swept regions and surfaces modeling and volumetric properties we consider swept regions w and swept hypersurfaces b in r n 1 and especially r 3 which are a disjoint union of subspaces w t w p t or b t b p t obtained from a varying family of affine subspaces p t t c we concentrate on the case where w and b are obtained from a skeletal structure m u this generalizes the blum medial axis m of a region w which consists of the centers of interior spheres tangent to the boundary b at two or more points with u denoting the vectors from the centers of the spheres to the points of tangency we extend methods developed for skeletal structures so that they can be deduced from the properties of the individual intersections w t or b t and a relative shape operator s r e l which we introduce to capture changes relative to the varying family p t we use these results to deduce modeling properties of the global b in terms of the individual b t and determine volumetric properties of regions w expressed as global integrals of functions g on w in terms of iterated integrals over the skeletal structure of w t which is then integrated over the parameter space c

a proper generalized decomposition for the solution of elliptic problems in abstract form by using a functional eckart young approach the proper generalized decomposition pgd is a methodology initially proposed for the solution of partial differential equations pde defined in tensor product spaces it consists in constructing a separated representation of the solution of a given pde in this paper we consider the mathematical analysis of this framework for a larger class of problems in an abstract setting in particular we introduce a generalization of eckart and young theorem which allows to prove the convergence of the so called progressive pgd for a large class of linear problems defined in tensor product hilbert spaces

finite cover method for linear and non linear analyses of heterogeneous solids we introduce the finite cover method fcm as a generalization of the finite element method fem and extend it to analyse the linear and non linear mechanical behaviour of heterogeneous solids and structures the name fcm is actually an alias for the manifold method mm and the basic idea of the method has already been established for linear analyses of structures with homogeneous materials after reviewing the concept of physical and mathematical covers for approximating functions in the fcm we present the formulation for the static equilibrium state of a structure with arbitrary physical boundaries including material interfaces the problem essentially involves the discontinuities in strains and possibly has the discontinuities in displacement caused by interfacial debonding or rupture of material interfaces we simulate such non linear mechanical behaviour after presenting simple numerical examples that demonstrate the equivalence between the approximation capabilities of the fcm and those of the fem copyright 2003 john wiley sons ltd

global superconvergence of finite element methods for biharmonic equations and blending surfaces abstract in this paper new numerical algorithms of finite element methods fem are reported for both biharmonic equations and 3d blending surfaces to achieve the global superconvergence o h3 o h4 in h2 norms this is significant compared with the optimal convergence o h2 the algorithms are simple because only an a posteriori interpolant solution is needed such a global convergence method was originated by lin and his colleagues in 1 3 for only the clamped boundary conditions recently we extended the global superconvergence to other boundary conditions such as the simple support condition the periodic boundary and the natural boundary condition moreover we apply in 4 6 this global superconvergence to the fem using the penalty techniques for biharmonic equations and blending problems also to reach o h3 and o h4 for quasiuniform and uniform ij respectively currently we develop in 7 8 and in this paper the fem using the penalty plus hybrid techniques to reduce the condition number down to o h 4 o h 5 of the associated matrix while retaining superconvergence o h3 o h4 since instability is severe for biharmonic equations any reduction of the condition number is crucial by the new algorithms in this paper not only can a great deal of cpu time be saved but also the complicated biharmonic equations and blending surfaces may be solved in double precision numerical experiments are carried out to support the theoretical conclusions

the accuracy of b spline finite element approximations to pde surfaces in computer aided design a developing design tool is the pde method where surfaces are generated by solving a partial differential equation with appropriate boundary conditions methods for generating b spline approximations to pde surfaces to enable data transfer to other cad systems have been considered and the fairness properties of the resulting b spline surfaces examined for some applications the accuracy of the approximations will be important this paper examines the effect on the accuracy of different finite element mesh sizes and degrees of b spline surface concluding with recommendations about how to change these parameters in order to achieve the required accuracy

multigrid accelerated tensor approximation of function related multidimensional arrays in this paper we describe and analyze a novel tensor approximation method for discretized multidimensional functions and operators in mathbb r d based on the idea of multigrid acceleration the approach stands on successive reiterations of the orthogonal tucker tensor approximation on a sequence of nested refined grids on the one hand it provides a good initial guess for the nonlinear iterations to find the approximating subspaces on finer grids on the other hand it allows us to transfer from the coarse to fine grids the important data structure information on the location of the so called most important fibers in directional unfolding matrices the method indicates linear complexity with respect to the size of data representing the input tensor in particular if the target tensor is given by using the rank r canonical model then our approximation method is proved to have linear scaling in the univariate grid size n and in the input rank r the method is tested by three dimensional 3d electronic structure calculations for the multigrid accelerated low tucker rank approximation of the all electron densities having strong nuclear cusps we obtain high resolution of their 3d convolution product with the newton potential the accuracy of order 10 6 in max norm is achieved on large n times n times n grids up to n 1 6 cdot10 4 with the time scale in several minutes

integrable discrete differential geometry of plated membranes in equilibrium recently it has been established that a well known system of classical shell theory descriptive of membranes in equilibrium is in fact amenable to the techniques of soliton theory here it is d

the partition of unity finite element approach with hp refinement for the stationary fokker planck equation abstract in this paper the stationary fokker planck equation fpe is solved for nonlinear dynamical systems using a local numerical technique based on the meshless partition of unity finite element method pufem the method is applied to stationary fpe for 2 3 and 4 d systems and is argued to be an excellent candidate for higher dimensional problems and the transient problem local refinement is applied by introducing higher order polynomials in selected subdomains local p refinement to keep the problem size small while ensuring high approximation accuracy various local approximations are blended using novel pasting functions that provide any desired order of continuity results are compared with existing global and local techniques local p refinement is touted as an important step towards breaking the curse of dimensionality in numerical solution of fpe

a priori model reduction through proper generalized decomposition for solving time dependent partial differential equations over the past years model reduction techniques have become a necessary path for the reduction of computational requirements in the numerical simulation of complex models a family of a priori model reduction techniques called proper generalized decomposition pgd methods are receiving a growing interest these methods rely on the a priori construction of separated variables representations of the solution of models defined in tensor product spaces they can be interpreted as generalizations of proper orthogonal decomposition pod for the a priori construction of such separated representations in this paper we introduce and study different definitions of pgd for the solution of time dependent partial differential equations we review classical definitions of pgd based on galerkin or minimal residual formulations and we propose and discuss several improvements for these classical definitions we give an interpretation of optimal decompositions as the solution of pseudo eigenproblems we also introduce a new definition of pgd called minimax pgd which can be interpreted as a petrov galerkin model reduction technique where test and trial reduced basis functions are related by an adjoint problem this new definition improves convergence properties of separated representations with respect to a chosen metric it coincides with a classical pod for degenerate time dependent partial differential equations for the numerical construction of each pgd we propose algorithms inspired from the solution of eigenproblems several numerical examples illustrate and compare the different definitions of pgd on transient advection diffusion reaction equations

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

probabilistic analysis on the splitting shooting method for image transformations abstract this paper explores a new topic in image processing where accuracy of images even in details is crucial and adopts a new methodology dealing with discrete topics by continuous mathematics and numerical approximation the key idea is that a pixel of images at different levels can be quantified by a greyness value which can then be regarded as the mean of an integral of continuous functions over a small pixel region and evaluated by numerical integration approximately hence new treatments of approximate integration and new discrete algorithms of images have been developed this paper also integrates different mathematics disciplines numerical analysis geometry probability and statistics to discrete images that can be applied to many areas in computer sciences image processing computer graphics computer vision geometric added designs and pattern recognition in this paper new error analysis in terms of probability theory is explored for the popular splitting shooting method ssm and the combination csim of the splitting shooting integrating methods proposed in 35 37 and convergence rates in probability of image greyness are proven to be o p 1 n 1 5 higher than o 1 n reported in 45 by strict error analysis moreover a new partial refinement technique of pixel partition is also proposed in this paper to achieve the convergence rate o p 1 n 2 in probability for ssm by the new study of this paper the ssm can be applied to real images with 256 greyness levels the numerical and graphical experiments are also provided to confirm the theoretical analysis made both the strict error bounds and the probabilistic error bounds with explicit constants are also derived for general norms as 1 and the countable probability inequalities for several sums of random variables are developed from probability theory which are more suited to numerical computations

an algorithm for isotonic regression for two or more independent variables algorithms for solving the isotonic regression problem in more than one dimension are difficult to implement because of the large number of lower sets present or because they involve search techniques which require a significant amount of checking and readjustment here a new algorithm for solving this problem based on a simple iterative technique is proposed and shown to converge to the correct solution

maximum likelihood estimation under order restrictions by the prior feedback method abstract algorithms for deriving isotonic regression estimators in order restricted linear models and more generally restricted maximum likelihood estimators are usually quite dependent on the particular problem considered we propose here an optimization method based on a sequence of formal bayes estimates whose variances converge to zero this method akin to simulated annealing can be applied universally that is as long as these bayes estimators can be derived by exact computation or markov chain monte carlo sampling approximation we then give an illustration of our method for two real life examples

proper generalized decomposition of time multiscale models summary models encountered in computational mechanics could involve many time scales when these time scales cannot be separated one must solve the evolution model in the entire time interval by using the finest time step that the model implies in some cases the solution procedure becomes cumbersome because of the extremely large number of time steps needed for integrating the evolution model in the whole time interval in this paper we considered an alternative approach that lies in separating the time axis one dimensional in nature in a multidimensional time space then for circumventing the resulting curse of dimensionality the proper generalized decomposition was applied allowing a fast solution with significant computing time savings with respect to a standard incremental integration copyright 2011 john wiley sons ltd

interpolation with slackness and continuity control and convexity preservation using singular blending this paper presents a new interpolation method that enables the construction of c2 cubic polynomial spline curves without solving a global system of equations while providing slackness continuity control and convexity preserving ability the basic idea is to blend a cubic b spline curve with a singularly parametrized sequence of connected line segments a global slackness parameter controls the tautness specifically the distance between the interpolating curve and the linear interpolant the order of continuity at each knot is controlled via multiple knot insertions so that cusps and straight line segments can be conveniently prescribed in addition a method for selecting local slackness values to produce g1 convexity preserving curve is presented with the low degree polynomials and direct computation of control vertices this local method is computationally simple and is useful for interactive shape design and computer graphics applications

sampling based approaches to calculating marginal densities abstract stochastic substitution the gibbs sampler and the sampling importance resampling algorithm can be viewed as three alternative sampling or monte carlo based approaches to the calculation of numerical estimates of marginal probability distributions the three approaches will be reviewed compared and contrasted in relation to various joint probability structures frequently encountered in applications in particular the relevance of the approaches to calculating bayesian posterior densities for a variety of structured models will be discussed and illustrated

the finite cell method for three dimensional problems of solid mechanics this article presents a generalization of the recently proposed finite cell method to three dimensional problems of linear elasticity the finite cell method combines ideas from embedding or fictitious domain methods with the p version of the finite element method besides supporting a fast simple generation of meshes it also provides high convergence rates mesh generation for a boundary representation of solids and for voxel based data obtained from ct scans is addressed in detail in addition the implementation of non homogeneous neumann boundary conditions and the computation of cell matrices based on a composed integration is presented the performance of the proposed method is demonstrated by three numerical examples including the elastostatic computation of a human bone biopsy

tensor decompositions and applications this survey provides an overview of higher order tensor decompositions their applications and available software a tensor is a multidimensional or n way array decompositions of higher order tensors i e n way arrays with n geq 3 have applications in psycho metrics chemometrics signal processing numerical linear algebra computer vision numerical analysis data mining neuroscience graph analysis and elsewhere two particular tensor decompositions can be considered to be higher order extensions of the matrix singular value decomposition candecomp parafac cp decomposes a tensor as a sum of rank one tensors and the tucker decomposition is a higher order form of principal component analysis there are many other tensor decompositions including indscal parafac2 candelinc dedicom and paratuck2 as well as nonnegative variants of all of the above the n way toolbox tensor toolbox and multilinear engine are examples of software packages for working with tensors

nonconjugate bayesian analysis of variance component models we consider the usual normal linear mixed model for variance components from a bayesian viewpoint with conjugate priors and balanced data gibbs sampling is easy to implement however simulating from full conditionals can become difficult for the analysis of unbalanced data with possibly nonconjugate priors thus leading one to consider alternative markov chain monte carlo schemes we propose and investigate a method for posterior simulation based on an independence chain the method is customized to exploit the structure of the variance component model and it works with arbitrary prior distributions as a default reference prior we use a version of jeffreys prior based on the integrated restricted likelihood we demonstrate the ease of application and flexibility of this approach in familiar settings involving both balanced and unbalanced data

on the verification of model reduction methods based on the proper generalized decomposition in this work we introduce a consistent error estimator for numerical simulations performed by means of the proper generalized decomposition pgd approximation this estimator which is based on the constitutive relation error enables to capture all error sources i e those coming from space and time numerical discretizations from the truncation of the pgd decomposition etc and leads to guaranteed bounds on the exact error the specificity of the associated method is a double approach i e a kinematic approach and a unusual static approach for solving the parameterized problem by means of pgd this last approach makes straightforward the computation of a statically admissible solution which is necessary for robust error estimation an attractive feature of the error estimator we set up is that it is obtained by means of classical procedures available in finite element codes it thus represents a practical and relevant tool for driving algorithms carried out in pgd being possibly used as a stopping criterion or as an adaptation indicator numerical experiments on transient thermal problems illustrate the performances of the proposed method for global error estimation

a least squares approximation of partial differential equations with high dimensional random inputs uncertainty quantification schemes based on stochastic galerkin projections with global or local basis functions and also stochastic collocation methods in their conventional form suffer from the so called curse of dimensionality the associated computational cost grows exponentially as a function of the number of random variables defining the underlying probability space of the problem in this paper to overcome the curse of dimensionality a low rank separated approximation of the solution of a stochastic partial differential spde with high dimensional random input data is obtained using an alternating least squares als scheme it will be shown that in theory the computational cost of the proposed algorithm grows linearly with respect to the dimension of the underlying probability space of the system for the case of an elliptic spde an a priori error analysis of the algorithm is derived finally different aspects of the proposed methodology are explored through its application to some numerical experiments

asymptotically exact a posteriori error estimators for first order div least squares methods in local and global l2 norm a new asymptotically exact a posteriori error estimator is developed for first order div least squares ls finite element methods let uh h be ls approximate solution for u au then e a1 2 h auh 0 is asymptotically exact a posteriori error estimator for a1 2 uuh 0 or a1 2 h 0 depending on the order of approximate spaces for and u for e to be asymptotically exact for a1 2 uuh 0 we require higher order approximation property for and vice versa when both au and are approximated in the same order of accuracy the estimator becomes an equivalent error estimator for both errors the underlying mesh is only required to be shape regular i e it does not require quasi uniform mesh nor any special structure for the underlying meshes confirming numerical results are provided and the performance of the estimator is explored for other choice of spaces for uh h

design and recovery of 2 d and 3 d shapes using rational gaussian curves and surfaces a new representation for parametric curves and surfaces is introduced here it is in rational form and uses rational gaussian bases this representation allows design of 2 d and 3 d shapes and makes recovery of shapes from noisy image data possible the standard deviations of gaussians in a curve or surface control the smoothness of a recovered shape the control points of a surface in this representation are not required to form a regular grid and a scattered set of control points is sufficient to reconstruct a surface examples of shape design shape recovery and image segmentation using the proposed representation are given

proper generalized decomposition for nonlinear convex problems in tensor banach spaces tensor based methods are receiving a growing interest in scientific computing for the numerical solution of problems defined in high dimensional tensor product spaces a family of methods called proper generalized decompositions pgd methods have been recently introduced for the a priori construction of tensor approximations of the solution of such problems in this paper we give a mathematical analysis of a family of progressive and updated pgds for a particular class of problems associated with the minimization of a convex functional over a reflexive tensor banach space

an error estimator for separated representations of highly multidimensional models fine modeling of the structure and mechanics of materials from the nanometric to the micrometric scales uses descriptions ranging from quantum to statistical mechanics most of these models consist of a partial differential equation defined in a highly multidimensional domain e g schrodinger equation fokker planck equations among many others the main challenge related to these models is their associated curse of dimensionality we proposed in some of our former works a new strategy able to circumvent the curse of dimensionality based on the use of separated representations also known as finite sums decomposition this technique proceeds by computing at each iteration a new sum that consists of a product of functions each one defined in one of the model coordinates the issue related to error estimation has never been addressed this paper presents a first attempt on the accuracy evaluation of such a kind of discretization techniques

transformations of quadrilateral lattices motivated by the classical studies on transformations of conjugate nets we develop the general geometric theory of transformations of their discrete analogs the multidimensional quadrilateral lattices i e lattices x zn rm n m whose elementary quadrilaterals are planar our investigation is based on the discrete analog of the theory of the rectilinear congruences which we also present in detail we study in particular the discrete analogs of the laplace combescure levy radial and fundamental transformations and their interrelations the composition of these transformations and their permutability is also investigated from a geometric point of view the deep connections between transformations and discretizations is also investigated for quadrilateral lattices we finally interpret these results within the formalism

algorithms for numerical analysis in high dimensions nearly every numerical analysis algorithm has computational complexity that scales exponentially in the underlying physical dimension the separated representation introduced previously allows many operations to be performed with scaling that is formally linear in the dimension in this paper we further develop this representation by i discussing the variety of mechanisms that allow it to be surprisingly efficient ii addressing the issue of conditioning iii presenting algorithms for solving linear systems within this framework and iv demonstrating methods for dealing with antisymmetric functions as arise in the multiparticle schrodinger equation in quantum mechanics numerical examples are given

tensor based methods for numerical homogenization from high resolution images we present a complete numerical strategy based on tensor approximation techniques for the solution of numerical homogenization problems with geometrical data coming from high resolution images we first introduce specific numerical treatments for the translation of image based homogenization problems into a tensor framework it includes the tensor approximations in suitable tensor formats of fields of material properties or indicator functions of multiple material phases recovered from segmented images we then introduce some variants of proper generalized decomposition pgd methods for the construction of tensor decompositions in different tensor formats of the solution of boundary value problems a new definition of pgd is introduced which allows the progressive construction of a tucker decomposition of the solution this tensor format is well adapted to the present application and improves convergence properties of tensor decompositions finally we use a dual based error estimator on quantities of interest which was recently introduced in the context of pgd we exhibit its specificities when it is used for assessing the error on the homogenized properties of the heterogeneous material we also provide a complete goal oriented adaptive strategy for the progressive construction of tensor decompositions of primal and dual solutions yielding to predictions of homogenized quantities with a prescribed accuracy

bayesian inference on order constrained parameters in generalized linear models summary in biomedical studies there is often interest in assessing the association between one or more ordered categorical predictors and an outcome variable adjusting for covariates for a k level predictor one typically uses either a k 1 degree of freedom df test or a single df trend test which requires scores for the different levels of the predictor in the absence of knowledge of a parametric form for the response function one can incorporate monotonicity constraints to improve the efficiency of tests of association this article proposes a general bayesian approach for inference on order constrained parameters in generalized linear models instead of choosing a prior distribution with support on the constrained space which can result in major computational difficulties we propose to map draws from an unconstrained posterior density using an isotonic regression transformation this approach allows flat regions over which increases in the level of a predictor have no effect bayes factors for assessing ordered trends can be computed based on the output from a gibbs sampling algorithm results from a simulation study are presented and the approach is applied to data from a time to pregnancy study

voronoi diagram and medial axis algorithm for planar domains with curved boundaries ii detailed algorithm description abstract details of algorithms to construct the voronoi diagrams and medial axes of planars domain bounded by free form polynomial or rational curve segments are presented based on theoretical foundations given in the first installment ramamurthy and farouki j comput appl math 1999 102 119 141 of this two part paper in particular we focus on key topological and computational issues that arise in these constructions the topological issues include i the data structures needed to represent various geometrical entities bisectors voronoi regions etc and ii the boolean operations i e union intersection and difference on planar sets required by the algorithm specifically representations for the voronoi polygons of boundary segments and for individual voronoi diagram or medial axis edges are proposed since these edges may be segments of a nonrational algebraic curves curve curve bisectors b rational curves point curve bisectors or c straight lines point point bisectors data structures tailored to each of these geometrical entities are introduced the computational issues addressed include the curve intersection algorithms required in the boolean operations and iterative schemes used to precisely locate bifurcation or n prong points n 3 of the voronoi diagram and medial axis a selection of computed voronoi diagram and medial axis examples is included to illustrate the capabilities of the algorithm

multi degree reduction of tensor product bezier surfaces with general boundary constraints abstract we propose an efficient approach to the problem of multi degree reduction of rectangular bezier patches with prescribed boundary control points we observe that the solution can be given in terms of constrained bivariate dual bernstein polynomials the complexity of the method is o mn 1 n 2 with m min m1 m2 where n1 n2 and m1 m2 is the degree of the input and output bezier surface respectively if the approximation with appropriate boundary constraints is performed for each patch of several smoothly joined rectangular bezier surfaces the result is a composite surface of global cr continuity with a prescribed r 0 in the detailed discussion we restrict ourselves to r 0 1 which is the most important case in practical application some illustrative examples are given

a fast numerical method to price american options under the bates model we consider the problem of pricing american options in the framework of a well known stochastic volatility model with jumps the bates model according to this model the price of an american option can be obtained as the solution of a linear complementarity problem governed by a partial integro differential equation in this paper a numerical method for solving such a problem is proposed in particular first of all using a bermudan approximation and a richardson extrapolation technique the linear complementarity problem is reduced to a set of standard linear partial differential problems see for example ballestra and sgarra 2010 chang et al 2007 2012 then these problems are solved using an ad hoc pseudospectral method which efficiently combines the chebyshev polynomial approximation an implicit explicit time stepping and an operator splitting technique numerical experiments are presented showing that the novel algorithm is very accurate and fast and significantly outperforms other methods that have recently been proposed for pricing american options under the bates model

confidence interval estimation subject to order restrictions this article deals with the construction of confidence intervals when the components of the location parameter mu of the random variable mathbf x which is elliptically symmetrically distributed are subject to order restrictions several domination results are proved by studying the derivative of the coverage probability of the confidence intervals centered at the improved point estimators consequently we strengthen the previously known results regarding the simple ordering and obtain several new results for other general forms of order restrictions including the simple tree ordering the umbrella ordering the simple and the double loop ordering and some combination of these these domination results are obtained under the assumption that sigma is a diagonal matrix when sigma is nondiagonal some new intervals are introduced which dominate the standard intervals centered at the unrestricted maximum likelihood estimator for various types of order restrictions similar results are obtained for scale parameters as well contrary to the location problems in case of the scale parameters satisfying the simple ordering we find that the restricted maximum likelihood estimator of the largest parameter fails to universally dominate the unrestricted maximum likelihood estimator a similar negative result is noted for simple tree order restriction

bayesian analysis of constrained parameter and truncated data problems using gibbs sampling abstract constrained parameter problems arise in a wide variety of applications including bioassay actuarial graduation ordinal categorical data response surfaces reliability development testing and variance component models truncated data problems arise naturally in survival and failure time studies ordinal data models and categorical data studies aimed at uncovering underlying continuous distributions in many applications both parameter constraints and data truncation are present the statistical literature on such problems is very extensive reflecting both the problems widespread occurrence in applications and the methodological challenges that they pose however it is striking that so little of this applied and theoretical literature involves a parametric bayesian perspective from a technical viewpoint this perhaps is not difficult to understand the fundamental tool for bayesian calculations in typical realistic models is multidimensional numerical integration which often is problem

efficient statistically accurate algorithms for the fokker planck equation in large dimensions abstract solving the fokker planck equation for high dimensional complex turbulent dynamical systems is an important and practical issue however most traditional methods suffer from the curse of dimensionality and have difficulties in capturing the fat tailed highly intermittent probability density functions pdfs of complex systems in turbulence neuroscience and excitable media in this article efficient statistically accurate algorithms are developed for solving both the transient and the equilibrium solutions of fokker planck equations associated with high dimensional nonlinear turbulent dynamical systems with conditional gaussian structures the algorithms involve a hybrid strategy that requires only a small number of ensembles here a conditional gaussian mixture in a high dimensional subspace via an extremely efficient parametric method is combined with a judicious non parametric gaussian kernel density estimation in the remaining low dimensional subspace particularly the parametric method provides closed analytical formulae for determining the conditional gaussian distributions in the high dimensional subspace and is therefore computationally efficient and accurate the full non gaussian pdf of the system is then given by a gaussian mixture different from traditional particle methods each conditional gaussian distribution here covers a significant portion of the high dimensional pdf therefore a small number of ensembles is sufficient to recover the full pdf which overcomes the curse of dimensionality notably the mixture distribution has significant skill in capturing the transient behavior with fat tails of the high dimensional non gaussian pdfs and this facilitates the algorithms in accurately describing the intermittency and extreme events in complex turbulent systems it is shown in a stringent set of test problems that the method only requires an order of o 100 ensembles to successfully recover the highly non gaussian transient pdfs in up to 6 dimensions with only small errors

a numerical solver for high dimensional transient fokker planck equation in modeling polymeric fluids in this paper a tensor decomposition approach combined with chebyshev spectral differentiation is presented to solve the high dimensional transient fokker planck equations fpe arising in the simulation of polymeric fluids via multi bead spring mbs model generalizing the authors previous work on the stationary fpe the transient solution is obtained in a single candecomp parafac decomposition cpd form for all times via the alternating least squares algorithm this is accomplished by treating the temporal dimension in the same manner as all other spatial dimensions thereby decoupling it from them as a result the transient solution is obtained without resorting to expensive time stepping schemes a new relaxed approach for imposing the vanishing boundary conditions is proposed improving the quality of the approximation the asymptotic behavior of the temporal basis functions is studied the proposed solver scales very well with the dimensionality of the mbs model numerical results for systems up to 14 dimensional state space are successfully obtained on a regular personal computer and compared with the corresponding matrix riccati differential equation for linear models or monte carlo simulations for nonlinear models

accurate approximations for posterior moments and marginal densities abstract this article describes approximations to the posterior means and variances of positive functions of a real or vector valued parameter and to the marginal posterior densities of arbitrary i e not necessarily positive parameters these approximations can also be used to compute approximate predictive densities to apply the proposed method one only needs to be able to maximize slightly modified likelihood functions and to evaluate the observed information at the maxima nevertheless the resulting approximations are generally as accurate and in some cases more accurate than approximations based on third order expansions of the likelihood and requiring the evaluation of third derivatives the approximate marginal posterior densities behave very much like saddle point approximations for sampling distributions the principal regularity condition required is that the likelihood times prior be unimodal

tensor train decomposition a simple nonrecursive form of the tensor decomposition in d dimensions is presented it does not inherently suffer from the curse of dimensionality it has asymptotically the same number of parameters as the canonical decomposition but it is stable and its computation is based on low rank approximation of auxiliary unfolding matrices the new form gives a clear and convenient way to implement all basic operations efficiently a fast rounding procedure is presented as well as basic linear algebra operations examples showing the benefits of the decomposition are given and the efficiency is demonstrated by the computation of the smallest eigenvalue of a 19 dimensional operator

numerical solution of high dimensional stationary fokker planck equations via tensor decomposition and chebyshev spectral differentiation abstract this paper focuses on the curse of dimensionality in the numerical solution of the stationary fokker planck equation for systems with state independent excitation a tensor decomposition approach is combined with chebyshev spectral differentiation to drastically reduce the number of degrees of freedom required to maintain accuracy as dimensionality increases following the enforcement of the normality condition via a penalty method the discretized system is solved using alternating least squares algorithm numerical results for a variety of systems including separable non separable systems linear nonlinear systems and systems with without closed form stationary solutions up to ten dimensional state space are presented to illustrate the effectiveness of the proposed method

boundary penalty finite element methods for blending surfaces iii superconvergence and stability and examples abstract this paper is part iii of the study on blending surfaces by partial differential equations pdes the blending surfaces in three dimensions 3d are taken into account by three parametric functions x r t y r t and z r t the boundary penalty techniques are well suited to the complicated tangent i e normal derivative boundary conditions in engineering blending by following the previous papers parts i and ii in li j comput math 16 1998 457 480 j comput appl math 110 1999 155 176 the corresponding theoretical analysis is made to discover that when the penalty power 2 3 or 3 5 and 0 1 5 in the boundary penalty finite element methods bp fems optimal convergence rates superconvergence and optimal numerical stability can be achieved respectively several interesting samples of 3d blending surfaces are provided to display the remarkable advantages of the proposed approaches in this paper unique solutions of blending surfaces optimal blending surfaces in minimum energy ease in handling the complicated boundary constraint conditions and less cpu time and computer storage needed this paper and li j comput math 16 1998 457 480 j comput appl math provide a foundation of blending surfaces by pde solutions a new trend of computer geometric design

improving the local solution accuracy of large scale digital image based finite element analyses digital image based finite element modeling dibfem has become a widely utilized approach for efficiently meshing complex biological structures such as trabecular bone while dibfem can provide accurate predictions of apparent mechanical properties its application to simulate local phenomena such as tissue failure or adaptation has been limited by high local solution errors at digital model boundaries furthermore refinement of digital meshes does not necessarily reduce local maximum errors the purpose of this study was to evaluate the potential to reduce local mean and maximum solution errors in digital meshes using a post processing filtration method the effectiveness of a three dimensional boundary specific filtering algorithm was found to be mesh size dependent mean absolute and maximum errors were reduced for meshes with more than five elements through the diameter of a cantilever beam considered representative of a single trabecula furthermore mesh refinement consistently decreased errors for filtered solutions but not necessarily for non filtered solutions models with more than five elements through the beam diameter yielded absolute mean errors of less than 15 for both von mises stress and maximum principal strain when applied to a high resolution model of trabecular bone microstructure boundary filtering produced a more continuous solution distribution and reduced the predicted maximum stress by 30 boundary specific filtering provides a simple means of improving local solution accuracy while retaining the model generation and numerical storage efficiency of the dibfem technique

a finite element method for the statistics of non linear random vibration the transitional probability density function for the random response of a certain class of non linear system satisfies the fokker planck kolmogorov equation this paper concerns the numerical solution of the stationary form of this equation yielding the stationary probability density function of response the weighted residual statement for the problem is integrated by parts to yield the weak form of the equations which are then solved by the finite element method the method is applied to a duffing oscillator and good agreement is found with the exact result and the method is compared favourably with a galerkin solution method given by bhandari and sherrer 1 also the method is applied to the ship rolling problem and good agreement is found with an approximate analytical result due to roberts 2

a unified approach for degree reduction of polynomials in the bernstein basis part i real polynomials the polynomials determined in the bernstein bezier basis enjoy considerable popularity and the algorithms for reducing their degree are of practical importance in computer aided design applications on the other hand the conversion between the bernstein and the power basis is ill conditioned thus only the degree reduction algorithms which may be carried out without using this conversion are of practical value our unified approach enables us to describe all the algorithms of this kind known in the literature to construct a number of new ones which are better conditioned and cheaper than some of the currently used ones and to study the errors of all of them in a simple homogeneous way all these algorithms can be applied componentwise to a vector valued polynomial representing a bezier curve consider the values of the derivatives whose orders vary successively from 1 to a given number i or k at the start and end point respectively of this curve the current algorithms allow us to preserve these points and values for i equal to k the new ones do that also without the latter constraint