estimation of distributions via multilevel monte carlo with stratified sampling abstract we design and implement a novel algorithm for computing a multilevel monte carlo mlmc estimator of the joint cumulative distribution function cdf of a vector valued quantity of interest in problems with random input parameters or initial conditions our approach combines mlmc with stratified sampling of the input sample space by replacing standard monte carlo at each level with stratified monte carlo initialized with proportionally allocated samples we show that the resulting stratified mlmc smlmc algorithm is more efficient than its standard mlmc counterpart due to the additional variance reduction provided by the stratification of the random parameter s domain especially at the coarsest levels additional computational cost savings are obtained by smoothing the indicator function with a gaussian kernel which proves to be an efficient and robust alternative to recently developed polynomial based techniques

tracing the equilibrium path beyond compound critical points a unified presentation of some popular continuation procedures used in the non linear finite element analysis of structural mechanics is introduced an extension of the elliptical constraint equation proposed by crisfield is given it is shown that in the proposed procedure real roots can always be obtained in solving the iterative change of the load parameter updated weighting factors are introduced in the constraint equation in order to get better convergence characteristics in the case when localized deformations occur for bifurcation points a modification of rheinboldt s branching procedure is presented post critical response after limit and bifurcation points is determined in some numerical examples

on the geometry of four qubit invariants the geometry of four qubit entanglement is investigated we replace some of the polynomial invariants for four qubits introduced recently by new ones of direct geometrical meaning it is shown that these invariants describe four points six lines and four planes in complex projective space cp3 for the generic entanglement class of stochastic local operations and classical communication they take a very simple form related to the elementary symmetric polynomials in four complex variables moreover suitable powers of their magnitudes are entanglement monotones that fit nicely into the geometric set of n qubit ones related to grassmannians of l planes found recently we also show that in terms of these invariants the hyperdeterminant of order 24 in the four qubit amplitudes takes a more instructive form than the previously published expressions available in the literature finally in order to understand two three and four qubit entanglement in geometric terms we propose a unified setting based on cp3 furnished with a fixed quadric

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

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

local convergence analysis of tensor methods for nonlinear equations tensor methods for nonlinear equations base each iteration upon a standard linear model augmented by a low rank quadratic term that is selected in such a way that the mode is efficient to form store and solve these methods have been shown to be very efficient and robust computationally especially on problems where the jacobian matrix at the root has a small rank deficiency this paper analyzes the local convergence properties of two versions of tensor methods on problems where the jacobian matrix at the root has a null space of rank one both methods augment the standard linear model by a rank one quadratic term we show under mild conditions that the sequence of iterates generated by the tensor method based upon an ideal tensor model converges locally and two step q superlinearly to the solution with q order 3 2 and that the sequence of iterates generated by the tensor method based upon a practial tensor model converges locally and three step q superlinearly to the solution with q order 3 2 in the same situation it is known that standard methods converge linearly with constant converging to 1 2 hence tensor methods have theoretical advantages over standard methods our analysis also confirms that tensor methods converge at least quadratically on problems where the jacobian matrix at the root is nonsingular

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

an accelerated newton method for equations with semismooth jacobians and nonlinear complementarity problems we discuss local convergence of newton s method to a singular solution x of the nonlinear equations f x 0 for f mathbb r n rightarrow mathbb r n it is shown that an existing proof of griewank concerning linear convergence to a singular solution x from a starlike domain around x for f twice lipschitz continuously differentiable and x satisfying a particular regularity condition can be adapted to the case in which f is only strongly semismooth at the solution further newton s method can be accelerated to produce fast linear convergence to a singular solution by overrelaxing every second newton step these results are applied to a nonlinear equations reformulation of the nonlinear complementarity problem ncp whose derivative is strongly semismooth when the function f arising in the ncp is sufficiently smooth conditions on f are derived that ensure that the appropriate regularity conditions are satisfied for the nonlinear equations reformulation of the ncp at x

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

geometry of discrete quantum computing conventional quantum computing entails a geometry based on the description of an n qubit state using 2n infinite precision complex numbers denoting a vector in a hilbert space such numbers are in general uncomputable using any real world resources and if we have the idea of physical law as some kind of computational algorithm of the universe we would be compelled to alter our descriptions of physics to be consistent with computable numbers our purpose here is to examine the geometric implications of using finite fields fp and finite complexified fields based on primes p congruent to 3 mod4 as the basis for computations in a theory of discrete quantum computing which would therefore become a computable theory because the states of a discrete n qubit system are in principle enumerable we are able to determine the proportions of entangled and unentangled states in particular we extend the hopf fibration that defines the irreducible state space of conventional continuous n qubit theories which is the complex projective space to an analogous discrete geometry in which the hopf circle for any n is found to be a discrete set of p 1 points the tally of unit length n qubit states is given and reduced via the generalized hopf fibration to the discrete analogue of the complex projective space which has irreducible states using a measure of entanglement the purity we explore the entanglement features of discrete quantum states and find that the n qubit states based on the complexified field have pn p 1 n unentangled states the product of the tally for a single qubit with purity 1 and they have pn 1 p 1 p 1 n 1 maximally entangled states with purity zero

the method of distributions for dispersive transport in porous media with uncertain hydraulic properties predictions of solute transport in subsurface environments are notoriously unreliable due to aquifer heterogeneity and uncertainty about the values of hydraulic parameters probabilistic framework which treats the relevant parameters and solute concentrations as random fields allows for quantification of this predictive uncertainty by providing deterministic equations for either probability density function or cumulative distribution function cdf of predicted concentrations the method of distributions enables one to estimate e g the probability of a contaminant s concentration exceeding a safe dose we derive a deterministic equation for the cdf of solute concentration which accounts for uncertainty in flow velocity and initial conditions the coefficients in this equation are expressed in terms of the mean and variance of concentration the accuracy and robustness of the cdf equations are analyzed by comparing their predictions with those obtained with monte carlo simulations and an assumed beta cdf

quantum and fisher information from the husimi and related distributions the two principal immediate influences which we seek to interrelate here upon the undertaking of this study are papers of yczkowski and slomczynski j phys a 34 6689 2001 and of petz and sudar j math phys 37 2262 1996 in the former work a metric the monge one specifically over generalized husimi distributions was employed to define a distance between two arbitrary density matrices in the petz sudar work completing a program of chentsov the quantum analog of the classically unique fisher information monotone metric of a probability simplex was extended to define an uncountable infinitude of riemannian also monotone metrics on the set of positive definite density matrices we pose here the questions of what is the specific unique fisher information metric for the classically defined husimi distributions and how does it relate to the infinitude of quantum metrics over the density matrices of petz and sudar we find a highly proximate small relative entropy relationship bet

hp discontinuous galerkin time stepping for parabolic problems abstract we consider the hp version of the discontinuous galerkin dg time stepping method for linear parabolic problems with non symmetric elliptic spatial operators we derive new analyticity estimates for the exact solutions by means of semigroup techniques these estimates allow us to show that the hp dg time stepping method can resolve start up singularities at exponential rates of convergence

geometry of entangled states bloch spheres and hopf fibrations we discuss a generalization of the standard bloch sphere representation for a single qubit to two qubits in the framework of hopf fibrations of highdimensional spheres by lower dimensional spheres the single qubit hilbert space is the three dimensional sphere s 3 t hes 2 base space of a suitably oriented s 3 hopf fibration is nothing but the bloch sphere while the circular fibres represent the overall qubit phase degree of freedom for the two qubits case the hilbert space is a seven dimensional sphere s 7 which also allows for a hopf fibration with s 3 fibres and a s 4 base the most striking result is that suitably oriented s 7 hopf fibrations are entanglement sensitive the relation with the standard schmidt decomposition is also discussed

approximation of probability density functions by the multilevel monte carlo maximum entropy method we develop a complete convergence theory for the maximum entropy method based on moment matching for a sequence of approximate statistical moments estimated by the multilevel monte carlo method under appropriate regularity assumptions on the target probability density function the proposed method is superior to the maximum entropy method with moments estimated by the monte carlo method new theoretical results are illustrated in numerical examples

multilevel monte carlo path simulation we show that multigrid ideas can be used to reduce the computational complexity of estimating an expected value arising from a stochastic differential equation using monte carlo path simulations in the simplest case of a lipschitz payoff and a euler discretisation the computational cost to achieve an accuracy of o e is reduced from o e 3 to o e 2 log e 2 the analysis is supported by numerical results showing significant computational savings

a reduced basis approach for variational problems with stochastic parameters application to heat conduction with variable robin coefficient in this work a reduced basis rb approach is used to solve a large number of boundary value problems bvps parametrized by a stochastic input expressed as a karhunen loeve expansion in order to compute outputs that are smooth functionals of the random solution fields the rb method proposed here for variational problems parametrized by stochastic coefficients bears many similarities to the rb approach developed previously for deterministic systems however the stochastic framework requires the development of new a posteriori estimates for statistical outputs such as the first two moments of integrals of the random solution fields these error bounds in turn permit efficient sampling of the input stochastic parameters and fast reliable computation of the outputs in particular in the many query context

a combinatorial approach to nonlocality and contextuality so far most of the literature on quantum contextuality and the kochen specker theorem seems either to concern particular examples of contextuality or be considered as quantum logic here we develop a general formalism for contextuality scenarios based on the combinatorics of hypergraphs which significantly refines a similar recent approach by cabello severini and winter csw in contrast to csw we explicitly include the normalization of probabilities which gives us a much finer control over the various sets of probabilistic models like classical quantum and generalized probabilistic in particular our framework specializes to quantum nonlocality in the case of bell scenarios which arise very naturally from a certain product of contextuality scenarios due to foulis and randall in the spirit of csw we find close relationships to several graph invariants the recently proposed local orthogonality principle turns out to be a special case of a general principle for contextuality scenarios related to the shannon capacity of graphs our results imply that it is strictly dominated by a low level of the navascues pironio acin hierarchy of semidefinite programs which we also apply to contextuality scenarios

analysis and modification of newton s method for algebraic riccati equations when newton s method is applied to find the maximal symmetric solution of an algebraic riccati equation convergence can be guaranteed under moderate conditions in particular the initial guess need not be close to the solution the convergence is quadratic if the frechet derivative is invertible at the solution in this paper we examine the behaviour of the newton iteration when the derivative is not invertible at the solution we find that a simple modification can improve the performance of the newton iteration dramatically

impact of parametric uncertainty on estimation of the energy deposition into an irradiated brain tumor abstract predictions of the total energy deposited into a brain tumor through x ray irradiation are notoriously error prone we investigate how this predictive uncertainty is affected by uncertainty in both the location of the region occupied by a dose enhancing iodinated contrast agent and the agent s concentration this is done within the probabilistic framework in which these uncertain parameters are modeled as random variables we employ the stochastic collocation sc method to estimate statistical moments of the deposited energy in terms of statistical moments of the random inputs and the global sensitivity analysis gsa to quantify the relative importance of uncertainty in these parameters on the overall predictive uncertainty a nonlinear radiation diffusion equation dramatically magnifies the coefficient of variation of the uncertain parameters yielding a large coefficient of variation for the predicted energy deposition this demonstrates that accurate prediction of the energy deposition requires a proper treatment of even small parametric uncertainty our analysis also reveals that sc outperforms standard monte carlo but its relative efficiency decreases as the number of uncertain parameters increases from one to three a robust gsa ameliorates this problem by reducing this number

the geometry of concurrence as a measure of entanglement it is widely recognized that concurrence can be regarded as a measure of two qubit entanglement this paper studies the geometry of concurrence for a two qubit system to show that the concurrence serves as a coordinate of the factor space where is the space of normalized two qubit states and where g u 1 su 2 su 2 any monotonically increasing function of the concurrence can serve as a measure for entanglement from the viewpoint of riemannian geometry a state with concurrence r is shown to be distant from the separable states by where r ranges over 0 r 1 in addition measures of entanglement for n qubits are discussed on the basis of a bipartite decomposition with m n they are invariant under the local unitary transformation group u 2 u 2m

worst case scenario analysis for elliptic problems with uncertainty this work studies linear elliptic problems under uncertainty the major emphasis is on the deterministic treatment of such uncertainty in particular this work uses the worst scenario approach for the characterization of uncertainty on functional outputs quantities of physical interest assuming that the input data belong to a given functional set eventually infinitely dimensional this work proposes numerical methods to approximate the corresponding uncertainty intervals for the quantities of interest numerical experiments illustrate the performance of the proposed methodology

active lattices determine aw algebras abstract we prove that operator algebras that have enough projections are completely determined by those projections their symmetries and the action of the latter on the former this includes all von neumann algebras and all aw algebras we introduce active lattices which are formed from these three ingredients more generally we prove that the category of aw algebras is equivalent to a full subcategory of active lattices crucial ingredients are an equivalence between the category of piecewise aw algebras and that of piecewise complete boolean algebras and a refinement of the piecewise algebra structure of an aw algebra that enables recovering its total structure

sublinear convergence of the chord method at singular points we give sufficient conditions for convergence of the chord method for a class of singular problems the rate of convergence is sublinear

geometry of the three qubit state entanglement and division algebras we present a generalization to three qubits of the standard bloch sphere representation for a single qubit and of the seven dimensional sphere representation for two qubits presented in mosseri et al mosseri r and dandoloff r 2001 j phys a math gen 34 10243 the hilbert space of the three qubit system is the 15 dimensional sphere s15 which allows for a natural last hopf fibration with s8 as base and s7 as fibre a striking feature is as in the case of one and two qubits that the map is entanglement sensitive and the two distinct ways of un entangling three qubits are naturally related to the hopf map we define a quantity that measures the degree of entanglement of the three qubit state conjectures on the possibility of generalizing the construction for higher qubit states are also discussed

stochastic properties of quadrature formulas the average error of suitable quadrature formulas and the stochastic error of monte carlo methods are both much smaller than the worst case error in many cases this depends however on the classf of functions which is considered and there are counterexamples as well

the geometry of entanglement metrics connections and the geometric phase using the natural connection equivalent to the su 2 yang mills instanton on the quaternionic hopf fibration of s7over the quaternionic projective space hp1 s4 with an su 2 s3 fibre the geometry of entanglement for two qubits is investigated the relationship between base and fibre i e the twisting of the bundle corresponds to the entanglement of the qubits the measure of entanglement can be related to the length of the shortest geodesic with respect to the mannoury fubini study metric on hp1 between an arbitrary entangled state and the separable state nearest to it using this result an interpretation of the standard schmidt decomposition in geometric terms is given schmidt states are the nearest and furthest separable ones obtained by parallel transport along the geodesic passing through the entangled state some examples showing the correspondence between the anholonomy of the connection and entanglement via the geometric phase are shown connections with important notions such as the bures metric and uhlmann s connection the hyperbolic structure for density matrices and anholonomic quantum computation are also pointed out

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

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

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

thomas rotation and the mixed state geometric phase it is shown that uhlmann s parallel transport of purifications along a path of mixed states represented by 2 2 density matrices is just the path ordered product of thomas rotations these rotations are invariant under hyperbolic translations inside the bloch sphere that can be regarded as the poincare ball model of hyperbolic geometry a general expression for the mixed state geometric phase for an arbitrary geodesic triangle in terms of the bures fidelities is derived the formula gives back the solid angle result well known from studies of the pure state geometric phase it is also shown that this mixed state anholonomy can be reinterpreted as the pure state non abelian anholonomy of entangled states living in a suitable restriction of the quaternionic hopf bundle in this picture uhlmann s parallel transport is just the pancharatnam transport of quaternionic spinors

a shamanskii like acceleration scheme for nonlinear equations at singular roots on applique une variante de la methode de shamanskii pour realiser une methode a convergence superlineaire de resolution d une classe d equations non lineaires ayant une derivee de frechet singuliere a la racine

verified error bounds for isolated singular solutions of polynomial systems case of breadth one in this paper we describe how to improve the performance of the symbolic numeric method in 19 20 for computing the multiplicity structure and refining approximate isolated singular solutions in the breadth one case by introducing a parameterized deflated system with smoothing parameters we generalize the algorithm in 33 to compute verified error bounds such that a slightly perturbed polynomial system is guaranteed to have a breadth one multiple root within the computed bounds

on solving elliptic stochastic partial differential equations abstract a model elliptic boundary value problem of second order with stochastic coefficients described by the karhunen loeve expansion is addressed this problem is transformed into an equivalent deterministic one the perturbation method and the method of successive approximations is analyzed rigorous error estimates in the framework of sobolev spaces are given

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

convergence rates for newton s method at singular points if newton s method is employed to find a root of a map from a banach space into itself and the derivative is singular at that root the convergence of the newton iterates to the root is linear rather than quadratic in this paper we give a detailed analysis of the linear convergence rates for several types of singular problems for some of these problems we describe modifications of newton s method which will restore quadratic convergence

on solving nonlinear equations with simple singularities or nearly singular solutions the paper begins with a review of the convergence theory for newton s method near simple and other regular singularities followed by a brief discussion of the inherent difficulty of singular problems and the effect of rounding errors on the achievable solution accuracy then several ways to accelerate the slow convergence of newton s method are described in some detail these are richardson extrapolation overrelaxation bordering by a singularity condition and appending the usual linear model by a quadratic term the last two approaches require limited second derivative information which can be obtained analytically by differencing or through secant updating numerical comparisons reported in the final sections lead to the conclusion that the collection of the additional information pays off in terms of speed and accuracy

convergence acceleration for newton s method at singular points if newton s method is employed to find a root of a map from a banach space into itself and the derivative is singular at that root linear convergence of the newton sequence to the root is the best

a new acceleration method for newton s method at singular points when newton s method is used to find a root of a map from a banach space into itself and the derivative is singular at that root convergence of the newton sequence is in general linear in this paper we give a modification of newton s method that under certain conditions converges superlinearly our method is applicable under more general conditions than other techniques in particular it may be used for certain quadratic problems and problems in which the dimension of the null space of the derivative is larger than one

the stochastic galerkin scaled boundary finite element method on random domain summary the research work extends the scaled boundary finite element method sbfem to non deterministic framework defined on random domain wherein random behaviour is exhibited in the presence of the random field uncertainties the aim is to blend the sbfem into the galerkin spectral stochastic methods which leads to a proficient procedure for handling the stress singularity problems and crack analysis the young s modulus of structures is considered to have random field uncertainty resulting in the stochastic behaviour of responses mathematical expressions and the solution procedure are derived to evaluate the statistical characteristics of responses displacement strain and stress and stress intensity factors of cracked structures the feasibility and effectiveness of the presented method are demonstrated by particularly chosen numerical examples this article is protected by copyright all rights reserved

stochastic modeling of uncertainties in computational structural dynamics recent theoretical advances this paper deals with a short overview on stochastic modeling of uncertainties we introduce the types of uncertainties the variability of real systems the types of probabilistic approaches the representations for the stochastic models of uncertainties the construction of the stochastic models using the maximum entropy principle the propagation of uncertainties the methods to solve the stochastic dynamical equations the identification of the prior and the posterior stochastic models the robust updating of the computational models and the robust design with uncertain computational models we present recent theoretical advances in this field concerning the parametric and the nonparametric probabilistic approaches of uncertainties in computational structural dynamics for the construction of the prior stochastic models of both the uncertainties on the computational model parameters and on the modeling uncertainties and for their identification with experimental data we also present the construction of the posterior stochastic model of uncertainties using the bayesian method

fourier mode analysis of multigrid methods for partial differential equations with random coefficients partial differential equations with random coefficients appear for example in reliability problems and uncertainty propagation models various approaches exist for computing the stochastic characteristics of the solution of such a differential equation in this paper we consider the spectral expansion approach this method transforms the continuous model into a large discrete algebraic system we study the convergence properties of iterative methods for solving this discretized system we consider one level and multi level methods the classical fourier mode analysis technique is extended towards the stochastic case this is done by taking the eigenstructure into account of a certain matrix that depends on the random structure of the problem we show how the convergence properties depend on the particulars of the algorithm on the discretization parameters and on the stochastic characteristics of the model numerical results are added to illustrate some of our theoretical findings

the bird s eye view on finite element method for structures with large stochastic variations abstract extensive motivation to do an additional work in the finite element method in stochastic problems femsp is discussed the qualitative comparison of femsp with the state of the art of the deterministic fem is given these observations and thoughts are then realized in several manners we first present the exact inverse femsp which however cannot serve as a general tool for stochastic analysis of complex structures the new variational principles for stochastic beams for the mean response function as well as response s auto correlation function are formulated and the fem based on the variational principles is presented finally a general non perturbative fem for stochastic problems with large variations is developed based on the element level flexibility it is concluded that much work needs to be done in order femsp to be at the level compared to that of the deterministic fem the advantage of the main methods presented here over the conventional ones lies in their non perturbational nature numerical examples are presented

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

symmetric four dimensional polytope and visualization method in four eight and sixteen dimensions using hopf maps inspired by and using methods of optimization derived from classical three dimensional electrostatics we have found a four dimensional polytope new to our knowledge with a high degree of symmetry in terms of the lengths of sides 64 of the 80 vertices have twelve nearest neighbors with the same four nearest neighbor distances and the other 16 vertices have ten nearest neighbors with distances that are two of the four nearest neighbor distances for the set of 64 vertices we give and illustrate a simple geometric method to visualize this configuration and other configurations in four eight and sixteen dimensions

finite dimensional models for random functions abstract truncated karhunen loeve kl representations are used to construct finite dimensional fd models for non gaussian functions with finite variances the second moment specification of the random coefficients of these representations are enhanced to full probabilistic characterization by using translation polynomial chaos and translated polynomial chaos models referred to as t pc and pct models following theoretical considerations on kl representations and t pc and pct models three numerical examples are presented to illustrate the implementation and performance of these models the pct models inherit the desirable features of both t and pc models it approximates accurately all quantities of interest considered in these examples

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

modified deflation algorithm for the solution of singular problems i a system of nonlinear algebraic equations abstract the behavior of the newton raphson method at the singular roots has been studied by a number of authors and the convergence of the newton raphson sequence has been shown to be linear in this paper a new method with symbolic and numerical manipulations termed the modified deflation algorithm is proposed for the singular root of a system of nonlinear algebraic equations the basic idea of the present method is to replace a part of the original equations by a set of new equations which pass through the singular root according to the method both convergency and accuracy can greatly be improved in addition it is often possible to obtain analytically the singular root from the new equations in order to show the effectiveness of the present method two illustrative examples are solved

geometry of quantum systems density states and entanglement various problems concerning the geometry of the space of hermitian operators on a hilbert space are addressed in particular we study the canonical poisson and riemann jordan tensors and the corresponding foliations into k hler submanifolds it is also shown that the space of density states on an n dimensional hilbert space is naturally a manifold stratified space with the stratification induced by the the rank of the state thus the space of rank k states k 1 n is a smooth manifold of real dimension 2nk k2 1 and this stratification is maximal in the sense that every smooth curve in viewed as a subset of the dual to the lie algebra of the unitary group at every point must be tangent to the strata it crosses for a quantum composite system i e for a hilbert space decomposition an abstract criterion of entanglement is proved

on iterative methods for the quadratic matrix equation with m matrix in this paper we consider newton s method and bernoulli s method for a quadratic matrix equation arising from an overdamped vibrating system by introducing m matrix to this equation we provide a sufficient condition for the existence of the primary solution moreover we show that newton s method and bernoulli s method with an initial zero matrix converge to the primary solvent under the proposed sufficient condition

adaptive weak approximation of stochastic differential equations adaptive time stepping methods based on the monte carlo euler method for weak approximation of ito stochastic differential equations are developed the main result is new expansions of the computational error with computable leading order term in a posteriori form based on stochastic flows and discrete dual backward problems the expansions lead to efficient and accurate computation of error estimates adaptive algorithms for either stochastic time steps or deterministic time steps are described numerical examples illustrate when stochastic and deterministic adaptive time steps are superior to constant time steps and when adaptive stochastic steps are superior to adaptive deterministic steps stochastic time steps use brownian bridges and require more work for a given number of time steps deterministic time steps may yield more time steps but require less work for example in the limit of vanishing error tolerance the ratio of the computational error and its computable estimate tends to 1 with negligible additional work to determine the adaptive deterministic time steps c 2001 john wiley sons inc

the geometry of multi qubit entanglement this paper a continuation of a previous one iwai 2007 j phys a math theor 40 1361 studies the geometry of multi qubit entanglement with respect to bipartite partitions an n qubit system c2 n is isomorphic with the linear space of 2l 2m complex matrices where l m n according to the isomorphism the local transformation group u 2l u 2m acts on the space of normalized states in let m and g denote the space of normalized states and the local transformation group acting on m respectively according to the orbit types m is stratified into strata among which a principal stratum and the sets of separable states and maximally entangled states will be identified for a function f c det i cc proves to serve as a measure of entanglement where i denotes the 2l 2l identity matrix the f c attains the minimal and the maximal values respectively on the sets of separable states and maximally entangled states and further takes no extremal values on the principal stratum the f c projects to a function on the factor space g m a naturally defined metric on m also projects to that on g m which serves to measure the distance between the separable states f 1 0 and the states f 1 k of prescribed value k of measure to be precise one has to restrict m to the principal stratum when projecting the metric three and four qubit systems will be studied intensively and then multi qubit systems discussed

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

optimal model management for multifidelity monte carlo estimation this work presents an optimal model management strategy that exploits multifidelity surrogate models to accelerate the estimation of statistics of outputs of computationally expensive high fidelity models existing acceleration methods typically exploit a multilevel hierarchy of surrogate models that follow a known rate of error decay and computational costs however a general collection of surrogate models which may include projection based reduced models data fit models support vector machines and simplified physics models does not necessarily give rise to such a hierarchy our multifidelity approach provides a framework to combine an arbitrary number of surrogate models of any type instead of relying on error and cost rates an optimization problem balances the number of model evaluations across the high fidelity and surrogate models with respect to error and costs we show that a unique analytic solution of the model management optimization problem exists under mild conditions on the models ou

the h p and h p versions of the finite element method in 1 dimension this paper is the first one in the series of three which are addressing in detail the properties of the three basic versions of the finite element method in the one dimensional setting the main emphasis is placed on the analysis when the exact solution has singularity of x type the first part analyzes thep version the second theh version and generalh p version and the final third part addresses the problems of the adaptiveh p version

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

effects of uncertainties in the domain on the solution of dirichlet boundary value problems a domain with possibly non lipschitz boundary is defined as a limit of monotonically expanding or shrinking domains with lipschitz boundary a uniquely solvable dirichlet boundary value problem dbvp is defined on each of the lipschitz domains and the limit of these solutions is investigated the limit function also solves a dbvp on the limit domain but the problem can depend on the sequences of domains if the limit domain is unstable with respect to the dbvp the core of the paper consists in estimates of the difference between the respective solutions of the dbvp on two close domains one of which is lipschitz and the other can be unstable estimates for starshaped as well as rather general domains are derived their numerical evaluation is possible and can be done in different ways

fe2 multiscale in linear elasticity based on parametrized microscale models using proper generalized decomposition abstract in recent years a tremendous growth of activity in multiscale modelling has been produced to get the mechanical response of highly heterogenous materials where the complexity of solving numerically all microscale details is not feasible in this paper we present a multiscale framework based on the decomposition of the displacement field into coarse macro and fine micro scales earlier proposed in the variational multiscale approach or the hp d method the novelty of this work lies in solving the microscale step where a multidimensional parametrized model of a generic rve is solved by means of the multidimensional model reduction technique named as proper generalized decomposition pgd as result of this previous and off line step the displacement field over the rve is obtained by simple algebraic operations for any combination of parameters boundary condition material properties loads etc allowing a significant reduction in computational cost when solving the macro scale problem for problems where the detailed structure of the localized displacement strain and stress fields are of interest recovery of the fine scale components can be performed immediately the basic concepts and several 1 d and 2 d linear elastic problems are presented to demonstrate the robustness of the formulation and computer time saving

propagation of probabilistic uncertainty in complex physical systems using a stochastic finite element approach abstract this paper presents an efficient procedure for characterizing the solution of evolution equations with stochastic coefficients these typically model the behavior of physical systems whose properties are modeled as spatially or temporally varying stochastic processes described within the framework of probability theory the concepts of projection orthogonality and weak convergence are exploited in a manner which directly mimics deterministic finite element solutions except that in the stochastic case inner products refer to expectation operations specifically the karhunen loeve expansion is used to discretize these processes into a denumerable set of random variables thus providing a denumerable function space in which the problem is cast the polynomial chaos expansion is then used to represent the solution in this space and the coefficients in the expansion are evaluated as generalized fourier coefficients via a galerkin procedure in the hilbert space of random variables

a kochen specker theorem for integer matrices and noncommutative spectrum functors abstract we investigate the possibility of constructing kochen specker uncolorable sets of idempotent matrices whose entries lie in various rings including the rational numbers the integers and finite fields most notably we show that there is no kochen specker coloring of the n n idempotent integer matrices for n 3 thereby illustrating that kochen specker contextuality is an inherent feature of pure matrix algebra we apply this to generalize recent no go results on noncommutative spectrum functors showing that any contravariant functor from rings to sets respectively topological spaces or locales that restricts to the zariski prime spectrum functor for commutative rings must assign the empty set respectively empty space or locale to the matrix ring m n r for any integer n 3 and any ring r an appendix by alexandru chirvasitu shows that kochen specker colorings of idempotents in partial subalgebras of m 3 f for a perfect field f can be extended to partial algebra morphisms into the algebraic closure of f

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

superlinearly convergent algorithms for solving singular equations and smooth reformulations of complementarity problems we propose a new algorithm for solving smooth nonlinear equations in the case where their solutions can be singular compared to other techniques for computing singular solutions a distinctive feature of our approach is that we do not employ second derivatives of the equation mapping in the algorithm and we do not assume their existence in the convergence analysis important examples of once but not twice differentiable equations whose solutions are inherently singular are smooth equation based reformulations of the nonlinear complementarity problems reformulations of complementarity problems serve both as illustration of and motivation for our approach and one of them we consider in detail we show that the proposed method possesses local superlinear quadratic convergence under reasonable assumptions we further demonstrate that these assumptions are in general not weaker and not stronger than regularity conditions employed in the context of other superlinearly convergent newton type algorithms for solving complementarity problems which are typically based on nonsmooth reformulations therefore our approach appears to be an interesting complement to the existing ones

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

on the geometry of a class of n qubit entanglement monotones a family of n qubit entanglement monotones invariant under stochastic local operations and classical communication slocc is defined this class of entanglement monotones includes the well known examples of the concurrence the 3 tangle and some of the four five and n qubit slocc invariants introduced recently the construction of these invariants is based on bipartite partitions of the hilbert space in the form with l 2n n l 2n such partitions can be given a nice geometrical interpretation in terms of grassmannians gr l l of l planes in cl that can be realized as the zero locus of quadratic polynomials in the complex projective space of suitable dimension via the plucker embedding the invariants are neatly expressed in terms of the plucker coordinates of the grassmannians

bloch sphere like construction of su 3 hamiltonians using unitary integration the bloch sphere is a familiar and useful geometrical picture of the time evolution of a single spin or a quantal two level system the analogous geometrical picture for three level systems is presented with several applications the relevant su 3 group and su 3 algebra are eight dimensional objects and are realized in our picture as two four dimensional manifolds that describe the time evolution operator the first called the base manifold is the counterpart of the s2 bloch sphere whereas the second called the fiber generalizes the single u 1 phase of a single spin now four dimensional it breaks down further into smaller objects depending on alternative representations that we discuss geometrical phases are also developed and presented for specific applications arbitrary time dependent couplings between three levels or between two spins qubits with su 3 hamiltonians can be conveniently handled through these geometrical objects

verified error bounds for isolated singular solutions of polynomial systems in this paper we generalize the algorithm described by rump and graillat as well as our previous work on verifying breadth one singular solutions of polynomial systems to compute verified and narrow error bounds such that a slightly perturbed system is guaranteed to possess an isolated singular solution within computed error bounds our verification method is based on deflation techniques using smoothing parameters we demonstrate the performance of the algorithm for polynomial systems with singular solutions of multiplicity up to hundreds

effects of uncertainties in the domain on the solution of neumann boundary value problems in two spatial dimensions an essential part of any boundary value problem is the domain on which the problem is defined the domain is often given by scanning or another digital image technique with limited resolution this leads to significant uncertainty in the domain definition the paper focuses on the impact of the uncertainty in the domain on the neumann boundary value problem nbvp it studies a scalar nbvp defined on a sequence of domains the sequence is supposed to converge in the set sense to a limit domain then the respective sequence of nbvp solutions is examined first it is shown that the classical variational formulation is not suitable for this type of problem as even a simple nbvp on a disk approximated by a pixel domain differs much from the solution on the original disk with smooth boundary a new definition of the nbvp is introduced to avoid this difficulty by means of reformulated natural boundary conditions then the convergence of solutions of the nbvp is demonstrated the uniqueness of the limit solution however depends on the stability property of the limit domain finally estimates of the difference between two nbvp solutions on two different but close domains are given

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

stochastic finite elements of discretely parameterized random systems on domains with boundary uncertainty the problem of representing random fields describing the material and boundary properties of the physical system at discrete points of the spatial domain is studied in the context of linear stochastic finite element method a randomly parameterized diffusion system with a set of independent identically distributed stochastic variables is considered the discretized parametric fields are interpolated within each element with multidimensional lagrange polynomials and integrated into the weak formulation the proposed discretized random field representation has been utilized to express the random fluctuations of the domain boundary with nodal position coordinates and a set of random variables the description of the boundary perturbation has been incorporated into the weak stochastic finite element formulation using a stochastic isoparametric mapping of the random domain to a deterministic master domain a method for obtaining the linear system of equations under the proposed mapping using generic finite element weak formulation and the stochastic spectral galerkin framework is studied in detail the treatment presents a unified way of handling the parametric uncertainty and random boundary fluctuations for dynamic systems the convergence behavior of the proposed methodologies has been demonstrated with numerical examples to establish the validity of the numerical scheme

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 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

stability of singular equilibria in quasilinear implicit differential equations abstract this paper addresses stability properties of singular equilibria arising in quasilinear implicit odes under certain assumptions local dynamics near a singular point may be described through a continuous or directionally continuous vector field this fact motivates a classification of geometric singularities into weak and strong ones stability in the weak case is analyzed through certain linear matrix equations a singular version of the lyapunov equation being especially relevant in the study weak stable singularities include singular zeros having a spherical domain of attraction which contains other singular points regarding strong equilibria stability is proved via a lyapunov schmidt approach under additional hypotheses the results are shown to be relevant in singular root finding problems

geometry of a two qubit state and intertwining quaternionic conformal mapping under local unitary transformations in this paper the geometry of two qubit systems under a local unitary group so 2 su 2 is discussed it is shown that the quaternionic conformal map intertwines between this local unitary subgroup of sp 2 and the quaternionic mobius transformation which is rather a generalization of the results of lee et al 2002 quantum inf process 1 129