the boundary of domains subjected to specified external forcing starting with a particular mathematical model for the mechanical behaviour of the specimen the unknown field to be identified is projected on an adapted functional basis such as that provided by a finite element discretization for each set of measurements of the displacement field along the boundary an inverse problem is formulated to calculate the corresponding optimal realization of the coefficients of the unknown random field on the adapted basis realizations of these coefficients are then used in conjunction with the maximum likelihood principle to set up and solve an optimization problem for the estimation of the coefficients in a polynomial chaos representation of the parameters of the sbvp

geometric maximizers of schatten norms of some convolution type integral operators abstract in this paper we prove that the ball is a maximizer of the schatten p norm of some convolution type integral operators with non increasing kernels among all domains of a given measure in r d we also show that the equilateral triangle has the largest schatten p norm among all triangles of a given area some physical motivations for our results are also presented

variational bayesian strategies for high dimensional stochastic design problems this paper is concerned with a lesser studied problem in the context of model based uncertainty quantification uq that of optimization design control under uncertainty the solution of such problems is hindered not only by the usual difficulties encountered in uq tasks e g the high computational cost of each forward simulation the large number of random variables but also by the need to solve a nonlinear optimization problem involving large numbers of design variables and potentially constraints we propose a framework that is suitable for a class of such problems and is based on the idea of recasting them as probabilistic inference tasks to that end we propose a variational bayesian vb formulation and an iterative vb expectation maximization scheme that is capable of identifying a local maximum as well as a low dimensional set of directions in the design space along which the objective exhibits the largest sensitivity we demonstrate the validity of the proposed approach in the context of two numerical examples involving thousands of random and design variables in all cases considered the cost of the computations in terms of calls to the forward model was of the order of 100 or less the accuracy of the approximations provided is assessed by information theoretic metrics

biorthogonal systems and quotient spaces via baire category methods we show that every banach space x of density smaller that the baire category number open image in new window admits a quotient with a long schauder basis that can be taken of length 1 if x is not separable so assuming that the baire category number open image in new window does not take its minimal possible value a banach space x is separable if and only if all biorthogonal systems of x are countable

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

comparison of approaches for random pde optimization problems based on different matching functionals in this article we consider an optimal control problem for an elliptic partial differential equation with random inputs to determine an applicable deterministic control f x we consider the four cases which we compare for efficiency and feasibility we prove the existence of optimal states adjoint states and optimality conditions for each cases we also derive the optimality systems for the four cases the optimality system is then discretized by a standard finite element method and sparse grid collocation method for physical space and probability space respectively the numerical experiments are performed for their efficiency and feasibility

a data driven stochastic collocation approach for uncertainty quantification in mems this work presents a data driven stochastic collocation approach to include the effect of uncertain design parameters during complex multi physics simulation of micro electromechanical systems mems the proposed framework comprises of two key steps first probabilistic characterization of the input uncertain parameters based on available experimental information and second propagation of these uncertainties through the predictive model to relevant quantities of interest the uncertain input parameters are modeled as independent random variables for which the distributions are estimated based on available experimental observations using a nonparametric diffusion mixing based estimator botev nonparametric density estimation via diffusion mixing technical report 2007 the diffusion based estimator derives from the analogy between the kernel density estimation kde procedure and the heat dissipation equation and constructs density estimates that are smooth and asymptotically consistent the diffusion model allows for the incorporation of the prior density and leads to an improved density estimate in comparison with the standard kde approach as demonstrated through several numerical examples following the characterization step the uncertainties are propagated to the output variables using the stochastic collocation approach based on sparse grid interpolation smolyak soviet math dokl 1963 4 240 243 the developed framework is used to study the effect of variations in young s modulus induced as a result of variations in manufacturing process parameters or heterogeneous measurements on the performance of a mems switch copyright 2010 john wiley sons ltd

stochastic reduced order models for inverse problems under uncertainty abstract this work presents a novel methodology for solving inverse problems under uncertainty using stochastic reduced order models sroms given statistical information about an observed state variable in a system unknown parameters are estimated probabilistically through the solution of a model constrained stochastic optimization problem the point of departure and crux of the proposed framework is the representation of a random quantity using a srom a low dimensional discrete approximation to a continuous random element that permits efficient and non intrusive stochastic computations characterizing the uncertainties with sroms transforms the stochastic optimization problem into a deterministic one the non intrusive nature of sroms facilitates efficient gradient computations for random vector unknowns and relies entirely on calls to existing deterministic solvers furthermore the method is naturally extended to handle multiple sources of uncertainty in cases where state variable data system parameters and boundary conditions are all considered random the new and widely applicable srom framework is formulated for a general stochastic optimization problem in terms of an abstract objective function and constraining model for demonstration purposes however we study its performance in the specific case of inverse identification of random material parameters in elastodynamics we demonstrate the ability to efficiently recover random shear moduli given material displacement statistics as input data we also show that the approach remains effective for the case where the loading in the problem is random as well

eigenfunction expansions of ultradifferentiable functions and ultradistributions in this paper we give a global characterisation of classes of ultradifferentiable functions and corresponding ultradistributions on a compact manifold x the characterisation is given in terms of the eigenfunction expansion of an elliptic operator on x this extends the result for analytic functions on compact manifolds by seeley in 1969 and the characterisation of gevrey functions and gevrey ultradistributions on compact lie groups and homogeneous spaces by the authors 2014

minimal multi element stochastic collocation for uncertainty quantification of discontinuous functions we propose a multi element stochastic collocation method that can be applied in high dimensional parameter space for functions with discontinuities lying along manifolds of general geometries the key feature of the method is that the parameter space is decomposed into multiple elements defined by the discontinuities and thus only the minimal number of elements are utilized on each of the resulting elements the function is smooth and can be approximated using high order methods with fast convergence properties the decomposition strategy is in direct contrast to the traditional multi element approaches which define the sub domains by repeated splitting of the axes in the parameter space such methods are more prone to the curse of dimensionality because of the fast growth of the number of elements caused by the axis based splitting the present method is a two step approach firstly a discontinuity detector is used to partition parameter space into disjoint elements in each of which the function is smooth the detector uses an efficient combination of the high order polynomial annihilation technique along with adaptive sparse grids and this allows resolution of general discontinuities with a smaller number of points when the discontinuity manifold is low dimensional after partitioning an adaptive technique based on the least orthogonal interpolant is used to construct a generalized polynomial chaos surrogate on each element the adaptive technique reuses all information from the partitioning and is variance suppressing we present numerous numerical examples that illustrate the accuracy efficiency and generality of the method when compared against standard locally adaptive sparse grid methods the present method uses many fewer number of collocation samples and is more accurate

an adaptive multi element generalized polynomial chaos method for stochastic differential equations we formulate a multi element generalized polynomial chaos me gpc method to deal with long term integration and discontinuities in stochastic differential equations we first present this method for legendre chaos corresponding to uniform random inputs and subsequently we generalize it to other random inputs the main idea of me gpc is to decompose the space of random inputs when the relative error in variance becomes greater than a threshold value in each subdomain or random element we then employ a generalized polynomial chaos expansion we develop a criterion to perform such a decomposition adaptively and demonstrate its effectiveness for odes including the kraichnan orszag three mode problem as well as advection diffusion problems the new method is similar to spectral element method for deterministic problems but with h p discretization of the random space

adaptive reduced basis generation for reduced order modeling for the solution of stochastic nondestructive evaluation problems a novel algorithm for creating a computationally efficient approximation of a system response that is defined by a boundary value problem is presented more specifically the approach presented is focused on substantially reducing the computational expense required to approximate the solution of a stochastic partial differential equation particularly for the purpose of estimating the solution to an associated nondestructive evaluation problem with significant system uncertainty in order to achieve this computational efficiency the approach combines reduced basis reduced order modeling with a sparse grid collocation surrogate modeling technique to estimate the response of the system of interest with respect to any designated unknown parameters provided the distributions are known the reduced order modeling component includes a novel algorithm for adaptive generation of a data ensemble based on a nested grid technique to then create the reduced order basis the capabilities and potential applicability of the approach presented are displayed through two simulated case studies regarding inverse characterization of material properties for two different physical systems involving some amount of significant uncertainty the first case study considered characterization of an unknown localized reduction in stiffness of a structure from simulated frequency response function based nondestructive testing then the second case study considered characterization of an unknown temperature dependent thermal conductivity of a solid from simulated thermal testing overall the surrogate modeling approach was shown through both simulated examples to provide accurate solution estimates to inverse problems for systems represented by stochastic partial differential equations with a fraction of the typical computational cost

a method for stochastic constrained optimization using derivative free surrogate pattern search and collocation recent advances in coupling novel optimization methods to large scale computing problems have opened the door to tackling a diverse set of physically realistic engineering design problems a large computational overhead is associated with computing the cost function for most practical problems involving complex physical phenomena such problems are also plagued with uncertainties in a diverse set of parameters we present a novel stochastic derivative free optimization approach for tackling such problems our method extends the previously developed surrogate management framework smf to allow for uncertainties in both simulation parameters and design variables the stochastic collocation scheme is employed for stochastic variables whereas kriging based surrogate functions are employed for the cost function this approach is tested on four numerical optimization problems and is shown to have significant improvement in efficiency over traditional monte carlo schemes problems with multiple probabilistic constraints are also discussed

a functional optimization approach to an inverse magneto convection problem a formulation and an fem implementation for the solution of an ill posed inverse design magneto convection problem is proposed in particular an incompressible viscous electrically conducting liquid material occupying a given domain is considered convection is driven by buoyancy effects as well as a lorentz force generated due to the applied magnetic field thermal boundary conditions are only prescribed in the part h0 of the boundary in addition the temperature distribution is also prescribed in the part i of the boundary h1 where the heat flux is known i e i is a boundary with overspecified thermal boundary conditions the inverse magneto convection problem is posed as an optimization problem in l2 i 0 tmax for the calculation of the boundary heat flux qo x t with x t h0 0 tmax the optimization scheme minimizes the discrepancy m x t x t qo l2 i 0 tmax between the temperature x t qo calculated from the solution of a direct problem for each flux qo and the desired or measured temperature m x t on the boundary i the exact l2 gradient of the cost functional is obtained from appropriately defined adjoint fields the adjoint problem is defined from the sensitivity operators obtained by linearization of the equations governing the direct problem the standard supg pspg stabilized fem formulation for incompressible fluid flow simulation is here extended to thermo magnetically driven flows the proposed fem formulation is used for the calculation of the direct adjoint and sensitivity thermal fluid flow and electric potential fields the entire optimization algorithm is solved using the conjugate gradient method finally the method is demonstrated through the solution of a few inverse problems with known results the need for regularization is identified in one of the example problems with uniformly distributed random errors in the temperature data m x t and a h1 regularized formulation is introduced to obtain stable solutions finally the numerical results are elucidated and potential applications are addressed

isoperimetric inequalities for schatten norms of riesz potentials in this note we prove that the ball is a maximiser for integer order schatten p norms of the riesz potential operators among all domains of a given measure in r d in particular the result is valid for the polyharmonic newton potential operator which is related to a nonlocal boundary value problem for the poly laplacian extending the one considered by m kac in the case of the laplacian so we obtain isoperimetric inequalities for its eigenvalues as well namely analogues of rayleigh faber krahn and hong krahn szego inequalities

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

on a problem of rolewicz about banach spaces that admit support sets we construct an example of a nonseparable banach space which does not admit a support set 2 it is a consistent and necessarily independent from the axioms of zfc example of a space c k of continuous functions on a compact hausdorff k with the supremum norm the construction depends on a construction of a boolean algebra with some combinatorial properties the space is also hereditarily lindelof in the weak topology but it doesn t have any nonseparable subspace nor any nonseparable quotient which is a c k space for k dispersed

probabilistic characterization of transport in heterogeneous media abstract the mechanics of transport and flow in a random porous medium are addressed in this paper the hydraulic properties of the porous medium are modeled as spatial random processes the random aspect of the problem is treated by introducing a new dimension along which spectral approximations are implemented thus the hydraulic processes are discretized using the spectral karhunen loeve expansion this expansion represents the random spatial functions as deterministic modes of fluctuation with random amplitudes these amplitudes form a basis in the manifold associated with the random processes the concentrations over the whole domain are also random processes with unknown probabilistic structure these processes are represented using the polynomial chaos basis this is a basis in the functional space described by all second order random variables the deterministic coefficients in this expansion are calculated via a weighted residual procedure with respect to the random measure and the inner product specified by the expectation operator once the spatio temporal variation of the concentrations has been specified in terms of the polynomial chaos expansion individual realizations can be readily computed

isoperimetric inequalities for the logarithmic potential operator abstract in this paper we prove that the disc is a maximiser of the schatten p norm of the logarithmic potential operator among all domains of a given measure in r 2 for all even integers 2 p we also show that the equilateral triangle has the largest schatten p norm among all triangles of a given area for the logarithmic potential operator on bounded open or triangular domains we also obtain analogies of the rayleigh faber krahn or polya inequalities respectively the logarithmic potential operator can be related to a nonlocal boundary value problem for the laplacian so we obtain isoperimetric inequalities for its eigenvalues as well

a posteriori error estimation and adaptivity in stochastic galerkin fem for parametric elliptic pdes beyond the affine case abstract we consider a linear elliptic partial differential equation pde with a generic uniformly bounded parametric coefficient the solution to this pde problem is approximated in the framework of stochastic galerkin finite element methods we perform a posteriori error analysis of galerkin approximations and derive a reliable and efficient estimate for the energy error in these approximations practical versions of this error estimate are discussed and tested numerically for a model problem with non affine parametric representation of the coefficient furthermore we use the error reduction indicators derived from spatial and parametric error estimators to guide an adaptive solution algorithm for the given parametric pde problem the performance of the adaptive algorithm is tested numerically for model problems with two different non affine parametric representations of the coefficient

optimal control with stochastic pde constraints and uncertain controls abstract the optimal control of problems that are constrained by partial differential equations with uncertainties and with uncertain controls is addressed the lagrangian that defines the problem is postulated in terms of stochastic functions with the control function possibly decomposed into an unknown deterministic component and a known zero mean stochastic component the extra freedom provided by the stochastic dimension in defining cost functionals is explored demonstrating the scope for controlling statistical aspects of the system response one shot stochastic finite element methods are used to find approximate solutions to control problems it is shown that applying the stochastic collocation finite element method to the formulated problem leads to a coupling between stochastic collocation points when a deterministic optimal control is considered or when moments are included in the cost functional thereby forgoing the primary advantage of the collocation method over the stochastic galerkin method for the considered problem the application of the presented methods is demonstrated through a number of numerical examples the presented framework is sufficiently general to also consider a class of inverse problems and numerical examples of this type are also presented

probabilistic collocation for period 1 limit cycle oscillations in this paper probabilistic collocation for limit cycle oscillations pclco is proposed probabilistic collocation pc is a non intrusive approach to compute the polynomial chaos description of uncertainty numerically polynomial chaos can require impractical high orders to approximate long term time integration problems due to the fast increase of required polynomial chaos order with time pclco is a pc formulation for modeling the long term stochastic behavior of dynamical systems exhibiting a periodic response i e a limit cycle oscillation lco in the pc method deterministic time series are computed at collocation points in probability space in pclco pc is applied to a time independent parametrization of the periodic response of the deterministic solves instead of to the time dependent functions themselves due to the time independent parametrization the accuracy of pclco is independent of time the approach is applied to period 1 oscillations with one main frequency subject to a random parameter numerical results are presented for the harmonic oscillator a two dof airfoil flutter model and the fluid structure interaction of an elastically mounted cylinder

on stability and monotonicity requirements of finite difference approximations of stochastic conservation laws with random viscosity on stability and monotonicity requirements of finite difference approximations of stochastic conservation laws with random viscosity

fast bayesian approach for parameter estimation this paper presents two techniques i e the proper orthogonal decomposition pod and the stochastic collocation method scm for constructing surrogate models to accelerate the bayesian inference approach for parameter estimation problems associated with partial differential equations pod is a model reduction technique that derives reduced order models using an optimal problem adapted basis to effect significant reduction of the problem size and hence computational cost scm is an uncertainty propagation technique that approximates the parameterized solution and reduces further forward solves to function evaluations the utility of the techniques is assessed on the non linear inverse problem of probabilistically calibrating scalar robin coefficients from boundary measurements arising in the quenching process and non destructive evaluation a hierarchical bayesian model that handles flexibly the regularization parameter and the noise level is employed and the posterior state space is explored by the markov chain monte carlo the numerical results indicate that significant computational gains can be realized without sacrificing the accuracy

a fast collocation method for solving stochastic integral equations based on sparse grid multiscale piecewise polynomial bases we develop a fast collocation method for solving fredholm integral equations of the second kind with stochastic loading terms it is proved that the proposed method preserves the optimal rate of convergence and has linear up to a logarithmic factor computational complexity the reduction in computational costs come from three aspects the adoption of the collocation principle the use of the sparse grid for the solution space and the employment of a truncation strategy for the coefficient matrix of the resulting linear system numerical experiments confirm the theoretical results on approximation accuracy and computational efficiency of the proposed method

difference equations and pseudo differential operators on zn abstract in this paper we develop the calculus of pseudo differential operators on the lattice z n which we can call pseudo difference operators an interesting feature of this calculus is that the global frequency space t n is compact so the symbol classes are defined in terms of the behaviour with respect to the lattice variable we establish formulae for composition adjoint transpose and for parametrix for the elliptic operators we also give conditions for the l 2 weighted l 2 and l p boundedness of operators and for their compactness on l p we describe a link to the toroidal quantization on the torus t n and apply it to give conditions for the membership in schatten classes on l 2 z n furthermore we discuss a version of fourier integral operators on the lattice and give conditions for their l 2 boundedness the results are applied to give estimates for solutions to difference equations on the lattice z n moreover we establish garding and sharp garding inequalities with an application to the unique solvability of parabolic equations on the lattice z n

generalized spectral decomposition for stochastic nonlinear problems we present an extension of the generalized spectral decomposition method for the resolution of nonlinear stochastic problems the method consists in the construction of a reduced basis approximation of the galerkin solution and is independent of the stochastic discretization selected polynomial chaos stochastic multi element or multi wavelets two algorithms are proposed for the sequential construction of the successive generalized spectral modes they involve decoupled resolutions of a series of deterministic and low dimensional stochastic problems compared to the classical galerkin method the algorithms allow for significant computational savings and require minor adaptations of the deterministic codes the methodology is detailed and tested on two model problems the one dimensional steady viscous burgers equation and a two dimensional nonlinear diffusion problem these examples demonstrate the effectiveness of the proposed algorithms which exhibit convergence rates with the number of modes essentially dependent on the spectrum of the stochastic solution but independent of the dimension of the stochastic approximation space

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

modeling uncertainty in flow simulations via generalized polynomial chaos we present a new algorithm to model the input uncertainty and its propagation in incompressible flow simulations the stochastic input is represented spectrally by employing orthogonal polynomial functionals from the askey scheme as trial basis to represent the random space a standard galerkin projection is applied in the random dimension to obtain the equations in the weak form the resulting system of deterministic equations is then solved with standard methods to obtain the solution for each random mode this approach can be considered as a generalization of the original polynomial chaos expansion first introduced by wiener am j math 60 1938 897 the original method employs the hermite polynomials one of the 13 members of the askey scheme as the basis in random space the algorithm is applied to micro channel flows with random wall boundary conditions and to external flows with random freestream efficiency and convergence are studied by comparing with exact solutions as well as numerical solutions obtained by monte carlo simulations it is shown that the generalized polynomial chaos method promises a substantial speed up compared with the monte carlo method the utilization of different type orthogonal polynomials from the askey scheme also provides a more efficient way to represent general non gaussian processes compared with the original wiener hermite expansions

a non linear dimension reduction methodology for generating data driven stochastic input models stochastic analysis of random heterogeneous media polycrystalline materials porous media functionally graded materials provides information of significance only if realistic input models of the topology and property variations are used this paper proposes a framework to construct such input stochastic models for the topology and thermal diffusivity variations in heterogeneous media using a data driven strategy given a set of microstructure realizations input samples generated from given statistical information about the medium topology the framework constructs a reduced order stochastic representation of the thermal diffusivity this problem of constructing a low dimensional stochastic representation of property variations is analogous to the problem of manifold learning and parametric fitting of hyper surfaces encountered in image processing and psychology denote by m the set of microstructures that satisfy the given experimental statistics a non linear dimension reduction strategy is utilized to map m to a low dimensional region a we first show that m is a compact manifold embedded in a high dimensional input space r n an isometric mapping f from m to a low dimensional compact connected set email protected r d email protected n is constructed given only a finite set of samples of the data the methodology uses arguments from graph theory and differential geometry to construct the isometric transformation f m a asymptotic convergence of the representation of m by a is shown this mapping f serves as an accurate low dimensional data driven representation of the property variations the reduced order model of the material topology and thermal diffusivity variations is subsequently used as an input in the solution of stochastic partial differential equations that describe the evolution of dependant variables a sparse grid collocation strategy smolyak algorithm is utilized to solve these stochastic equations efficiently we showcase the methodology by constructing low dimensional input stochastic models to represent thermal diffusivity in two phase microstructures this model is used in analyzing the effect of topological variations of two phase microstructures on the evolution of temperature in heat conduction processes

inversion of robin coefficient by a spectral stochastic finite element approach this paper investigates a variational approach to the nonlinear stochastic inverse problem of probabilistically calibrating the robin coefficient from boundary measurements for the steady state heat conduction the problem is formulated into an optimization problem and mathematical properties relevant to its numerical computations are investigated the spectral stochastic finite element method using polynomial chaos is utilized for the discretization of the optimization problem and its convergence is analyzed the nonlinear conjugate gradient method is derived for the optimization system numerical results for several two dimensional problems are presented to illustrate the accuracy and efficiency of the stochastic finite element method

a robust bi orthogonal dynamically orthogonal method using the covariance pseudo inverse with application to stochastic flow problems abstract we develop a new robust methodology for the stochastic navier stokes equations based on the dynamically orthogonal do and bi orthogonal bo methods 1 2 3 both approaches are variants of a generalized karhunen loeve kl expansion in which both the stochastic coefficients and the spatial basis evolve according to system dynamics hence capturing the low dimensional structure of the solution the do and bo formulations are mathematically equivalent 3 but they exhibit computationally complimentary properties specifically the bo formulation may fail due to crossing of the eigenvalues of the covariance matrix while both bo and do become unstable when there is a high condition number of the covariance matrix or zero eigenvalues to this end we combine the two methods into a robust hybrid framework and in addition we employ a pseudo inverse technique to invert the covariance matrix the robustness of the proposed method stems from addressing the following issues in the do bo formulation i eigenvalue crossing we resolve the issue of eigenvalue crossing in the bo formulation by switching to the do near eigenvalue crossing using the equivalence theorem and switching back to bo when the distance between eigenvalues is larger than a threshold value ii ill conditioned covariance matrix we utilize a pseudo inverse strategy to invert the covariance matrix iii adaptivity we utilize an adaptive strategy to add remove modes to resolve the covariance matrix up to a threshold value in particular we introduce a soft threshold criterion to allow the system to adapt to the newly added removed mode and therefore avoid repetitive and unnecessary mode addition removal when the total variance approaches zero we show that the do bo formulation becomes equivalent to the evolution equation of the optimally time dependent modes 4 we demonstrate the capability of the proposed methodology with several numerical examples namely i stochastic burgers equation we analyze the performance of the method in the presence of eigenvalue crossing and zero eigenvalues ii stochastic kovasznay flow we examine the method in the presence of a singular covariance matrix and iii we examine the adaptivity of the method for an incompressible flow over a cylinder where for large stochastic forcing thirteen do bo modes are active

uncertainty propagation in finite deformations a spectral stochastic lagrangian approach this work proposes a method for quantifying uncertainty propagation in finite deformation problems using the spectral stochastic finite element method ssfem a spectral expansion of the current configuration of a deforming body is proposed to compute the stochastic deformation gradient which is in turn used to compute the stochastic analogs of the various quantities which appear in large deformation analysis such as strain and stress measures and consistent moduli a total lagrangian approach to the stochastic large deformation problem is presented model problems in large deformation elasto plasticity are considered highlighting the features of the methodology developed rigorous comparisons with monte carlo solutions are presented it is shown that the proposed approach can estimate the probability density function and response statistics of the field variables with satisfactory accuracy

mid frequency structural dynamics with parameter uncertainty abstract in this paper a number of frequency domain dynamic analysis procedures of randomly disordered structural systems in the medium frequency range are integrated into the stochastic finite element method in all cases frequency domain model reduction strategies are used to minimize the computational effort in the mid frequency range firstly an energy operator approach eoa is investigated in this procedure an energy operator adapted to a fixed medium frequency band is defined whose dominant eigensubspace is used to construct a reduced model using a ritz galerkin method secondly the proper orthogonal decomposition method is used to extract the spatial dominant coherent structures in the vibration wave field in the mid frequency band the coherent structures are the eigenvectors corresponding to the dominant eigenvalues of the spatial autocorrelation function of the system response consequently a close relationship between the energy operator approach and the proper orthogonal decomposition method is identified although these two approaches are not mathematically identical the proper orthogonal decomposition method appears to be more straightforward compared to the energy operator method from the viewpoint of numerical implementation thirdly another approach namely the dynamic element method based on frequency dependent finite element shape functions is considered a stochastic reduction method is then utilized to represent the uncertain parameters modelled as stochastic processes in terms of their dominant scales of fluctuation the karhunen loeve and the polynomial chaos decompositions are used to that effect in the context of a stochastic finite element formalism the methodology adopted in the paper thus integrates efficient dynamical reduction techniques with a reduction scheme of stochastic processes for the mid frequency vibration of linear random systems the formulation is exemplified by its application to the analysis of the dynamics of a coupled uncertain rod assembly subjected to an external excitation the example is also used to highlight some of the relative features of the three dynamic reduction strategies

a scalable framework for the solution of stochastic inverse problems using a sparse grid collocation approach experimental evidence suggests that the dynamics of many physical phenomena are significantly affected by the underlying uncertainties associated with variations in properties and fluctuations in operating conditions recent developments in stochastic analysis have opened the possibility of realistic modeling of such systems in the presence of multiple sources of uncertainties these advances raise the possibility of solving the corresponding stochastic inverse problem the problem of designing estimating the evolution of a system in the presence of multiple sources of uncertainty given limited information a scalable parallel methodology for stochastic inverse design problems is developed in this article the representation of the underlying uncertainties and the resultant stochastic dependant variables is performed using a sparse grid collocation methodology a novel stochastic sensitivity method is introduced based on multiple solutions to deterministic sensitivity problems the stochastic inverse design problem is transformed to a deterministic optimization problem in a larger dimensional space that is subsequently solved using deterministic optimization algorithms the design framework relies entirely on deterministic direct and sensitivity analysis of the continuum systems thereby significantly enhancing the range of applicability of the framework for the design in the presence of uncertainty of many other systems usually analyzed with legacy codes various illustrative examples with multiple sources of uncertainty including inverse heat conduction problems in random heterogeneous media are provided to showcase the developed framework

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

on the schatten von neumann properties of some pseudo differential operators abstract we obtain a number of explicit estimates for quasi norms of pseudo differential operators in the schatten von neumann classes s q with 0 q 1 the estimates are applied to derive semi classical bounds for operators with smooth or non smooth symbols

towards essential improvement for the parareal tr and parareal gauss4 algorithms parareal is an iterative algorithm and is characterized by two propagators g and f which are respectively associated with large step size t and small step size t where t j t and j 2 is an integer for symmetric positive definite spd system u t a u t g t arising from semi discretizing time dependent pdes if we fix the g propagator to the backward euler method and choose for f some l stable time integrator it can be proven that the convergence factors of the corresponding parareal algorithms satisfy 1 3 j 2 and a 0 where a is the spectrum of the matrix a however this result does not hold when time integrators that lack l stability such as the trapezoidal rule and the 4th order gauss rk method are chosen as the f propagator the parareal algorithms using these two methods for the f propagator are denoted by parareal tr and parareal gauss4 in this paper we propose a strategy to let these two parareal algorithms possess such a uniform convergence property the idea is to choose an l stable propagator f and on each coarse time interval t n t n 1 we perform first two steps of f then followed by j 2 steps of f precisely for the trapezoidal rule we select the 2nd order sdirk method as the f propagator and for the 4th order gauss rk method we select the 4th order lobatto iii c method as the f propagator numerical results are given to support our theoretical conclusions

a bayesian inference approach to the ill posed cauchy problem of steady state heat conduction this paper studies a bayesian inference approach to the cauchy problem in steady state heat conduction of probabilistically calibrating the boundary temperature the prior modeling is achieved via the markov random field and its regularizing property is investigated a hierarchical bayesian model is adopted for selecting the regularization parameter and detecting the noise level automatically the posterior state space is explored using the markov chain monte carlo for obtaining relevant statistics two augmented tikhonov regularization methods that could determine the regularization parameter and the noise level are proposed and analyzed numerical results indicate that the bayesian inference approach could yield an accurate estimate of the solution with its uncertainties quantified and the augmented tikhonov regularization methods are accurate and flexible copyright 2008 john wiley sons ltd

a practical polynomial chaos kalman filter implementation using nonlinear error projection on a reduced polynomial chaos expansion summary the polynomial chaos kalman filter pckf has been gaining popularity as a computationally efficient and robust alternative to sampling methods in sequential data assimilation settings the pckf s sampling free scheme and attractive structure to represent non gaussian uncertainties makes it a promising approach for data filtering techniques in nonlinear and non gaussian frameworks however the accuracy of pckf is dependent on the dimension and order of the polynomial chaos expansion pce employed to represent all sources of uncertainty in the system thus when independent sources of errors like process noise and time independent sensors errors are incorporated in the system the curse of dimensionality hinders the efficiency and the applicability of pckf this study sheds light on this issue and presents a practical framework to maintain an acceptable accuracy of pckf without scarifying the computational efficiency of the filter the robustness and efficiency of the presented implementation is demonstrated on three typical numerical examples to illustrate its ability to achieve considerable accuracy at a low computational tax

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

directed sets and cofinal types we show that 1 s 1 x x xl and l are the only cofinal types of directed sets of size sl but that there exist many cofinal types of directed sets of size continuum a partially ordered set d is directed if every two elements of d have an upper bound in d in this note we consider some basic problems concerning directed sets which have their origin in the theory of moore smith convergence in topology 12 3 19 9 one such problem is to determine all essential kind of directed sets needed for defining the closure operator in a given class of spaces 3 p 47 concerning this problem the following important notion was introduced by j tukey 19 two directed sets d and e are cofinally similar if there is a partially ordered set c in which both can be embedded as cofinal subsets he showed that this is an equivalence relation and that d and e are cofinally similar iff there is a convergent map from d into e and also a convergent map from e into d the equivalence classes of this relation are called cofinal types this concept has been extensively studied since then by various authors 4 13 7 8 already from the first introduction of this concept it has been known that 1 1 x x 1 and x1 represent different cofinal types of directed sets of size btl but no more than five such types were known the main result of this paper shows that 1 1 x x 1 and x1 are the only cofinal types of convergence in spaces of character 81 which can be constructed without additional set theoretic assumptions on the other hand we shall construct many different cofinal types of directed sets of size continuum this gives a solution to problem 1 of j isbell 7 the paper also contains several results about the structure of the class of all cofinal types as well as a result about decomposing arbitrary partially ordered sets into directed sets the results of this note were proved in february march 1982 and presented to the asl in january 1983 1 a decomposition theorem in this section we show that an arbitrary partially ordered set can be decomposed into a number of its directed subsets depending on the sizes of its antichains this result is connected with an unpublished problem of f galvin concerning the dilworth decomposition theorem 6 and it generalizes a similar result of e milner and k prikry 11 the transitivity condition of a partial ordering is not used in our proof so we state our result so as to apply to an arbitrary received by the editors december 3 1984 1980 mathpmaties subjeet classification primary 03e05 03e35 secondary 06a10 18b35 54a15 c1985 american mathematical society 0002 9947 85 1 00 25 per page

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

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

using stochastic analysis to capture unstable equilibrium in natural convection a stabilized stochastic finite element implementation for the natural convection system of equations under boussinesq assumptions with uncertainty in inputs is considered the stabilized formulations are derived using the variational multiscale framework assuming a one step trapezoidal time integration rule the stabilization parameters are shown to be functions of the time step size provision is made for explicit tracking of the subgrid scale solution through time a support space stochastic galerkin approach and the generalized polynomial chaos expansion gpce approach are considered for input output uncertainty representation stochastic versions of standard rayleigh benard convection problems are used to evaluate the approach it is shown that for simulations around critical points the gpce approach fails to capture the highly non linear input uncertainty propagation whereas the support space approach gives fairly accurate results a summary of the results and findings is provided

green s formula for integro differential operators abstract in this report we establish green s formula for an integro differential operator and apply it to describe a class of self adjoint fractional order differential operators a found symmetric fractional order caputo riemann liouville type operator can be considered as a fractional analogue of the classical sturm liouville operator in some sense

a stochastic computational method for evaluation of global and local behavior of random elastic media a stochastic computational method is developed to evaluate global effective properties and local probabilistic behavior of random elastic media the elasticity solution for random media is addressed with a method that combines a fast iterative numerical method with stochastic decomposition techniques stochastic homogenization problems are mathematically formulated with an ensemble average approximation scheme based on a concept of stochastic representative volume element srve probabilistic descriptors of local strain are generated based on convolution equations derived from the equilibrium equations of elasticity with the application of the galerkin formulation in probability space and the fourier transform of the convolution equations an efficient numerical scheme is implemented using an iterative algorithm that takes advantage of fast fourier transform two algorithms are provided for gaussian and non gaussian random media respectively with corresponding sufficient conditions for convergence derived finally numerical experiments are conducted for a gaussian and a non gaussian random medium respectively with verifications obtained from monte carlo simulation

identification of boundary conditions by solving cauchy problem in linear elasticity with material uncertainties this work is a contribution to non destructive testing in the context of uncertainties it consists in identifying boundary conditions on an inaccessible part of a solid body boundary from the knowledge of over specified data given on an accessible part of this boundary this problem is well known as cauchy problem the material properties are considered uncertain polynomial chaos expansion is used in order to solve this cauchy problem in the random context a stochastic constrained optimization problem is formulated and solved numerical experiments are presented

backward uncertainty propagation method in flow problems application to the prediction of rarefaction shock waves abstract a computational method for taking into account backward uncertainty propagation in flow problems is presented and applied to the study of rarefaction shock waves rsw in a dense gas shock tube previous theoretical and numerical studies have shown that a rsw is relatively weak and that the prediction of its occurrence and intensity are highly sensitive to uncertainties on the initial flow conditions and on the fluid thermodynamic model the objective of this work is to introduce an innovative flexible and efficient algorithm combining computational fluid dynamics cfd uncertainty quantification uq tools and metamodel based optimization in order to obtain a reliable estimate for the rsw probability of occurrence and to prescribe the experimental accuracy requirements ensuring the reproducibility of the measurements with sufficient confidence

schatten classes on compact manifolds kernel conditions abstract in this paper we give criteria on integral kernels ensuring that integral operators on compact manifolds belong to schatten classes a specific test for nuclearity is established as well as the corresponding trace formulae in the special case of compact lie groups kernel criteria in terms of locally and globally hypoelliptic operators are also given

schatten von neumann properties in the weyl calculus let opt a for t r be the pseudo differential operator f x 2 n a 1 t x ty f y ei x y dyd and let ip be the set of schatten von neumann operators of order p 1 on l2 we are especially concerned with the weyl case i e when t 1 2 we prove that if m and g are appropriate metrics and weight functions respectively hg is the planck s function hgk 2m lp for some k 0 and a s m g then opt a ip iff a lp consequently if 0 1 and a s r then opt a is bounded on l2 iff a l

a computational inverse method for identification of non gaussian random fields using the bayesian approach in very high dimension this paper is devoted to the identification of bayesian posteriors for the random coefficients of the high dimension polynomial chaos expansions of non gaussian tensor valued random fields using partial and limited experimental data the experimental data sets correspond to an observation vector which is the response of a stochastic boundary value problem depending on the tensor valued random field which has to be identified so an inverse stochastic problem must be solved to perform the identification of the random field a complete methodology is proposed to solve this very challenging problem in high dimension which consists in using the first four steps introduced in a previous paper followed by the identification of the posterior model the steps of the methodology are the following 1 introduction of a family of prior algebraic stochastic model pasm 2 identification of an optimal pasm in the constructed family using the partial experimental data 3 construction of a statistical reduced order optimal pasm 4 construction in high dimension of the polynomial chaos expansion with deterministic vector valued coefficients of the reduced order optimal pasm 5 substitution of these deterministic vector valued coefficients by random vector valued coefficients in order to extend the capability of the polynomial chaos expansion to represent the experimental data and for which the joint probability distribution must be identified 6 construction of the prior probability model of these random vector valued coefficients and finally 7 identification of the posterior probability model of these random vector valued coefficients using partial and limited experimental data through the stochastic boundary value problem two methods are proposed to carry out the identification of the posterior model the first one is based on the use of the classical bayesian method the second one is a new approach derived from the bayesian method which is more efficient in high dimension an application is presented for which several millions of random coefficients are identified

comparative study of projection schemes for stochastic finite element analysis we present a comparison of subspace projection schemes for stochastic finite element analysis in terms of accuracy and computational efficiency more specifically we compare the polynomial chaos projection scheme with reduced basis projection schemes based on the preconditioned stochastic krylov subspace numerical studies are presented for two problems 1 static analysis of a plate with random young s modulus and 2 settlement of a foundation supported on a randomly heterogeneous soil monte carlo simulation results based on exact structural analysis are used to generate benchmark results against which the projection schemes are compared we show that stochastic reduced basis methods require significantly less computer memory and execution time compared to the polynomial chaos approach particularly for large scale problems with many random variables for the class of problems considered we find that stochastic reduced basis methods can be up to orders of magnitude faster while providing results of comparable or better accuracy

stochastic collocation for optimal control problems with stochastic pde constraints we discuss the use of stochastic collocation for the solution of optimal control problems which are constrained by stochastic partial differential equations spde thereby the constraining spde depends on data which is not deterministic but random assuming a deter ministic control randomness within the states of the input data will propagate to the states of the system for the solution of spdes there has recently been an increasing effort in the development of efficient numerical schemes based upon the mathematical concept of generalized polynomial chaos modal based stochastic galerkin and nodal based stochastic collocation versions of this methodol ogy exist both of which rely on a certain level of smoothness of the solution in the random space to yield accelerated convergence rates in this paper we apply the stochastic collocation method to develop a gradient descent as well as a sequential quadratic program sqp for the minimization of objective functions constrained by an spde the stochastic function involves several higher order moments of the random states of the system as well as classical regularization of the control in particular we discuss several objective functions of tracking type numerical examples are presented to demonstrate the performance of our new stochastic collocation minimization approach

irredundant sets in boolean algebras it is shown that every uncountable boolean algebra a contains an uncountable subset i such that no a of i is in the subalgebra generated by i a using an additional axiom of set theory it is also shown that a use of some such axiom is necessary

emulation of numerical models with over specified basis functions abstractmathematical models are frequently used to explore physical systems but can be computationally expensive to evaluate in such settings an emulator is used as a surrogate in this work we propose a basis function approach for computer model emulation to combine field observations with a collection of runs from the numerical model we use the proposed emulator within the kennedy o hagan framework of model calibration a novel feature of the approach is the use of an over specified set of basis functions where number of bases used and their inclusion probabilities are treated as unknown quantities the new approach is found to have smaller predictive uncertainty and computational efficiency than the standard gaussian process approach to emulation and calibration along with several simulation examples focusing on different model characteristics we also use the method to analyze a dataset on laboratory experiments related to astrophysics

complex powers of hypoelliptic pseudodifferential operators abstract complex powers of a class of hypoelliptic pseudodifferential operators in r n as well as their heat kernels are studied an application to the schatten von neumann property of pseudodifferential operators is given

a bayesian inference approach to identify a robin coefficient in one dimensional parabolic problems this paper investigates a nonlinear inverse problem associated with the heat conduction problem of identifying a robin coefficient from boundary temperature measurement a bayesian inference approach is presented for the solution of this problem the prior modeling is achieved via the markov random field mrf the use of a hierarchical bayesian method for automatic selection of the regularization parameter in the function estimation inverse problem is discussed the markov chain monte carlo mcmc algorithm is used to explore the posterior state space numerical results indicate that mrf provides an effective prior regularization and the bayesian inference approach can provide accurate estimates as well as uncertainty quantification to the solution of the inverse problem

bayesian estimates of parameter variability in the k e turbulence model in this paper we are concerned with obtaining estimates for the error in reynolds averaged navier stokes rans simulations based on the launder sharma email protected turbulence closure model for a limited class of flows in particular we search for estimates grounded in uncertainties in the space of model closure coefficients for wall bounded flows at a variety of favorable and adverse pressure gradients in order to estimate the spread of closure coefficients which reproduces these flows accurately we perform 13 separate bayesian calibrations each at a different pressure gradient using measured boundary layer velocity profiles and a statistical model containing a multiplicative model inadequacy term in the solution space the results are 13 joint posterior distributions over coefficients and hyper parameters to summarize this information we compute highest posterior density hpd intervals and subsequently represent the total solution uncertainty with a probability box p box this p box represents both parameter variability across flows and epistemic uncertainty within each calibration a prediction of a new boundary layer flow is made with uncertainty bars generated from this uncertainty information and the resulting error estimate is shown to be consistent with measurement data

a stochastic variational multiscale method for diffusion in heterogeneous random media a stochastic variational multiscale method with explicit subgrid modelling is provided for numerical solution of stochastic elliptic equations that arise while modelling diffusion in heterogeneous random media the exact solution of the governing equations is split into two components a coarse scale solution that can be captured on a coarse mesh and a subgrid solution a localized computational model for the subgrid solution is derived for a generalized trapezoidal time integration rule for the coarse scale solution the coarse scale solution is then obtained by solving a modified coarse formulation that takes into account the subgrid model the generalized polynomial chaos method combined with the finite element technique is used for the solution of equations resulting from the coarse formulation and subgrid models finally various numerical examples are considered for evaluating the method

identification of high dimension polynomial chaos expansions with random coefficients for non gaussian tensor valued random fields using partial and limited experimental data this paper is devoted to the identification of high dimension polynomial chaos expansions with random coefficients for non gaussian tensor valued random fields using partial and limited experimental data the experimental data sets correspond to partial experimental data made up of an observation vector which is the response of a stochastic boundary value problem depending on the tensor valued random field which has to be identified so an inverse stochastic problem has to be solved to carry out the identification of the random field a complete methodology is proposed to solve this challenging problem and consists in introducing a family of prior probability models in identifying an optimal prior model in the constructed family using the experimental data in constructing a statistical reduced order optimal prior model in constructing the polynomial chaos expansion with deterministic vector valued coefficients of the reduced order optimal prior model and finally in constructing the probability distribution of random coefficients of the polynomial chaos expansion and in identifying the parameters using experimental data an application is presented for which several millions of random coefficients are identified solving an inverse stochastic problem c 2010 elsevier b v all rights reserved

computational aspects for constructing realizations of polynomial chaos in high dimension this paper deals with computational aspects related to the construction of realizations of polynomial chaos expansion in high dimension the method proposed consists of 1 constructing the realizations of the multivariate monomials using a generator of independent realizations of the germs whose probability distribution is the given arbitrary measure and 2 performing an orthogonalization of the realizations of the multivariate monomials with an algorithm different from the gram schmidt orthogonalization algorithm which is not stable in high dimension a brief review of polynomial chaos expansion with arbitrary measure is given the statistically independent realizations of multivariate monomials are introduced the centered statistically independent realizations of orthonormal multivariate polynomials are developed finally a quantification of the errors induced by the usual methods is given

numerical studies of the stochastic korteweg de vries equation we present numerical solutions of the stochastic korteweg de vries equation for three cases corresponding to additive time dependent noise multiplicative space dependent noise and a combination of the two we employ polynomial chaos for discretization in random space and discontinuous galerkin and finite difference for discretization in physical space the accuracy of the stochastic solutions is investigated by comparing the first two moments against analytical and monte carlo simulation results of particular interest is the interplay of spatial discretization error with the stochastic approximation error which is examined for different orders of spatial and stochastic approximation

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

a measure theoretic computational method for inverse sensitivity problems i method and analysis we consider the inverse sensitivity analysis problem of quantifying the uncertainty of inputs to a deterministic map given specified uncertainty in a linear functional of the output of the map this is a version of the model calibration or parameter estimation problem for a deterministic map we assume that the uncertainty in the quantity of interest is represented by a random variable with a given distribution and we use the law of total probability to express the inverse problem for the corresponding probability measure on the input space assuming that the map from the input space to the quantity of interest is smooth we solve the generally ill posed inverse problem by using the implicit function theorem to derive a method for approximating the set valued inverse that provides an approximate quotient space representation of the input space we then derive an efficient computational approach to compute a measure theoretic approximation of the probability measure on the input space imparted by the approximate set valued inverse that solves the inverse problem

global functional calculus for operators on compact lie groups abstract in this paper we develop the functional calculus for elliptic operators on compact lie groups without the assumption that the operator is a classical pseudo differential operator consequently we provide a symbolic descriptions of complex powers of such operators as an application we give a constructive symbolic proof of the garding inequality for operators in classes in the setting of compact lie groups

uncertainty quantification in hybrid dynamical systems uncertainty quantification uq techniques are frequently used to ascertain output variability in systems with parametric uncertainty traditional algorithms for uq are either system agnostic and slow such as monte carlo or fast with stringent assumptions on smoothness such as polynomial chaos and quasi monte carlo in this work we develop a fast uq approach for hybrid dynamical systems by extending the polynomial chaos methodology to these systems to capture discontinuities we use a wavelet based wiener haar expansion we develop a boundary layer approach to propagate uncertainty through separable reset conditions we also introduce a transport theory based approach for propagating uncertainty through hybrid dynamical systems here the expansion yields a set of hyperbolic equations that are solved by integrating along characteristics the solution of the partial differential equation along the characteristics allows one to quantify uncertainty in hybrid or switching dynamical systems the above methods are demonstrated on example problems

propagation of mechanical waves through a stochastic medium with spherical symmetry abstract we theoretically analyze the propagation of outgoing mechanical waves through an infinite isotropic elastic medium possessing spherical symmetry whose lame coefficients and density are spatial random functions characterized by well defined statistical parameters we derive the differential equation that governs the average displacement for a system whose properties depend on the radial coordinate we show that such an equation is an extended version of the well known bessel differential equation whose perturbative additional terms contain coefficients that depend directly on the squared noise intensities and the autocorrelation lengths in an exponential decay fashion we numerically solve the second order differential equation for several values of noise intensities and autocorrelation lengths and compare the corresponding displacement profiles with that of the exact analytic solution for the case of absent inhomogeneities

a concurrent model reduction approach on spatial and random domains for the solution of stochastic pdes a methodology is introduced for rapid reduced order solution of stochastic partial differential equations on the random domain a generalized polynomial chaos expansion gpce is used to generate a reduced subspace gpce involves expansion of the random variable as a linear combination of basis functions defined using orthogonal polynomials from the askey series a proper orthogonal decomposition pod approach coupled with the method of snapshots is used to generate a reduced solution space from the space spanned by the finite element basis functions on the spatial domain pod methods have been extremely popular in fluid mechanics applications and have subsequently been applied to other interesting areas they have been shown to be capable of representing complicated phenomena with a handful of degrees of freedom this concurrent model reduction on the random and spatial domains is applied to stochastic partial differential equations pdes in natural convection processes involving randomness in the porosity of the medium and the rayleigh number the results indicate that owing to the multiplicative nature of the concurrent model reduction extremely large computational gains are realized without significant loss of accuracy