bayesian guided pattern search for robust local optimization optimization for complex systems in engineering often involves the use of expensive computer simulation by combining statistical emulation using treed gaussian processes with pattern search optimization we are able to perform robust local optimization more efficiently and effectively than when using either method alone our approach is based on the augmentation of local search patterns with location sets generated through improvement prediction over the input space we further develop a computational framework for asynchronous parallel implementation of the optimization algorithm we demonstrate our methods on two standard test problems and our motivating example of calibrating a circuit device simulator

an iterative substructuring algorithm for equilibrium equations the topic of iterative substructuring methods and more generally domain decomposition methods has been extensively studied over the past few years and the topic is well advanced with respect to first and second order elliptic problems however relatively little work has been done on more general constrained least squares problems or equivalent formulations involving equilibrium equations such as those arising for example in realistic structural analysis applications the potential is good for effective use of iterative algorithms on these problems but such methods are still far from being competitive with direct methods in industrial codes the purpose of this paper is to investigate an order reducing preconditioned conjugate gradient method proposed by barlow nichols and plemmons for solving problems of this type the relationships between this method and nullspace methods such as the force method for structures and the dual variable method for fluids are examined convergence properties are discussed in relation to recent optimality results for varga s theory ofp cyclic sor we suggest a mixed approach for solving equilibrium equations consisting of both direct reduction in the substructures and the conjugate gradient iterative algorithm to complete the computations

emulation of utility functions over a set of permutations sequencing reliability growth tasks abstractwe consider bayesian design of experiments problems in which we maximise the prior expectation of a utility function over a set of permutations for example when sequencing a number of tasks to perform when the number of tasks is large and the expected utility is expensive to compute it may be unreasonable or infeasible to evaluate the expected utility of all permutations we propose an approach to emulate the expected utility using a surrogate function based on a parametric probabilistic model for permutations the surrogate function is fitted by maximising the correlation with the expected utility over a set of training points we propose a suitable transformation of the expected utility to improve the fit we provide results linking the correlation between the two functions and the number of expected utility evaluations to undertake the approach is applied to the sequencing of reliability growth tasks in the development of hardware systems in which there is a large number of potential tasks

post fisherian experimentation from physical to virtual fisher s pioneering work in design of experiments has inspired further work with broader applications especially in industrial experimentation this article discusses three topics in physical experiments principles of effect hierarchy sparsity and heredity for factorial designs a new method called conditional main effect cme for de aliasing aliased effects and robust parameter design i also review the recent emergence of virtual experiments on a computer some major challenges in computer experiments which must go beyond fisherian principles are outlined

multivariate gaussian process emulators with nonseparable covariance structures the gaussian process regression model is a popular type of emulator used as a fast surrogate for computationally expensive simulators deterministic computer models for simulators with multivariate output common practice is to specify a separable covariance structure for the gaussian process though computationally convenient this can be too restrictive leading to poor performance of the emulator particularly when the different simulator outputs represent different physical quantities also treating the simulator outputs as independent can lead to inappropriate representations of joint uncertainty we develop nonseparable covariance structures for gaussian process emulators based on the linear model of coregionalization and convolution methods using two case studies we compare the performance of these covariance structures both with standard separable covariance structures and with emulators that assume independence between the outputs in each case study we find that only emulators with nons

estimation of a density from an imperfect simulation model uncertainty quantification of a technical system can be done using density estimation we usually start with a stochastic model which is fitted to the technical system and the density estimation is done using data from this stochastic model however in any application such a stochastic model will not be perfect and the estimation of the density should take into account the inadequacy of the stochastic model in this paper we show how observed data of the real system together with an imperfect simulation model can be used to derive confidence bands for the density of the technical system our main result is that the newly introduced confidence bands allow to derive lower and upper bounds on the probability of intervals in the technical system furthermore we present an upper bound on the area of the confidence band in case of a smooth density the results are illustrated by applying the estimates to simulated and real data

a bayesian analysis of kriging this article is concerned with predicting for gaussian random fields in a way that appropriately deals with uncertainty in the covariance function to this end we analyze the best linear unbiased prediction procedure within a bayesian framework particular attention is paid to the treatment of parameters in the covariance structure and their effect on the quality both real and perceived of the prediction these ideas are implemented using topographical data from davis

bayesian calibration of inexact computer models bayesian calibration is used to study computer models in the presence of both a calibration parameter and model bias the parameter in the predominant methodology is left undefined this results in an issue where the posterior of the parameter is suboptimally broad there has been no generally accepted alternatives to date this article proposes using bayesian calibration where the prior distribution on the bias is orthogonal to the gradient of the computer model problems associated with bayesian calibration are shown to be mitigated through analytic results in addition to examples supplementary materials for this article are available online

bayesian prediction of deterministic functions with applications to the design and analysis of computer experiments abstract this article is concerned with prediction of a function y t over a multidimensional domain t given the function values at a set of sites t 1 t 2 t n in t and with the design that is with the selection of those sites the motivating application is the design and analysis of computer experiments where t determines the input to a computer model of a physical or behavioral system and y t is a response that is part of the output or is calculated from it following a bayesian formulation prior uncertainty about the function y is expressed by means of a random function y which is taken here to be a gaussian stochastic process the mean of the posterior process can be used as the prediction function t and the variance can be used as a measure of uncertainty this kind of approach has been used previously in bayesian interpolation and is strongly related to the kriging methods used in geostatistics here emphasis is placed on product linear and product cubic correlation func

large sample properties of simulations using latin hypercube sampling latin hypercube sampling mckay conover and beckman 1979 is a method of sampling that can be used to produce input values for estimation of expectations of functions of output variables the asymptotic variance of such an estimate is obtained the estimate is also shown to be asymptotically normal asymptotically the variance is less than that obtained using simple random sampling with the degree of variance reduction depending on the degree of additivity in the function being integrated a method for producing latin hypercube samples when the components of the input variables are statistically dependent is also described these techniques are applied to a simulation of the performance of a printer actuator

bayesian inference of nonlinear unsteady aerodynamics from aeroelastic limit cycle oscillations a bayesian model selection and parameter estimation algorithm is applied to investigate the influence of nonlinear and unsteady aerodynamic loads on the limit cycle oscillation lco of a pitching airfoil in the transitional reynolds number regime at small angles of attack laminar boundary layer trailing edge separation causes negative aerodynamic damping leading to the lco the fluid structure interaction of the rigid but elastically mounted airfoil and nonlinear unsteady aerodynamics is represented by two coupled nonlinear stochastic ordinary differential equations containing uncertain parameters and model approximation errors several plausible aerodynamic models with increasing complexity are proposed to describe the aeroelastic system leading to lco the likelihood in the posterior parameter probability density function pdf is available semi analytically using the extended kalman filter for the state estimation of the coupled nonlinear structural and unsteady aerodynamic model the posterior parameter pdf is sampled using a parallel and adaptive markov chain monte carlo mcmc algorithm the posterior probability of each model is estimated using the chib jeliazkov method that directly uses the posterior mcmc samples for evidence marginal likelihood computation the bayesian algorithm is validated through a numerical study and then applied to model the nonlinear unsteady aerodynamic loads using wind tunnel test data at various reynolds numbers

inference for deterministic simulation models the bayesian melding approach abstract deterministic simulation models are used in many areas of science engineering and policy making typically these are complex models that attempt to capture underlying mechanisms in considerable detail and they have many user specified inputs the inputs are often specified by some form of trial and error approach in which plausible values are postulated the corresponding outputs inspected and the inputs modified until plausible outputs are obtained here we address the issue of more formal inference for such models a probabilistic approach called bayesian synthesis was shown to suffer from the borel paradox according to which the results can depend on the parameterization of the model we propose a modified approach called bayesian melding which takes into full account information and uncertainty about both inputs and outputs to the model while avoiding the borel paradox this is done by recognizing the existence of two priors one implicit and one explicit on each input and output

error analysis and implementation aspects of deferred correction for equality constrained least squares problems the deferred correction procedure recently proposed by van loan on the method of weighting for equality constrained least squares problems siam j numer anal 22 1985 pp 851 864 solves the problem of minimizing f ex 2 subject to cx g by repeatedly solving the related least squares problem of minimizing left left begin array 20 c tau delta g delta f end array right left begin array 20 c tau c e end array right delta x right 2 for some large weight tau the convergence behavior of the procedure is shown to be greatly dependent upon error analysis considerations an analysis of its convergence with an approximation factorization of left begin array 20 c tau c e end array right is given an implementation improvement to the procedure that makes it more robust is also given van loan s conjecture that the method converges for all but ill conditioned problems is shown to be true

order reducing conjugate gradients versus block aor for constrained least squares problems abstract we compare the convergence properties of two iterative algorithms for solving equality constrained least squares problems the first algorithm due to barlow nichols and plemmons applies a variation of the conjugate gradient algorithm to a symmetric positive definite system which is smaller than the original problem the second block accelerated overrelaxation is a two parameter generalization of block sor barlow nichols and plemmons have proven that their order reducing conjugate gradient algorithm converges faster than block sor we extend their result to show that the algorithm is also superior to block aor numerical experiments on some structural engineering problems support the analysis

advances in modeling model discrepancy comment on wu and browne 2015 wu and browne psychometrika 79 2015 have proposed an innovative approach to modeling discrepancy between a covariance structure model and the population that the model is intended to represent their contribution is related to ongoing developments in the field of uncertainty quantification uq on modeling and quantifying effects of model discrepancy we provide an overview of basic principles of uq and some relevant developments and we examine the wu browne work in that context we view the wu browne contribution as a seminal development providing a foundation for further work on the critical problem of model discrepancy in statistical modeling in psychological research

minimum norm solutions to linear elastic analysis problems en analyse d elasticite lineaire un probleme consiste a trouver les vecteurs de contraintes et de deformations etant donnes un modele d elements finis d une structure et un systeme de charges externes on montre que ce probleme est un cas particulier du probleme de norme minimum pour des systemes indetermines d equations lineaires

an optimal algorithm for approximate nearest neighbor searching fixed dimensions consider a set of s of n data points in real d dimensional space r d where distances are measured using any minkowski metric in nearest neighbor searching we preprocess s into a data structure so that given any query point q r d is the closest point of s to q can be reported quickly given any positive real e data point p is a 1 e approximate nearest neighbor of q if its distance from q is within a factor of 1 e of the distance to the true nearest neighbor we show that it is possible to preprocess a set of n points in r d in o dn log n time and o dn space so that given a query point q r d and e 0 a 1 e approximate nearest neighbor of q can be computed in o c d e log n time where c d e d 1 6d e d is a factor depending only on dimension and e in general we show that given an integer k 1 1 e approximations to the k nearest neighbors of q can be computed in additional o kd log n time

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

optimal successive overrelaxation iterative methods for p cyclic matrices we consider linear systems whose associated block jacobi matricesj p are weakly cyclic of indexp in a recent paper pierce hadjidimos and plemmons 13 proved that the block two cyclic successive overrelaxation sor iterative method is numerically more effective than the blockq cyclic sor method 2 2 finally as a counterpart to the result of pierce hadjidimos and plemmons we show that under this more general assumption the two cyclic sor method is not always superior to theq cyclic sor method 2 q p

an information theoretic approach to use high fidelity codes to calibrate low fidelity codes for many simulation models it can be prohibitively expensive or physically infeasible to obtain a complete set of experimental data to calibrate model parameters in such cases one can alternatively employ validated higher fidelity codes to generate simulated data which can be used to calibrate the lower fidelity code in this paper we employ an information theoretic framework to determine the reduction in parameter uncertainty that is obtained by evaluating the high fidelity code at a specific set of design conditions these conditions are chosen sequentially based on the amount of information that they contribute to the low fidelity model parameters the goal is to employ bayesian experimental design techniques to minimize the number of high fidelity code evaluations required to accurately calibrate the low fidelity model we illustrate the performance of this framework using heat and diffusion examples a 1 d kinetic neutron diffusion equation and a particle transport model and include initial results from the integration of the high fidelity thermal hydraulics code hydra th with a low fidelity exponential model for the friction correlation factor

a class of iterative methods for solving saddle point problems we consider the numerical solution of indefinite systems of linear equations arising in the calculation of saddle points we are mainly concerned with sparse systems of this type resulting from certain discretizations of partial differential equations we present an iterative method involving two levels of iteration similar in some respects to the uzawa algorithm we relate the rates of convergence of the outer and inner iterations proving that under natural hypotheses the outer iteration achieves the rate of convergence of the inner iteration the technique is applied to finite element approximations of the stokes equations

estimating mutual information we present two classes of improved estimators for mutual information m x y from samples of random points distributed according to some joint probability density mu x y in contrast to conventional estimators based on binnings they are based on entropy estimates from k nearest neighbor distances this means that they are data efficient with k 1 we resolve structures down to the smallest possible scales adaptive the resolution is higher where data are more numerous and have minimal bias indeed the bias of the underlying entropy estimates is mainly due to nonuniformity of the density at the smallest resolved scale giving typically systematic errors which scale as functions of k n for n points numerically we find that both families become exact for independent distributions i e the estimator m x y vanishes up to statistical fluctuations if mu x y mu x mu y this holds for all tested marginal distributions and for all dimensions of x and y in addition we give estimators for redundancies between more than two random variables we compare our algorithms in detail with existing algorithms finally we demonstrate the usefulness of our estimators for assessing the actual independence of components obtained from independent component analysis ica for improving ica and for estimating the reliability of blind source separation

ensemble evaluation of hydrological model hypotheses it is demonstrated for the first time how model parameter structural and data uncertainties can be accounted for explicitly and simultaneously within the generalized likelihood uncertainty estimation glue methodology as an example application 72 variants of a single soil moisture accounting store are tested as simplified hypotheses of runoff generation at six experimental grassland field scale lysimeters through model rejection and a novel diagnostic scheme the fields designed as replicates exhibit different hydrological behaviors which yield different model performances for fields with low initial discharge levels at the beginning of events the conceptual stores considered reach their limit of applicability conversely one of the fields yielding more discharge than the others but having larger data gaps allows for greater flexibility in the choice of model structures as a model learning exercise the study points to a leaking of the fields not evident from previous field experiments it is discussed how understanding observational uncertainties and incorporating these into model diagnostics can help appreciate the scale of model structural error

a comparison of three methods for selecting values of input variables in the analysis of output from a computer code two types of sampling plans are examined as alternatives to simple random sampling in monte carlo studies these plans are shown to be improvements over simple random sampling with respect to variance for a class of estimators which includes the sample mean and the empirical distribution function

a framework for equilibrium equations the equations of equilibrium in a wide class of applications share a common structure the discrete case leads to a positive definite symmetric matrix a t ca in which a is rectangular and c is often diagonal the continuous case linear or nonlinear repeats that pattern there is a minimum principle for the potentials and a dual principle for the flow variables we try to clarify these different expressions of the same law

factorial sampling plans for preliminary computational experiments a computational model is a representation of some physical or other system of interest first expressed mathematically and then implemented in the form of a computer program it may be viewed as a function of inputs that when evaluated produces outputs motivation for this article comes from computational models that are deterministic complicated enough to make classical mathematical analysis impractical and that have a moderate to large number of inputs the problem of designing computational experiments to determine which inputs have important effects on an output is considered the proposed experimental plans are composed of individually randomized one factor at a time designs and data analysis is based on the resulting random sample of observed elementary effects those changes in an output due solely to changes in a particular input advantages of this approach include a lack of reliance on assumptions of relative sparsity of important inputs monotonicity of outputs with respect to inputs or ad

inverse regression based uncertainty quantification algorithms for high dimensional models many uncertainty quantification uq approaches suffer from the curse of dimensionality that is their computational costs become intractable for problems involving a large number of uncertainty parameters in these situations the classic monte carlo often remains the preferred method of choice because its convergence rate o n 1 2 where n is the required number of model simulations does not depend on the dimension of the problem however many high dimensional uq problems are intrinsically low dimensional because the variation of the quantity of interest qoi is often caused by only a few latent parameters varying within a low dimensional subspace known as the sufficient dimension reduction sdr subspace in the statistics literature motivated by this observation we propose two inverse regression based uq algorithms iruq for high dimensional problems both algorithms use inverse regression to convert the original high dimensional problem to a low dimensional one which is then efficiently solved by building a response surface for the reduced model for example via the polynomial chaos expansion the first algorithm which is for the situations where an exact sdr subspace exists is proved to converge at rate o n 1 hence much faster than mc the second algorithm which doesn t require an exact sdr employs the reduced model as a control variate to reduce the error of the mc estimate the accuracy gain could still be significant depending on how well the reduced model approximates the original high dimensional one iruq also provides several additional practical advantages it is non intrusive it does not require computing the high dimensional gradient of the qoi and it reports an error bar so the user knows how reliable the result is

a physiologically based criterion of muscle force prediction in locomotion abstract the inversely nonlinear relationship of muscle contraction force and the possible contraction duration i e endurance is utilized in a method to mathematically predict individual muscle forces the method uses a nonlinear optimization scheme to solve the redundant distribution problem at the joints of interest the method is demonstrated at the elbow during isometric contraction and in the lower extremity during locomotion during gait the observed muscle activity pattern in the lower extremity as determined by emg shows substantial agreement with that activity pattern predicted when endurance is used as the optimization criterion the importance of selecting muscle prediction criteria based on sound physiological bases rather than on an arbitrary or mathematically convenient basis is emphasized

convergence of a direct iterative method for large scale least squares problems abstract in 1975 chen and gentleman suggested a 3 block sor method for solving least squares problems based on a partitioning scheme for the observation matrix a into a a 1 a 2 where a1 is square and nonsingular in many cases a1 is obvious from the nature of the problem this combined direct iterative method was discussed further and applied to angle adjustment problems in geodesy where a1 is easily formed and is large and sparse by plemmons in 1979 recently niethammer de pillis and varga have rekindled interest in this method by correcting and extending the sor convergence interval the purpose of our paper is to discuss an alternative formulation of the problem leading to a 2 block sor method for this formulation it is shown that the resulting direct iterative method always converges for sufficiently small sor parameter in contrast to the 3 block formulation formulas for the optimum sor parameter and the resulting asymptotic convergence factor based upon a2a 11 2 are given furthermore it is shown that this 2 cyclic block sor method always gives better convergence results than the 3 cyclic one for the same amount of work per iteration the direct part of the algorithm requires only a sparse matrix factorization of a1 our purpose here is to establish theoretical convergence results in line with the purpose of the recent paper by niethammer de pillis and varga practical considerations of choosing a1 in certain applications and of estimating the resulting a2a 11 2 will be addressed elsewhere

experimental design and observation for large systems large systems require new methods of experimental designs suitable for the highly adaptive models which are employed to cope with complex non linear responses and high dimensionality of input spaces the area of computer experiments has started to provide such designs especially latin hypercube and lattice designs system decomposition prevalent in several branches of engineering can be employed to decrease complexity a combination of system decomposition using a sparse matrix method experimental design and modelling is applied to one example of an electrical circuit simulator producing a usable emulator of the circuit for use in optimization and sensitivity analysis

a note on two block sor methods for sparse least squares problems abstract we compare two recently proposed block sor methods for the solution of large least squares problems with the conjugate gradient algorithm for solving the specially preconditioned normal equations by proving that all three methods are based on the same krylov sequence we show that the conjugate gradient algorithm is generally preferable to the two sor approaches

predicting the output from a complex computer code when fast approximations are available s we consider prediction and uncertainty analysis for complex computer codes which can be run at different levels of sophistication in particular we wish to improve efficiency by combining expensive runs of the most complex versions of the code with relatively cheap runs from one or more simpler approximations a bayesian approach is described in which prior beliefs about the codes are represented in terms of gaussian processes an example is presented using two versions of an oil reservoir simulator 1 c complex mathematical models implemented in large computer codes have been used to study real systems in many areas of scientific research sacks et al 1989 usually because physical experimentation is too costly and sometimes impossible as in the case of large environmental systems a computer experiment involves running the code with various input values for the purpose of learning something about the real system often a simulator can be run at different levels of complexity with versions ranging from the most sophisticated high level code to the most basic for example in 4 we consider two codes which simulate oil pressure at a well of a hydrocarbon reservoir both codes use finite element analysis in which the rocks comprising the reservoir are represented by small interacting grid blocks the flow of oil within the reservoir can be simulated by considering the interaction between the blocks the two codes differ in the resolution of the grid so that we have a very accurate slow version using many small blocks and a crude approximation using large blocks which runs much faster alternatively a mathematical model could be expanded to include more of the scientific laws underlying the physical processes simple fast versions of the code may well include the most important features and are useful for preliminary investigations in real time applications the number of runs from a high level simulator may be limited by expense then there is a need to trade off the complexity of the expensive code with the availability of the simpler approximations the purpose of the current paper is to explore ways in which runs from several levels of a code can be used to make inference about the output from the most complex code we may also have uncertainty about values for the input parameters which apply in any given application uncertainty analysis of computer codes describes how this uncertainty on the inputs affects our uncertainty about the output

the correlation between metabolic and individual leg mechanical power during walking at different slopes and velocities during level ground walking mechanical work from each leg is required to redirect and accelerate the center of mass previous studies show a linear correlation between net metabolic power and the rate of step to step transition work during level ground walking with changing step lengths however correlations between metabolic power and individual leg power during step to step transitions while walking on uphill downhill slopes and at different velocities are not known this basic understanding of these relationships between metabolic demands and biomechanical tasks can provide important information for design and control of biomimetic assistive devices such as leg prostheses and orthoses thus we compared changes in metabolic power and mechanical power during step to step transitions while 19 subjects walked at seven slopes 0 3 6 and 9 and three velocities 1 00 1 25 and 1 50 m s a quadratic model explained more of the variance r2 0 58 0 61 than a linear model r2 0 37 0 52 between metabolic power and individual leg mechanical power during step to step transitions across all velocities a quadratic model explained more of the variance r2 0 57 0 76 than a linear model r2 0 52 0 59 between metabolic power and individual leg mechanical power during step to step transitions at each velocity for all slopes and explained more of the variance r2 0 12 0 54 than a linear model r2 0 07 0 49 at each slope for all velocities our results suggest that it is important to consider the mechanical function of each leg in the design of biomimetic assistive devices aimed at reducing metabolic costs when walking at different slopes and velocities

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

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

convergence of block iterative methods applied to sparse least squares problems abstract recently special attention has been given in the mathematical literature to the problems of accurately computing the least squares solutions of very large scale overdetermined systems of linear equations such as those arising in geodetical network problems in particular it has been suggested that one solve such problems iteratively by applying the block sor successive overrelaxation iterative method to a consistently ordered block jacobi matrix that is weakly cyclic of index 3 here we obtain new results theorem 1 giving the exact convergence and divergence domains for such iterative applications it is then shown how these results extend and correct the literature on such applications in addition analogous results theorem 2 are given for the case when the eigenvalues of the associated block jacobi matrix are nonnegative

bayesian design and analysis of computer experiments use of derivatives in surface prediction this article is concerned with the problem of predicting a deterministic response function yo over a multidimensional domain t given values of yo and all of its first derivatives at a set of design sites points in t the intended application is to computer experiments in which yo is an output from a computer model of a physical system and each point in t represents a particular configuration of the input parameters it is assumed that the first derivatives are already available e g from a sensitivity analysis or can be produced by the code that implements the model a bayesian approach in which the random function that represents prior uncertainty about yo is taken to be a stationary gaussian stochastic process is used the calculations needed to update the prior given observations of yo and its first derivatives at the design sites are given and are illustrated in a small example the issue of experimental design is also discussed in particular the criterion of maximizing the reduction in entropy

emulation of numerical models with over specified basis functions abstractmathematical models are frequently used to explore physical systems but can be computationally expensive to evaluate in such settings an emulator is used as a surrogate in this work we propose a basis function approach for computer model emulation to combine field observations with a collection of runs from the numerical model we use the proposed emulator within the kennedy o hagan framework of model calibration a novel feature of the approach is the use of an over specified set of basis functions where number of bases used and their inclusion probabilities are treated as unknown quantities the new approach is found to have smaller predictive uncertainty and computational efficiency than the standard gaussian process approach to emulation and calibration along with several simulation examples focusing on different model characteristics we also use the method to analyze a dataset on laboratory experiments related to astrophysics

quantifying adventitious error in a covariance structure as a random effect we present an approach to quantifying errors in covariance structures in which adventitious error identified as the process underlying the discrepancy between the population and the structured model is explicitly modeled as a random effect with a distribution and the dispersion parameter of this distribution to be estimated gives a measure of misspecification analytical properties of the resultant procedure are investigated and the measure of misspecification is found to be related to the root mean square error of approximation an algorithm is developed for numerical implementation of the procedure the consistency and asymptotic sampling distributions of the estimators are established under a new asymptotic paradigm and an assumption weaker than the standard pitman drift assumption simulations validate the asymptotic sampling distributions and demonstrate the importance of accounting for the variations in the parameter estimates due to adventitious error two examples are also given as illustrations

forcing function diagnostics for nonlinear dynamics this article investigates the problem of model diagnostics for systems described by nonlinear ordinary differential equations odes i propose modeling lack of fit as a time varying correction to the right hand side of a proposed differential equation this correction can be described as being a set of additive forcing functions estimated from data representing lack of fit in this manner allows us to graphically investigate model inadequacies and to suggest model improvements i derive lack of fit tests based on estimated forcing functions model building in partially observed systems of odes is particularly difficult and i consider the problem of identification of forcing functions in these systems the methods are illustrated with examples from computational neuroscience

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

simultaneous positive and negative external mechanical work in human walking in human walking the center of mass motion is similar to an inverted pendulum viewing double support as a transition from one inverted pendulum to the next we hypothesized that the leading leg performs negative work to redirect the center of mass velocity while simultaneously the trailing leg performs positive work to replace the lost energy to test this hypothesis we developed a method to quantify the external mechanical work performed by each limb individual limbs method traditional measures of external mechanical work use the sum of the ground reaction forces acting on the limbs combined limbs method allowing for the mathematical cancellation of simultaneous positive and negative work during multiple support periods we expected to find that the traditional combined limbs method underestimates external mechanical work by a substantial amount we used both methods to measure the external mechanical work performed by humans walking over a range of speeds we found that during double support the legs perform a substantial amount of positive and negative external work simultaneously the combined limbs measures of positive and negative external work were approximately 33 less than those calculated using the individual limbs method at all speeds the trailing leg performs greater than 97 of the double support positive work while the leading leg performs greater than 94 of the double support negative work

manifold learning for the emulation of spatial fields from computational models repeated evaluations of expensive computer models in applications such as design optimization and uncertainty quantification can be computationally infeasible for partial differential equation pde models the outputs of interest are often spatial fields leading to high dimensional output spaces although emulators can be used to find faithful and computationally inexpensive approximations of computer models there are few methods for handling high dimensional output spaces for gaussian process gp emulation approximations of the correlation structure and or dimensionality reduction are necessary linear dimensionality reduction will fail when the output space is not well approximated by a linear subspace of the ambient space in which it lies manifold learning can overcome the limitations of linear methods if an accurate inverse map is available in this paper we use kernel pca and diffusion maps to construct gp emulators for very high dimensional output spaces arising from pde model simulations for diffusion maps we develop a new inverse map approximation several examples are presented to demonstrate the accuracy of our approach

calibration of computational models with categorical parameters and correlated outputs via bayesian smoothing spline anova it has become commonplace to use complex computer models to predict outcomes in regions where data do not exist typically these models need to be calibrated and validated using some experimental data which often consists of multiple correlated outcomes in addition some of the model parameters may be categorical in nature such as a pointer variable to alternate models or submodels for some of the physics of the system here we present a general approach for calibration in such situations where an emulator of the computationally demanding models and a discrepancy term from the model to reality are represented within a bayesian smoothing spline bss anova framework the bss anova framework has several advantages over the traditional gaussian process including ease of handling categorical inputs and correlated outputs and improved computational efficiency finally this framework is then applied to the problem that motivated its design a calibration of a computational fluid dynamics cfd model of

optimality relationships forp cyclic sor the optimality question for blockp cyclic sor iterations discussed in young and varga is answered under natural conditions on the spectrum of the block jacobi matrix in particular it is shown that repartitioning a blockp cyclic matrix into a blockq cyclic form q p results in asymptotically faster sor convergence for the same amount of work per iteration as a consequence block 2 cyclic sor is optimal under these conditions