bayesian design of experiments using approximate coordinate exchange abstractthe construction of decision theoretical bayesian designs for realistically complex nonlinear models is computationally challenging as it requires the optimization of analytically intractable expected utility functions over high dimensional design spaces we provide the most general solution to date for this problem through a novel approximate coordinate exchange algorithm this methodology uses a gaussian process emulator to approximate the expected utility as a function of a single design coordinate in a series of conditional optimization steps it has flexibility to address problems for any choice of utility function and for a wide range of statistical models with different numbers of variables numbers of runs and randomization restrictions in contrast to existing approaches to bayesian design the method can find multi variable designs in large numbers of runs without resorting to asymptotic approximations to the posterior distribution or expected utility the methodology is demonstrated on

error estimation problem of nonlinear integral equations in the space l p p 1 the main purpose of this paper is to intoduce and study approximate solution method for nonlinear volterra hammerstein integral equations with an arbitrary smooth nonlinear kernel function at the end of this paper numerical example is treated

is gauss quadrature better than clenshaw curtis we compare the convergence behavior of gauss quadrature with that of its younger brother clenshaw curtis seven line matlab codes are presented that implement both methods and experiments show that the supposed factor of 2 advantage of gauss quadrature is rarely realized theorems are given to explain this effect first following o hara and smith in the 1960s the phenomenon is explained as a consequence of aliasing of coefficients in chebyshev expansions then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of log z 1 z 1 in the complex plane gauss quadrature corresponds to pade approximation at z infty clenshaw curtis quadrature corresponds to an approximation whose order of accuracy at z infty is only half as high but which is nevertheless equally accurate near 1 1

fast bayesian experimental design laplace based importance sampling for the expected information gain abstract in calculating expected information gain in optimal bayesian experimental design the computation of the inner loop in the classical double loop monte carlo requires a large number of samples and suffers from underflow if the number of samples is small these drawbacks can be avoided by using an importance sampling approach we present a computationally efficient method for optimal bayesian experimental design that introduces importance sampling based on the laplace method to the inner loop we derive the optimal values for the method parameters in which the average computational cost is minimized for a specified error tolerance we use three numerical examples to demonstrate the computational efficiency of our method compared with the classical double loop monte carlo and a single loop monte carlo method that uses the laplace approximation of the return value of the inner loop the first demonstration example is a scalar problem that is linear in the uncertain parameter the second example is a nonlinear scalar problem the third example deals with the optimal sensor placement for an electrical impedance tomography experiment to recover the fiber orientation in laminate composites

on the solution of a mixed nonlinear integral equation in this paper we consider a mixed nonlinear integral equation of the second kind in position and time the existence of a unique solution of this equation is discussed and proved a numerical method is used to obtain a system of harmmerstein integral equations of the second kind in position then the modified toeplitz matrix method as a numerical method is used to obtain a nonlinear algebraic system many important theorems related to the existence and uniqueness solution to the produced nonlinear algebraic system are derived the rate of convergence of the total error is discussed finally numerical examples when the kernel of position takes a logarithmic and carleman forms are presented and the error estimate in each case is calculated

new approach for numerical solution of hammerstein integral equations in this paper we use a numerical procedure for solving a class of nonlinear integral equations of the hammerstein type using the globally defined sinc basis functions properties of the sinc procedure are utilized to reduce the computation of the hammerstein equations to some algebraic equations we used some numerical examples to illustrate the accuracy and the implementation of the method

discrete legendre spectral projection methods for fredholm hammerstein integral equations in this paper we discuss the discrete legendre galerkin and discrete legendre collocation methods for fredholm hammerstein integral equations with smooth kernel using sufficiently accurate numerical quadrature rule we obtain optimal convergence rates for both discrete legendre galerkin and discrete legendre collocation solutions in both infinity and l 2 norm numerical examples are given to illustrate the theoretical results

on the solution of linear and nonlinear integral equation the purpose of this paper is to establish the solution of fredholm volterra integral equation of the second kind in the space l 2 w xc 0 t considering the following when fredholm integral term in li 2 w and volterra integral term in 0 t are linear also each of fredholm or volterra integral term is linear while the other term is nonlinear here w is defined as the domain of integration with respect to position while t is consider as the time email protected 0 t t also the fredholm volterra integral equation of the first kind is established as special case of these work many special in one two and three dimensional problems of integral equation are considered representing the kernel of fredholm integral term in the form of logarithmic function carleman function potential and generalized potential function and macdonald function are considered

about a numerical method of successive interpolations for functional hammerstein integral equations a new and robust numerical method for hammerstein functional integral equations is proposed the convergence and the numerical stability of the method are mathematically proved and tested on some examples

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

markov chains for exploring posterior distributions several markov chain methods are available for sampling from a posterior distribution two important examples are the gibbs sampler and the metropolis algorithm in addition several strategies are available for constructing hybrid algorithms this paper outlines some of the basic methods and strategies and discusses some related theoretical and practical issues on the theoretical side results from the theory of general state space markov chains can be used to obtain convergence rates laws of large numbers and central limit theorems for estimates obtained from markov chain methods these theoretical results can be used to guide the construction of more efficient algorithms for the practical use of markov chain methods standard simulation methodology provides several variance reduction techniques and also give guidance on the choice of sample size and allocation

gaussian process surrogates for failure detection an important task of uncertainty quantification is to identify the probability of undesired events in particular system failures caused by various sources of uncertainties in this work we consider the construction of gaussian process surrogates for failure detection and failure probability estimation in particular we consider the situation that the underlying computer models are extremely expensive and in this setting determining the sampling points in the state space is of essential importance we formulate the problem as an optimal experimental design for bayesian inferences of the limit state i e the failure boundary and propose an efficient numerical scheme to solve the resulting optimization problem in particular the proposed limit state inference method is capable of determining multiple sampling points at a time and thus it is well suited for problems where multiple computer simulations can be performed in parallel the accuracy and performance of the proposed method is demonstrated by both academic and practical examples

maximum entropy sampling and optimal bayesian experimental design when shannon entropy is used as a criterion in the optimal design of experiments advantage can be taken of the classical identity representing the joint entropy of parameters and observations as the sum of the marginal entropy of the observations and the preposterior conditional entropy of the parameters following previous work in which this idea was used in spatial sampling the method is applied to standard parameterized bayesian optimal experimental design under suitable conditions which include non linear as well as linear regression models it is shown in a few steps that maximizing the marginal entropy of the sample is equivalent to minimizing the preposterior entropy the usual bayesian criterion thus avoiding the use of conditional distributions it is shown using this marginal formulation that under normality assumptions every standard model which has a two point prior distribution on the parameters gives an optimal design supported on a single point other results include a new asymptotic formula which applies as the error variance is large and bounds on support size

sensor placement for calibration of spatially varying model parameters abstract this paper presents a sensor placement optimization framework for the calibration of spatially varying model parameters to account for the randomness of the calibration parameters over space and across specimens the spatially varying parameter is represented as a random field based on this representation bayesian calibration of spatially varying parameter is investigated to reduce the required computational effort during bayesian calibration the original computer simulation model is substituted with kriging surrogate models based on the singular value decomposition svd of the model response and the karhunen loeve expansion kle of the spatially varying parameters a sensor placement optimization problem is then formulated based on the bayesian calibration to maximize the expected information gain measured by the expected kullback leibler k l divergence the optimization problem needs to evaluate the expected k l divergence repeatedly which requires repeated calibration of the spatially varying parameter and this significantly increases the computational effort of solving the optimization problem to overcome this challenge an approximation for the posterior distribution is employed within the optimization problem to facilitate the identification of the optimal sensor locations using the simulated annealing algorithm a heat transfer problem with spatially varying thermal conductivity is used to demonstrate the effectiveness of the proposed method

optimal bayesian design by inhomogeneous markov chain simulation we consider decision problems defined by a utility function and an underlying probability model for all unknowns the utility function quantifies the decision maker s preferences over consequences the optimal decision maximizes the expected utility function where the expectation is taken with respect to all unknowns that is future data and parameters in many problems the solution is not analytically tractable for example the utility function might involve moments that can be computed only by numerical integration or simulation also the nature of the decision space i e the set of all possible actions might have a shape or dimension that complicates the maximization the motivating application for this discussion is the choice of a monitoring network when the optimization is performed over the high dimensional set of all possible locations of monitoring stations possibly including choice of the number of locations we propose an approach to optimal bayesian design based on inhomogeneous markov

optimal bayesian experimental design for priors of compact support with application to shock tube experiments for combustion kinetics summary the analysis of reactive systems in combustion science and technology relies on detailed models comprising many chemical reactions that describe the conversion of fuel and oxidizer into products and the formation of pollutants shock tube experiments are a convenient setting for measuring the rate parameters of individual reactions the temperature pressure and concentration of reactants are chosen to maximize the sensitivity of the measured quantities to the rate parameter of the target reaction in this study we optimize the experimental setup computationally by optimal experimental design in a bayesian framework we approximate the posterior probability density functions pdf using truncated gaussian distributions in order to account for the bounded domain of the uniform prior pdf of the parameters the underlying gaussian distribution is obtained in the spirit of the laplace method more precisely the mode is chosen as the maximum a posteriori map estimate and the covariance is chosen as the negative inverse of the hessian of the misfit function at the map estimate the model related entities are obtained from a polynomial surrogate the optimality quantified by the information gain measures can be estimated efficiently by a rejection sampling algorithm against the underlying gaussian probability distribution rather than against the true posterior this approach offers a significant error reduction when the magnitude of the invariants of the posterior covariance are comparable with the size of the bounded domain of the prior we demonstrate the accuracy and superior computational efficiency of our method for shock tube experiments aiming to measure the model parameters of a key reaction which is part of the complex kinetic network describing the hydrocarbon oxidation in the experiments the initial temperature and fuel concentration are optimized with respect to the expected information gain in the estimation of the parameters of the target reaction rate we show that the expected information gain surface can change its shape dramatically according to the level of noise introduced into the synthetic data the information that can be extracted from the data saturates as a logarithmic function of the number of experiments and few experiments are needed when they are conducted at the optimal experimental design conditions furthermore inversion of the legacy data indicates the validity and robustness of our designs copyright 2016 john wiley sons ltd

simulation based optimization with stochastic approximation using common random numbers the method of common random numbers is a technique used to reduce the variance of difference estimates in simulation optimization problems these differences are commonly used to estimate gradients of objective functions as part of the process of determining optimal values for parameters of a simulated system asymptotic results exist which show that using the common random numbers method in the iterative finite difference stochastic approximation optimization algorithm fdsa can increase the optimal rate of convergence of the algorithm from the typical rate of k 1 3 to the faster k 1 2 where k is the algorithm s iteration number simultaneous perturbation stochastic approximation spsa is a newer and often much more efficient optimization algorithm and we will show that this algorithm too converges faster when the common random numbers method is used we will also provide multivariate asymptotic covariance matrices for both the spsa and fdsa errors

nelder mead simplex modifications for simulation optimization when the nelder mead method is used to optimize the expected response of a stochastic system e g an output of a discrete event simulation model the simplex resizing steps of the method introduce risks of inappropriate termination we give analytical and empirical results describing the performance of nelder mead when it is applied to a response function that incorporates an additive white noise error and we use these results to develop new modifications of nelder mead that yield improved estimates of the optimal expected response compared to nelder mead the best performance was obtained by a modified method rs s9 in which a the best point in the simplex is reevaluated at each shrink step and b the simplex is reduced by 10 rather than 50 at each shrink step in a suite of 18 test problems that were adapted from the minpack collection of netlib the expected response at the estimated optimal point obtained by rs s9 had errors that averaged 15 less than at the original method s estimated optimal point at an average cost of three times as many function evaluations two well known existing modifications for stochastic responses the n 3 rule and the next to worst rule were found to be inferior to the new modification rs s9

optimal design via curve fitting of monte carlo experiments abstract this article explores numerical methods for stochastic optimization with special attention to bayesian design problems a common and challenging situation occurs when the objective function in bayesian applications the expected utility is very expensive to evaluate perhaps because it requires integration over a space of very large dimensionality our goal is to explore a class of optimization algorithms designed to gain efficiency in such situations by exploiting smoothness of the expected utility surface and borrowing information from neighboring design points the central idea is that of implementing stochastic optimization by curve fitting of monte carlo samples this is done by simulating draws from the joint parameter sample space and evaluating the observed utilities fitting a smooth surface through these simulated points serves as estimate for the expected utility surface the optimal design can then be found deterministically in this article we introduce a general algorithm for cu

reduced wiener chaos representation of random fields via basis adaptation and projection abstract a new characterization of random fields appearing in physical models is presented that is based on their well known homogeneous chaos expansions we take advantage of the adaptation capabilities of these expansions where the core idea is to rotate the basis of the underlying gaussian hilbert space in order to achieve reduced functional representations that concentrate the induced probability measure in a lower dimensional subspace for a smooth family of rotations along the domain of interest the uncorrelated gaussian inputs are transformed into a gaussian process thus introducing a mesoscale that captures intermediate characteristics of the quantity of interest

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

uncertainty quantification in chemical systems summary we demonstrate the use of multiwavelet spectral polynomial chaos techniques for uncertainty quantification in non isothermal ignition of a methane air system we employ bayesian inference for identifying the probabilistic representation of the uncertain parameters and propagate this uncertainty through the ignition process we analyze the time evolution of moments and probability density functions of the solution we also examine the role and significance of dependence among the uncertain parameters we finish with a discussion of the role of non linearity and the performance of the algorithm copyright q 2009 john wiley sons ltd

piecewise continuous collocation for integral equations the collocation method for integral equations of the second kind is surveyed and analyzed for the case in which the approximate solutions are only piecewise continuous difficulties with the usual function space setting of l infty d are discussed and a satisfactory sense of point evaluation is given for elements of l infty d other approaches which are discussed include i reformulation as a degenerate kernel method ii the prolongation restriction framework of noble iii other function space settings and iv reformulation as a continuous approximation problem by iterating the piecewise continuous approximate solution in the original integral equation

simulation based optimal bayesian experimental design for nonlinear systems the optimal selection of experimental conditions is essential to maximizing the value of data for inference and prediction particularly in situations where experiments are time consuming and expensive to conduct we propose a general mathematical framework and an algorithmic approach for optimal experimental design with nonlinear simulation based models in particular we focus on finding sets of experiments that provide the most information about targeted sets of parameters our framework employs a bayesian statistical setting which provides a foundation for inference from noisy indirect and incomplete data and a natural mechanism for incorporating heterogeneous sources of information an objective function is constructed from information theoretic measures reflecting expected information gain from proposed combinations of experiments polynomial chaos approximations and a two stage monte carlo sampling method are used to evaluate the expected information gain stochastic approximation algorithms are then used to make optimization feasible in computationally intensive and high dimensional settings these algorithms are demonstrated on model problems and on nonlinear parameter inference problems arising in detailed combustion kinetics

asymptotic preserving particle in cell method for the vlasov poisson system near quasineutrality this paper deals with the numerical resolution of the vlasov poisson system in a nearly quasineutral regime by particle in cell pic methods in this regime classical pic methods are subject to stability constraints on the time and space steps related to the small debye length and large plasma frequency here we propose an asymptotic preserving pic scheme which is not subjected to these limitations additionally when the plasma period and debye length are small compared to the time and space steps this method provides a consistent pic discretization of the quasineutral vlasov equation we perform several one dimensional numerical experiments which provide a solid validation of the method and its underlying concepts and compare the method with classical pic and direct implicit methods

modelling of heat transport in magnetised plasmas using non aligned coordinates we present discretisation schemes for the heat diffusion equation in strongly magnetised plasmas in 2 d cartesian and 3 d cylindrical geometry the algorithms described do not use field aligned coordinates but show nevertheless a strong reduction of the pollution of perpendicular heat flux caused by parallel heat conduction compared to standard finite element or other finite difference formulations results for test examples show independence of the numerical error from g for values of g g up to 10 9 in plane and 10 1 2 in periodic cylinder geometry simulating toroidal connection the algorithms appear particularly well suited for dynamic mhd calculations where the effort of using exactly aligned coordinates becomes prohibitive

monotone finite volume schemes for diffusion equations on polygonal meshes we construct a nonlinear finite volume fv scheme for diffusion equation on star shaped polygonal meshes and prove that the scheme is monotone i e it preserves positivity of analytical solutions for strongly anisotropic and heterogeneous full tensor coefficients our scheme has only cell centered unknowns and it treats material discontinuities rigorously and offers an explicit expression for the normal flux numerical results are presented to show how our scheme works for preserving positivity on various distorted meshes for both smooth and non smooth highly anisotropic solutions and numerical results show that our scheme appears to be approximate second order accuracy for the solution and first order accuracy for the flux

superconvergence of the iterated collocation methods for hammerstein equations in this paper we analyse the iterated collocation method for hammerstein equations with smooth and weakly singular kernels the paper expands the study which began in 16 concerning the superconvergence of the iterated galerkin method for hammerstein equations we obtain in this paper a similar superconvergence result for the iterated collocation method for hammerstein equations we also discuss the discrete collocation method for weakly singular hammerstein equations some discrete collocation methods for hammerstein equations with smooth kernels were given previously in 3 18

an asymptotic preserving method for strongly anisotropic diffusion equations based on field line integration in magnetized plasma the magnetic field confines the particles around the field lines the anisotropy intensity in the viscosity and heat conduction may reach the order of 1012 when the boundary conditions are periodic or neumann the strong diffusion leads to an ill posed limiting problem to remove the ill conditionedness in the highly anisotropic diffusion equations we introduce a simple but very efficient asymptotic preserving reformulation in this paper the key idea is that instead of discretizing the neumann boundary conditions locally we replace one of the neumann boundary condition by the integration of the original problem along the field line the singular 1 terms can be replaced by o 1 terms after the integration which yields a well posed problem small modifications to the original code are required and no change of coordinates nor mesh adaptation are needed uniform convergence with respect to the anisotropy strength 1 can be observed numerically and the condition number does not scale with the anisotropy

superconvergent nystrom and degenerate kernel methods for hammerstein integral equations in a recent paper we introduced new methods called superconvergent nystrom and degenerate kernel methods for approximating the solution of fredholm integral equations of the second kind with a smooth kernel in this paper these methods are applied to numerically solve the hammerstein equations by using an interpolatory projection at gauss points onto the space of discontinuous piecewise polynomials of degree r 1 we prove that as for fredholm integral equations the proposed methods exhibit convergence orders 3r and 4r for the iterated version several numerical examples are given to demonstrate the effectiveness of the current methods

gaussian process regression and conditional polynomial chaos for parameter estimation abstract we present a new approach for constructing a data driven surrogate model and using it for parameter estimation in partial differential equation pde models we first use parameter observations and gaussian process regression to condition the karhunen loeve kl expansion of the unknown space dependent parameters and then build the conditional generalized polynomial chaos gpc surrogate model of the pde states next we estimate the unknown parameters by computing coefficients in the kl expansion by minimizing the square difference between the gpc predictions and measurements of the states our approach addresses two major challenges in the parameter estimation first it reduces dimensionality of the parameter space and replaces expensive direct solutions of pdes with the conditional gpc surrogates second the estimated parameter field exactly matches the parameter measurements in addition we show that the conditional gpc surrogate can be used to estimate the state variance that in turn can be used to guide data acquisition we demonstrate that our approach improves the accuracy of parameter estimation with application to one and two dimensional darcy equations that have unknown space dependent hydraulic conductivity we also discuss the effect of hydraulic conductivity and head locations on the accuracy of the hydraulic conductivity estimations

quasi monte carlo integration the standard monte carlo approach to evaluating multidimensional integrals using pseudo random integration nodes is frequently used when quadrature methods are too difficult or expensive to implement as an alternative to the random methods it has been suggested that lower error and improved convergence may be obtained by replacing the pseudo random sequences with more uniformly distributed sequences known as quasi random in this paper quasi random halton sobol and faure and pseudo random sequences are compared in computational experiments designed to determine the effects on convergence of certain properties of the integrand including variance variation smoothness and dimension the results show that variation which plays an important role in the theoretical upper bound given by the koksma hlawka inequality does not affect convergence while variance the determining factor in random monte carlo is shown to provide a rough upper bound but does not accurately predict performance in general quasi monte carlo methods are superior to random monte carlo but the advantage may be slight particularly in high dimensions or for integrands that are not smooth for discontinuous integrands we derive a bound which shows that the exponent for algebraic decay of the integration error from quasi monte carlo is only slightly larger than 1 2 in high dimensions more 21 refs 6 figs 5 tabs less

on approximate solution of hammerstein integral equations in the space l p p 1 this paper deals with an approximate method for solving hammerstein integral equation this method is based on the replacement of the kernel of the given equation by degenerate kernel and which it has a solution equivalent to the solution of a system of nonlinear algebraic equations suitable numerical technique can be used to obtain the results of that problem at the end of this paper numerical examples were presented

finite element and higher order difference formulations for modelling heat transport in magnetised plasmas we present a finite element analogue to the second order finite difference scheme for the solution of the heat diffusion equation in strongly magnetised plasmas given in gunter et al s gunter et al j comp phys 209 2005 354 compared to standard finite element or finite difference formulations it strongly reduces the pollution of perpendicular heat fluxes by parallel ones even without resorting to field aligned coordinates we present both bi linear and bi quadratic versions of this scheme as well as a fourth order extension of the original difference scheme of gunter et al 2005 in the second part of the paper we address the formulation of the boundary conditions at walls with an oblique incidence of field lines and the treatment of the coordinate singularity at r 0 in cylindrical or topologically equivalent coordinates with the reduced pollution finite difference scheme all tests shown indicate that both the finite difference and the finite element versions of the scheme should substantially alleviate the requirement for field alignment of the coordinates over the realistic range of in toroidal magnetic confinement devices

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

on existence of unique solution to non linear integral equations of the second kind this paper we employ the technique of degenerate kernel method by using the linear polynomial operator which plays an important role in study of numerical technique can be used to obtain the results of that problem a numerical example is reported at the end of this paper to illustrate our technique

convergence of simultaneous perturbation stochastic approximation for nondifferentiable optimization we consider simultaneous perturbation stochastic approximation for function minimization the standard assumption for convergence is that the function be three times differentiable although weaker assumptions have been used for special cases however all work that we are aware of at least requires differentiability we relax the differentiability requirement and prove convergence using convex analysis

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

asymptotic preserving scheme for highly anisotropic non linear diffusion equations heat transfer in magnetically confined plasmas is a process characterized by non linear and extremely high anisotropic diffusion phenomena standard numerical methods successfully employed in the numerical treatment of classical diffusion problems are generally inefficient or even prone to break down when such high anisotropies come into play leading thus to the need of new numerical techniques suitable for this kind of problems in the present paper the authors propose a numerical scheme based on an asymptotic preserving ap reformulation of this non linear evolution problem generalizing the ideas introduced in a previous paper for the case of elliptic anisotropic problems p degond a lozinski j narski c negulescu an asymptotic preserving method for highly anisotropic elliptic equations based on a micro macro decomposition j comput phys 231 7 2012 2724 2740 the performances of the here proposed ap scheme are tested numerically in particular it is shown that the scheme is capable to deal with problems characterized by a high degree of anisotropy thus proving to be suitable for the study of anisotropic diffusion in magnetically confined plasmas

fixed point method for solving nonlinear quadratic volterra integral equations we consider a paper of banas and sadarangani 2008 11 which deals with monotonicity properties of the superposition operator and their applications an application of the monotonicity properties is to study the solvability of a quadratic volterra integral equation in this paper we prepare an efficient numerical technique based on the fixed point method and quadrature rules to approximate a solution for quadratic volterra integral equation then convergence of numerical scheme is proved by some theorems and some numerical examples are given to show applicability and accuracy of the numerical method and guarantee the theoretical results

an asymptotic preserving method for highly anisotropic elliptic equations based on a micro macro decomposition the concern of the present work is the introduction of a very efficient asymptotic preserving scheme for the resolution of highly anisotropic diffusion equations the characteristic features of this scheme are the uniform convergence with respect to the anisotropy parameter 0 e 1 the applicability on cartesian grids to cases of non uniform and non aligned anisotropy fields b and the simple extension to the case of a non constant anisotropy intensity 1 e the mathematical approach and the numerical scheme are different from those presented in the previous work p degond f deluzet a lozinski j narski c negulescu duality based asymptotic preserving method for highly anisotropic diffusion equations communications in mathematical sciences 10 1 2012 1 31 and its considerable advantages are pointed out

a discrete collocation type method for hammerstein equations in recent papers we have considered the numerical solution of the hammerstein equation y t f t int a b k t s g s y s ds quad t in a b by a method that first applies the standard collocation procedure to an equivalent equation for z t g t y t and then obtains an approximation to y by use of the equation y t f t int a b k t s z s ds quad t in a b we study here a discretized version of the above method this arises when numerical quadrature is used to approximate the definite integrals occurring in the said method we consider the use of interpolatory quadrature rules and we seek the discrete collocation approximation to z in certain piecewise polynomial function spaces our principal result gives the precision required of a quadrature rule to guarantee the best possible super convergence rate for the discrete approximation to y in the case where the kernel k is sufficiently smooth this rate is of the same order as the rate for an approximation to y obtain

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

optimal experimental design for inverse problems with state constraints in this paper we address the challenging problem of optimal experimental design oed of inverse problems with state constraints we consider two oed formulations that allow us to reduce the exper

a new collocation type method for hammerstein integral equations on presente une methode du type de collocation pour la resolution numerique des equations integrales d hammerstein on etablit la convergence de l approximation desiree vers la solution exacte

fast multilevel augmentation methods for solving hammerstein equations fast multilevel augmentation methods based on projections for solving hammerstein equations are developed each of these methods requires availability of a multilevel decomposition of the solution space and a projection from the solution space onto a finite dimensional subspace a subspace at a level is obtained from the subspace at the previous level by adding a difference subspace accordingly the projection equation at the present level is obtained by augmenting the projection equation at the previous level using this idea recursively we solve the nonlinear equation by an iteration method only at an initial lower resolution level while obtaining its solution at a higher resolution level with the help of the multiscale analysis we separate the procedure of solving the nonlinear operator equation at a high level into two major components 1 solving the nonlinear equation only at an initial lower level and 2 compensating the error by solving a linear system at the high level we prove that the proposed methods require only linear up to a logarithmic factor computational complexity and have the optimal convergence order a relationship between the proposed method and the multigrid method is discussed two specific fast methods based on the galerkin projection and the collocation projection are developed numerical results are presented to confirm the theoretical estimates with comparisons to the two grid method

multivariate stochastic approximation using a simultaneous perturbation gradient approximation the problem of finding a root of the multivariate gradient equation that arises in function minimization is considered when only noisy measurements of the function are available a stochastic approximation sa algorithm for the general kiefer wolfowitz type is appropriate for estimating the root the paper presents an sa algorithm that is based on a simultaneous perturbation gradient approximation instead of the standard finite difference approximation of keifer wolfowitz type procedures theory and numerical experience indicate that the algorithm can be significantly more efficient than the standard algorithms in large dimensional problems

a chebyshev approximation for solving optimal control problems abstract this paper presents a numerical solution for solving optimal control problems and the controlled duffing oscillator a new chebyshev spectral procedure is introduced control variables and state variables are approximated by chebyshev series then the system dynamics is transformed into systems of algebraic equations the optimal control problem is reduced to a constrained optimization problem results and comparisons are given at the end of the paper

an anisotropic mesh adaptation method for the finite element solution of heterogeneous anisotropic diffusion problems heterogeneous anisotropic diffusion problems arise in the various areas of science and engineering including plasma physics petroleum engineering and image processing standard numerical methods can produce spurious oscillations when they are used to solve those problems a common approach to avoid this difficulty is to design a proper numerical scheme and or a proper mesh so that the numerical solution validates the discrete counterpart dmp of the maximum principle satisfied by the continuous solution a well known mesh condition for the dmp satisfaction by the linear finite element solution of isotropic diffusion problems is the non obtuse angle condition that requires the dihedral angles of mesh elements to be non obtuse in this paper a generalization of the condition the so called anisotropic non obtuse angle condition is developed for the finite element solution of heterogeneous anisotropic diffusion problems the new condition is essentially the same as the existing one except that the dihedral angles are now measured in a metric depending on the diffusion matrix of the underlying problem several variants of the new condition are obtained based on one of them two metric tensors for use in anisotropic mesh generation are developed to account for dmp satisfaction and the combination of dmp satisfaction and mesh adaptivity numerical examples are given to demonstrate the features of the linear finite element method for anisotropic meshes generated with the metric tensors

monotone finite volume schemes of nonequilibrium radiation diffusion equations on distorted meshes we construct a monotone finite volume scheme on distorted meshes for multimaterial nonequilibrium radiation diffusion problems which are described by the coupled radiation diffusion and material conduction equations moreover we prove theoretically that the scheme is monotone numerical results are presented to show that our scheme preserves positivity of solution on various distorted meshes and the contours of numerical solution obtained by our scheme on distorted meshes accord with that on rectangular meshes moreover numerical tests indicate that our monotone scheme is more computationally efficient than the nine point scheme these results show that our nonlinear monotone finite volume scheme is a practical and attractive method for solving nonlinear diffusion equations on distorted meshes

on locking and robustness in the finite element method a numerical scheme for the approximation of a parameter dependent problem is said to exhibit locking if the accuracy of the approximations deteriorates as the parameter tends to a limiting value a robust numerical scheme for the problem is one that is essentially uniformly convergent for all values of the parameter precise mathematical definitions for these terms are developed their quantitative characterization is given and some general theorems involving locking and robustness are proven a model problem involving heat transfer is analyzed in detail using this mathematical framework and various related computational results are described applications to some different problems involving locking are presented

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

delayed rejection in reversible jump metropolis hastings summary in a metropolis hastings algorithm rejection of proposed moves is an intrinsic part of ensuring that the chain converges to the intended target distribution however persistent rejection perhaps in particular parts of the state space may indicate that locally the proposal distribution is badly calibrated to the target as an alternative to careful off line tuning of state dependent proposals the basic algorithm can be modified so that on rejection a second attempt to move is made a different proposal can be generated from a new distribution that is allowed to depend on the previously rejected proposal we generalise this idea of delaying the rejection and adapting the proposal distribution due to tierney mira 1999 to generate a more flexible class of methods that applies in particular to a variable dimension setting the approach is illustrated by two pedagogical examples and a more realistic application to a changepoints analysis for point processes some key wor ds adaptive changepoint efficiency of markov chain monte carlo estimation integrated autocorrelation time peskun ordering

a meshless method for solving nonlinear two dimensional integral equations of the second kind on non rectangular domains using radial basis functions with error analysis in this paper we present a numerical method for solving two dimensional nonlinear fredholm integral equations of the second kind on a non rectangular domain the method utilizes radial basis functions rbfs constructed on scattered points as a basis in the discrete collocation method the proposed scheme is meshless since it does not need any domain element and so it is independent of the geometry of the domain the method reduces the solution of the two dimensional nonlinear integral equation to the solution of a nonlinear system of algebraic equations error analysis is presented for this method finally numerical examples are included to show the validity and efficiency of the new technique

a fast multiscale solver for modified hammerstein equations abstract this paper presents a fast solver called the multilevel augmentation method for modified nonlinear hammerstein equations when we utilize the method to solve a large scale problem most of components of the solution can be computed directly and lower frequency components can be obtained by solving a fixed low order algebraic nonlinear system the advantage of using the algorithm to modified equations is that it leads to reduce the cost of numerical integrations greatly the optimal error estimate of the method is established and the nearly linear computational complexity is proved finally numerical examples are presented to confirm the theoretical results and illustrate the efficiency of the method

fast estimation of expected information gains for bayesian experimental designs based on laplace approximations shannon type expected information gain can be used to evaluate the relevance of a proposed experiment subjected to uncertainty the estimation of such gain however relies on a double loop integration moreover its numerical integration in multi dimensional cases e g when using monte carlo sampling methods is therefore computationally too expensive for realistic physical models especially for those involving the solution of partial differential equations in this work we present a new methodology based on the laplace approximation for the integration of the posterior probability density function pdf to accelerate the estimation of the expected information gains in the model parameters and predictive quantities of interest we obtain a closed form approximation of the inner integral and the corresponding dominant error term in the cases where parameters are determined by the experiment such that only a single loop integration is needed to carry out the estimation of the expected information gain to deal with the issue of dimensionality in a complex problem we use a sparse quadrature for the integration over the prior pdf we demonstrate the accuracy efficiency and robustness of the proposed method via several nonlinear numerical examples including the designs of the scalar parameter in a one dimensional cubic polynomial function the design of the same scalar in a modified function with two indistinguishable parameters the resolution width and measurement time for a blurred single peak spectrum and the boundary source locations for impedance tomography in a square domain