stochastic multiscale flux basis for stokes darcy flows abstract three algorithms are developed for uncertainty quantification in modeling coupled stokes and darcy flows the porous media may consist of multiple regions with different properties the permeability is modeled as a non stationary stochastic variable with its log represented as a sum of local karhunen loeve kl expansions the problem is approximated by stochastic collocation on either tensor product or sparse grids coupled with a multiscale mortar mixed finite element method for the spatial discretization a non overlapping domain decomposition algorithm reduces the global problem to a coarse scale mortar interface problem which is solved by an iterative solver for each stochastic realization in the traditional domain decomposition implementation each subdomain solves a local dirichlet or neumann problem in every interface iteration to reduce this cost two additional algorithms based on deterministic or stochastic multiscale flux basis are introduced the basis consists of the local flux or velocity trace responses from each mortar degree of freedom it is computed for each subdomain independently before the interface iteration begins the use of the multiscale flux basis avoids the need for subdomain solves on each iteration the deterministic basis is computed at each stochastic collocation and used only at this realization the stochastic basis is formed by further looping over all local realizations of a subdomain s kl region before the stochastic collocation begins it is reused over multiple realizations numerical tests are presented to illustrate the performance of the three algorithms with the stochastic multiscale flux basis showing significant savings in computational cost compared to the other two algorithms

efficient geostatistical inverse methods for structured and unstructured grids 1 the geostatistical inverse approach of kitanidis 1995 requires the covariance matrix of a discretized spatial random variable and matrix multiplications of the covariance matrix the explicit computation of the full covariance matrix is prohibitive for large scale problems discrete spectral methods which help to reduce computational costs dramatically are restricted to regular grids to make the geostatistical inverse approach applicable to unstructured grids which may be of great interest for practical applications we use the functional parameterization of the spatial random field by the karhunen loeve kl expansion in which the spatial variable is parameterized by weighted base functions that are derived from the covariance function in the inverse approach the estimation of the spatial variable becomes estimating the weights of base functions the prior covariance matrix of the weights is the identity matrix the base functions are continuous in space and can be discretized in any fashion for the derivation of the base functions we embed the domain into a larger unit cell of a periodic domain then the base functions are sinusoidal and can be derived by spectral methods which are independent of the discretization used for forward and inverse modeling for illustration we present case studies using synthetic data to estimate hydraulic conductivities on regular and unstructured grids the results show that we can efficiently obtain a good estimate using the inverse method in conjunction with the kl expansion we also demonstrate how to estimate structural parameters and to generate conditional realizations in the framework of our method

an efficient monte carlo interior penalty discontinuous galerkin method for elastic wave scattering in random media abstract this paper develops and analyzes an efficient monte carlo interior penalty discontinuous galerkin mcip dg method for elastic wave scattering in random media the method is constructed based on a multi modes expansion of the solution of the governing random partial differential equations it is proved that the mode functions satisfy a three term recurrence system of partial differential equations pdes which are nearly deterministic in the sense that the randomness only appears in the right hand side source terms not in the coefficients of the pdes moreover the same differential operator applies to all mode functions a proven unconditionally stable and optimally convergent ip dg method is used to discretize the deterministic pde operator an efficient numerical algorithm is proposed based on combining the monte carlo method and the ip dg method with the l u direct linear solver it is shown that the algorithm converges optimally with respect to both the mesh size h and the sampling number m and practically its total computational complexity only amounts to solving a few deterministic elastic helmholtz equations using a gaussian elimination direct linear solver numerical experiments are also presented to demonstrate the performance and key features of the proposed mcip dg method

a stochastic lagrangian approach for geometrical uncertainties in electrostatics this work proposes a general framework to quantify uncertainty arising from geometrical variations in the electrostatic analysis the uncertainty associated with geometry is modeled as a random field which is first expanded using either polynomial chaos or karhunen loeve expansion in terms of independent random variables the random field is then treated as a random displacement applied to the conductors defined by the mean geometry to derive the stochastic lagrangian boundary integral equation the surface charge density is modeled as a random field and is discretized both in the random dimension and space using polynomial chaos and classical boundary element method respectively various numerical examples are presented to study the effect of uncertain geometry on relevant parameters such as capacitance and net electrostatic force the results obtained using the proposed method are verified using rigorous monte carlo simulations it has been shown that the proposed method accurately predicts the statistics and probability density functions of various relevant parameters

efficient robust conditional random fields conditional random fields crfs are a flexible yet powerful probabilistic approach and have shown advantages for popular applications in various areas including text analysis bioinformatics and computer vision traditional crf models however are incapable of selecting relevant features as well as suppressing noise from noisy original features moreover conventional optimization methods often converge slowly in solving the training procedure of crfs and will degrade significantly for tasks with a large number of samples and features in this paper we propose robust crfs rcrfs to simultaneously select relevant features an optimal gradient method ogm is further designed to train rcrfs efficiently specifically the proposed rcrfs employ the ell 1 norm of the model parameters to regularize the objective used by traditional crfs therefore enabling discovery of the relevant unary features and pairwise features of crfs in each iteration of ogm the gradient direction is determined jointly by the current gradient together with the historical gradients and the lipschitz constant is leveraged to specify the proper step size we show that an ogm can tackle the rcrf model training very efficiently achieving the optimal convergence rate o 1 k vphantom r r 2 where k is the number of iterations this convergence rate is theoretically superior to the convergence rate o 1 k of previous first order optimization methods extensive experiments performed on three practical image segmentation tasks demonstrate the efficacy of ogm in training our proposed rcrfs

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

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 nonlocal tv based variational method for polsar data speckle reduction in this paper we propose a nonlocal total variation nltv based variational model for polarimetric synthetic aperture radar polsar data speckle reduction this model named wisnltv is obtained based on the wishart fidelity term and the nltv regularization defined for the complex valued fourth order tensor data since the proposed model is non convex an equivalent bi convex model is obtained using the property of conjugate functions then an efficient iteration algorithm is developed to solve the equivalent bi convex model based on the alternating minimization and the forward backward operator splitting technique the proposed iteration algorithm is proved to be convergent under certain conditions theoretically and numerically experimental results on both synthetic and real polsar data demonstrate that the proposed method can effectively reduce speckle noise and meanwhile better preserve the details and the repetitive structures such as textures and edges and the polarimetric scattering characteristics compared with the other methods

multiplicative noise removal using variable splitting and constrained optimization multiplicative noise also known as speckle noise models are central to the study of coherent imaging systems such as synthetic aperture radar and sonar and ultrasound and laser imaging these models introduce two additional layers of difficulties with respect to the standard gaussian additive noise scenario 1 the noise is multiplied by rather than added to the original image 2 the noise is not gaussian with rayleigh and gamma being commonly used densities these two features of multiplicative noise models preclude the direct application of most state of the art algorithms which are designed for solving unconstrained optimization problems where the objective has two terms a quadratic data term log likelihood reflecting the additive and gaussian nature of the noise plus a convex possibly nonsmooth regularizer e g a total variation or wavelet based regularizer prior in this paper we address these difficulties by 1 converting the multiplicative model into an additive one by taking logarithms as proposed by some other authors 2 using variable splitting to obtain an equivalent constrained problem and 3 dealing with this optimization problem using the augmented lagrangian framework a set of experiments shows that the proposed method which we name midal multiplicative image denoising by augmented lagrangian yields state of the art results both in terms of speed and denoising performance

an improved variational model for denoising magnetic resonance images abstract this paper presents an improved variational model for denoising mr images degraded by rician noise an additional data fidelity based on image gradient is introduced which can retain more details and edges in this model an automatic estimation of standard deviation of rician noise is designed to enhance the quality of denoised results the alternating direction method of multipliers admm is adopted to implement numerical schedule and gains the robust solution experimental results demonstrate that the proposed method is efficient and has better denoising capability than the state of the art models

a variational approach to removing multiplicative noise this paper focuses on the problem of multiplicative noise removal we draw our inspiration from the modeling of speckle noise by using a map estimator we can derive a functional whose minimizer corresponds to the denoised image we want to recover although the functional is not convex we prove the existence of a minimizer and we show the capability of our model on some numerical examples we study the associated evolution problem for which we derive existence and uniqueness results for the solution we prove the convergence of an implicit scheme to compute the solution

a new multiplicative denoising variational model based on m th root transformation in coherent imaging systems such as the synthetic aperture radar sar the observed images are contaminated by multiplicative noise due to the edge preserving feature of the total variation tv variational models with tv regularization have attracted much interest in removing multiplicative noise however the fidelity term of the variational model based on maximum a posteriori estimation is not convex and so it is usually difficult to find a global solution hence the logarithmic function is used to transform the nonconvex variational model to the convex one in this paper instead of using the log we exploit the th root function to relax the nonconvexity of the variational model an algorithm based on the augmented lagrangian function which has been applied to solve the log transformed convex variational model can be applied to solve our proposed model however this algorithm requires solving a subproblem which does not have a closed form solution at each iteration hence we propose to adapt the linearized proximal alternating minimization algorithm which does not require inner iterations for solving the subproblems in addition the proposed method is very simple and highly parallelizable thus it is efficient to remove multiplicative noise in huge sar images the proposed model for multiplicative noise removal shows overall better performance than the convex model based on the log transformation

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

solution of stochastic partial differential equations using galerkin finite element techniques this paper presents a framework for the construction of galerkin approximations of elliptic boundary value problems with stochastic input data a variational formulation is developed which allows among others numerical treatment by the finite element method a theory of a posteriori error estimation and corresponding adaptive approaches based on practical experience can be utilized the paper develops a foundation for treating stochastic partial differential equations pdes which can be further developed in many directions

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

computational uncertainty analysis in multiresolution materials via stochastic constitutive theory abstract a stochastic constitutive theory is proposed in this work to propagate microstructure uncertainties in computational multiscale continuum models to bulk multiresolution material behavior ubiquitous fine resolution uncertainty sources influencing prediction of material properties based on their structures are categorized in detail and this research transmits these uncertainties to coarser material resolutions by introducing a stochastic constitutive theory deduced from volume element simulations to implement the stochastic upscaling process two advanced uncertainty quantification methods are examined statistical copula functions and random process polynomial chaos expansion both methods confront the mathematical difficulty in randomizing constitutive laws by capturing the marked correlation among constitutive parameters seen in complex materials thus the results proffer a more accurate probabilistic estimation of constitutive material behavior the contribution of this work is twofold uncertainty is propagated from heterogeneous material structure to material property via the stochastic constitutive theory and rigorous data driven mathematics are formalized to represent complicated dependence structures in multivariate statistical distributions to the authors knowledge this is the first work in multiresolution mechanics that presents an approach to computationally derive correlation functions from numerical experiments as opposed for instance to assuming one a priori the method put forth in this research though quite general is applied to a mathematical example and plastic high strength steel alloy for demonstration results include stochastic constitutive curve confidence intervals for the material stress strain response and qualitative comparisons of the two stochastic methods detailed herein

nonconvex sparse regularizer based speckle noise removal this paper focuses on the problem of speckle noise removal a new variational model is proposed for this task in the model a nonconvex regularizer rather than the classical convex total variation is used to preserve edges details of images the advantage of the nonconvex regularizer is pointed out in the sparse framework in order to solve the model a new fast iteration algorithm is designed in the algorithm to overcome the disadvantage of the nonconvexity of the model both the augmented lagrange multiplier method and the iteratively reweighted method are introduced to resolve the original nonconvex problem into several convex ones from the algorithm we can obtain restored images as well as edge indicator of the images comprehensive experiments are conducted to measure the performance of the algorithm in terms of visual evaluation and a variety of quantitative indices for the task of speckle noise removal

stochastic analysis of transport in tubes with rough walls flow and transport in tubes with rough surfaces play an important role in a variety of applications often the topology of such surfaces cannot be accurately described in all of its relevant details due to either insufficient data or measurement errors or both in such cases this topological uncertainty can be efficiently handled by treating rough boundaries as random fields so that an underlying physical phenomenon is described by deterministic or stochastic differential equations in random domains to deal with this class of problems we use a computational framework which is based on stochastic mappings to transform the original deterministic stochastic problem in a random domain into a stochastic problem in a deterministic domain the latter problem has been studied more extensively and existing analytical numerical techniques can be readily applied in this paper we employ both a generalized polynomial chaos and monte carlo simulations to solve the transformed stochastic problem we use our approach to describe transport of a passive scalar in stokes flow and to quantify the corresponding predictive uncertainty

speckle reduction via higher order total variation approach multiplicative noise also known as speckle reduction is a prerequisite for many image processing tasks in coherent imaging systems such as the synthetic aperture radar one approach extensively used in this area is based on total variation tv regularization which can recover significantly sharp edges of an image but suffers from the staircase like artifacts in order to overcome the undesirable deficiency we propose two novel models for removing multiplicative noise based on total generalized variation tgv penalty the tgv regularization has been mathematically proven to be able to eliminate the staircasing artifacts by being aware of higher order smoothness furthermore an efficient algorithm is developed for solving the tgv based optimization problems numerical experiments demonstrate that our proposed methods achieve state of the art results both visually and quantitatively in particular when the image has some higher order smoothness our methods outperform the tv based algorithms

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

total nuclear variation and jacobian extensions of total variation for vector fields we explore a class of vectorial total variation vtv measures formed as the spatial sum of a pixel wise matrix norm of the jacobian of a vector field we give a theoretical treatment that indicates that while color smearing and affine coupling bias often reported as gray scale bias are typically cited as drawbacks for vtv these are actually fundamental to smoothing vector direction i e smoothing hue and saturation in color images additionally we show that encouraging different vector channels to share a common gradient direction is equivalent to minimizing jacobian rank we thus propose total nuclear variation tnv and since nuclear norm is the convex envelope of matrix rank we argue that tnv is the optimal convex regularizer for enforcing shared directions we also propose extended jacobians which use larger neighborhoods than the conventional finite difference operator and we discuss efficient vtv optimization algorithms in simple color image denoising experiments tnv outperformed other common vtv regularizers and was further improved by using extended jacobians tnv was also competitive with the method of non local means often outperforming it by 0 25 to 2 db when using extended jacobians

random field finite elements the probabilistic finite element method pfem is formulated for linear and non linear continua with inhomogeneous random fields analogous to the discretization of the displacement field in finite element methods the random field is also discretized the formulation is simplified by transforming the correlated variables to a set of uncorrelated variables through an eigenvalue orthogonalization furthermore it is shown that a reduced set of the uncorrelated variables is sufficient for the second moment analysis based on the linear formulation of the pfem the method is then extended to transient analysis in non linear continua the accuracy and efficiency of the method is demonstrated by application to a one dimensional elastic plastic wave propagation problem and a two dimensional plane stress beam bending problem the moments calculated compare favourably with those obtained by monte carlo simulation also the procedure is amenable to implementation in deterministic fem based computer programs

a nonlinear primal dual method for total variation based image restoration we present a new method for solving total variation tv minimization problems in image restoration the main idea is to remove some of the singularity caused by the nondifferentiability of the quantity nabla u in the definition of the tv norm before we apply a linearization technique such as newton s method this is accomplished by introducing an additional variable for the flux quantity appearing in the gradient of the objective function which can be interpreted as the normal vector to the level sets of the image u our method can be viewed as a primal dual method as proposed by conn and overton a primal dual interior point method for minimizing a sum of euclidean norms preprint 1994 and andersen ph d thesis odense university denmark 1995 for the minimization of a sum of euclidean norms in addition to possessing local quadratic convergence experimental results show that the new method seems to be globally convergent

an efficient nonconvex regularization for wavelet frame and total variation based image restoration in order to improve the quality of the image many works impose explicit priors on the solution to regularize the ill posed inverse problem in this paper we propose a hybrid variational model which takes advantages of the wavelet tight frame model and the total variation model for image restoration the core of the method is a new nonconvex penalty function that is designed for efficient minimization by means of the firm thresholding and soft shrinkage operations we address the proposed optimization problem by converting it to a constrained problem with variable splitting and using the alternating direction method of multipliers numerical examples for image restoration are given to show that the proposed method outperforms some existing methods in terms of the peak signal to noise ratio and structural similarity index

efficient uncertainty quantification for permeability of three dimensional porous media through image analysis and pore scale simulations in this paper we propose an efficient coupled approach for uncertainty quantification of permeability for randomly reconstructed three dimensional 3d pore images where the porosity and two point correlations of a realistic sandstone sample are honored the joshi quiblier adler approach and karhunen loeve expansion are utilized for quick reconstruction of 3d pore images with reduced random dimensionality the eigenvalue problem for the covariance matrix of 3d intermediate gaussian random fields is solved equivalently by a kernel method then the lattice boltzmann method is adopted to simulate fluid flow in reconstructed pore space and evaluate permeability lastly the sparse polynomial chaos expansion sparse pce integrated with a feature selection method is employed to predict permeability distributions incurred by the randomness in microscopic pore structures the feature selection process which is intended to discard redundant basis functions is carried out by the least absolute shrinkage and selection operator modified least angle regression along with cross validation the competence of our proposed approach is validated by the results from monte carlo simulation it reveals that a small number of samples is sufficient for sparse pce with feature selection to produce convincing results then we utilize our method to quantify the uncertainty of permeability under different porosities and correlation parameters it is found that the predicted permeability distributions for reconstructed 3d pore images are close to experimental measurements of berea sandstones in the literature in addition the results show that porosity and correlation length are the critical influence factors for the uncertainty of permeability

an efficient dimension adaptive uncertainty propagation approach abstract in recent years there has been a growing interest in uncertainty propagation and a wide variety of uncertainty propagation methods exist in literature in this paper an uncertainty propagation approach is developed by using high dimensional model representation hdmr and dimension reduction dr method 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 to save the computational cost a dimension adaptive version of the additive decomposition is proposed to detect the important component functions to reduce the terms the proposed method requires neither the calculation of partial derivatives of response as in commonly used taylor expansion perturbation methods nor the inversion of random matrices as in the neumann expansion method two numerical examples show the efficiency and accuracy of the proposed method

linear optimal estimation for discrete time systems with measurement delay and packet dropping this paper is concerned with the remote estimation problem for measurement delay system over a packet dropping network the packet dropping phenomenon in every communication channel is described by an independent and identically distributed bernoulli process an optimal linear filter is provided in the linear minimum mean square error sense by applying the reorganized innovation analysis approach the solution to the filter is given in terms of two riccati difference equations and one lyapunov difference equation further the infinite horizon filter is investigated under the condition of system stability in the end we supply a numerical example to show the effectiveness of our proposed estimation approach

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

fast linearized alternating direction minimization algorithm with adaptive parameter selection for multiplicative noise removal owing to the edge preserving ability and low computational cost of the total variation tv variational models with the tv regularizer have been widely investigated in the field of multiplicative noise removal the key points of the successful application of these models lie in the optimal selection of the regularization parameter which balances the data fidelity term with the tv regularizer the efficient algorithm to compute the solution in this paper we propose two fast algorithms based on the linearized technique which are able to estimate the regularization parameter and recover the image simultaneously in the iteration step of the proposed algorithms the regularization parameter is adjusted by a special discrepancy function defined for multiplicative noise the convergence properties of the proposed algorithms are proved under certain conditions and numerical experiments demonstrate that the proposed algorithms overall outperform some state of the art methods in the psnr values and computational time

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

sequential experimental design based generalised anova over the last decade surrogate modelling technique has gained wide popularity in the field of uncertainty quantification optimization model exploration and sensitivity analysis this approach relies on experimental design to generate training points and regression interpolation for generating the surrogate in this work it is argued that conventional experimental design may render a surrogate model inefficient in order to address this issue this paper presents a novel distribution adaptive sequential experimental design da sed the proposed da sed has been coupled with a variant of generalised analysis of variance g anova developed by representing the component function using the generalised polynomial chaos expansion moreover generalised analytical expressions for calculating the first two statistical moments of the response which are utilized in predicting the probability of failure have also been developed the proposed approach has been utilized in predicting probability of failure of three structural mechanics problems it is observed that the proposed approach yields accurate and computationally efficient estimate of the failure probability

primal dual algorithm based on gauss seidel scheme with application to multiplicative noise removal due to the strong edge preserving ability and low computational cost the total variation tv regularization has been developed as one promising approach to solve the multiplicative denoising problem in recent years many efficient algorithms have been proposed for computing the numerical solution of tv based convex variational models among these methods the linearized augmented lagrangian algorithm alm and the primal dual hybrid gradient pdhg algorithm are two of the most effective and most widely used techniques in this paper inspired by the connection of the alm and pdhg algorithms we develop an improved primal dual algorithm for multiplicative noise removal in the proposed algorithm an auxiliary variable which is updated by the gauss seidel scheme is introduced to accelerate the original primal dual framework the global convergence property of the proposed algorithm is also investigated numerical experiments on the multiplicative denoising show that the proposed algorithm outperforms the current state of the art 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

parallel domain decomposition methods for stochastic elliptic equations we present parallel schwarz type domain decomposition preconditioned recycling krylov subspace methods for the numerical solution of stochastic elliptic problems whose coefficients are assumed to be a random field with finite variance karhunen loeve kl expansion and double orthogonal polynomials are used to reformulate the stochastic elliptic problem into a large number of related but uncoupled deterministic equations the key to an efficient algorithm lies in recycling computed subspaces based on a careful analysis of the kl expansion we propose and test a grouping algorithm that tells us when to recycle and when to recompute some components of the expensive computation we show theoretically and experimentally that the schwarz preconditioned recycling gmres method is optimal for the entire family of linear systems a fully parallel implementation is provided and scalability results are reported in the paper

efficient solvers for a linear stochastic galerkin mixed formulation of diffusion problems with random data we introduce a stochastic galerkin mixed formulation of the steady state diffusion equation and focus on the efficient iterative solution of the saddle point systems obtained by combining standard finite element discretizations with two distinct types of stochastic basis functions so called mean based preconditioners based on fast solvers for scalar diffusion problems are introduced for use with the minimum residual method we derive eigenvalue bounds for the preconditioned system matrices and report on the efficiency of the chosen preconditioning schemes with respect to all the discretization parameters

h div preconditioning for a mixed finite element formulation of the diffusion problem with random data we study h div preconditioning for the saddle point systems that arise in a stochastic galerkin mixed formulation of the steady state diffusion problem with random data the key ingredient is a multigrid v cycle for an h div operator with random weight function acting on a certain tensor product space of random fields with finite variance we build on the arnold falk winther multigrid algorithm presented in 1997 by varying the spatial discretization from grid to grid whilst keeping the stochastic discretization fixed we extend the deterministic analysis to accommodate the modified h div operator and establish spectral equivalence bounds with a new multigrid v cycle operator that are independent of the spatial and stochastic discretization parameters we implement multigrid within a block diagonal preconditioner for the full saddle point problem derive eigenvalue bounds for the preconditioned system matrices and investigate the impact of all the discretization parameters on the convergence rate of preconditioned minre s

signal recovery with certain involved convex data fidelity constraints this paper proposes an optimization framework that can efficiently deal with convex data fidelity constraints onto which the metric projections are difficult to compute although such an involved data fidelity constraint is expected to play an important role in signal recovery under non gaussian noise contamination the said difficulty precludes existing algorithms from solving convex optimization problems with the constraint to resolve this dilemma we introduce a fixed point set characterization of involved data fidelity constraints based on a certain computable quasi nonexpansive mapping this characterization enables us to mobilize the hybrid steepest descent method to solve convex optimization problems with such a constraint the proposed framework can handle a variety of involved data fidelity constraints in a unified manner without geometric approximation to them in addition it requires no computationally expensive procedure such as operator inversion and inner loop as applications of the proposed framework we provide image restoration under several types of non gaussian noise contamination with illustrative examples

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

spatially adapted total variation model to remove multiplicative noise multiplicative noise removal based on total variation tv regularization has been widely researched in image science in this paper inspired by the spatially adapted methods for denoising gaussian noise we develop a variational model which combines the tv regularizer with local constraints it is also related to a tv model with spatially adapted regularization parameters the automated selection of the regularization parameters is based on the local statistical characteristics of some random variable the corresponding subproblem can be efficiently solved by the augmented lagrangian method numerical examples demonstrate that the proposed algorithm is able to preserve small image details whereas the noise in the homogeneous regions is sufficiently removed as a consequence our method yields better denoised results than those of the current state of the art methods with respect to the signal to noise ratio values

multi element stochastic reduced basis methods this paper presents multi element stochastic reduced basis methods me srbms for solving linear stochastic partial differential equations in me srbms the domain of definition of the random inputs is decomposed into smaller subdomains or random elements stochastic reduced basis methods srbms are employed in each random element to evaluate the response statistics these elemental statistics are assimilated to compute the overall statistics the effectiveness of the method is demonstrated by solving the stochastic steady state heat transfer equation on two geometries involving different types of boundary conditions numerical studies are conducted to investigate the h convergence rates of global and local preconditioning strategies

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

three dimensional stochastic finite element method for elasto plastic bodies a new stochastic finite element method sfem is formulated for three dimensional softening elasto plastic bodies with random material properties the method is based on the karhunen loeve and polynomial chaos expansions and able to efficiently estimate complete probabilistic characteristics of the response such as moments or pdfs to reduce the computational complexity in the three dimensional setting two alterations are made with respect to the two dimensional sfem proposed earlier by the authors first a variability preserving modification of the karhunen loeve expansion is rigorously derived and applied in the stochastic discretization of random fields representing material properties second an efficient algorithm for parallel processing is developed with time consumption being the same order as for an ordinary fem rendering the proposed sfem an effective alternative to monte carlo simulation the applicability of the proposed method to stochastic analysis of strain localization is examined using monte carlo simulation then it is applied to a fault formation problem which is a recent concern of earthquake engineering ground surface layers are modelled by a softening elasto plastic body and the evolution of probabilistic characteristics of the rupture process is analysed in detail some practical observations are made regarding the nature of the fault formation from the stochastic viewpoint copyright 2001 john wiley sons ltd

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

gradient histogram estimation and preservation for texture enhanced image denoising natural image statistics plays an important role in image denoising and various natural image priors including gradient based sparse representation based and nonlocal self similarity based ones have been widely studied and exploited for noise removal in spite of the great success of many denoising algorithms they tend to smooth the fine scale image textures when removing noise degrading the image visual quality to address this problem in this paper we propose a texture enhanced image denoising method by enforcing the gradient histogram of the denoised image to be close to a reference gradient histogram of the original image given the reference gradient histogram a novel gradient histogram preservation ghp algorithm is developed to enhance the texture structures while removing noise two region based variants of ghp are proposed for the denoising of images consisting of regions with different textures an algorithm is also developed to effectively estimate the reference gradient histogram from the noisy observation of the unknown image our experimental results demonstrate that the proposed ghp algorithm can well preserve the texture appearance in the denoised images making them look more natural

linear estimation for measurement delay systems with periodic coefficients and multiplicative noise this technical note is concerned with the remote estimation problem for measurement delay systems with multiplicative noise and periodic coefficients the multiplicative noise in every communication channel is described by an independent and identically distributed i i d bernoulli process the quality of the estimation achieved by an admissible filter is measured by a performance criterion described by the cesaro limit of the mean square of the deviation between the remote signal and the estimated signal the filter is designed in terms of the unique periodic solution of a lyapunov equation together with the periodic stabilizing solution of a riccati equation in the end we supply a numerical example to show the effectiveness of our proposed estimation approach

probability density functions of hydraulic head and velocity in three dimensional heterogeneous porous media 1 in this study we assess probability density functions of hydraulic heads and specific discharges in three dimensional bounded heterogeneous porous media by monte carlo mc simulation we discuss their empirical shapes and demonstrate that the intuitive use of obvious information on boundedness leads to parametric distribution functions which fit surprisingly well on the basis of statistical moments of hydraulic heads and velocities up to fourth order we discuss the spatial dependence of the empirical distributions and their dependence on the variance of log conductivity comparison of the first and second central moment to the results from classical numerical first order second moment fosm analysis reveals that fosm predicts these moments surprisingly close for hydraulic heads on the basis of this fact we demonstrate that fitting the chosen parametric distributions for hydraulic heads to fosm moments is promising for the sake of estimating exceedance probabilities our mc scenarios vary in variance of log conductivity 0 125 to 5 0 in the type of multivariate dependence in correlation scale and types of boundary conditions our study illustrates that in contrast to the common assumption fosm is a reasonable choice for evaluating multivariate and univariate moments for heads if used in conjunction with additional information on distribution shapes in the absence of utilizable additional information we demonstrate that second moment methods are mostly inadequate for assessing distributions accurately significant deviations from gaussian distributions occurred for discharge components even at a variance of log conductivity as low as 0 125 and we found that the distributions of transverse discharge components are extremely fat tailed the observed non gaussianity questions the results of approximate approaches in solute flux and dispersion studies where velocity fields are assumed to be multi gaussian and then directly represented by or generated from their covariances the main implication is to apply more accurate schemes such as exact non local methods extensive mc or higher order stochastic galerkin approaches and to include higher order moments at least if no additional assumptions on the shape of distributions are available or justifiable

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

a new nonconvex approach for image restoration with gamma noise abstract in this paper a new total generalized variational tgv model for restoring images with multiplicative noise is proposed which contains a nonconvex fidelity term and a tgv term we use a difference of convex functions algorithm dca to deal with the proposed model for multiplicative noise removal there exist many models and algorithms most of which focus on convex approximation so that numerical algorithms with guaranteed convergence can be designed unlike these algorithms we use the dca algorithm to remove multiplicative noise by numerical experiments it is shown that the proposed approach leads to a better solution compared with the gradient projection algorithm for solving the classic multiplicative noise removal models we prove that the sequence generated by the dca algorithm converges to a stationary point which satisfies the first order optimality condition finally we demonstrate the performance of our whole scheme by numerical examples a comparison with other methods is provided as well numerical results demonstrate that the proposed algorithm significantly outperforms some previous methods for multiplicative gamma noise removal

multiplicative noise removal via a learned dictionary multiplicative noise removal is a challenging image processing problem and most existing methods are based on the maximum a posteriori formulation and the logarithmic transformation of multiplicative denoising problems into additive denoising problems sparse representations of images have shown to be efficient approaches for image recovery following this idea in this paper we propose to learn a dictionary from the logarithmic transformed image and then to use it in a variational model built for noise removal extensive experimental results suggest that in terms of visual quality peak signal to noise ratio and mean absolute deviation error the proposed algorithm outperforms state of the art methods

efficient stochastic galerkin methods for random diffusion equations we discuss in this paper efficient solvers for stochastic diffusion equations in random media we employ generalized polynomial chaos gpc expansion to express the solution in a convergent series and obtain a set of deterministic equations for the expansion coefficients by galerkin projection although the resulting system of diffusion equations are coupled we show that one can construct fast numerical methods to solve them in a decoupled fashion the methods are based on separation of the diagonal terms and off diagonal terms in the matrix of the galerkin system we examine properties of this matrix and show that the proposed method is unconditionally stable for unsteady problems and convergent for steady problems with a convergent rate independent of discretization parameters numerical examples are provided for both steady and unsteady random diffusions to support the analysis

finite elements for stochastic media problems we consider discretization methods for stochastic partial differential equations which are used to model media with stochastic properties the wick product formulation of the stochastic pde is used and it is shown how it may be discretized both spatially and in the stochastic part a stochastic variational principle serves as the guideline for the discretization procedure it is shown that both the projection onto the wiener chaos and the hermite transform lead to the same fully discrete equations the application examples show that there are other advantageous bases besides the karhunen loeve expansion and that the methods may be used in a variety of settings

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

an alternating direction algorithm for total variation reconstruction of distributed parameters augmented lagrangian variational formulations and alternating optimization have been adopted to solve distributed parameter estimation problems the alternating direction method of multipliers admm is one of such formulations optimization methods very recently the number of applications of the admm or variants of it to solve inverse problems in image and signal processing has increased at an exponential rate the reason for this interest is that admm decomposes a difficult optimization problem into a sequence of much simpler problems in this paper we use the admm to reconstruct piecewise smooth distributed parameters of elliptical partial differential equations from noisy and linear blurred observations of the underlying field the distributed parameters are estimated by solving an inverse problem with total variation tv regularization the proposed instance of the admm solves in each iteration an and a decoupled optimization problems an operator splitting is used to simplify the treatment of the tv regularizer avoiding its smooth approximation and yielding a simple yet effective admm reconstruction method compared with previously proposed approaches the competitiveness of the proposed method with respect to the state of the art is illustrated in simulated 1 d and 2 d elliptical equation problems which are representative of many real applications

restoration of poissonian images using alternating direction optimization much research has been devoted to the problem of restoring poissonian images namely for medical and astronomical applications however the restoration of these images using state of the art regularizers such as those based upon multiscale representations or total variation is still an active research area since the associated optimization problems are quite challenging in this paper we propose an approach to deconvolving poissonian images which is based upon an alternating direction optimization method the standard regularization or maximum a posteriori map restoration criterion which combines the poisson log likelihood with a nonsmooth convex regularizer log prior leads to hard optimization problems the log likelihood is nonquadratic and nonseparable the regularizer is nonsmooth and there is a nonnegativity constraint using standard convex analysis tools we present sufficient conditions for existence and uniqueness of solutions of these optimization problems for several types of regularizers total variation frame based analysis and frame based synthesis we attack these problems with an instance of the alternating direction method of multipliers admm which belongs to the family of augmented lagrangian algorithms we study sufficient conditions for convergence and show that these are satisfied either under total variation or frame based analysis and synthesis regularization the resulting algorithms are shown to outperform alternative state of the art methods both in terms of speed and restoration accuracy

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

alternating minimization algorithm for speckle reduction with a shifting technique speckles multiplicative noise in synthetic aperture radar sar make it difficult to interpret the observed image due to the edge preserving feature of total variation tv variational models with tv regularization have attracted much interest in reducing speckles algorithms based on the augmented lagrangian function have been proposed to efficiently solve speckle reduction variational models with tv regularization however these algorithms require inner iterations or inverses involving the laplacian operator at each iteration in this paper we adapt tseng s alternating minimization algorithm with a shifting technique to efficiently remove the speckle without any inner iterations or inverses involving the laplacian operator the proposed method is very simple and highly parallelizable therefore it is very efficient to despeckle huge size sar images numerical results show that our proposed method outperforms the state of the art algorithms for speckle reduction variational models with a tv regularizer in terms of central processing unit time

uncertainty quantification methodology for hyperbolic systems with application to blood flow in arteries abstract we present a stochastic finite volume ader sfv ader methodology for uncertainty quantification uq in the general framework of systems of hyperbolic balance laws with uncertainty in parameters the resulting method is weakly intrusive meaning that deterministic solvers need minor modifications to include random contributions and has no theoretical accuracy barrier an illustration of the second order version for the viscous burgers equation with uncertain viscosity coefficient is first given in detail under different deterministic initial conditions the attainment of the theoretically expected convergence rate is demonstrated empirically results and features of our scheme are discussed we then extend the sfv ader method to a non linear hyperbolic system with source terms that models one dimensional blood flow in arteries assuming uncertainties in a set of parameters of the problem results are then compared with measurements in a well defined 1 1 experimental replica of the network of largest arteries in the human systemic circulation for a 99 confidence level severe influence of parameter fluctuations is seen in pressure profiles while secondary effects on flow rate profiles become visible particularly in terminal branches results suggest that the proposed methodology could successfully be applied to other physical problems involving hyperbolic balance laws

a new twist two step iterative shrinkage thresholding algorithms for image restoration iterative shrinkage thresholding 1st algorithms have been recently proposed to handle a class of convex unconstrained optimization problems arising in image restoration and other linear inverse problems this class of problems results from combining a linear observation model with a nonquadratic regularizer e g total variation or wavelet based regularization it happens that the convergence rate of these 1st algorithms depends heavily on the linear observation operator becoming very slow when this operator is ill conditioned or ill posed in this paper we introduce two step 1st twist algorithms exhibiting much faster convergence rate than 1st for ill conditioned problems for a vast class of nonquadratic convex regularizers lscrp norms some besov norms and total variation we show that twist converges to a minimizer of the objective function for a given range of values of its parameters for noninvertible observation operators we introduce a monotonic version of twist mtwist although the convergence proof does not apply to this scenario we give experimental evidence that mtwist exhibits similar speed gains over ist the effectiveness of the new methods are experimentally confirmed on problems of image deconvolution and of restoration with missing samples

a variational model for polsar data speckle reduction based on the wishart distribution in this paper we propose a variational model for polarimetric synthetic aperture radar polsar data speckle reduction which is based on the complex wishart distribution of the covariance or coherency matrix and multichannel total variation tv regularization defined for complex valued matrices by assuming the tv regularization to be a prior and taking the statistical distribution of the covariance matrix in each resolution element into account the variational model for polsar covariance data speckle suppression named wistv c is derived from the maximum a posteriori estimate a similar variational model for polsar coherency data speckle reduction named wistv t is also obtained as far as we know this is the first variational model for the whole polsar covariance or coherency matrix data despeckling since the model is nonconvex a convex relaxation iterative algorithm is designed to solve the variational problem based on the variable splitting and alternating minimization techniques experimental results on both simulated and real polsar data demonstrate that the proposed approach notably removes speckles in the extended uniform areas and meanwhile better preserves the spatial resolution the details such as edges and point scatterers and the polarimetric scattering characteristics compared with other methods

filtered stochastic galerkin methods for hyperbolic equations abstract uncertainty quantification for nonlinear hyperbolic problems becomes a challenging task in the vicinity of shocks standard intrusive methods such as stochastic galerkin sg lead to oscillatory solutions and can result in non hyperbolic moment systems the intrusive polynomial moment ipm method guarantees hyperbolicity but comes at higher numerical costs in this paper we filter the generalized polynomial chaos gpc coefficients of the sg approximation which allows a numerically cheap reduction of oscillations the derived filter is based on lasso regression which sets small gpc coefficients of high order to zero we adaptively and automatically choose the filter strength to obtain a zero valued highest order moment the filtered sg method is tested for burgers and the euler equations results show a reduction of oscillations at shocks which leads to an improved approximation of expectation values and the variance compared to sg and ipm

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

a convex relaxation method for computing exact global solutions for multiplicative noise removal we propose a convex relaxation technique for computing global solutions for the nonconvex multiplicative noise model the method is based on functional lifting by introducing an additional dimension we employ a primal dual based gradient type algorithm in numerical implementations to overcome the nondifferentiability of the total variation term numerical results show that our algorithm is highly efficient furthermore global solutions of the original model can be obtained with no dependence on the initial guess

a modified stochastic perturbation method for stochastic hyperbolic heat conduction problems abstract the stochastic hyperbolic heat conduction problem is addressed the random field is approximated by using the karhunen loeve expansion truncated after a finite numbers of terms and a modified stochastic perturbation method mspm is proposed for the stochastic hyperbolic heat conduction problems the mspm is based on the stochastic perturbation theory and can provide the mean and covariance of the random temperature including some third and forth order terms compared with the original stochastic perturbation method the mspm is more accurate and is more convenient for the numerical implementation and hence is more applicable for the stochastic hyperbolic heat conduction problem two numerical examples demonstrate that the proposed method can provide very accurate results and can be used for the stochastic hyperbolic heat conduction problems

impact of geometric uncertainty on hemodynamic simulations using machine learning abstract in the cardiovascular system blood flow rates blood velocities and blood pressures can be modeled using the navier stokes equations inputs to the system are typically uncertain such as a the geometry of the arterial tree b clinically measured blood pressure and viscosity c boundary resistances among others due to a large number of such parameters efficient quantification of uncertainty in solution fields in this multi parameter space is challenging we use an adaptive stochastic collocation method to quantify the impact of uncertainty in geometry in patient specific models we develop a novel subdivision method to define the stochastic space of geometries to accelerate convergence and make the problem tractable we use a machine learning approach to approximate the simulation based solution towards this a reduced order model of the navier stokes equations is developed using a segmental resistance analog boundary conditions ratio of pressure to flow using an offline database of pre computed solutions we compute a map rule from the features to solution fields we achieve significant speed up of a few orders of magnitude by approximating the simulation based solution using a machine learning predictor a bootstrap aggregated decision tree was found to be the best predictor among many candidate regressors correlation coefficient of training set was 0 94 we demonstrate stochastic space convergence using the adaptive stochastic collocation method and also show robustness to the choice of geometry parameterization the sensitivities to geometry obtained using machine learning had a correlation coefficient of 0 92 with the values obtained using finite element simulations segments with significant disease in the larger arteries had the highest sensitivities terminal segments are more sensitive to dilation and proximal healthy segments are more sensitive to erosion sensitivity to geometry is highest when geometric resistance is comparable to net downstream resistance

multi resolution analysis of wiener type uncertainty propagation schemes a multi resolution analysis mra is applied to an uncertainty propagation scheme based on a generalized polynomial chaos pc representation the mra relies on an orthogonal projection of uncertain data and solution variables onto a multi wavelet basis consisting of compact piecewise smooth polynomial functions the coefficients of the expansion are computed through a galerkin procedure the mra scheme is applied to the simulation of the lorenz system having a single random parameter the convergence of the solution with respect to the resolution level and expansion order is investigated in particular results are compared to two monte carlo sampling strategies demonstrating the superiority of the mra for more complex problems however the mra approach may require excessive cpu times adaptive methods are consequently developed in order to overcome this drawback two approaches are explored the first is based on adaptive refinement of the multi wavelet basis while the second is based on adaptive block partitioning of the space of random variables computational tests indicate that the latter approach is better suited for large problems leading to a more efficient flexible and parallelizable scheme

covariance kernel representations of multidimensional second order stochastic processes the dynamics of stationary stochastic processes in space is not exactly analogous to that of stationary stochastic processes in the time domain this is due to the unilateral nature of the time series that is only influenced by past values as opposed to the dependence in all directions of the spatial process in this work we unfold the connection that exits between the covariance kernel of a multi dimensional second order autoregressive random process and its underlying discrete random dynamical system starting from a discrete random dynamical system we show that the random process satisfying that system is governed by the modified helmholtz equation in the continuous limit we establish the dependence of the correlation constant on the grid size of the discretization we also show that the random forcing term in the continuous case turns out to be a white noise process a number of covariance functions are worked out for simple and more complex geometrical domains with various boundary conditions in multi dimensions we use both the discrete and the continuous systems in our computations

sampling free linear bayesian update of polynomial chaos representations we present a fully deterministic approach to a probabilistic interpretation of inverse problems in which unknown quantities are represented by random fields or processes described by possibly non gaussian distributions the description of the introduced random fields is given in a white noise framework which enables us to solve the stochastic forward problem through galerkin projection onto polynomial chaos with the help of such a representation the probabilistic identification problem is cast in a polynomial chaos expansion setting and the baye s linear form of updating by introducing the hermite algebra this becomes a direct purely algebraic way of computing the posterior which is comparatively inexpensive to evaluate in addition we show that the well known kalman filter is the low order part of this update the proposed method is here tested on a stationary diffusion equation with prescribed source terms characterised by an uncertain conductivity parameter which is then identified from limited and noisy data obtained by a measurement of the diffusing quantity

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

noise level estimation for natural images based on scale invariant kurtosis and piecewise stationarity noise level estimation is crucial in many image processing applications such as blind image denoising in this paper we propose a novel noise level estimation approach for natural images by jointly exploiting the piecewise stationarity and a regular property of the kurtosis in bandpass domains we design a k means based algorithm to adaptively partition an image into a series of non overlapping regions each of whose clean versions is assumed to be associated with a constant but unknown kurtosis throughout scales the noise level estimation is then cast into a problem to optimally fit this new kurtosis model in addition we develop a rectification scheme to further reduce the estimation bias through noise injection mechanism extensive experimental results show that our method can reliably estimate the noise level for a variety of noise types and outperforms some state of the art techniques especially for non gaussian noises

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

proximal linearized alternating direction method for multiplicative denoising multiplicative noise naturally appears in coherent imaging systems such as synthetic aperture radar due to the edge preserving feature of total variation tv variational models with tv regularization show good performance for multiplicative denoising recently proposed augmented lagrangian frameworks with the gauss seidel method are efficient for solving multiplicative denoising variational models with tv but recent algorithms based on the augmented lagrangian require inner iterations or an inverse involving the laplacian operator at each iteration in this paper we propose efficient algorithms for tv regularized convex variational models our proposed algorithms do not require any inner iteration or the inverse involving the laplacian operator by using a linearization scheme and a proximal function we establish global convergence under a lipschitz continuous assumption on the gradient of the fidelity term when a correction step is used numerical results show that our proposed algorithms overall

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

exact bernoulli euler dynamic stiffness matrix for a range of tapered beams bernoulli euler theory and bessel functions are used to obtain explicit expressions for the exact dynamic stiffnesses for axial torsional and flexural vibrations of any beam which is tapered such that a varies as yn and gj and i both vary as y n 2 where a gj and i have their usual meanings y cx l 1 c is a constant such that c 1 l is the length of the beam and x is the distance from one end of the beam numerical checks give better than seven figure agreement with the stiffnesses obtained by extrapolation from stepped beams with 400 and 500 uniform elements a procedure is given for calculating the number of natural frequencies exceeded by any trial frequency when the ends of the member are clamped this enables an existing algorithm to be used to obtain the natural frequencies of structures which contain tapered members

linear quadratic regulation problem for discrete time systems with multi channel multiplicative noise abstract this paper investigates the linear quadratic regulation lqr problem for discrete time systems with multiplicative noise multiplicative noise is usually assumed to be a scalar in existing literature works motivated by recent applications of networked control systems and mimo communication technology we consider multi channel multiplicative noise represented by a diagonal matrix we first show that the finite horizon lqr problem can be solved using a generalized riccati equation we then prove the convergence of the generalized riccati equation under the conditions of stabilization and exact observability and obtain the solution to the infinite horizon lqr problem finally we provide a numerical example to demonstrate the proposed approach

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

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

a convex total generalized variation regularized model for multiplicative noise and blur removal multiplicative noise and blur corruptions usually happen in coherent imaging systems such as the synthetic aperture radar total variation regularized multiplicative noise and blur removal models have been widely studied in the literature which can preserve sharp edges of the recovered images however the images recovered from the total variation based models usually suffer from staircase effects to overcome this deficiency we propose a total generalized variation regularized convex optimization model the resulting objective function involves the total generalized variation regularization term the map based data fitting term and a quadratic penalty term which is based on the statistical property of the noise indeed the map estimated data fitting term in the multiplicative noise and blur removal model is nonconvex under a mild condition the quadratic penalty term makes the objective function convex a primal dual algorithm is developed to solve the minimization problem numerical experiments show that the proposed method outperforms some state of the art methods

a hybrid optimization method for multiplicative noise and blur removal the main contribution of this paper is to propose a new hybrid optimization method for the multiplicative noise and blur removal problem a degraded image can often be recovered efficiently by minimizing an objective function which consists of a data fidelity term and a regularization term in the paper we apply the quadratic penalty function method combined with the alternating direction method to minimize the corresponding objective function numerical experiments are presented to demonstrate the effectiveness of the proposed method experimental results illustrate the state of the art performance of the proposed method to handle multiplicative noise and blur removal problem