the closure properties on real numbers under limits and computable operators in effective analysis various classes of real numbers are discussed for example the classes of computable semi computable weakly computable recursively approximable real numbers etc all these classes correspond to some kind of weak computability of the real numbers in this paper we discuss mathematical closure properties of these classes under the limit effective limit and computable function among others we show that the class of weakly computable real numbers is not closed under effective limit and partial computable functions while the class of recursively approximable real numbers is closed under effective limit and partial computable functions

randomness and recursive enumerability one recursively enumerable real alpha dominates another one beta if there are nondecreasing recursive sequences of rational numbers a n n in omega approximating alpha and b n n in omega approximating beta and a positive constant c such that for all n c alpha a n geq beta b n see r m solovay draft of a paper or series of papers on chaitin s work manuscript ibm thomas j watson research center yorktown heights ny 1974 p 215 and g j chaitin ibm j res develop 21 1977 pp 350 359 we show that every recursively enumerable random real dominates all other recursively enumerable reals we conclude that the recursively enumerable random reals are exactly the omega numbers g j chaitin ibm j res develop 21 1977 pp 350 359 second we show that the sets in a universal martin lof test for randomness have random measure and every recursively enumerable random number is the sum of the measures represented in a universal martin lof test

fuzzy stochastic fem based homogenization framework for materials with polymorphic uncertainties in the microstructure uncertainties in the macroscopic response of heterogeneous materials result from two sources the natural variability in the microstructure s geometry and the lack of sufficient knowledge regarding the microstructure the first type of uncertainty is denoted aleatoric uncertainty and may be characterized by a known probability density function the second type of uncertainty is denoted epistemic uncertainty this kind of uncertainty cannot be described using probabilistic methods models considering both sources of uncertainties are called polymorphic in the case of polymorphic uncertainties some combination of stochastic methods and fuzzy arithmetic should be used thus in the current work we examine a fuzzy stochastic finite element method based homogenization framework for materials with random inclusion sizes we analyze an experimental radii distribution of inclusions and develop a stochastic representative volume element the stochastic finite element method is used to obtain the material response in the case of random inclusion radii due to unavoidable noise in experimental data an insufficient number of samples and limited accuracy of the fitting procedure the radii distribution density cannot be obtained exactly thus it is described in terms of fuzzy location and scale parameters the influence of fuzzy input on the homogenized stress measures is analyzed

a stochastic galerkin method for first order quasilinear hyperbolic systems with uncertainty this paper is concerned with generalized polynomial chaos gpc approximation for first order quasilinear hyperbolic systems with uncertainty the one dimensional 1d hyperbolic system is first symmetrized with the aid of left eigenvector matrix of the jacobian matrix then the gpc stochastic galerkin method is applied to derive a provably symmetrically hyperbolic equations for the gpc expansion coefficients the resulting deterministic gpc galerkin system is discretized by a path conservative finite volume weno scheme in space and a third order total variation diminishing runge kutta method in time the method is further extended to two dimensional 2d quasilinear hyperbolic system with uncertainty where the symmetric hyperbolicity of the one dimensional gpc galerkin system is carried over via an operator splitting technique several numerical experiments are conducted to demonstrate the accuracy and effectiveness of the proposed gpc stochastic galerkin method

variance reduction through robust design of boundary conditions for stochastic hyperbolic systems of equations we consider a hyperbolic system with uncertainty in the boundary and initial data our aim is to show that dierent boundary conditions gives dierent convergence rates of the variance of the solution this means that we can with the same knowledge of data get a more or less accurate description of the uncertainty in the solution a variety of boundary conditions are compared and both analytical and numerical estimates of the variance of the solution is presented as applications we study the eect of this technique on maxwell s equations as well as on a subsonic outow boundary for the euler equations

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

computing interface motion in compressible gas dynamics a hamilton jacobi level set formulation of the equations of motion for propagating interfaces has been introduced recently by osher and sethian this formulation allows fronts to self intersect develop singularities and change topology the numerical algorithms based on this approach handle topological merging and breaking naturally work in any number of space dimensions and do not require that the moving front be written as a function instead the moving front is embedded as a particular level set in a fixed domain partial differential equation numerical techniques borrowed from hyperbolic conservation laws are then used to accurately capture the complicated surface motion that satisfies the global entropy condition for propagating fronts given by sethian in this paper we analyze the coupling of this level set formulation to a system of conservation laws for compressible gas dynamics we study both conservative and non conservative differencing of the level set function and compare the two approaches to illustrate the capability of the method we study the compressible rayleigh taylor and kelvin helmholtz instabilities for air air and air helium boundaries we perform numerical convergence studies of the method over a range of parameters and analyze the accuracy of this approach applied to these problems

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

a lattice boltzmann scheme for incompressible multiphase flow and its application in simulation of rayleigh taylor instability in this paper we propose a new lattice boltzmann scheme for simulation of multiphase flow in the nearly incompressible limit the new scheme simulates fluid flows based on distribution functions the interfacial dynamics such as phase segregation and surface tension are modeled by incorporating molecular interactions the lattice boltzmann equations are derived from the continuous boltzmann equation with appropriate approximations suitable for incompressible flow the numerical stability is improved by reducing the effect of numerical errors in calculation of molecular interactions an index function is used to track interfaces between different phases simulations of the two dimensional rayleigh taylor instability yield satisfactory results the interface thickness is maintained at 3 4 grid spacings throughout simulations without artificial reconstruction steps

multi fidelity non intrusive polynomial chaos based on regression abstract in this paper we present a multi fidelity mf extension of non intrusive polynomial chaos based on regression point collocation for uncertainty quantification purposes the proposed method uses the principle of a global correction function from a previous similar method that uses spectral projection to estimate the coefficients due to its usage of regression to estimate the coefficients the present method offers high flexibility in the sampling and generation of the polynomial basis the method takes advantage of a nested sampling plan to create the samples for the low fidelity lf and correction expansions where the high fidelity hf samples are a subset of the lf ones to build the polynomial basis a total order or hyperbolic truncation strategy is used with a highly flexible combination of the lf and correction polynomial expansions the method is demonstrated on some artificial test problems and aerodynamic problems of the euler flow around an airfoil and common three dimensional research models in order to derive the strategies for successful mf approximation the effect of the correlation and the errors between the lf and hf functions is also studied the results show that high correlation and moderately low errors are important to improve the mf approximation s accuracy on a common research model problem the mf approach with partially converged simulations as the lf samples can successfully reduce the computational cost to about 40 for similar accuracy compared to an approach using a single hf expansion

domain decomposition using spectral expansions of steklov poincare operators ii a matrix formulation we describe a matrix formulation of the iterative domain decomposition method in natarajan siam j sci comput 16 1995 pp 470 495 from one point of view this method can be regarded as a preconditioning technique for the interface schur complement operator obtained from a decomposition into nonoverlapping subdomains from another point of view it can be viewed as a method of the schwarz type for overlapping subdomains but with an overlap between the physical space in one subdomain and the leading components of the eigenspace induced by the steklov poincare operator in the complementary domain

experimental study of compressible turbulent mixing layers compressible turbulent mixing layers have been investigated experimentally using pressure mesurements schlieren photographs and velocity measurements with a two component laser doppler velocimeter system seven mixing layer cases were examined with relative mach numbers ranging from 0 40 to 1 97 which spans the region of significant compressibility effects both the spatial development and similarty of the mixing layers were considered

a reduced spectral function approach for the stochastic finite element analysis the stochastic finite element analysis of elliptic type partial differential equations with non gaussian random fields are considered a novel approach by projecting the solution of the discretized equation into a reduced finite dimensional orthonormal vector basis is investigated it is shown that the solution can be obtained using a finite series comprising functions of random variables and orthonormal vectors these functions called as the spectral functions can be expressed in terms of the spectral properties of the deterministic coefficient matrices arising due to the discretization of the governing partial differential equation based on the projection in a reduced orthonormal vector basis a galerkin error minimization approach is proposed the constants appearing in the galerkin method are solved from a system of linear equations which has much smaller dimension compared to the original discretized equation a hybrid analytical and simulation based computational approach is proposed to obtain the moments and probability density function of the solution the method is illustrated using the stochastic nanomechanics of a zinc oxide zno nanowire deflected under the atomic force microscope afm tip the results are compared with the results obtained using direct monte carlo simulation classical neumann expansion and polynomial chaos approach for different correlation lengths and strengths of randomness

cost reduction of stochastic galerkin method by adaptive identification of significant polynomial chaos bases for elliptic equations abstract one widely used and computationally efficient method for uncertainty quantification using spectral stochastic finite element is the stochastic galerkin method here the solution is represented in polynomial chaos expansion and the residual of the discretized governing equation is projected on the polynomial chaos bases this results in a system of deterministic algebraic equations with the polynomials chaos coefficients as unknown however one impediment for its large scale applications is the curse of dimensionality that is the exponential growth of the number of polynomial chaos bases with the stochastic dimensionality and degree of expansion here for a stochastic elliptic problem an adaptive selection of polynomial chaos bases is proposed accordingly during the first few iterations in the preconditioned conjugate gradient method for solving the system of linear algebraic equations the chaos bases with maximal contribution in an appropriately defined metric to the solution are first identified subsequently only these bases are retained for further iterations until convergence is achieved using numerical studies a three times cost saving over the existing method is observed furthermore for enhancing the computational cost gain the stochastic galerkin method is reformulated as a generalized sylvester equation this step allowed efficient usage of the sparsity of moments of product of polynomial chaos bases through numerical studies on problems with large stochastic dimensionality an additional cost saving of up to one order of magnitude twenty times is observed this amounts to sixty times speedup over the existing method when adaptive selection and generalized sylvester equation formulation are used together the proposed methodology can be easily incorporated in an existing standard stochastic galerkin method solver for elliptic problems

a stochastic mixed finite element heterogeneous multiscale method for flow in porous media a computational methodology is developed to efficiently perform uncertainty quantification for fluid transport in porous media in the presence of both stochastic permeability and multiple scales in order to capture the small scale heterogeneity a new mixed multiscale finite element method is developed within the framework of the heterogeneous multiscale method hmm in the spatial domain this new method ensures both local and global mass conservation starting from a specified covariance function the stochastic log permeability is discretized in the stochastic space using a truncated karhunen loeve expansion with several random variables due to the small correlation length of the covariance function this often results in a high stochastic dimensionality therefore a newly developed adaptive high dimensional stochastic model representation technique hdmr is used in the stochastic space this results in a set of low stochastic dimensional subproblems which are efficiently solved using the adaptive sparse grid collocation method asgc numerical examples are presented for both deterministic and stochastic permeability to show the accuracy and efficiency of the developed stochastic multiscale method

forbidden information godel incompleteness theorem leaves open a way around it vaguely perceived for a long time but not clearly identified thus godel believed informal arguments can answer any math question closing this loophole does not seem obvious and involves kolmogorov complexity this is unrelated to well studied before complexity quantifications of the usual godel effects i consider extensions u of the universal partial recursive predicate or say peano arithmetic i prove that any u either leaves an n bit input statement unresolved or contains nearly all information about the n bit prefix of any r e real which is n bits for some i argue that creating significant information about a specific math sequence is impossible regardless of the methods used similar problems and answers apply to other unsolvability results for tasks allowing multiple solutions for example nonrecursive tilings

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

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

asynchronous space time domain decomposition method with localized uncertainty quantification abstract the computational cost associated with uncertainty quantification of engineering problems featuring localized phenomenon can be reduced by confining the random variability of the model parameters within a region of interest in this case a localized treatment of mesh and time resolutions is required to capture the effect of the confined material uncertainty on the global response we present a computational approach for localized uncertainty quantification with the capability of asynchronous treatment of mesh and time resolutions in particular we allow each subdomain to have its local uncertainty representation and the corresponding mesh and time resolutions as a result computing resources can be directed toward a small region of interest where a model with high spatial and temporal resolutions is required to verify the numerical implementation we consider elastic wave propagation in an axially loaded beam moreover we perform convergence studies with respect to the spatial and temporal discretizations as well as the size of an uncertain subdomain a projectile impacting a composite sandwich plate is considered as an engineering application for the proposed method

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

a domain decomposition method of stochastic pdes recent advances in high performance computing systems and sensing technologies motivate computational simulations with extremely high resolution models with capabilities to quantify uncertainties for credible numerical predictions a two level domain decomposition method is reported in this investigation to devise a linear solver for the large scale system in the galerkin spectral stochastic finite element method ssfem in particular a two level scalable preconditioner is introduced in order to iteratively solve the large scale linear system in the intrusive ssfem using an iterative substructuring based domain decomposition solver the implementation of the algorithm involves solving a local problem on each subdomain that constructs the local part of the preconditioner and a coarse problem that propagates information globally among the subdomains the numerical and parallel scalabilities of the two level preconditioner are contrasted with the previously developed one level preconditioner for two dimensional flow through porous media and elasticity problems with spatially varying non gaussian material properties a distributed implementation of the parallel algorithm is carried out using mpi and petsc parallel libraries the scalabilities of the algorithm are investigated in a linux cluster

working with strong reducibilities above totally omega c e and array computable degrees we investigate the connections between the complexity of a c e set as calibrated by its strength as an oracle for turing computations of functions in the ershov hierarchy and how strong reducibilities allow us to compute such sets for example we prove that a c e degree is totally w c e iff every set in it is weak truth table reducible to a hypersimple or ranked set we also show that a c e degree is array computable iff every left c e real of that degree is reducible in a computable lipschitz way to a random left c e real an number

a note on stochastic elliptic models there exist two types of commonly studied stochastic elliptic models in literature i a x u x f x and ii a x u x f x where indicates randomness the wick product and a x is a positive random process model i is widely used in engineering and physical applications while model ii is usually studied from the mathematical point of view the difference between the above two stochastic elliptic models has not been fully clarified in this work we discuss the difference between models i and ii when a x is a log normal random process we show that the difference between models i and ii is mainly characterized by a scaling factor which is an exponential function of the degree of perturbation of a x we then construct a new stochastic elliptic model iii a 1 1 u x f x which has the same scaling factor as model i after removing the divergence from the scaling factor models i and iii can be highly comparable for many cases we demonstrate this by a numerical study for a one dimensional problem

a feti preconditioned conjugate gradient method for large scale stochastic finite element problems in the spectral stochastic finite element method for analyzing an uncertain system the uncertainty is represented by a set of random variables and a quantity of interest such as the system response is considered as a function of these random variables consequently the underlying galerkin projection yields a block system of deterministic equations where the blocks are sparse but coupled the solution of this algebraic system of equations becomes rapidly challenging when the size of the physical system and or the level of uncertainty is increased this paper addresses this challenge by presenting a preconditioned conjugate gradient method for such block systems where the preconditioning step is based on the dual primal finite element tearing and interconnecting method equipped with a krylov subspace reusage technique for accelerating the iterative solution of systems with multiple and repeated right hand sides preliminary performance results on a linux cluster suggest that the proposed solution method is numerically scalable and demonstrate its potential for making the uncertainty quantification of realistic systems tractable

random matrix theory for modeling uncertainties in computational mechanics this paper deals with data uncertainties and model uncertainties issues in computational mechanics if data uncertainties can be modeled by parametric probabilistic methods for a given mean model a nonparametric probabilistic approach can be used for modeling model uncertainties the first part is devoted to random matrix theory for which we summarize previous published results and for which two new ensembles of random matrices useful for the nonparametric models are introduced in a second part the nonparametric probabilistic approach of random uncertainties is presented for linear dynamical systems and for nonlinear dynamical systems constituted of a linear part with additional localized nonlinearities in a third part a new method is proposed for estimating the parameters of the nonparametric approach from experiments finally examples with experimental comparisons are given

a method of finite element tearing and interconnecting and its parallel solution algorithm a novel domain decomposition approach for the parallel finite element solution of equilibrium equations is presented the spatial domain is partitioned into a set of totally disconnected subdomains each assigned to an individual processor lagrange multipliers are introduced to enforce compatibility at the interface nodes in the static case each floating subdomain induces a local singularity that is resolved in two phases first the rigid body modes are eliminated in parallel from each local problem and a direct scheme is applied concurrently to all subdomains in order to recover each partial local solution next the contributions of these modes are related to the lagrange multipliers through an orthogonality condition a parallel conjugate projected gradient algorithm is developed for the solution of the coupled system of local rigid modes components and lagrange multipliers which completes the solution of the problem when implemented on local memory multiprocessors this proposed method of tearing and interconnecting requires less interprocessor communications than the classical method of substructuring it is also suitable for parallel vector computers with shared memory moreover unlike parallel direct solvers it exhibits a degree of parallelism that is not limited by the bandwidth of the finite element system of equations

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

on kurtz randomness kurtz randomness is a notion of algorithmic randomness for real numbers in particular a real x is called kurtz random or weakly random iff it is contained in every computably enumerable set u of lebesgue measure 1 we prove a number of characterizations of this notion relating it to other notions of randomness such as the well known notions of computable randomness martin lof randomness and schnorr randomness for the first time we give machine characterizations of kurtz randomness whereas the turing degree of every martin lof random c e real is the complete degree and the degrees of schnorr random c e reals are all high we show that kurtz random c e reals occur in every non zero c e degree additionally we show that the sets that are low for kurtz randomness are all hyperimmune and include those that are low for schnorr randomness characterized previously by terwijn and email protected 2004 elsevier b v all rights reserved

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

scalable domain decomposition solvers for stochastic pdes in high performance computing abstract stochastic spectral finite element models of practical engineering systems may involve solutions of linear systems or linearized systems for non linear problems with billions of unknowns for stochastic modeling it is therefore essential to design robust parallel and scalable algorithms that can efficiently utilize high performance computing to tackle such large scale systems domain decomposition based iterative solvers can handle such systems although these algorithms exhibit excellent scalabilities significant algorithmic and implementational challenges exist to extend them to solve extreme scale stochastic systems using emerging computing platforms intrusive polynomial chaos expansion based domain decomposition algorithms are extended here to concurrently handle high resolution in both spatial and stochastic domains using an in house implementation sparse iterative solvers with efficient preconditioners are employed to solve the resulting global and subdomain level local systems through multi level iterative solvers parallel sparse matrix vector operations are used to reduce the floating point operations and memory requirements numerical and parallel scalabilities of these algorithms are presented for the diffusion equation having spatially varying diffusion coefficient modeled by a non gaussian stochastic process scalability of the solvers with respect to the number of random variables is also investigated

reduced basis anova methods for partial differential equations with high dimensional random inputs in this paper we present a reduced basis anova approach for partial deferential equations pdes with random inputs the anova method combined with stochastic collocation methods provides model reduction in high dimensional parameter space through decomposing high dimensional inputs into unions of low dimensional inputs in this work to further reduce the computational cost we investigate spatial low rank structures in the anova collocation method and develop efficient spatial model reduction techniques using hierarchically generated reduced bases we present a general mathematical framework of the methodology validate its accuracy and demonstrate its efficiency with numerical experiments

addressing the curse of dimensionality in ssfem using the dependence of eigenvalues in kl expansion on domain size this work is aimed at reducing the dimensionality in the spectral stochastic finite element method ssfem thus the computational cost through a domain decomposition dd method this reduction hinges on some new mathematical results on domain size dependence of the karhunen loeve kl expansion it has been reported in the literature that for few covariance kernels a lower domain size leads to a faster convergence of the kl eigenvalues this observation leads to an early truncation of the kl expansion and this reduction in stochastic dimensionality brings down the total computational cost in this work first we mathematically show the generalization of this faster convergence that is for any arbitrary covariance kernel this is achieved via developing a bound on eigenvalues as a function of the domain size then we prove that for a chosen number of terms in the kl expansion with any kernel for a one dimensional process the approximation error in the trace norm reduces with the domain size based on this domain size dependence we propose an algorithm for solving a stochastic elliptic equation in a dd framework the computational cost gain is demonstrated by a numerical study and is observed that the serial implementation of the proposed algorithm is about an order of magnitude faster compared to the existing method the cost saving increases with the stochastic dimensionality in the global domain the structure of this algorithm provides scope for parallelization which would help in efficiently solving large scale problems the sharpness of the proposed eigenvalue bounds is also tested for gaussian and exponential kernels the generalization opens avenues for developing further dd based ssfem solvers c 2016 elsevier b v all rights reserved

a stochastic deterministic coupling method for continuum mechanics in this paper we present a novel approach that allows to couple a deterministic continuum model with a stochastic continuum one the coupling strategy is performed in the arlequin framework which is based on a volume coupling and a partition of the energy a suitable functional space is chosen for the weak enforcement of the continuity between the two models the choice of this space ensures that the mean of the stochastic solution equals the deterministic solution point wise and enforces appropriate boundary conditions on the stochastic dimension the proof of the existence of the solution of the mixed problem is provided the numerical strategy is also reviewed in particular with a view at the monte carlo method finally examples show the interest of the method and possible strategies for use in adaptive modeling

a domain decomposition approach for uncertainty analysis this paper proposes a decomposition approach for uncertainty analysis of systems governed by partial differential equations pdes the system is split into local components using domain decomposition our domain decomposed uncertainty quantification dduq approach performs uncertainty analysis independently on each local component in an offline phase and then assembles global uncertainty analysis results using precomputed local information in an online phase at the heart of the dduq approach is importance sampling which weights the precomputed local pde solutions appropriately so as to satisfy the domain decomposition coupling conditions to avoid global pde solves in the online phase a proper orthogonal decomposition reduced model provides an efficient approximate representation of the coupling functions

schwarz preconditioners for stochastic elliptic pdes abstract increasingly the spectral stochastic finite element method ssfem has become a popular computational tool for uncertainty quantification in numerous practical engineering problems for large scale problems however the computational cost associated with solving the arising linear system in the ssfem still poses a significant challenge the development of efficient and robust preconditioned iterative solvers for the ssfem linear system is thus of paramount importance for uncertainty quantification of large scale industrially relevant problems in the context of high performance computing the preconditioner must scale to a large number of processors therefore in this paper a two level additive schwarz preconditioner is described for the iterative solution of the ssfem linear system the proposed preconditioner can be viewed as a generalization of the mean based block diagonal preconditioner commonly used in the literature for the numerical illustrations two dimensional steady state diffusion and elasticity problems with spatially varying random coefficients are considered the performance of the algorithm is investigated with respect to the geometric parameters strength of randomness dimension and order of the stochastic expansion

a dynamically bi orthogonal method for time dependent stochastic partial differential equations ii adaptivity and generalizations this is part ii of our paper in which we propose and develop a dynamically bi orthogonal method dybo to study a class of time dependent stochastic partial differential equations spdes whose solutions enjoy a low dimensional structure in part i of our paper 9 we derived the dybo formulation and proposed numerical algorithms based on this formulation some important theoretical results regarding consistency and bi orthogonality preservation were also established in the first part along with a range of numerical examples to illustrate the effectiveness of the dybo method in this paper we focus on the computational complexity analysis and develop an effective adaptivity strategy to add or remove modes dynamically our complexity analysis shows that the ratio of computational complexities between the dybo method and a generalized polynomial chaos method gpc is roughly of order o m n p 3 for a quadratic nonlinear spde where m is the number of mode pairs used in the dybo method and n p is the number of elements in the polynomial basis in gpc the effective dimensions of the stochastic solutions have been found to be small in many applications so we can expect m is much smaller than n p and computational savings of our dybo method against gpc are dramatic the adaptive strategy plays an essential role for the dybo method to be effective in solving some challenging problems another important contribution of this paper is the generalization of the dybo formulation for a system of time dependent spdes several numerical examples are provided to demonstrate the effectiveness of our method including the navier stokes equations and the boussinesq approximation with brownian forcing

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

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

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

how to prevent pressure oscillations in multicomponent flow calculations a new numerical scheme for computing multicomponent flows is presented it is able to handle strong shocks is pressure oscillation free through the contact discontinuities and guarantees the posi tivity of mass fractions some numerical experiments on shock tubes indicate the convergence to the correct weak solution comparisons with classical finite volume schemes are also proposed

a multiscale method with patch for the solution of stochastic partial differential equations with localized uncertainties abstract we here propose a multiscale numerical method for the solution of stochastic parametric partial differential equations with localized uncertainties described with a finite number of random variables it is based on a multiscale domain decomposition method that exploits the localized side of uncertainties and incidentally improves the conditioning of the problem by operating a separation of scales an efficient iterative algorithm is proposed that requires the solution of a sequence of simple global problems at a macro scale involving a deterministic operator and local problems at a micro scale for which we have the possibility to use fine approximation spaces global and local problems are solved using tensor approximation methods allowing the representation of high dimensional stochastic parametric solutions convergence properties of these tensor based methods which are closely related to spectral decompositions benefit from the separation of scales different types of uncertainties are considered at the micro level they may be associated with some variability in the operator or source terms or even with some geometrical variability in the latter case specific reformulations of local problems using fictitious domain methods are introduced

computing halting probabilities from other halting probabilities the halting probability of a turing machine is the probability that the machine will halt if it starts with a random stream written on its one way input tape when the machine is universal this probability is referred to as chaitin s omega number and is the most well known example of a real which is random in the sense of martin lof although omega numbers depend on the underlying universal turing machine they are robust in the sense that they all have the same turing degree namely the degree of the halting problem this means that given two universal prefix free machines u v the halting probability u of u computes the halting probability v of v if this computation uses at most the first n g n bits of u for the computation of the first n bits of v we say that u computes v with redundancy g in this paper we give precise bounds on the redundancy growth rate that is generally required for the computation of an omega number from another omega number we show that for each 1 any pair of omega numbers compute each other with redundancy log n on the other hand this is not true for 1 in fact we show that for each omega number u there exists another omega number which is not computable from u with redundancy log n this latter result improves an older result of frank stephan

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

a stochastic collocation method for elliptic partial differential equations with random input data in this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms input data of the model the input data are assumed to depend on a finite number of random variables the method consists in a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it can be seen as a generalization of the stochastic galerkin method proposed in i babuscka r tempone and g e zouraris siam j numer anal 42 2004 pp 800 825 and allows one to treat easily a wider range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction in the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method

a compressible navier stokes flow solver with scalar transport this paper concerns the extension of the gas kinetic scheme for the compressible navier stokes equations to the flow calculation with interfaces and mixing the objective of the current research targets mainly on the accurate capturing of navier stokes diffusive interfaces where the thickness can be resolved by the cell size firstly a new bgk ns scheme coupling with the level set type scalar function transport is constructed even though the scalar function is directly incorporated into the gas distribution function and it evolves according to the gas kinetic equation it likes more or less a color function which has no direct contribution for the time evolution of conservative flow variables such as mass momentum and energy due to the coupling of the scalar function into the gas kinetic formulation the governing equations for the scalar function turns out to be an advection diffusion equations and the diffusive coefficient can be controlled by the particle collision time which makes the current scheme suitable for the gas mixing problems with a controllable diffusion coefficients however for the non mixing or sharp interface problems such as the interface between gas and liquid the current method can be used as a scheme similar to the level set method where the interface location can be identified with a fixed level set value such as q 0 the current method is applied to a few examples from the simple square wave propagation and diffusion to the 3d rayleigh taylor instability the supersonic mixing layer and the shock helium bubble interaction case show clearly the convergence of the current navier stokes solver for the flow problems with mixing of components and interface once the interface thickness can be well resolved by the cell size in the case of shock hitting sf6 cylinder the computation predicts the experimental measure very well in the current scheme the schmidt number can be freely chosen according to the physical reality

dual primal domain decomposition method for uncertainty quantification abstract the spectral stochastic finite element method ssfem may offer an efficient alternative to the traditional monte carlo simulations mcs for uncertainty quantification of large scale numerical simulations in the framework of the intrusive ssfem the main computational challenge involves solving a coupled set of deterministic linear systems for large scale numerical models the computational efficiency of the intrusive ssfem primarily depends on the solution techniques employed to tackle the resulting coupled linear systems in this paper we report a probabilistic version of the dual primal domain decomposition method for the intrusive ssfem in order to exploit high performance computing platforms for uncertainty quantification in particular we formulate a probabilistic version of the dual primal finite element tearing and interconnect feti dp technique to solve the large scale linear systems in the intrusive ssfem in the probabilistic setting the operator of the dual interface system in the dual primal approach contains a coarse problem the introduction of the coarse problem in the probabilistic setting leads to a scalable performance of the dual primal iterative substructuring method for uncertainty quantification of large scale computational models the convergence properties numerical and parallel scalabilities of the probabilistic feti dp method and the recently developed probabilistic version of the balancing domain decomposition by constraints bddc method are contrasted for numerical illustrations we consider flow through porous media and linear elasticity problems with spatially varying system parameters modelled as non gaussian random processes the algorithms are implemented on a linux cluster using mpi and petsc parallel libraries

a new algorithm for high dimensional uncertainty quantification based on dimension adaptive sparse grid approximation and reduced basis methods in this work we develop an adaptive and reduced computational algorithm based on dimension adaptive sparse grid approximation and reduced basis methods for solving high dimensional uncertainty quantification uq problems in order to tackle the computational challenge of curse of dimensionality commonly faced by these problems we employ a dimension adaptive tensor product algorithm 16 and propose a verified version to enable effective removal of the stagnation phenomenon besides automatically detecting the importance and interaction of different dimensions to reduce the heavy computational cost of uq problems modelled by partial differential equations pde we adopt a weighted reduced basis method 7 and develop an adaptive greedy algorithm in combination with the previous verified algorithm for efficient construction of an accurate reduced basis approximation the efficiency and accuracy of the proposed algorithm are demonstrated by several numerical experiments

sparse tensor discretization of elliptic spdes we propose and analyze sparse deterministic stochastic tensor galerkin finite element methods sparse sgfems for the numerical solution of elliptic partial differential equations pdes with random coefficients in a physical domain d subset mathbb r d in tensor product sgfems the variational solution to the boundary value problem is approximated in tensor product finite element spaces v gamma otimes v d where v gamma and v d denote suitable finite dimensional subspaces of the stochastic and deterministic function spaces respectively these approaches lead to sgfem algorithms of complexity o n gamma n d where n gamma dim v gamma and n d dim v d in this work we use hierarchic sequences v gamma 1 subset v gamma 2 subset ldots and v d 1 subset v d 2 subset ldots of finite dimensional spaces to approximate the law of the random solution the hierarchies of approximation spaces allow us to define sparse tensor product spaces v gamma ell hat otimes v d ell ell 1 2 dots yielding algorithms of o n gamma log n d n d log n gamma work and memory we estimate the convergence rate of sgfem for algebraic decay of the input random field karhunen loeve coefficients we give an algorithm for an input adapted a priori selection of deterministic and stochastic discretization spaces the convergence rate in terms of the total number of degrees of freedom of the proposed method is superior to monte carlo approximations numerical examples illustrate the theoretical results and demonstrate superiority of the sparse tensor product discretization proposed here versus the full tensor product approach

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

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

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

natural halting probabilities partial randomness and zeta functions we introduce the zeta number natural halting probability and natural complexity of a turing machine and we relate them to chaitin s omega number halting probability and program size complexity a classification of turing machines according to their zeta numbers is proposed divergent convergent and tuatara we prove the existence of universal convergent and tuatara machines various results on algorithmic randomness and partial randomness are proved for example we show that the zeta number of a universal tuatara machine is c e and random a new type of partial randomness asymptotic randomness is introduced finally we show that in contrast to classical algorithmic randomness which cannot be naturally characterised in terms of plain complexity asymptotic randomness admits such a characterisation

the two level feti method part ii extension to shell problems parallel implementation and performance results the two level feti method presented in part i of this work is a mathematically optimal and scalable substructuring algorithm for solving iteratively large scale systems of equations arising from the finite element discretization of static and dynamic plate bending problems in this paper we extend the two level feti method to shell problems describe a variant approach revisit the preconditioning problem address the efficient implementation of the corresponding iterative solvers on massively parallel processors highlight the computational price of mathematical optimality and discuss its consequences and report some impressive performance results on two paragon xp s and ibm sp2 massively parallel processors for several realistic plate and shell problems

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

a third order gas kinetic cpr method for the euler and navier stokes equations on triangular meshes abstract a third order accurate gas kinetic scheme based on the correction procedure via reconstruction cpr framework is developed for the euler and navier stokes equations on triangular meshes the scheme combines the accuracy and efficiency of the cpr formulation with the multidimensional characteristics and robustness of the gas kinetic flux solver comparing with high order finite volume gas kinetic methods the current scheme is more compact and efficient by avoiding wide stencils on unstructured meshes unlike the traditional cpr method where the inviscid and viscous terms are treated differently the inviscid and viscous fluxes in the current scheme are coupled and computed uniformly through the kinetic evolution model in addition the present scheme adopts a fully coupled spatial and temporal gas distribution function for the flux evaluation achieving high order accuracy in both space and time within a single step numerical tests with a wide range of flow problems from nearly incompressible to supersonic flows with strong shocks for both inviscid and viscous problems demonstrate the high accuracy and efficiency of the present scheme

comparison of the generalized riemann solver and the gas kinetic scheme for inviscid compressible flow simulations the generalized riemann problem grp scheme for the euler equations and gas kinetic scheme gks for the boltzmann equation are two high resolution shock capturing schemes for fluid simulations the difference is that one is based on the characteristics of the inviscid euler equations and their wave interactions and the other is based on the particle transport and collisions the similarity between them is that both methods can use identical muscl type initial reconstructions around a cell interface and the spatial slopes on both sides of a cell interface involve in the gas evolution process and the construction of a time dependent flux function although both methods have been applied successfully to the inviscid compressible flow computations their performances have never been compared since both methods use the same initial reconstruction any difference is solely coming from different underlying mechanism in their flux evaluation therefore such a comparison is important to help us to understand the correspondence between physical modeling and numerical performances since grp is so faithfully solving the inviscid euler equations the comparison can be also used to show the validity of solving the euler equations itself the numerical comparison shows that the grp exhibits a slightly better computational efficiency and has comparable accuracy with gks for the euler solutions in 1d case but the gks is more robust than grp for the 2d high mach number flow simulations the gks is absent from the shock instability and converges to the steady state solutions faster than the grp the grp has carbuncle phenomena likes a cloud hanging over exact riemann solvers the grp and gks use different physical processes to describe the flow motion starting from a discontinuity one is based on the assumption of equilibrium state with infinite number of particle collisions and the other starts from the non equilibrium free transport process to evolve into an equilibrium one through particle collisions the different mechanism in the flux evaluation deviates their numerical performance through this study we may conclude scientifically that it may not be valid to use the euler equations as governing equations to construct numerical fluxes in a discretized space with limited cell resolution to adapt the navier stokes ns equations is not valid either because the ns equations describe the flow behavior on the hydrodynamic scale and have no any corresponding physics starting from a discontinuity this fact alludes to the consistency of the euler and navier stokes equations with the continuum assumption and the necessity of a direct modeling of the physical process in the discretized space in the construction of numerical scheme when modeling very high mach number flows the development of numerical algorithm is similar to the modeling process in deriving the governing equations but the control volume here cannot be shrunk to zero

on the multidimensional gas kinetic bgk scheme this paper concerns the performance difference among the multidimensional md gas kinetic bgk scheme the corresponding quasi one dimensional q1d extension and the directional splitting ds scheme with kinetic boundary conditions the md scheme includes tangential slopes in the flux calculation which are absent in a q1d or ds method in spite of taking more computational time the md scheme is found to be able to capture the characteristics of rarefied flow having a boundary with curvature or a nonuniform temperature such as the inverted velocity distribution in rarefied cylindrical couette flow and the weak flow field induced by the temperature gradient of a body where the q1d fails the md scheme can also yield clearly better results in high speed microchannel flow and power law fluid flow between concentric rotating cylinders when compared with direct simulation monte carlo studies and analytic solutions in the low reynolds number flow around a naca0012 airfoil case the md scheme and the q1d method predict wall heat flux distribution with obvious difference the ds method predicts results nearly identical to those from q1d in these steady flow cases except that it can hardly give reasonable solutions in power law fluid case with viscous exponential factor away from unity correct prediction of the stress and the wall temperature gradient is considered responsible for this better performance in simulations of the scalar convection diffusion equation it is found that the inclusion of tangential slopes in the flux computation can clearly improve the temporal accuracy of the scheme in this case the ds scheme also shows good performance the present study suggests caution in adopting of the q1d extension or ds method in multidimensional flow when it is sensitive to the accuracy of stress or wall variable gradient calculations it is better to use the md scheme for high performance simulation

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

a scalable lagrange multiplier based domain decomposition method for time dependent problems we present a new efficient and scalable domain decomposition method for solving implicitly linear and non linear time dependent problems in computational mechanics the method is derived by adding a coarse problem to the recently proposed transient feti substructuring algorithm in order to propagate the error globally and accelerate convergence it is proved that in the limit for large time steps the new method converges toward the feti algorithm for time independent problems computational results confirm that the optimal convergence properties of the time independent feti method are preserved in the time dependent case we employ an iterative scheme for solving efficiently the coarse problem on massively parallel processors and demonstrate the effective scalability of the new transient feti method with the large scale finite element dynamic analysis on the paragon xp s and ibm sp2 systems of several diffraction grating finite element structural models we also show that this new domain decomposition method outperforms the popular direct skyline solver the coarse problem presented herein is applicable and beneficial to a large class of lagrange multiplier based substructuring algorithms for time dependent problems including the fictitious domain decomposition method

unified solver for rarefied and continuum flows with adaptive mesh and algorithm refinement this paper describes a unified flow solver ufs for rarefied and continuum gas flows the ufs separates the rarefied and continuum flow domains and selects appropriate solvers to combine the efficiency of continuum models with the accuracy of kinetic models the direct numerical solution of the boltzmann transport equation is used in rarefied regions while kinetic schemes of continuum fluid dynamics are used elsewhere using similar computational techniques for the kinetic and continuum solvers and employing intelligent domain decomposition algorithms attain the efficiency and numerical stability of the ufs solutions of test problems are presented to illustrate the capabilities of the ufs for high and low speed flows it is shown that the ufs can dynamically adapt the computational mesh and automatically introduce and remove kinetic patches to provide significant savings by limiting molecular scale solutions only to the regions where they are needed

application of gas kinetic scheme with kinetic boundary conditions in hypersonic flow this work focuses on the application of the gas kinetic scheme based on the bhatnagar gross krook particle collision model in the hypersonic flow simulations kinetic boundary conditions with different accommodation coefficients are constructed and implemented in the scheme the numerical study is based on a laminar shock wave boundary layer interaction in a mach 10 flow passing through a hollow cylinder flare model the grid independent numerical results agree well with experimental measurements the utilization of the kinetic slip boundary condition is necessary to improve the agreement between the current results experimental measurements as well as direct simulation monte carlo solutions the effect of the variation of the accommodation coefficient on the flow solutions is quantitatively evaluated

acceleration of the spectral stochastic fem using pod and element based discrete empirical approximation for a micromechanical model of heterogeneous materials with random geometry abstract the spectral stochastic fem with local basis functions in the stochastic domain sl fem is one of the most flexible and accurate stochastic methods however also the most computationally expensive these expenses are traditionally associated with the extra large tangent stiffness matrix and a huge number of elements which need to be re integrated in every iteration in this work we incorporate the proper orthogonal decomposition pod into the sl fem thus performing a drastic reduction of the stiffness matrix in order to reduce the integration costs by hyperreduction a novel element based modification of the discrete empirical interpolation the so called element based empirical approximation method edeam is developed and combined with the pod particular advantages of the sl fem for order reduction and hyperreduction compared to other stochastic techniques are discussed the new reduced order sl fem is applied to the computational homogenization of materials with random geometry of the microstructure i e to a general class of problems exhibiting strongly nonlinear non smooth and sometimes discontinuous dependency of the solution on some random parameters the reduced order sl fem demonstrates a high accuracy and a high solution speed whereby the solution time for the reduced order sl fem is comparable to the solution time of only one single monte carlo sample

efficient iterative algorithms for the stochastic finite element method with application to acoustic scattering in this study we describe the algebraic computations required to implement the stochastic finite element method for solving problems in which uncertainty is restricted to right hand side data coming from forcing functions or boundary conditions we show that the solution can be represented in a compact outer product form which leads to efficiencies in both work and storage and we demonstrate that block iterative methods for algebraic systems with multiple right hand sides can be used to advantage to compute this solution we also show how to generate a variety of statistical quantities from the computed solution finally we examine the behavior of these statistical quantities in one setting derived from a model of acoustic scattering

domain decomposition of stochastic pdes theoretical formulations we present a novel theoretical framework for the domain decomposition of uncertain systems defined by stochastic partial differential equations the methodology involves a domain decomposition method in the geometric space and a functional decomposition in the probabilistic space the probabilistic decomposition is based on a version of stochastic finite elements based on orthogonal decompositions and projections of stochastic processes the spatial decomposition is achieved through a schur complement based domain decomposition the methodology aims to exploit the full potential of high performance computing platforms by reducing discretization errors with high resolution numerical model in conjunction to giving due regards to uncertainty in the system the mathematical formulation is numerically validated with an example of waves in random media copyright 2008 john wiley sons ltd

chaitin ω numbers solovay machines and godel incompleteness computably enumerable c e reals can be coded by chaitin machines through their halting probabilities tuning solovay s construction of a chaitin universal machine for which zfc if arithmetically sound cannot determine any single bit of the binary expansion of its halting probability we show that every c e random real is the halting probability of a universal chaitin machine for which zfc cannot determine more than its initial block of 1 bits as soon as you get a 0 it is all over finally a constructive version of chaitin information theoretic incompleteness theorem is proven

a multidimensional gas kinetic bgk scheme for hypersonic viscous flow this paper concerns the development and application of a gas kinetic scheme based on the bhatnagar gross krook bgk model for the navier stokes equations in the study of hypersonic viscous flow firstly we extend the gas kinetic navier stokes solver k xu a gas kinetic bgk scheme for the navier stokes equations and its connection with artificial dissipation and godunov method j comput phys 171 2001 289 335 by implementing a multidimensional particle propagation mechanism in the flux evaluation where the gradients of flow variables in both normal and tangential directions of a cell interface are explicitly included with the construction of a time averaged flux function an implicit bgk scheme with lu sgs method is constructed the main purpose of the current research is to pave the way to extend the current approach directly to the flow computation with unstructured mesh where the flow gradients in both parallel and perpendicular directions around a cell interface can be explicitly taken into account in a viscous flow computation in the numerical parts we concentrate on the computation of heat flux in laminar hypersonic viscous flows where complicated flow phenomena i e shock boundary layer interaction flow separation and viscous inviscid interaction will be encountered the cases studied include the type iv shock shock interaction around a circular cylinder and hypersonic flow passing through a double cone geometry in the hypersonic viscous flow in comparison with the capturing of velocity and pressure fields the accurate computation of stress and heat flux bears large difficulties in all cases studied here the heat fluxes obtained across body surfaces have good agreement with the experimental measurements

parameter and state model reduction for large scale statistical inverse problems a greedy algorithm for the construction of a reduced model with reduction in both parameter and state is developed for an efficient solution of statistical inverse problems governed by partial differential equations with distributed parameters large scale models are too costly to evaluate repeatedly as is required in the statistical setting furthermore these models often have high dimensional parametric input spaces which compounds the difficulty of effectively exploring the uncertainty space we simultaneously address both challenges by constructing a projection based reduced model that accepts low dimensional parameter inputs and whose model evaluations are inexpensive the associated parameter and state bases are obtained through a greedy procedure that targets the governing equations model outputs and prior information the methodology and results are presented for groundwater inverse problems in one and two dimensions

risk averse structural topology optimization under random fields using stochastic expansion methods abstract this work proposes a level set based approach for solving risk averse structural topology optimization problems considering random field loading and material uncertainty the use of random fields increases the dimensionality of the stochastic domain which poses several computational challenges related to the minimization of the excess probability as a measure of risk awareness this problem is addressed both from the theoretical and numerical viewpoints first an existence result under a typical geometrical constraint on the set of admissible shapes is proved second a level set continuous approach to find the numerical solution of the problem is proposed since the considered cost functional has a discontinuous integrand the numerical approximation of the functional and its sensitivity combine an adaptive anisotropic polynomial chaos pc approach with a monte carlo mc sampling method for uncertainty propagation furthermore to address the increment of dimensionality induced by the random field an anisotropic sparse grid stochastic collocation method is used for the efficient computation of the pc coefficients a key point is that the non intrusive nature of such an approach facilitates the use of high performance computing hpc to alleviate the computational burden of the problem several numerical experiments including random field loading and material uncertainty are presented to show the feasibility of the proposal

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

things that can be made into themselves one says that a property p of sets of natural numbers can be made into itself iff there is a numbering a 0 a 1 of all left r e sets such that the index set e a e satisfies p has the property p as well for example the property of being martin lof random can be made into itself herein we characterize those singleton properties which can be made into themselves a second direction of the present work is the investigation of the structure of left r e sets under inclusion modulo a finite set in contrast to the corresponding structure for r e sets which has only maximal but no minimal members both minimal and maximal left r e sets exist moreover our construction of minimal and maximal left r e sets greatly differs from friedberg s classical construction of maximal r e sets finally we investigate whether the properties of minimal and maximal left r e sets can be made into themselves

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

on the construction of kinetic schemes a general method for the construction of kinetic schemes of evolutionary equations is illustrated with the simple example of the linear advection equation where the role of the collision effect is clarified theoretically and numerically the application to the compressible euler equation and navier stokes equation is explained the theoretical backgrounds of the existing kinetic schemes such as the pullin scheme for the euler equation and chou baganoff scheme and modified prendergast xu scheme for the navier stokes equation are unified and their intrinsic errors are revealed

universal recursively enumerable sets of strings the main topics of the present work are universal machines for plain and prefix free description complexity and their domains it is characterised when an r e set w is the domain of a universal plain machine in terms of the description complexity of the spectrum function s w mapping each non negative integer n to the number of all strings of length n in w furthermore a characterisation of the same style is given for supersets of domains of universal plain machines similarly the prefix free sets which are domains or supersets of domains of universal prefix free machines are characterised furthermore it is shown that the halting probability w v of an r e prefix free set v containing the domain of a universal prefix free machine is martin lof random while v may not be the domain of any universal prefix free machine itself based on these investigations the question whether every domain of a universal plain machine is the superset of the domain of some universal prefix free machine is discussed a negative answer to this question had been presented at cie 2010 by mikhail andreev ilya razenshteyn and alexander shen while this paper was under review

finite element error analysis of elliptic pdes with random coefficients and its application to multilevel monte carlo methods we consider a finite element approximation of elliptic partial differential equations with random coefficients such equations arise for example in uncertainty quantification in subsurface flow modeling models for random coefficients frequently used in these applications such as log normal random fields with exponential covariance have only very limited spatial regularity and lead to variational problems that lack uniform coercivity and boundedness with respect to the random parameter in our analysis we overcome these challenges by a careful treatment of the model problem almost surely in the random parameter which then enables us to prove uniform bounds on the finite element error in standard bochner spaces these new bounds can then be used to perform a rigorous analysis of the multilevel monte carlo method for these elliptic problems that lack full regularity and uniform coercivity and boundedness to conclude we give some numerical results that confirm the new bounds

tensor train decomposition a simple nonrecursive form of the tensor decomposition in d dimensions is presented it does not inherently suffer from the curse of dimensionality it has asymptotically the same number of parameters as the canonical decomposition but it is stable and its computation is based on low rank approximation of auxiliary unfolding matrices the new form gives a clear and convenient way to implement all basic operations efficiently a fast rounding procedure is presented as well as basic linear algebra operations examples showing the benefits of the decomposition are given and the efficiency is demonstrated by the computation of the smallest eigenvalue of a 19 dimensional operator

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

a kronecker product preconditioner for stochastic galerkin finite element discretizations the discretization of linear partial differential equations with random data by means of the stochastic galerkin finite element method results in general in a large coupled linear system of equations using the stochastic diffusion equation as a model problem we introduce and study a symmetric positive definite kronecker product preconditioner for the galerkin matrix we compare the popular mean based preconditioner with the proposed preconditioner which in contrast to the mean based construction makes use of the entire information contained in the galerkin matrix we report on results of test problems where the random diffusion coefficient is given in terms of a truncated karhunen loeve expansion or is a lognormal random field