estimation of nonstationary spatial covariance structure summary we introduce a method for estimating nonstationary spatial covariance structure from space time data and apply the method to an analysis of sydney wind patterns our method constructs a process honouring a given spatial covariance matrix at observing stations and uses one or more stationary processes to describe conditional behaviour given observing site values the stationary processes give a localised description of the spatial covariance structure the method is computationally attractive and can be extended to the assessment of covariance for multivariate processes the technique is illustrated for data describing the east west component of sydney winds for this example our own methods are contrasted with a geometrically appealing though computationally intensive technique which describes spatial correlation via an isotropic process and a deformation of the geographical space

fast algorithms for bayesian uncertainty quantification in large scale linear inverse problems based on low rank partial hessian approximations we consider the problem of estimating the uncertainty in large scale linear statistical inverse problems with high dimensional parameter spaces within the framework of bayesian inference when the noise and prior probability densities are gaussian the solution to the inverse problem is also gaussian and is thus characterized by the mean and covariance matrix of the posterior probability density unfortunately explicitly computing the posterior covariance matrix requires as many forward solutions as there are parameters and is thus prohibitive when the forward problem is expensive and the parameter dimension is large however for many ill posed inverse problems the hessian matrix of the data misfit term has a spectrum that collapses rapidly to zero we present a fast method for computation of an approximation to the posterior covariance that exploits the low rank structure of the preconditioned by the prior covariance hessian of the data misfit analysis of an infinite dimensional model convection diffusion problem and numerical experiments on large scale three dimensional convection diffusion inverse problems with up to 1 5 million parameters demonstrate that the number of forward pde solves required for an accurate low rank approximation is independent of the problem dimension this permits scalable estimation of the uncertainty in large scale ill posed linear inverse problems at a small multiple independent of the problem dimension of the cost of solving the forward problem

computer model calibration using high dimensional output this work focuses on combining observations from field experiments with detailed computer simulations of a physical process to carry out statistical inference of particular interest here is determining uncertainty in resulting predictions this typically involves calibration of parameters in the computer simulator as well as accounting for inadequate physics in the simulator the problem is complicated by the fact that simulation code is sufficiently demanding that only a limited number of simulations can be carried out we consider applications in characterizing material properties for which the field data and the simulator output are highly multivariate for example the experimental data and simulation output may be an image or may describe the shape of a physical object we make use of the basic framework of kennedy and o hagan however the size and multivariate nature of the data lead to computational challenges in implementing the framework to overcome these challenges we make use of basis repre

scaling properties of scale free evolving networks continuous approach the scaling behavior of scale free evolving networks arising in areas such as communications scientific citations collaborations etc is studied we derive universal scaling relations describing properties of such networks and indicate the limits of their validity we show that the main properties of scale free evolving networks may be described in the framework of a simple continuous approach the simplest models of networks growing according to a mechanism of preferential attachment of links to nodes are used we consider different forms of this preference and demonstrate that the range of preferential attachments producing scale free networks is wide we also obtain scaling relations for networks with nonlinear accelerating growth and describe the temporal evolution of the arising distributions size effects char22 the cutoffs of these distributions char22 introduce restrictions for the observation of power law dependences mainly we discuss the so called degree distribution i e the distribution of the number of connections of nodes a scaling form of the distribution of links between pairs of individual nodes for a growing network of citations is also studied we describe the effects of differences between nodes the aging of nodes changes the exponents of the distributions the appearance of a single node with high fitness changes the degree distribution of a network dramatically if its fitness exceeds some threshold value this node captures a finite part of all links of the network we show that permanent random damage to a growing scale free network char22 a permanent deletion of some links char22 radically changes the values of the scaling exponents results of other kinds of permanent damage are described

a stochastic newton mcmc method for large scale statistical inverse problems with application to seismic inversion we address the solution of large scale statistical inverse problems in the framework of bayesian inference the markov chain monte carlo mcmc method is the most popular approach for sampling the posterior probability distribution that describes the solution of the statistical inverse problem mcmc methods face two central difficulties when applied to large scale inverse problems first the forward models typically in the form of partial differential equations that map uncertain parameters to observable quantities make the evaluation of the probability density at any point in parameter space very expensive and second the high dimensional parameter spaces that arise upon discretization of infinite dimensional parameter fields make the exploration of the probability density function prohibitive the challenge for mcmc methods is to construct proposal functions that simultaneously provide a good approximation of the target density while being inexpensive to manipulate here we present a so called stoch

uncertainty in prior elicitations a nonparametric approach summary a key task in the elicitation of expert knowledge is to construct a distribution from the finite and usually small number of statements that have been elicited from the expert these statements typically specify some quantiles or moments of the distribution such statements are not enough to identify the expert s probability distribution uniquely and the usual approach is to fit some member of a convenient parametric family there are two clear deficiencies in this solution first the expert s beliefs are forced to fit the parametric family secondly no account is then taken of the many other possible distributions that might have fitted the elicited statements equally well we present a nonparametric approach which tackles both of these deficiencies we also consider the issue of the imprecision in the elicited probability judgements

information based objective functions for active data selection learning can be made more efficient if we can actively select particularly salient data points within a bayesian learning framework objective functions are discussed that measure the expected informativeness of candidate measurements three alternative specifications of what we want to gain information about lead to three different criteria for data selection all these criteria depend on the assumption that the hypothesis space is correct which may prove to be their main weakness

emulation of higher order tensors in manifold monte carlo methods for bayesian inverse problems the bayesian approach to inverse problems relies predominantly on markov chain monte carlo methods for posterior inference the typical nonlinear concentration of posterior measure observed in many such inverse problems presents severe challenges to existing simulation based inference methods motivated by these challenges the exploitation of local geometric information in the form of covariant gradients metric tensors levi civita connections and local geodesic flows have been introduced to more effectively locally explore the configuration space of the posterior measure however obtaining such geometric quantities usually requires extensive computational effort and despite their effectiveness affects the applicability of these geometrically based monte carlo methods in this paper we explore one way to address this issue by the construction of an emulator of the model from which all geometric objects can be obtained in a much more computationally feasible manner the main concept is to approximate the geometric quantities using a gaussian process emulator which is conditioned on a carefully chosen design set of configuration points which also determines the quality of the emulator to this end we propose the use of statistical experiment design methods to refine a potentially arbitrarily initialized design online without destroying the convergence of the resulting markov chain to the desired invariant measure the practical examples considered in this paper provide a demonstration of the significant improvement possible in terms of computational loading suggesting this is a promising avenue of further development

analyzing nonstationary spatial data using piecewise gaussian processes in many problems in geostatistics the response variable of interest is strongly related to the underlying geology of the spatial location in these situations there is often little correlation in the responses found in different rock strata so the underlying covariance structure shows sharp changes at the boundaries of the rock types conventional stationary and nonstationary spatial methods are inappropriate because they typically assume that the covariance between points is a smooth function of distance in this article we propose a generic method for the analysis of spatial data with sharp changes in the underlying covariance structure our method works by automatically decomposing the spatial domain into disjoint regions within which the process is assumed to be stationary but the data are assumed independent across regions uncertainty in the number of disjoint regions their shapes and the model within regions is dealt with in a fully bayesian fashion we illustrate our approach on a previously

parallel lagrange newton krylov schur methods for pde constrained optimization part i the krylov schur solver large scale optimization of systems governed by partial differential equations pdes is a frontier problem in scientific computation reduced quasi newton sequential quadratic programming sqp methods are state of the art approaches for such problems these methods take full advantage of existing pde solver technology and parallelize well however their algorithmic scalability is questionable for certain problem classes they can be very slow to converge in this two part article we propose a new method for steady state pde constrained optimization based on the idea of using a full space newton solver combined with an approximate reduced space quasi newton sqp preconditioner the basic components of the method are newton solution of the first order optimality conditions that characterize stationarity of the lagrangian function krylov solution of the karush kuhn tucker kkt linear systems arising at each newton iteration using a symmetric quasi minimum residual method preconditioning of the kkt system using an approximate state decision variable decomposition that replaces the forward pde jacobians by their own preconditioners and the decision space schur complement the reduced hessian by a bfgs approximation initialized by a two step stationary method accordingly we term the new method it lagrange newton krylov schur lnks it is fully parallelizable exploits the structure of available parallel algorithms for the pde forward problem and is locally quadratically convergent in part i of this two part article we investigate the effectiveness of the kkt linear system solver we test our method on two optimal control problems in which the state constraints are described by the steady state stokes equations the objective is to minimize dissipation or the deviation from a given velocity field the control variables are the boundary velocities numerical experiments on up to 256 cray t3e processors and on an sgi origin 2000 include scalability and performance assessment of the lnks algorithm and comparisons with reduced sqp for up to 1 000 000 state and 50 000 decision variables in part ii of the article we address globalization and inexactness issues and apply lnks to the optimal control of the steady incompressible navier stokes equations

certified reduced basis model validation a frequentistic uncertainty framework abstract we introduce a frequentistic validation framework for assessment acceptance or rejection of the consistency of a proposed parametrized partial differential equation model with respect to noisy experimental data from a physical system our method builds upon the hotelling t2 statistical hypothesis test for bias first introduced by balci and sargent in 1984 and subsequently extended by mcfarland and mahadevan 2008 our approach introduces two new elements a spectral representation of the misfit which reduces the dimensionality and variance of the underlying multivariate gaussian but without introduction of the usual regression assumptions a certified verified reduced basis approximation reduced order model which greatly accelerates computational performance but without any loss of rigor relative to the full finite element discretization we illustrate our approach with examples from heat transfer and acoustics both based on synthetic data we demonstrate that we can efficiently identify possibility regions that characterize parameter uncertainty furthermore in the case that the possibility region is empty we can deduce the presence of unmodeled physics such as cracks or heterogeneities

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

vector coherent state representation theory a vector coherent state theory is formulated as a natural extension of standard coherent state theory it is shown that the godement representations and the coherent state representations of the sp n r groups of rowe and of deenen and quesne are special cases of this more general theory

model reduction for large scale systems with high dimensional parametric input space a model constrained adaptive sampling methodology is proposed for the reduction of large scale systems with high dimensional parametric input spaces our model reduction method uses a reduced basis approach which requires the computation of high fidelity solutions at a number of sample points throughout the parametric input space a key challenge that must be addressed in the optimization control and probabilistic settings is the need for the reduced models to capture variation over this parametric input space which for many applications will be of high dimension we pose the task of determining appropriate sample points as a pde constrained optimization problem which is implemented using an efficient adaptive algorithm that scales well to systems with a large number of parameters the methodology is demonstrated using examples with parametric input spaces of dimension 11 and 21 which describe thermal analysis and design of a heat conduction fin and compared with statistically based sampling methods for these examples the model constrained adaptive sampling leads to reduced models that for a given basis size have error several orders of magnitude smaller than that obtained using the other methods

nonparametric estimation of nonstationary spatial covariance structure abstract estimation of the covariance structure of spatial processes is a fundamental prerequisite for problems of spatial interpolation and the design of monitoring networks we introduce a nonparametric approach to global estimation of the spatial covariance structure of a random function z x t observed repeatedly at times ti i 1 t at a finite number of sampling stations xi i 1 2 n in the plane our analyses assume temporal stationarity but do not assume spatial stationarity or isotropy we analyze the spatial dispersions var z xi t z xj t as a natural metric for the spatial covariance structure and model these as a general smooth function of the geographic coordinates of station pairs xi xj the model is constructed in two steps first using nonmetric multidimensional scaling mds we compute a two dimensional representation of the sampling stations for which a monotone function of interpoint distances ij approximates the spatial dispersions mds transforms the problem

bayesian inference for non stationary spatial covariance structure via spatial deformations summary in geostatistics it is common practice to assume that the underlying spatial process is stationary and isotropic i e the spatial distribution is unchanged when the origin of the index set is translated and under rotation about the origin however in environmental problems such assumptions are not realistic since local influences in the correlation structure of the spatial process may be found in the data the paper proposes a bayesian model to address the anisot ropy problem following sampson and guttorp we define the correlation function of the spatial process by reference to a latent space denoted by d where stationarity and isotropy hold the space where the gauged monitoring sites lie is denoted by g we adopt a bayesian approach in which the mapping between g and d is represented by an unknown function d a gaussian process prior distribution is defined for d unlike the sampson guttorp approach the mapping of both gauged and ungauged sites is handled in a single framework and predictive inferences take explicit account of uncertainty in the mapping markov chain monte carlo methods are used to obtain samples from the posterior distributions two examples are discussed a simulated data set and the solar radiation data set that also was analysed by sampson and guttorp

spr based markov chain method for degree distributions of evolving networks abstract in this paper we develop a stochastic process rules spr based markov chain method to calculate the degree distributions of evolving networks this new approach overcomes two shortcomings of shi chen and liu s use of the markov chain method shi et al 2005 21 in addition we show how an spr based markov chain method can be effectively used to calculate degree distributions of random birth and death networks which we believe to be novel first spr are introduced to replace traditional evolving rules tr making it possible to compute degree distributions in one sample space then the spr based markov chain method is introduced and tested by using it to calculate two kinds of evolving network finally and most importantly the spr based method is applied to the problem of calculating the degree distributions of random birth and death networks

dynamical group of microscopic collective states i one dimensional case in the present series of papers it is intended to determine the nature and study various realizations of the dynamical group of microscopic collective states for an a nucleon system defined as those a particle states invariant under the orthogonal group o n associated with the n a 1 jacobi vectors the present paper discusses the case of a hypothetical one dimensional space simple invariance considerations show that the dynamical group of collective states is then the group spc 2 r which is the restriction to the collective subspace of the group sp 2 r of linear canonical transformations in n dimensions conserving the o n symmetry in addition to the well known realization of the dynamical group in the schrodinger representation based upon the dzublik zickendraht transformation two new realizations are proposed the first acts in a barut hilbert space which is the subspace of a bargmann hilbert space of analytic functions left invariant by o n a unitary mapping is established between the ordi

the modeling of scale free networks abstract in order to explore further the mechanism responsible for scale free networks we introduce two extended models of the ba model the model a where the system incorporates the addition of new links between existing nodes a new node with new links and the rewiring of some links at every time step all sites are born with some initial attractiveness we calculate analytically the degree distribution the system self organizes into a scale free network the scaling exponent 2 the model b is a new model we consider that some old links are deleted with the anti preferential probability the result indicates that the system evolves itself into a scale free network the scaling exponent varies from 2 to 3

near optimal sensor placements in gaussian processes theory efficient algorithms and empirical studies when monitoring spatial phenomena which can often be modeled as gaussian processes gps choosing sensor locations is a fundamental task there are several common strategies to address this task for example geometry or disk models placing sensors at the points of highest entropy variance in the gp model and a d or e optimal design in this paper we tackle the combinatorial optimization problem of maximizing the mutual information between the chosen locations and the locations which are not selected we prove that the problem of finding the configuration that maximizes mutual information is np complete to address this issue we describe a polynomial time approximation that is within 1 1 e of the optimum by exploiting the submodularity of mutual information we also show how submodularity can be used to obtain online bounds and design branch and bound search procedures we then extend our algorithm to exploit lazy evaluations and local structure in the gp yielding significant speedups we also extend our approach to find placements which are robust against node failures and uncertainties in the model these extensions are again associated with rigorous theoretical approximation guarantees exploiting the submodularity of the objective function we demonstrate the advantages of our approach towards optimizing mutual information in a very extensive empirical study on two real world data sets

stochastic spectral methods for efficient bayesian solution of inverse problems we present a reformulation of the bayesian approach to inverse problems that seeks to accelerate bayesian inference by using polynomial chaos pc expansions to represent random variables evaluation of integrals over the unknown parameter space is recast more efficiently as monte carlo sampling of the random variables underlying the pc expansion we evaluate the utility of this technique on a transient diffusion problem arising in contaminant source inversion the accuracy of posterior estimates is examined with respect to the order of the pc representation the choice of pc basis and the decomposition of the support of the prior the computational cost of the new scheme shows significant gains over direct sampling

markov chains theory for scale free networks abstract this paper proposes a markov chain method to predict the growth dynamics of the individual nodes in scale free networks and uses this to calculate numerically the degree distribution we first find that the degree evolution of a node in the ba model is a nonhomogeneous markov chain an efficient algorithm to calculate the degree distribution is developed by the theory of markov chains the numerical results for the ba model are consistent with those of the analytical approach a directed network with the logarithmic growth is introduced the algorithm is applied to calculate the degree distribution for the model the numerical results show that the system self organizes into a scale free network

a highly clustered scale free network evolved by random walking abstract in present paper we propose a highly clustered weighted network model that incorporates the addition of a new node with some links new links between existing nodes and the edge s weight dynamical evolution based on weight dependent walks at each time step the analytical approach and numerical simulation show that the system grows into a weighted network with the power law distributions of strength weight and degree the weight dependent walk length l will not influence the strength distribution but the clustering coefficient of the network is sensitive to l particularly the clustering coefficient is especially high and almost independent of the network size when l 2

dimensionality reduction and polynomial chaos acceleration of bayesian inference in inverse problems we consider a bayesian approach to nonlinear inverse problems in which the unknown quantity is a spatial or temporal field endowed with a hierarchical gaussian process prior computational challenges in this construction arise from the need for repeated evaluations of the forward model e g in the context of markov chain monte carlo and are compounded by high dimensionality of the posterior we address these challenges by introducing truncated karhunen loeve expansions based on the prior distribution to efficiently parameterize the unknown field and to specify a stochastic forward problem whose solution captures that of the deterministic forward model over the support of the prior we seek a solution of this problem using galerkin projection on a polynomial chaos basis and use the solution to construct a reduced dimensionality surrogate posterior density that is inexpensive to evaluate we demonstrate the formulation on a transient diffusion equation with prescribed source terms inferring the spatially varying diffusivity of the medium from limited and noisy data

adaptive hessian based nonstationary gaussian process response surface method for probability density approximation with application to bayesian solution of large scale inverse problems we develop an adaptive hessian based non stationary gaussian process gp response surface method for approximating a probability density function pdf that exploits its structure particularly the hessian of its negative logarithm of particular interest to us are pdfs that arise from the bayesian solution of large scale inverse problems which imply very expensive to evaluate pdfs the method can be considered as a piecewise adaptive gaussian approximation in which a gaussian tailored to the local hessian of the negative log probability density is constructed for each subregion in high dimensional parameter space the task of efficiently partitioning the parameter space into subregions is done implicitly through hessian informed membership probability functions the gp machinery is then employed to glue all local gaussian approximations into a global analytical response surface that is far cheaper to evaluate than the original expensive probability density the resulting response surface is also equipp

vector coherent state theory and its application to the orthogonal groups vector coherent state theory is developed and presented in a form that explicitly exhibits its general applicability to the ladder representations of all semisimple lie groups and their lie algebras it is shown that in a suitable basis the vector coherent state inner product can be inferred algebraically by k matrix theory and changed to a simpler bargmann inner product thereby facilitating the explicit calculation of the matrix representaions of lie algebras applications are made to the even and odd orthogonal lie algebras

geometric mcmc for infinite dimensional inverse problems bayesian inverse problems often involve sampling posterior distributions on infinite dimensional function spaces traditional markov chain monte carlo mcmc algorithms are characterized by deteriorating mixing times upon mesh refinement when the finite dimensional approximations become more accurate such methods are typically forced to reduce step sizes as the discretization gets finer and thus are expensive as a function of dimension recently a new class of mcmc methods with mesh independent convergence times has emerged however few of them take into account the geometry of the posterior informed by the data at the same time recently developed geometric mcmc algorithms have been found to be powerful in exploring complicated distributions that deviate significantly from elliptic gaussian laws but are in general computationally intractable for models defined in infinite dimensions in this work we combine geometric methods on a finite dimensional subspace with mesh independent infinite dimensional approaches our objective is to speed up mcmc mixing times without significantly increasing the computational cost per step for instance in comparison with the vanilla preconditioned crank nicolson pcn method this is achieved by using ideas from geometric mcmc to probe the complex structure of an intrinsic finite dimensional subspace where most data information concentrates while retaining robust mixing times as the dimension grows by using pcn like methods in the complementary subspace the resulting algorithms are demonstrated in the context of three challenging inverse problems arising in subsurface flow heat conduction and incompressible flow control the algorithms exhibit up to two orders of magnitude improvement in sampling efficiency when compared with the pcn method

scale free characteristics of random networks the topology of the world wide web the world wide web forms a large directed graph whose vertices are documents and edges are links pointing from one document to another here we demonstrate that despite its apparent random character the topology of this graph has a number of universal scale free characteristics we introduce a model that leads to a scale free network capturing in a minimal fashion the self organization processes governing the world wide web c 2000 elsevier science b v all rights reserved pacs 84 35 i 64 60 fr 87 23 ge

bayesian calibration of computer models we consider prediction and uncertainty analysis for systems which are approximated using complex mathematical models such models implemented as computer codes are often generic in the sense that by a suitable choice of some of the model s input parameters the code can be used to predict the behaviour of the system in a variety of specific applications however in any specific application the values of necessary parameters may be unknown in this case physical observations of the system in the specific context are used to learn about the unknown parameters the process of fitting the model to the observed data by adjusting the parameters is known as calibration calibration is typically effected by ad hoc fitting and after calibration the model is used with the fitted input values to predict the future behaviour of the system we present a bayesian calibration technique which improves on this traditional approach in two respects first the predictions allow for all sources of uncertainty including the remaining uncertainty over the fitted parameters second they attempt to correct for any inadequacy of the model which is revealed by a discrepancy between the observed data and the model predictions from even the best fitting parameter values the method is illustrated by using data from a nuclear radiation release at tomsk and from a more complex simulated nuclear accident exercise

mean field theory for scale free random networks random networks with complex topology are common in nature describing systems as diverse as the world wide web or social and business networks recently it has been demonstrated that most large networks for which topological information is available display scale free features here we study the scaling properties of the recently introduced scale free model that can account for the observed power law distribution of the connectivities we develop a mean eld method to predict the growth dynamics of the individual vertices and use this to calculate analytically the connectivity distribution and the scaling exponents the mean eld method can be used to address the properties of two variants of the scale free model that do not display power law scaling c 1999 elsevier science b v all rights reserved