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

on kawai theorem for orbifold riemann surfaces we prove a generalization of kawai theorem for the case of orbifold riemann surface the computation is based on a formula for the differential of a holomorphic map from the cotangent bundle of the teichmuller space to the mathrm psl 2 mathbb c character variety which allows to evaluate explicitly the pullback of goldman symplectic form in the spirit of riemann bilinear relations as a corollary we obtain a generalization of goldman s theorem that the pullback of goldman symplectic form on the mathrm psl 2 mathbb r character variety is a symplectic form of the weil petersson metric on the teichmuller space

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

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

ranking of fuzzy sets based on the concept of existence abstract various approaches have been proposed for the comparison or ranking of fuzzy sets however due to the complexity of the problem a general method which can be used for any situation still does not exist this paper formalizes the concept of existence for the ranking of fuzzy sets many of the existing fuzzy ranking methods are shown to be some application of this concept an improved fuzzy ranking method is then introduced based on this concept this newly introduced method is expanded for treating both normal and nonnormal convex and nonconvex fuzzy sets emphasis is placed on the use of the subjectivity of the decision maker such as the optimistic or the pessimistic view points an improved procedure for obtaining linguistic conclusions is also developed finally some numerical examples are given to illustrate the approach

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

on the symplectic geometry of deformations of a hyperbolic surface let r be a riemann surface in this manuscript we consider a geometry on the moduli space x r for r which we regard as the space of equivalence classes of constant curvature metrics on the underlying smooth manifold of r classically the space of flat metrics for a torus is the locally symmetric space 0 2 sl 2 r sl 2 z we shall describe a symplectic geometry for the space of hyperbolic metrics on a surface of negative euler characteristic the teichmiiller space t r a covering of the moduli space is a complex kihler manifold a kuhler metric for t r defined in terms of the petersson product for automorphic forms was introduced by weil 1 the weil petersson metric is invariant under the covering transformations and so projects to the moduli space x r the metric provides a link between the function theory of r and the geometry of x r in the fenchel nielsen manuscript 8 a deformation based on an amalgamation construction for fuchsian groups is introduced the deformation is defined geometrically by cutting the surface along a simple closed geodesic rotating one side of the cut relative to the other and attaching the sides in their new position the hyperbolic metric in the complement of the cut extends to a hyperbolic metric on the new surface choose a free homotopy class a on the surface r then for each marked surface r realize a by the closed geodesic ar the fenchel nielsen deformations for the athen define a 1 parameter group of diffeomorphisms of t r whose infinitesimal generator by definition is the fenchel nielsen vector field ta in 21 the fenchel nielsen deformation was described in terms of quasiconformal mappings and an investigation of the vector fields t was begun the fenchel nielsen vector fields were found to be related to the geodesic length functions 1 introduced by fricke klein to provide coordinates for t r

group cohomology and the symplectic structure on the moduli space of representations the space of equivalence classes of irreducibleb representations of the fundamental group of a compact oriented surface of genus at least 2 in a lie group has a natural symplectic form in lab atiyah and bott described this symplectic structure using methods from gauge theory goldman g constructed the skew symmetric pairing algebraically using methods from group cohomology using poincar6 duality goldman showed that the pairing is nondegenerate and identified it with the symplectic structure given by gauge theory

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

investors preference order of fuzzy numbers nowadays greater and greater realistic financial problems are modeled by using the stochastic programming in the fuzzy environment hence ranking a set of fuzzy numbers that is consistent with the investors preference becomes important for modelling a realistic problem in this paper we will provide a new ranking procedure that is consistent with the preference of the conservative investors our ranking procedure satisfies the axioms of three order relations for the separable fuzzy numbers or the triangle fuzzy numbers we found that our ranking procedure has a better capability of discriminating the order of two fuzzy numbers for the lr type fuzzy numbers our ranking procedure reduces the computational time substantially

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

triangulated riemann surfaces with boundary and the weil petersson poisson structure given a hyperbolic surface with geodesic boundary s the lengths of a maximal system of disjoint simple geodesic arcs on s that start and end at s perpendicularly are coordinates on the teichmuller space t s we express the weil petersson poisson structure of t s in this system of coordinates and we prove that it limits pointwise to the piecewise linear poisson structure defined by kontsevich on the arc complex of s at the same time we obtain a formula for the first order variation of the distance between two closed geodesics under fenchel nielsen deformation

characteristic classes and representations of discrete subgroups of lie groups a volume invariant is used to characterize those representations of a countable group into a connected semisimple lie group g which are injective and whose image is a discrete cocompact subgroup of g let it be a discrete cocompact subgroup of g and consider the analytic variety hom 7r g consisting of homomorphisms 0 tt g denote by k a maximal compact subgroup of g and x k g the associated symmetric space let m be the orbit space x tt for convenience we shall henceforth assume that n is torsionfree by selberg s lemma 12 this may be accomplished by replacing tt by a subgroup of finite index this insures that m is a compact smooth manifold having tt as its fundamental group the case when n has torsion follows from the torsionfree case with minor modifications but these modifications need not concern us here to every representation 0 g hom 7r g we associate a foliated bundle e over m with fibre x and structure group g see e g 6 if co is a closed ginvariant differential fc form on x then we may spread co over the fibres of e copies of x to obtain a closed fc form co on e we define co 0 fj f where is any section of e moreover co 0 is independent of the choice of section for example taking co to be the g invariant volume form on x we obtain a real number u 0 which depends on 0 when x is even dimensional the chern gauss bonnet theorem implies that u 0 may be described as an euler number i e the self intersection number of any section which is a topological invariant of e when x is odd dimensional this volume invariant is related to a more recent kind of topological invariant based on bounded cohomology and due to gromov 3 and is constant on the connected components of hom 7r g the volume invariant satisfies an inequality

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

possibilistic linear regression analysis for fuzzy data abstract fuzzy data given by expert knowledge can be regarded as a possibility distribution by which possibilistic linear systems are defined recently it has become important to deal with fuzzy data in connection with expert knowledge three formulations of possibilistic linear regression analysis are proposed here to deal with fuzzy data since our formulations can be reduced to linear programming problems the merit of our formulations is to be able to obtain easily fuzzy parameters in possibilistic linear models and to add other constraint conditions which might be obtained from expert knowledge of fuzzy parameters this approach can be regarded as a fuzzy interval analysis in a fuzzy environment

fuzzy data analysis by possibilistic linear models abstract since fuzzy data can be regarded as distribution of possibility fuzzy data analysis by possibilistic linear models is proposed in this paper possibilistic linear systems are defined by the extension principle fuzzy parameter estimations are discussed in possibilistic linear systems and possibilistic linear models are employed for fuzzy data analysis with non fuzzy inputs and fuzzy outputs defined by fuzzy numbers the estimated possibilistic linear system can be obtained by solving a linear programming problem this approach can be regarded as fuzzy interval analysis

the symplectic nature of fundamental groups of surfaces soit 7b g la categorie des groupes de lie g avec une forme bilineaire symetrique non singuliere b sur l algebre de lie soit le groupe fondamental d une surface orientee close de genre 1 et soit 7b s la categorie des out espaces symplectiques alors la correspondance g b hom g g b definit un foncteur 7b g 7b s ou b est la structure symplectique sur hom g g definie par b

the fenchel nielsen deformation the uniformization theorem provides that a riemann surface s of negative euler characteristic has a metric of constant curvature 1 a hyperbolic structure can be understood in terms of its deformations unfortunately the variation of the hyperbolic metric arising from a deformation is not determined by local data if z is a conformal coordinate for s and a ii h ii 00 1 a beltrami differential then dw dz p dj defines a new conformal coordinate and structure si the tensor p may vanish on the open set 0 c s and yet the hyperbolic metrics of s and si do not coincide on this phenomenon presents a s erious difficulty in the deformation theory of hyperbolic structures the fenchel nielsen twist deformation does not share this defect the deformation is defined by cutting the surface along a simple closed geodesic rotating one side of the cut relative to the other and attaching the sides in their new position a geodesic intersecting the cut is deformed to a broken geodesic the hyperbolic structure in the complement of the cut extends naturally to a hyperbolic structure on the new surface the deformation is concentrated at the cut the basic idea for the deformation appears in the work of fricke klein dehn and fenchel nielsen 71 8 in the fenchel nielsen manuscript the deformation was considered extensively coordinates for teichmiller space are defined in terms of the deformation the proper definition is given by a fuchsian group construction calculations are made in terms of unimodular matrices this approach is often cumbersome our basic objective is to give a description of the deformation in terms of quasiconformal mappings the bers embedding of teichmiiller space is central to the analytic theory of deformations using our characterization we calculate the first variation of the bers embedding for the fenchel nielsen deformation the result is a poincare series 6 which already appears in the work of petersson 16

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

fuzzy ranking and quadratic fuzzy regression abstract three different ranking methods namely the overall existence ranking index oeri the approach proposed by diamond 1 and a new two step method based on oeri are used to estimate the distance between two fuzzy numbers this distance parameter is then used in the least square or quadratic regression nonlinear programming is used to solve the resulting quadratic regression equations with constraints and simulation is used to evaluate the performance of the approaches the criterion used to evaluate the performance is the average of the absolute difference between the estimated and the observed values it appears that the two step oeri obtains better results for the case of small sample size and diamond s approach gets better as the sample size increases

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

left and right centers in quasi poisson geometry of moduli spaces abstract we introduce left central and right central functions and left and right leaves in quasi poisson geometry generalizing central or casimir functions and symplectic leaves from poisson geometry they lead to a new type of quasi poisson reduction which is both simpler and more general than known quasi hamiltonian reductions we study these notions in detail for moduli spaces of flat connections on surfaces where the quasi poisson structure is given by an intersection pairing on homology

evaluation of fuzzy linear regression models abstract fuzzy linear regression provides means for tackling regression problems lacking a significant amount of data for determining regression models and with vague relationships between the dependent variable and independent variables in the paper a two stage construction of a linear regression model is proposed using an enhancement of a minimal vagueness criterion already discussed in fuzzy regression analysis two different experiments are set up and used for evaluating both models the choice of the fuzzy regression model should be made in the context of a practical application

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

an area preserving action of the modular group on cubic surfaces and the painleve vi equation we construct an area preserving action of the modular group on a general 4 parameter family of affine cubic surfaces we present a geometrical background behind this construction that is a natural symplectic structure on a moduli space of rank two linear monodromy representations over the 2 dimensional sphere with four punctures and a natural symplectic action upon it of the braid group on three strings studying this action as a discrete dynamical system will be important in discussing the monodromy of the painleve vi equation

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

linear fuzzy regression abstract this paper presents an approach which is useful for regression analysis in the case of heterogeneity of a set of observations for which regression is evaluated the proposed procedure consists of two stages first for a set of observations fuzzy classification is determined due to this homogenous classes of observations which are of hyperellipsoidal shape are obtained then for each fuzzy class the so called linear fuzzy regression is evaluated in the paper the method of calculating linear fuzzy regression coefficients is given it is a generalized version of the least squares method the formula for the values of coefficients is given some properties of linear fuzzy regression are analyzed it is proved that in one and two dimensional cases the formulae are analogous to those for usual regression a measure of goodness of fit and the method of determination of the number of fuzzy classes are also given presented examples indicate the superiority of fuzzy regression in comparison to usual regression in the case of heterogenous observations

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

fuzzy least squares abstract several models for simple least squares fitting of fuzzy valued data are developed criteria are given for when fuzzy data sets can be fitted to the models and analogues of the normal equations are derived

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