approximate likelihood for large irregularly spaced spatial data likelihood approaches for large irregularly spaced spatial datasets are often very difficult if not infeasible to implement due to computational limitations even when we can assume normality exact calculations of the likelihood for a gaussian spatial process observed at n locations requires o n3 operations we present a version of whittle s approximation to the gaussian log likelihood for spatial regular lattices with missing values and for irregularly spaced datasets this method requires o nlog2n operations and does not involve calculating determinants we present simulations and theoretical results to show the benefits and the performance of the spatial likelihood approximation method presented here for spatial irregularly spaced datasets and lattices with missing values we apply these methods to estimate the spatial structure of sea surface temperatures using satellite data with missing values

existence of nontrivial solutions for a semilinear elliptic problem with supercritical exponent let n 3 p 2 2n n 2 and rn be a bounded domain with a boundary our purpose in this paper is to consider the existence of solutions of problem p u up 1on u 0on u 0on

edge effects and efficient parameter estimation for stationary random fields summary we consider the estimation of the parameters of a stationary random field on d dimensional lattice by minimizing the classical whittle approximation to the gaussian log likelihood if the usual biased sample covariances are used the estimate is efficient only in one dimension to remove this edge effect we introduce data tapers and show that the resulting modified estimate is efficient also in two and three dimensions this avoids the use of the unbiased sample covariances which are in general not positive definite

hierarchical spatial process models for multiple traits in large genetic trials this article expands upon recent interest in bayesian hierarchical models in quantitative genetics by developing spatial process models for inference on additive and dominance genetic variance within the context of large spatially referenced trial datasets of multiple traits of interest direct application of such multivariate models to large spatial datasets is often computationally infeasible because of cubic order matrix algorithms involved in estimation the situation is even worse in markov chain monte carlo mcmc contexts where such computations are performed for several thousand iterations here we discuss approaches that help obviate these hurdles without sacrificing the richness in modeling for genetic effects we demonstrate how an initial spectral decomposition of the relationship matrices negates the expensive matrix inversions required in previously proposed mcmc methods for spatial effects we discuss a multivariate predictive process that reduces the computational burden by projecting th

a class of parallel two level nonlinear schwarz preconditioned inexact newton algorithms we propose and test a new class of two level nonlinear additive schwarz preconditioned inexact newton algorithms aspin the two level aspin combines a local nonlinear additive schwarz preconditioner and a global linear coarse preconditioner this approach is more attractive than the two level method introduced in cai keyes and marcinkowski nonlinear additive schwarz preconditioners and applications in computational fluid dynamics international journal numerical methods in fluids 40 2002 1463 1470 which is nonlinear on both levels since the coarse part of the global function evaluation requires only the solution of a linear coarse system rather than a nonlinear coarse system derived from the discretization of original partial dierential equations the overall computational cost is reduced considerably our parallel numerical results based on an incompressible lid driven flow problem show that the new two level aspin is quite scalable with respect to the number of processors and the fine mesh size when the coarse mesh size is fine enough and in addition the convergence is not sensitive to the reynolds numbers

radial solutions for the brezis nirenberg problem involving large nonlinearities abstract let us consider the problem 0 1 u a x u u p in b 1 u 0 in b 1 u 0 on b 1 where b 1 is the unit ball in r n n 3 and a x 0 is a smooth radial function under some suitable assumptions on the regular part of the green function of the operator u n 1 r u a r u we prove the existence of a radial solution to 0 1 for p large enough

overlapping schwarz methods for mixed linear elasticity and stokes problems overlapping schwarz preconditioners are introduced and studied for saddle point problems with a penalty term such as stokes equations and mixed formulations of linear elasticity these preconditioners are based on the solution of local saddle point problems on overlapping subdomains and the solution of a coarse saddle point problem numerical experiments show that these are parallel and scalable preconditioners since the rate of convergence of the preconditioned operator is independent of the mesh size h the number of subdomains n and the penalty parameter

iterative substructuring methods for spectral element discretizations of elliptic systems ii mixed methods for linear elasticity and stokes flow iterative substructuring methods are introduced and analyzed for saddle point problems with a penalty term two examples of saddle point problems are considered the mixed formulation of the linear elasticity system and the generalized stokes system in three dimensions these problems are discretized with mixed spectral element methods the resulting stiffness matrices are symmetric and indefinite the interior unknowns of each element are first implicitly eliminated by using exact local solvers the resulting saddle point schur complement is solved with a krylov space method with block preconditioners the velocity block can be approximated by a domain decomposition method e g of wire basket type which is constructed from a local solver for each face of the elements and a coarse solver related to the wire basket of the elements the condition number of the preconditioned operator is independent of the number of spectral elements and is bounded from above by the product of the square of the logarithm of the spectral degree and the inverse of the discrete inf sup constant of the problem

joint composite estimating functions in spatiotemporal models summary modelling of spatiotemporal processes has received considerable attention in recent statistical research however owing to the high dimensionality of the data the joint modelling of spatial and temporal processes presents a great computational challenge in both likelihood based and bayesian approaches we propose a joint composite estimating function approach to estimating spatiotemporal covariance structures this substantially reduces the computational complexity and is more efficient than existing composite likelihood methods the novelty of the proposed joint composite estimating function is rooted in the construction of three sets of estimating functions from spatial temporal and spatiotemporal cross pairs which results in overidentified estimating functions thus we form a joint inference function in a spirit that is similar to hansen s generalized method of moments we show that under practical scenarios the estimator proposed is consistent and asymptotically normal simulation studies prove that our method performs well in finite samples finally we illustrate the joint composite estimating function method by estimating the spatiotemporal dependence structure of airborne particulates pm10 in the north eastern usa over a 32 month period

nonparametric estimation of the variogram and its spectrum in the study of intrinsically stationary spatial processes a new nonparametric variogram estimator is proposed through its spectral representation the methodology is based on estimation of the variogram s spectrum by solving a regularized inverse problem through quadratic programming the estimated variogram is guaranteed to be conditionally negative definite simulation shows that our estimator is flexible and generally has smaller mean integrated squared error than the parametric estimator under model misspecification our methodology is applied to a spatial dataset of decadal temperature changes copyright 2011 oxford university press

spectral methods for nonstationary spatial processes summary we propose a nonstationary periodogram and various parametric approaches for estimating the spectral density of a nonstationary spatial process we also study the asymptotic properties of the proposed estimators via shrinking asymptotics assuming the distance between neighbouring observations tends to zero as the size of the observation region grows without bound with this type of asymptotic model we can uniquely determine the spectral density avoiding the aliasing problem we also present a new class of nonstationary processes based on a convolution of local stationary processes this model has the advantage that the model is simultaneously defined everywhere unlike moving window approaches but it retains the attractive property that locally in small regions it behaves like a stationary spatial process applications include the spatial analysis and modelling of air pollution data provided by the us environmental protection agency

two level newton and hybrid schwarz preconditioners for fluid structure interaction we introduce and study numerically a two level schwarz preconditioner for newton krylov methods for fluid structure interaction with special consideration of the application area of simulating blood flow our approach monolithically couples the fluid to the structure on both fine and coarse grids and in the subdomain solves insuring that there is multiphysics coupling during all aspects of the algorithm the fluid structure system is discretized on unstructured nonnested meshes with an overlapping additive domain decomposition on both coarse and fine levels and multiplicative schwarz preconditioning between levels we investigate the effect of different coarse discretization sizes solver stopping criteria and overlap size and we demonstrate that the method is robust to physical parameters including the structure s young s modulus and the timestep size finally we show effective preconditioning of the complicated coupled system with nearly perfect weak scaling to a thousand processors and millions of unknowns

parameter estimation for a stationary process on a d dimensional lattice summary we study asymptotic properties of various estimation procedures for a general stationary regular process on a d dimensional lattice differences between d 1 time series and d 2 spatial processes are pointed out we suppose that the process is observed on a set pn with cn points which tends to infinity with the same speed in all directions the relative edge effect is of order cn lid increasing with d this effect is without statistical importance if d 1 but is important if d 2 we give a lcn consistent asymptotically normal estimator of the underlying parameter the procedure being constructed by a modification of whittle s approximation to the log likelihood in the gaussian case this procedure is asymptotically efficient

covariance tapering for likelihood based estimation in large spatial data sets maximum likelihood is an attractive method of estimating covariance parameters in spatial models based on gaussian processes but calculating the likelihood can be computationally infeasible for large data sets requiring o n3 calculations for a data set with n observations this article proposes the method of covariance tapering to approximate the likelihood in this setting in this approach covariance matrixes are tapered or multiplied element wise by a sparse correlation matrix the resulting matrixes can then be manipulated using efficient sparse matrix algorithms we propose two approximations to the gaussian likelihood using tapering one of these approximations simply replaces the model covariance with a tapered version whereas the other is motivated by the theory of unbiased estimating equations focusing on the particular case of the matern class of covariance functions we give conditions under which estimators maximizing the tapering approximations are like the maximum likelihood estimat

spectral estimation of irregularly sampled multidimensional processes by generalized prolate spheroidal sequences a nonparametric spectral estimation method is presented for bandlimited random processes that have been sampled at arbitrary points in one or more dimensions the method makes simultaneous use of several weight sequences that depend on the set of sampling point the signal band and the frequency band being analyzed these sequences are solutions to a generalized matrix eigenvalue problem and are termed generalized prolate spheroidal sequences being extensions of the familiar discrete prolate spheroidal sequences statistics of the estimator are derived and the tradeoff among bias variance and resolution is quantified the method avoids several problems typically associated with irregularly sampled data and multidimensional processes a related method is suggested that has nearly as good performance while requiring significantly fewer computations

bddc algorithms for incompressible stokes equations the purpose of this paper is to extend the balancing domain decomposition by constraints bddc algorithm to saddle point problems that arise when mixed finite element methods are used to approximate the system of incompressible stokes equations the bddc algorithms are iterative substructuring methods which form a class of domain decomposition methods based on the decomposition of the domain of the differential equations into nonoverlapping subdomains they are defined in terms of a set of primal continuity constraints which are enforced across the interface between the subdomains and which provide a coarse space component of the preconditioner sets of such constraints are identified for which bounds on the rate of convergence can be established that are just as strong as previously known bounds for the elliptic case in fact the preconditioned operator is effectively positive definite which makes the use of a conjugate gradient method possible a close connection is also established between the bddc

positive solutions of elliptic equations involving supercritical growth positive radial solutions of elliptic equation involving supercritical growth are analysed as their supremum norm tends to infinity it is shown that they converge uniformly away from the origin as well as in h 1 to the unique singular solution

an overlapping schwarz method for spectral element solution of the incompressible navier stokes equations efficient solution of the navier stokes equations in complex domains is dependent upon the availability of fast solvers for sparse linear systems for unsteady incompressible flows the pressure operator is the leading contributor to stiffness as the characteristic propagation speed is infinite in the context of operator splitting formulations it is the pressure solve which is the most computationally challenging despite its elliptic origins we examine several preconditioners for the consistent l2poisson operator arising in the pn pn 2spectral element formulation of the incompressible navier stokes equations we develop a finite element based additive schwarz preconditioner using overlapping subdomains plus a coarse grid projection operator which is applied directly to the pressure on the interior gauss points for large two dimensional problems this approach can yield as much as a fivefold reduction in simulation time over previously employed methods based upon deflation

matern cross covariance functions for multivariate random fields we introduce a flexible parametric family of matrix valued covariance functions for multivariate spatial random fields where each constituent component is a matern process the model parameters are interpretable in terms of process variance smoothness correlation length and colocated correlation coefficients which can be positive or negative both the marginal and the cross covariance functions are of the matern type in a data example on error fields for numerical predictions of surface pressure and temperature over the north american pacific northwest we compare the bivariate matern model to the traditional linear model of coregionalization

nonexistence results for elliptic problems with supercritical nonlinearity in nontrivial domains let be a bounded domain in rn with n 3 and assume p 2n n 2 we study problems of the type u u p 2u 0 in u 0 on we prove that for suitable p 2n n 2 there exists some bounded domain with nontrivial topology in the sense of bahri and coron such that the above problem has only the solution u 0

an explicit link between gaussian fields and gaussian markov random fields the stochastic partial differential equation approach continuously indexed gaussian fields gfs are the most important ingredient in spatial statistical modelling and geostatistics the specification through the covariance function gives an intuitive interpretation of the field properties on the computational side gfs are hampered with the big n problem since the cost of factorizing dense matrices is cubic in the dimension although computational power today is at an all time high this fact seems still to be a computational bottleneck in many applications along with gfs there is the class of gaussian markov random fields gmrfs which are discretely indexed the markov property makes the precision matrix involved sparse which enables the use of numerical algorithms for sparse matrices that for fields in r 2 only use the square root of the time required by general algorithms the specification of a gmrf is through its full conditional distributions but its marginal properties are not transparent in such a parameterization we show that using an approximate stochastic weak solution to linear stochastic partial differential equations we can for some gfs in the matern class provide an explicit link for any triangulation of r d between gfs and gmrfs formulated as a basis function representation the consequence is that we can take the best from the two worlds and do the modelling by using gfs but do the computations by using gmrfs perhaps more importantly our approach generalizes to other covariance functions generated by spdes including oscillating and non stationary gfs as well as gfs on manifolds we illustrate our approach by analysing global temperature data with a non stationary model defined on a sphere less

the cost of continuity performance of iterative solvers on isogeometric finite elements in this paper we study how the use of a more continuous set of basis functions affects the cost of solving systems of linear equations resulting from a discretized galerkin weak form specifically we compare performance of linear solvers when discretizing using c 0 b splines which span traditional finite element spaces and c p 1 b splines which represent maximum continuity we provide theoretical estimates for the increase in cost of the matrix vector product as well as for the construction and application of black box preconditioners we accompany these estimates with numerical results and study their sensitivity to various grid parameters such as element size h and polynomial order of approximation p in addition to the aforementioned continuity of the basis finally we present timing results for a range of preconditioning options for the laplace problem we conclude that the matrix vector product operation is at most left 33p 2 middle 8 right times more expensive for the more cont

multi peak solutions for super critical elliptic problems in domains with small holes abstract this paper deals with the slightly super critical elliptic problem u u n 2 n 2 e in u 0 in u 0 on 1 where e 0 is a small parameter and r n is a bounded domain with smooth boundary assuming that the domain exhibits k sufficiently small holes multiple solutions are constructed by gluing double spike patterns located near each of the holes

a domain decomposition method for isogeometric multi patch problems with inexact local solvers abstract in isogeometric analysis the computational domain is often described as multi patch where each patch is given by a tensor product spline nurbs parametrization in this work we propose a feti like solver where local inexact solvers exploit the tensor product structure at the patch level to this purpose we extend to the isogeometric framework the so called all floating variant of feti that allows us to use the fast diagonalization method at the patch level we construct then a preconditioner for the whole system and prove its quasi robustness with respect to the local mesh size h and patch size h precisely the condition number of the preconditioned system is bounded by the square of the logarithm of h h our numerical tests confirm the theory and also show a favourable dependence of the computational cost of the method from the spline degree p

balancing neumann neumann methods for incompressible stokes equations this paper describes the unerlying mathematical model and the balancing neumann neumann methods are introduced and studied for incompressible stokes equations discretized with mixed finite or spectral elements with discontinuous pressures after decomposing the original domain of the problem into nonoverlapping subdomains the interior unknowns which are the interior velocity component and all except the constant pressure component of each subdomain problem are implicitly eliminated the resulting saddle point schur complement is solved with a krylov space method with a balancing neumann neumann preconditioner based on the solution of a coarse stokes problem with a few degrees of freedom per subdomain and on the solution of local stokes problems with natural neumann velocity and essential boundary conditions on the subdomains this preconditioner is of hybrid form in which the coarse problem is treated multiplicatively while the local problems are treated additively the condition number of the preconditioned operator is independent of the number of subdomains and is bounded from above by the product of the square of the logarithm of the local number of unknowns in each subdomain and the inverse of the inf sup constants of the discrete problem and of the coarse subproblem numerical results show that the method is quite fast they are also fully consistent with the theory

isogeometric bddc preconditioners with deluxe scaling a balancing domain decomposition by constraints bddc preconditioner with a novel scaling introduced by dohrmann for problems with more than one variable coefficient and here denoted as deluxe scaling is extended to isogeometric analysis of scalar elliptic problems this new scaling turns out to be more powerful than the standard rho and stiffness scalings considered in a previous isogeometric bddc study our h analysis shows that the condition number of the resulting deluxe bddc preconditioner is scalable with a quasi optimal polylogarithmic bound which is also independent of coefficient discontinuities across subdomain interfaces extensive numerical experiments support the theory and show that the deluxe scaling yields a remarkable improvement over the older scalings in particular for large isogeometric polynomial degree and high regularity

a multi resolution approximation for massive spatial datasets abstractautomated sensing instruments on satellites and aircraft have enabled the collection of massive amounts of high resolution observations of spatial fields over large spatial regions if these datasets can be efficiently exploited they can provide new insights on a wide variety of issues however traditional spatial statistical techniques such as kriging are not computationally feasible for big datasets we propose a multi resolution approximation m ra of gaussian processes observed at irregular locations in space the m ra process is specified as a linear combination of basis functions at multiple levels of spatial resolution which can capture spatial structure from very fine to very large scales the basis functions are automatically chosen to approximate a given covariance function which can be nonstationary all computations involving the m ra including parameter inference and prediction are highly scalable for massive datasets crucially the inference algorithms can also be parallelize

isogeometric block feti dp preconditioners for the stokes and mixed linear elasticity systems abstract the aim of this work is to construct and analyze a feti dp type domain decomposition preconditioner for isogeometric discretizations of the stokes and mixed linear elasticity systems this method extends to the isogeometric analysis context the preconditioner previously proposed by tu and li 2015 for finite element discretizations of the stokes system the resulting isogeometric feti dp algorithm is proven to be scalable in the number of subdomains and has a quasi optimal convergence rate bound which is polylogarithmic in the ratio of subdomain and element sizes extensive two dimensional numerical experiments validate the theory investigate the behavior of the preconditioner with respect to both the spline polynomial degree and regularity and show its robustness with respect to domain deformation material incompressibility and presence of elastic coefficient discontinuities across subdomain interfaces

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

fourier analysis of irregularly spaced data on rd summary the purpose of the paper is to propose a frequency domain approach for irregularly spaced data on rd we extend the original definition of a periodogram for time series to that for irregularly spaced data and define non parametric and parametric spectral density estimators in a way that is similar to the classical approach introduction of the mixed asymptotics which are one of the asymptotics for irregularly spaced data makes it possible to provide asymptotic theories to the spectral estimators the asymptotic result for the parametric estimator is regarded as a natural extension of the classical result for regularly spaced data to that for irregularly spaced data empirical studies are also included to illustrate the frequency domain approach in comparisons with the existing spatial and frequency domain approaches

approximating likelihoods for large spatial data sets summary likelihood methods are often difficult to use with large irregularly sited spatial data sets owing to the computational burden even for gaussian models exact calculations of the likelihood for n observations require o n3 operations since any joint density can be written as a product of conditional densities based on some ordering of the observations one way to lessen the computations is to condition on only some of the past observations when computing the conditional densities we show how this approach can be adapted to approximate the restricted likelihood and we demonstrate how an estimating equations approach allows us to judge the efficacy of the resulting approximation previous work has suggested conditioning on those past observations that are closest to the observation whose conditional density we are approximating through theoretical numerical and practical examples we show that there can often be considerable benefit in conditioning on some distant observations as well

fixed rank kriging for very large spatial data sets summary spatial statistics for very large spatial data sets is challenging the size of the data set n causes problems in computing optimal spatial predictors such as kriging since its computa tional cost is of order a73 in addition a large data set is often defined on a large spatial domain so the spatial process of interest typically exhibits non stationary behaviour over that domain a flexible family of non stationary covariance functions is defined by using a set of basis functions that is fixed in number which leads to a spatial prediction method that we call fixed rank kriging specifically fixed rank kriging is kriging within this class of non stationary covariance functions it relies on computational simplifications when n is very large for obtaining the spatial best linear unbiased predictor and its mean squared prediction error for a hidden spatial process a method based on minimizing a weighted frobenius norm yields best estimators of the covari ance function parameters which are then substituted into the fixed rank kriging equations the new methodology is applied to a very large data set of total column ozone data observed over the entire globe where n is of the order of hundreds of thousands

gaussian predictive process models for large spatial data sets with scientific data available at geocoded locations investigators are increasingly turning to spatial process models for carrying out statistical inference over the last decade hierarchical models implemented through markov chain monte carlo methods have become especially popular for spatial modelling given their flexibility and power to fit models that would be infeasible with classical methods as well as their avoidance of possibly inappropriate asymptotics however fitting hierarchical spatial models often involves expensive matrix decompositions whose computational complexity increases in cubic order with the number of spatial locations rendering such models infeasible for large spatial data sets this computational burden is exacerbated in multivariate settings with several spatially dependent response variables it is also aggravated when data are collected at frequent time points and spatiotemporal process models are used with regard to this challenge our contribution is to work with what we call predictive process models for spatial and spatiotemporal data every spatial or spatiotemporal process induces a predictive process model in fact arbitrarily many of them the latter models project process realizations of the former to a lower dimensional subspace thereby reducing the computational burden hence we achieve the flexibility to accommodate non stationary non gaussian possibly multivariate possibly spatiotemporal processes in the context of large data sets we discuss attractive theoretical properties of these predictive processes we also provide a computational template encompassing these diverse settings finally we illustrate the approach with simulated and real data sets

robust and optimal multi iterative techniques for iga galerkin linear systems we consider fast solvers for the large linear systems coming from the isogeometric analysis iga collocation approximation based on b splines of full elliptic d dimensional partial differential equations pdes we are interested in designing iterative algorithms which are optimal and robust the former property implies that the computational cost is linear with respect to the number of degrees of freedom i e the matrix size the latter property means that the convergence rate is completely independent of or only mildly dependent on all the relevant parameters in our setting we can mention the coefficients of the pde the dimensionality d the geometric map g describing the physical domain the matrix size related to the fineness parameters and the spline degrees associated with the iga approximation order our approach is based on the spectral symbol which describes the global eigenvalue behavior of the iga collocation matrices it is precisely the spectral information contained in the symbol that is exploited for the design of an optimal and robust multi iterative solver of multigrid type several numerical experiments are presented and discussed in view of recent theoretical findings concerning the symbol and its properties

isogeometric schwarz preconditioners for linear elasticity systems isogeometric schwarz preconditioners are constructed and analyzed for both compressible elasticity in primal formulation and almost incompressible elasticity in mixed formulation these preconditioners require the solution of local elasticity problems on overlapping subdomains forming a decomposition of the problem domain and the solution of a coarse elasticity problem associated with the subdomain coarse mesh an h analysis of the preconditioner for the primal formulation of compressible elasticity yields an optimal convergence rate bound that is scalable in the number of subdomains and is linear in the ratio between subdomain and overlap sizes extensive numerical experiments in 2d and 3d confirm this theoretical bound and show that an analogous bound holds for the mixed formulation of almost incompressible elasticity the numerical tests also show the good preconditioner performance with respect to the polynomial degree p and regularity k of the isogeometric basis functions as well as with respect to the presence of discontinuous elastic coefficients in composite materials and to domain deformation

the debiased whittle likelihood the whittle likelihood is a widely used and computationally efficient pseudolikelihood however it is known to produce biased parameter estimates with finite sample sizes for large classes of models we propose a method for debiasing whittle estimates for second order stationary stochastic processes the debiased whittle likelihood can be computed in the same o n log n operations as the standard whittle approach we demonstrate the superior performance of our method in simulation studies and in application to a large scale oceanographic dataset where in both cases the debiased approach reduces bias by up to two orders of magnitude achieving estimates that are close to those of the exact maximum likelihood at a fraction of the computational cost we prove that the method yields estimates that are consistent at an optimal convergence rate of n 1 2 for gaussian processes and for certain classes of non gaussian or nonlinear processes this is established under weaker assumptions than in the standard theory and in particular the power spectral density is not required to be continuous in frequency we describe how the method can be readily combined with standard methods of bias reduction such as tapering and differencing to further reduce bias in parameter estimates

a full scale approximation of covariance functions for large spatial data sets summary gaussian process models have been widely used in spatial statistics but face tremendous computational challenges for very large data sets the model fitting and spatial prediction of such models typically require o n3 operations for a data set of size n various approximations of the covariance functions have been introduced to reduce the computational cost however most existing approximations cannot simultaneously capture both the large and the small scale spatial dependence a new approximation scheme is developed to provide a high quality approximation to the covariance function at both the large and the small spatial scales the new approximation is the summation of two parts a reduced rank covariance and a compactly supported covariance obtained by tapering the covariance of the residual of the reduced rank approximation whereas the former part mainly captures the large scale spatial variation the latter part captures the small scale local variation that is unexplained by the former part by combining the reduced rank representation and sparse matrix techniques our approach allows for efficient computation for maximum likelihood estimation spatial prediction and bayesian inference we illustrate the new approach with simulated and real data sets

estimation and model identification for continuous spatial processes summary formal parameter estimation and model identification procedures for continuous domain spatial processes are introduced the processes are assumed to be adequately described by a linear model with residuals that follow a second order stationary gaussian random field and data are assumed to consist of noisy observations of the process at arbitrary sampling locations a general class of two dimensional rational spectral density functions with elliptic contours is used to model the spatial covariance function an iterative estimation procedure alleviates many of the computational difficulties of conventional maximum likelihood estimation for non lattice data the procedure is applied to several generated data sets and to an actual ground water data set

overlapping schwarz methods for isogeometric analysis we construct and analyze an overlapping schwarz preconditioner for elliptic problems discretized with isogeometric analysis the preconditioner is based on partitioning the domain of the problem into overlapping subdomains solving local isogeometric problems on these subdomains and solving an additional coarse isogeometric problem associated with the subdomain mesh we develop an h analysis of the preconditioner showing in particular that the resulting algorithm is scalable and its convergence rate depends linearly on the ratio between subdomain and overlap sizes for fixed polynomial degree p and regularity k of the basis functions numerical results in two and three dimensional tests show the good convergence properties of the preconditioner with respect to the isogeometric discretization parameters h p k number of subdomains n overlap size and also jumps in the coefficients of the elliptic operator

asymptotic behaviour of positive solutions of elliptic equations with critical and supercritical growth ii the nonradial case abstract we consider positive solutions ue of a family of elliptic boundary value problems ie as e 0 they approach in some sense the problem i0 which has a nonlinearity with critical or supercritical growth it is shown how the solution ue concentrates at a point how this happens asymptotically and where in the domain it happens

estimating space and space time covariance functions for large data sets a weighted composite likelihood approach in this article we propose two methods for estimating space and space time covariance functions from a gaussian random field based on the composite likelihood idea the first method relies on the maximization of a weighted version of the composite likelihood function while the second one is based on the solution of a weighted composite score equation this last scheme is quite general and could be applied to any kind of composite likelihood an information criterion for model selection based on the first estimation method is also introduced the methods are useful for practitioners looking for a good balance between computational complexity and statistical efficiency the effectiveness of the methods is illustrated through examples simulation experiments and by analyzing a dataset on ozone measurements

scaling multidimensional inference for structured gaussian processes exact gaussian process gp regression has cal o n 3 runtime for data size n making it intractable for large n many algorithms for improving gp scaling approximate the covariance with lower rank matrices other work has exploited structure inherent in particular covariance functions including gps with implied markov structure and inputs on a lattice both enable cal o n or cal o n log n runtime however these gp advances have not been well extended to the multidimensional input setting despite the preponderance of multidimensional applications this paper introduces and tests three novel extensions of structured gps to multidimensional inputs for models with additive and multiplicative kernels first we present a new method for inference in additive gps showing a novel connection between the classic backfitting method and the bayesian framework we extend this model using two advances a variant of projection pursuit regression and a laplace approximation for non gaussian observations lastly for multiplicative kernel structure we present a novel method for gps with inputs on a multidimensional grid we illustrate the power of these three advances on several data sets achieving performance equal to or very close to the naive gp at orders of magnitude less cost

fast sampling of gaussian markov random fields this paper demonstrates how gaussian markov random fields conditional autoregressions can be sampled quickly by using numerical techniques for sparse matrices the algorithm is general and efficient and expands easily to various forms for conditional simulation and evaluation of normalization constants we demonstrate its use by constructing efficient block updates in markov chain monte carlo algorithms for disease mapping

non nested testing of spatial correlation we develop non nested tests in a general spatial spatio temporal or panel data context the spatial aspect can be interpreted quite generally in either a geographical sense or employing notions of economic distance or when parametric modelling arises in part from a common factor or other structure in the former case observations may be regularly spaced across one or more dimensions as is typical with much spatio temporal data or irregularly spaced across all dimensions both isotropic models and non isotropic models can be considered and a wide variety of correlation structures in the second case models involving spatial weight matrices are covered such as spatial autoregressive models the setting is sufficiently general to potentially cover other parametric structures such as certain factor models and vector valued observations and here our preliminary asymptotic theory for parameter estimates is of some independent value the test statistic is based on a gaussian pseudo likelihood ratio and is shown to have an asymptotic standard normal distribution under the null hypothesis that one of the two models is correct this limit theory rests strongly on a central limit theorem for the gaussian pseudo maximum likelihood parameter estimates a small monte carlo study of finite sample performance is included

a dual primal feti method for incompressible stokes equations in this paper a dual primal feti method is developed for incompressible stokes equations approximated by mixed finite elements with discontinuous pressures the domain of the problem is decomposed into nonoverlapping subdomains and the continuity of the velocity across the subdomain interface is enforced by introducing lagrange multipliers by a schur complement procedure the solution of an indefinite stokes problem is reduced to solving a symmetric positive definite problem for the dual variables i e the lagrange multipliers this dual problem is solved by the conjugate gradient method with a dirichlet preconditioner in each iteration step both subdomain problems and a coarse level problem are solved by a direct method it is proved that the condition number of this preconditioned dual problem is independent of the number of subdomains and bounded from above by the square of the product of the inverse of the inf sup constant of the discrete problem and the logarithm of the number of unknowns in the individual subdomains numerical experiments demonstrate the scalability of this new method

overlapping schwarz methods for isogeometric analysis based on generalized b splines abstract generalized b splines gb splines are a special class of tchebycheff b splines that are smooth piecewise function with sections in more general spaces gb splines allow for an exact representation of conic sections as well as transcendental curves and thus they become very attractive for geometrical modeling and numerical simulation in this paper we present overlapping schwarz preconditioners for elliptic problems discretized with isogeometric analysis based on gb splines an h analysis of the proposed preconditioners shows an optimal convergence rate bound that is scalable in the number of subdomains and that is linear in the ratio between subdomain and overlap sizes numerical results in two and three dimensional tests confirm this analysis and also illustrate the good convergence properties of the preconditioner with respect to the discretization parameters the domain deformation and the jumping coefficients

a feti dp type domain decomposition algorithm for three dimensional incompressible stokes equations the feti dp dual primal finite element tearing and interconnecting algorithms proposed by the authors in siam j numer anal 51 2013 pp 1235 1253 and internat j numer methods engrg 94 2013 pp 128 149 for solving incompressible stokes equations are extended to three dimensional problems a new analysis of the condition number bound for using the dirichlet preconditioner is given the algorithm and analysis are valid for mixed finite elements with both continuous and discontinuous pressures an advantage of this new analysis is that the numerous coarse level velocity components required in the previous analysis to enforce the divergence free subdomain boundary velocity conditions are no longer needed this greatly reduces the size of the coarse level problem in the algorithm especially for three dimensional problems the coarse level velocity space can be chosen as simple as those coarse spaces for solving scalar elliptic problems corresponding to each velocity component both th

multigrid methods for isogeometric discretization abstract we present geometric multigrid methods for isogeometric discretization of scalar second order elliptic problems the smoothing property of the relaxation method and the approximation property of the intergrid transfer operators are analyzed these properties when used in the framework of classical multigrid theory imply uniform convergence of two grid and multigrid methods supporting numerical results are provided for the smoothing property the approximation property convergence factor and iterations count for v w and f cycles and the linear dependence of v cycle convergence on the smoothing steps for two dimensions numerical results include the problems with variable coefficients simple multi patch geometry a quarter annulus and the dependence of convergence behavior on refinement levels l whereas for three dimensions only the constant coefficient problem in a unit cube is considered the numerical results are complete up to polynomial order p 4 and for c 0 and c p 1 smoothness

existence of positive solutions for a class of semilinear and quasilinear elliptic equations with supercritical case abstract in this paper we consider a class of semilinear elliptic dirichlet problems in a bounded regular domain with cylindrical symmetry involving concave convex nonlinearities with supercritical growth using a new sobolev embedding theorem and variational method we show the existence of two positive solutions of the problem additionally we study the quasilinear elliptic equation and obtain a similar result

approximate bayesian inference for latent gaussian models by using integrated nested laplace approximations summary structured additive regression models are perhaps the most commonly used class of models in statistical applications it includes among others generalized linear models generalized additive models smoothing spline models state space models semiparametric regression spatial and spatiotemporal models log gaussian cox processes and geostatistical and geoadditive models we consider approximate bayesian inference in a popular subset of structured additive regression models latent gaussian models where the latent field is gaussian controlled by a few hyperparameters and with non gaussian response variables the posterior marginals are not available in closed form owing to the non gaussian response variables for such models markov chain monte carlo methods can be implemented but they are not without problems in terms of both convergence and computational time in some practical applications the extent of these problems is such that markov chain monte carlo sampling is simply not an appropriate tool for routine analysis we show that by using an integrated nested laplace approximation and its simplified version we can directly compute very accurate approximations to the posterior marginals the main benefit of these approximations is computational where markov chain monte carlo algorithms need hours or days to run our approximations provide more precise estimates in seconds or minutes another advantage with our approach is its generality which makes it possible to perform bayesian analysis in an automatic streamlined way and to compute model comparison criteria and various predictive measures so that models can be compared and the model under study can be challenged

bddc preconditioners for spectral element discretizations of almost incompressible elasticity in three dimensions balancing domain decomposition by constraints bddc algorithms are constructed and analyzed for the system of almost incompressible elasticity discretized with gauss lobatto legendre spectral elements in three dimensions initially mixed spectral elements are employed to discretize the almost incompressible elasticity system but a positive definite reformulation is obtained by eliminating all pressure degrees of freedom interior to each subdomain into which the spectral elements have been grouped appropriate sets of primal constraints can be associated with the subdomain vertices edges and faces so that the resulting bddc methods have a fast convergence rate independent of the almost incompressibility of the material in particular the condition number of the bddc preconditioned operator is shown to depend only weakly on the polynomial degree n the ratio h h of subdomain and element diameters and the inverse of the inf sup constants of the subdomains and the underlying mixed formulation while being scalable i e independent of the number of subdomains and robust i e independent of the poisson ratio and young s modulus of the material considered these results also apply to the related dual primal finite element tearing and interconnect feti dp algorithms defined by the same set of primal constraints numerical experiments carried out on parallel computing systems confirm these results

balancing neumann neumann preconditioners for mixed approximations of heterogeneous problems in linear elasticity balancing neumann neumann methods are extented to mixed formulations of the linear elasticity system with discontinuous coefficients discretized with mixed finite or spectral elements with discontinuous pressures these domain decomposition methods implicitly eliminate the degrees of freedom associated with the interior of each subdomain and solve iteratively the resulting saddle point schur complement using a hybrid preconditioner based on a coarse mixed elasticity problem and local mixed elasticity problems with natural and essential boundary conditions a polylogarithmic bound in the local number of degrees of freedom is proven for the condition number of the preconditioned operator in the constant coefficient case parallel and serial numerical experiments confirm the theoretical results indicate that they still hold for systems with discontinuous coefficients and show that our algorithm is scalable parallel and robust with respect to material heterogeneities the results on heterogeneous general problems are also supported in part by our theory

positive definite balancing neumann neumann preconditioners for nearly incompressible elasticity in this paper a positive definite balancing neumann neumann bnn solver for the linear elasticity system is constructed and analyzed the solver implicitly eliminates the interior degrees of freedom in each subdomain and solves iteratively the resulting schur complement involving only interface displacements using a bnn preconditioner based on the solution of a coarse elasticity problem and local elasticity problems with natural and essential boundary conditions while the schur complement becomes increasingly ill conditioned as the materials becomes almost incompressible the bnn preconditioned operator remains well conditioned the main theoretical result of the paper shows that the proposed bnn method is scalable and quasi optimal in the constant coefficient case this bound holds for material parameters arbitrarily close to the incompressible limit while this result is due to an underlying mixed formulation of the problem both the interface problem and the preconditioner are positive definite numerical results in two and three dimensions confirm these good convergence properties and the robustness of the methods with respect to the almost incompressibility of the material

a frequency domain empirical likelihood method for irregularly spaced spatial data this paper develops empirical likelihood methodology for irregularly spaced spatial data in the frequency domain unlike the frequency domain empirical likelihood fdel methodology for time series on a regular grid the formulation of the spatial fdel needs special care due to lack of the usual orthogonality properties of the discrete fourier transform for irregularly spaced data and due to presence of nontrivial bias in the periodogram under different spatial asymptotic structures a spatial fdel is formulated in the paper taking into account the effects of these factors the main results of the paper show that wilks phenomenon holds for a scaled version of the logarithm of the proposed empirical likelihood ratio statistic in the sense that it is asymptotically distribution free and has a chi squared limit as a result the proposed spatial fdel method can be used to build nonparametric asymptotically correct confidence regions and tests for covariance parameters that are defined through spectral estimating equations for irregularly spaced spatial data in comparison to the more common studentization approach a major advantage of our method is that it does not require explicit estimation of the standard error of an estimator which is itself a very difficult problem as the asymptotic variances of many common estimators depend on intricate interactions among several population quantities including the spectral density of the spatial process the spatial sampling density and the spatial asymptotic structure results from a numerical study are also reported to illustrate the methodology and its finite sample properties

autoregressive spatial spectral estimates abstract nonparametric spectral density estimates find many uses in econometrics for stationary random fields on a regular spatial lattice we propose an autoregressive nonparametric spectral density estimate that is guaranteed positive even when suitable edge effect correction is employed and is simple to compute using least squares our estimate is based on truncating a true half plane infinite autoregressive representation while also allowing the truncation length to diverge in all dimensions to avoid the potential bias due to truncation at a fixed lag length uniform consistency of the proposed estimate is established and new criteria for order selection are also suggested and studied in practical settings the asymptotic distribution of the estimate is shown to be zero mean normal and independent at fixed distinct frequencies mirroring the behaviour for time series a small monte carlo experiment examines finite sample performance technically the key to the results is the covariance structure of stationary random fields defined on regularly spaced lattices we show the covariance matrix to satisfy a generalization of the toeplitz property familiar from time series analysis

hybrid domain decomposition algorithms for compressible and almost incompressible elasticity overlapping schwarz methods are considered for mixed finite element approximations of linear elasticity with discontinuous pressure spaces as well as for compressible elasticity approximated by standard conforming finite elements the coarse components of the preconditioners are based on spaces with a number of degrees of freedom per subdomain which are uniformly bounded which are similar to those previously developed for scalar elliptic problems and domain decomposition methods of iterative substructuring type i e methods based on nonoverlapping decompositions of the domain the local components of the new preconditioners are based on solvers on a set of overlapping subdomains in the current study the dimension of the coarse spaces is smaller than in recently developed algorithms in the compressible case all independent face degrees of freedom have been eliminated while in the almost incompressible case five out of six are not needed in many cases this will result in a reduction of the dimension of the coarse space by about one half compared with that of the algorithm previously considered in addition in spite of using overlapping subdomains to define the local components of the preconditioner values of the residual and the approximate solution need only to be retained on the interface between the subdomains in the iteration of the new hybrid schwarz algorithm the use of discontinuous pressures makes it possible to work exclusively with symmetric positive definite problems and the standard preconditioned conjugate gradient method bounds are established for the condition number of the preconditioned operators the bound for the almost incompressible case grows in proportion to the square of the logarithm of the number of degrees of freedom of individual subdomains and the third power of the relative overlap between the overlapping subdomains and it is independent of the poisson ratio as well as jumps in the lame parameters across the interface between the subdomains numerical results illustrate the findings copyright 2009 john wiley sons ltd

a resampling based stochastic approximation method for analysis of large geostatistical data the gaussian geostatistical model has been widely used in modeling of spatial data however it is challenging to computationally implement this method because it requires the inversion of a large covariance matrix particularly when there is a large number of observations this article proposes a resampling based stochastic approximation method to address this challenge at each iteration of the proposed method a small subsample is drawn from the full dataset and then the current estimate of the parameters is updated accordingly under the framework of stochastic approximation since the proposed method makes use of only a small proportion of the data at each iteration it avoids inverting large covariance matrices and thus is scalable to large datasets the proposed method also leads to a general parameter estimation approach maximum mean log likelihood estimation which includes the popular maximum log likelihood estimation mle approach as a special case and is expected to play an important role

overlapping schwarz methods with a standard coarse space for almost incompressible linear elasticity low order finite element discretizations of the linear elasticity system suffer increasingly from locking effects and ill conditioning when the material approaches the incompressible limit if only the displacement variables are used mixed finite elements using both displacement and pressure variables provide a well known remedy but they yield larger and indefinite discrete systems for which the design of scalable and efficient iterative solvers is challenging two level overlapping schwarz preconditioners for the almost incompressible system of linear elasticity discretized by mixed finite elements with discontinuous pressures are constructed and analyzed the preconditioned systems are accelerated either by a gmres generalized minimum residual method applied to the resulting discrete saddle point problem or by a pcg preconditioned conjugate gradient method applied to a positive definite although extremely ill conditioned reformulation of the problem obtained by eliminating all pressure variab

an overlapping schwarz algorithm for almost incompressible elasticity overlapping schwarz methods are extended to mixed finite element approximations of linear elasticity which use discontinuous pressure spaces the coarse component of the preconditioner is based on a low dimensional space previously developed for scalar elliptic problems and a domain decomposition method of iterative substructuring type i e a method based on nonoverlapping decompositions of the domain while the local components of the preconditioner are based on solvers on a set of overlapping subdomains a bound is established for the condition number of the algorithm which grows in proportion to the logarithm of the number of degrees of freedom in individual subdomains and essentially to the third power of the relative overlap between the overlapping subdomains and which is independent of the poisson ratio as well as jumps in the lame parameters across the interface between the subdomains a positive definite reformulation of the discrete problem makes the use of the standard preconditioned conjugate gradient method straightforward numerical results which include a comparison with problems of compressible elasticity illustrate the findings

stochastic partial differential equation based modelling of large space time data sets summary increasingly larger data sets of processes in space and time ask for statistical models and methods that can cope with such data we show that the solution of a stochastic advection diffusion partial differential equation provides a flexible model class for spatiotemporal processes which is computationally feasible also for large data sets the gaussian process defined through the stochastic partial differential equation has in general a non separable covariance structure its parameters can be physically interpreted as explicitly modelling phenomena such as transport and diffusion that occur in many natural processes in diverse fields ranging from environmental sciences to ecology to obtain computationally efficient statistical algorithms we use spectral methods to solve the stochastic partial differential equation this has the advantage that approximation errors do not accumulate over time and that in the spectral space the computational cost grows linearly with the dimension the total computational cost of bayesian or frequentist inference being dominated by the fast fourier transform the model proposed is applied to post processing of precipitation forecasts from a numerical weather prediction model for northern switzerland in contrast with the raw forecasts from the numerical model the post processed forecasts are calibrated and quantify prediction uncertainty moreover they outperform the raw forecasts in the sense that they have a lower mean absolute error

restricted spatial regression methods implications for inference the issue of spatial confounding between the spatial random effect and the fixed effects in regression analyses has been identified as a concern in the statistical literature multiple authors have

bpx preconditioning for isogeometric analysis we consider elliptic pdes partial differential equations in the framework of isogeometric analysis i e we treat the physical domain by means of a b spline or nurbs mapping which we assume to be regular the numerical solution of the pde is computed by means of tensor product b splines mapped onto the physical domain we construct additive multilevel preconditioners and show that they are asymptotically optimal i e the spectral condition number of the resulting stiffness matrix is independent of h together with a nested iteration scheme this enables an iterative solution scheme of optimal linear complexity the theoretical results are substantiated by numerical examples in two and three space dimensions

a bddc algorithm for the stokes problem with weak galerkin discretizations abstract the bddc balancing domain decomposition by constraints methods have been applied successfully to solve the large sparse linear algebraic systems arising from conforming finite element discretizations of second order elliptic and stokes problems in this paper the stokes equations are discretized using the weak galerkin method a newly developed nonconforming finite element method a bddc algorithm is designed to solve the linear system such obtained edge face velocity interface average and mean subdomain pressure are selected for the coarse problem the condition number bounds of the bddc preconditioned operator are analyzed and the same rate of convergence is obtained as for conforming finite element methods numerical experiments are conducted to verify the theoretical results

fixed domain asymptotics for spatial periodograms abstract the periodogram for a spatial process observed on a lattice is often used to estimate the spectral density the bases for such estimators are two asymptotic properties that periodograms commonly possess 1 the periodogram at a particular frequency is approximately unbiased for the spectral density and 2 the correlation of the periodogram at distinct frequencies is approximately zero for spatial data it is often appropriate to use fixed domain asymptotics in which the observations get increasingly dense in some fixed region as their number increases using fixed domain asymptotics this article shows that standard asymptotic results for periodograms do not apply and that using the periodogram of the raw data can yield highly misleading results but by appropriately filtering the data before computing the periodogram it is possible to obtain results similar to the standard asymptotic results for spatial periodograms

spectral density estimation for random fields via periodic embeddings we introduce methods for estimating the spectral density of a random field on a formula see text dimensional lattice from incomplete gridded data data are iteratively imputed onto an expanded lattice according to a model with a periodic covariance function the imputations are convenient computationally in that circulant embedding and preconditioned conjugate gradient methods can produce imputations in formula see text time and formula see text memory however these so called periodic imputations are motivated mainly by their ability to produce accurate spectral density estimates in addition we introduce a parametric filtering method that is designed to reduce periodogram smoothing bias the paper contains theoretical results on properties of the imputed data periodogram and numerical and simulation studies comparing the performance of the proposed methods to existing approaches in a number of scenarios we present an application to a gridded satellite surface temperature dataset with missing values

a nonoverlapping domain decomposition method for incompressible stokes equations with continuous pressures a nonoverlapping domain decomposition algorithm is proposed to solve the linear system arising from mixed finite element approximation of incompressible stokes equations a continuous finite element space for the pressure is used in the proposed algorithm lagrange multipliers are used to enforce continuity of the velocity component across the subdomain boundary the continuity of the pressure component is enforced in the primal form i e neighboring subdomains share the same pressure degrees of freedom on the subdomain interface and no lagrange multipliers are needed after eliminating all velocity variables and the independent subdomain interior parts of the pressures a symmetric positive semidefinite linear system for the subdomain boundary pressures and the lagrange multipliers is formed and solved by a preconditioned conjugate gradient method a lumped preconditioner is studied and the condition number bound of the preconditioned operator is proved to be independent of the number of subdomains fo