multi resolution analysis scheme for uncertainty quantification in chemical systems this paper presents a multi resolution approach for the propagation of parametric uncertainty in chemical systems it is motivated by previous studies where galerkin formulations of wiener hermite expansions were found to fail in the presence of steep dependences of the species concentrations with regard to the reaction rates the multi resolution scheme is based on representation of the uncertain concentration in terms of compact polynomial multi wavelets allowing for the control of the convergence in terms of polynomial order and resolution level the resulting representation is shown to greatly improve the robustness of the galerkin procedure in presence of steep dependences however this improvement comes with a higher computational cost which drastically increases with the number of uncertain reaction rates to overcome this drawback an adaptive strategy is proposed to control locally in the parameter space and in time the resolution level the efficiency of the method is demonstrated for an uncertain chemical system having eight random parameters

an efficient algorithm for decomposing a polygon into star shaped polygons abstract in this paper we show how a theorem in plane geometry can be converted into a o n log n algorithm for decomposing a polygon into star shaped subsets the computational efficiency of this new decomposition contrasts with the heavy computational burden of existing methods

a morphological signature transform for shape description abstract a novel shape description method based on the multiresolution morphological image processing by non convex multiple structuring elements is presented a binary image which contains the object shape to be described is represented by means of a multiresolution pyramid a set of primary structuring elements along with their rotated versions is used the method is based on the successive morphological erosions of the input image at different resolutions by primary and rotated structuring elements the areas of successively eroded images are computed for each structuring element at each pyramid level the obtained set of numbers is arranged into vectors ordered and used as a shape descriptor results are included to show that the method is robust against noise and invariant to both rotation and scale change

morphological decomposition of 2 d binary shapes into conditionally maximal convex polygons mathematical morphology is an approach to image analysis based on the geometric concept of shape and size naturally it provides a very effective tool for shape analysis in this paper we present a morphological shape segmentation algorithm that decomposes a 2 d two dimensional binary shape into a collection of restricted convex polygons we use simple morphological operations to extract shape information from different skeleton subsets to form a group of conditionally maximal convex polygonal shape components then a set of more meaningful and natural convex polygonal components are constructed from these maximal components the resulting algorithm is very simple and the decomposition is always unique the components produced have well defined mathematical characterizations and they seem to be in good agreement with the nature structures of the given shapes the shape segments produced can be used to construct structural shape description and for other shape analysis purposes

uniform asymptotic solutions of a system of two schrodinger equations with potential curve crossing point a formal uniform asymptotic solution of the system of differential equations h2 d2u1 dz2 phi 1u1 alpha u2 h2 d2u2 dz2 phi 2u2 alpha u1 for z in d and for real and small h is obtained when d contains a curve crossing point asymptotic approximations for the solution are constructed in terms of parabolic cylinder functions analytical properties of the expansion coefficients are investigated

segmenting dot patterns by voronoi diagram concavity this correspondence defines a signed distance called internal concavity on paths of the voronoi diagram of a dot pattern an algorithm using internal concavity to segment dot patterns is described the segmentation algorithm produces subsets of the dirichlet tessellation delaunay triangulation of the dot pattern

stochastic models for closed boundary analysis representation and reconstruction the analysis of closed boundaries of arbitrary shapes on a plane is discussed specifically the problems of representation and reconstruction are considered a one to one correspondence between the given closed boundary and a univariate or multivariate sequence of real numbers is set up univariate or multivariate circular autoregressive models are suggested for the representation of the sequence of numbers derived from the closed boundary the stochastic model representing the closed boundary is invariant to transformations like sealing translation choice of starting point and rotation over angles that are multiples of 2 pi n where n is the number of observations methods for estimating the unknown parameters of the model are given and a decision rule for choosing the appropriate number of coefficients is included constraints on the estimates are derived so that the estimates are invariant to the transformations of the boundaries the stochastic model enables the reconstruction of a dosed boundary using fft algorithms results of simulations are included and the application to contour coding is discussed

non intrusive low rank separated approximation of high dimensional stochastic models this work proposes a sampling based non intrusive approach within the context of low rank separated representations to tackle the issue of curse of dimensionality associated with the solution of models e g pdes odes with high dimensional random inputs under some conditions discussed in details the number of random realizations of the solution required for a successful approximation grows linearly with respect to the number of random inputs the construction of the separated representation is achieved via a regularized alternating least squares regression together with an error indicator to estimate model parameters the computational complexity of such a construction is quadratic in the number of random inputs the performance of the method is investigated through its application to three numerical examples including two ode problems with high dimensional random inputs

near optimal mst based shape description using genetic algorithm abstract a new method for the selection of the optimal structuring element for shape description and matching based on the morphological signature transform mst is presented in this paper for a given class of shapes the optimal structuring element for mst method is selected by means of a genetic algorithm the optimization criteria is formulated to enable a robust shape matching experiments have been performed on a class of model shapes the proposed optimal shape description method is applied to the problem of shape matching which evolves in many object recognition applications here an unknown object is matched to a set of known objects in order to classify it into one of finite number of classes experimental results are presented and discussed

polynomial meta models with canonical low rank approximations the growing need for uncertainty analysis of complex computational models has led to an expanding use of meta models across engineering and sciences the efficiency of meta modeling techniques relies on their ability to provide statistically equivalent analytical representations based on relatively few evaluations of the original model polynomial chaos expansions pce have proven a powerful tool for developing meta models in a wide range of applications the key idea thereof is to expand the model response onto a basis made of multivariate polynomials obtained as tensor products of appropriate univariate polynomials the classical pce approach nevertheless faces the curse of dimensionality namely the exponential increase of the basis size with increasing input dimension to address this limitation the sparse pce technique has been proposed in which the expansion is carried out on only a few relevant basis terms that are automatically selected by a suitable algorithm an alternative for developing meta models with polynomial functions in high dimensional problems is offered by the newly emerged low rank approximations lra approach by exploiting the tensor product structure of the multivariate basis lra can provide polynomial representations in highly compressed formats through extensive numerical investigations we herein first shed light on issues relating to the construction of canonical lra with a particular greedy algorithm involving a sequential updating of the polynomial coefficients along separate dimensions specifically we examine the selection of optimal rank stopping criteria in the updating of the polynomial coefficients and error estimation in the sequel we confront canonical lra to sparse pce in structural mechanics and heat conduction applications based on finite element solutions canonical lra exhibit smaller errors than sparse pce in cases when the number of available model evaluations is small with respect to the input dimension a situation that is often encountered in real life problems by introducing the conditional generalization error we further demonstrate that canonical lra tend to outperform sparse pce in the prediction of extreme model responses which is critical in reliability analysis

uniform asymptotic solutions of boundary and eigenvalue problems for a second order ordinary differential equation with singular and turning points abstract a uniform asymptotic solution to the boundary value problem u x 2 r x u 2 g x u z i u z i i i 1 2 and x i z1 z2 is developed by langer s method r x can have poles at the end points of i the case in which r x e u x where e is an unknown eigenvalue and u has poles in i is also discussed the asymptotic nature of the formal solution to the boundary value problem is investigated and specific examples are solved

polynomial chaos expansions for dependent random variables abstract polynomial chaos expansions pce are well suited to quantifying uncertainty in models parameterized by independent random variables the assumption of independence leads to simple strategies for building multivariate orthonormal bases and for sampling strategies to evaluate pce coefficients in contrast the application of pce to models of dependent variables is much more challenging three approaches can be used to construct pce of models of dependent variables the first approach uses mapping methods where measure transformations such as the nataf and rosenblatt transformation can be used to map dependent random variables to independent ones however we show that this can significantly degrade performance since the jacobian of the map must be approximated a second strategy is the class of dominating support methods in these approaches a pce is built using independent random variables whose distributional support dominates the support of the true dependent joint density we provide evidence that this approach appears to produce approximations with suboptimal accuracy a third approach the novel method proposed here uses gram schmidt orthogonalization gso to numerically compute orthonormal polynomials for the dependent random variables this approach has been used successfully when solving differential equations using the intrusive stochastic galerkin method and in this paper we use gso to build pce using a non intrusive stochastic collocation method the stochastic collocation method treats the model as a black box and builds approximations of the input output map from a set of samples building pce from samples can introduce ill conditioning which does not plague stochastic galerkin methods to mitigate this ill conditioning we generate weighted leja sequences which are nested sample sets to build accurate polynomial interpolants we show that our proposed approach gso with weighted leja sequences produces pce which are orders of magnitude more accurate than pce constructed using mapping or dominating support methods

khatri rao space time codes space time st coding techniques exploit the spatial diversity afforded by multiple transmit and receive antennas to achieve reliable transmission in scattering rich environments st block codes are capable of realizing full diversity and spatial coding gains at relatively low rates st trellis codes can achieve better rate diversity tradeoffs at the cost of high complexity on the other hand v blast supports high rates but has no built in spatial coding and does not work well with fewer receive than transmit antennas we propose a novel linear block coding scheme based on the khatri rao matrix product the proposed scheme offers flexibility for achieving full rate or full diversity or a desired rate diversity tradeoff and it can handle any transmit receive antenna configuration or signal constellation the proposed codes are shown to have numerous desirable properties including guaranteed unique linear decodability built in blind channel identifiability and efficient near maximum likelihood decoding

high order collocation methods for differential equations with random inputs recently there has been a growing interest in designing efficient methods for the solution of ordinary partial differential equations with random inputs to this end stochastic galerkin methods appear to be superior to other nonsampling methods and in many cases to several sampling methods however when the governing equations take complicated forms numerical implementations of stochastic galerkin methods can become nontrivial and care is needed to design robust and efficient solvers for the resulting equations on the other hand the traditional sampling methods e g monte carlo methods are straightforward to implement but they do not offer convergence as fast as stochastic galerkin methods in this paper a high order stochastic collocation approach is proposed similar to stochastic galerkin methods the collocation methods take advantage of an assumption of smoothness of the solution in random space to achieve fast convergence however the numerical implementation of stochastic collocation is trivial as it requires only repetitive runs of an existing deterministic solver similar to monte carlo methods the computational cost of the collocation methods depends on the choice of the collocation points and we present several feasible constructions one particular choice based on sparse grids depends weakly on the dimensionality of the random space and is more suitable for highly accurate computations of practical applications with large dimensional random inputs numerical examples are presented to demonstrate the accuracy and efficiency of the stochastic collocation methods

stochastic model reduction for chaos representations abstract this paper addresses issues of model reduction of stochastic representations and computational efficiency of spectral stochastic galerkin schemes for the solution of partial differential equations with stochastic coefficients in particular an algorithm is developed for the efficient characterization of a lower dimensional manifold occupied by the solution to a stochastic partial differential equation spde in the hilbert space spanned by wiener chaos a description of the stochastic aspect of the problem on two well separated scales is developed to enable the stochastic characterization on the fine scale using algebraic operations on the coarse scale with such algorithms at hand the solution of spde s becomes both computationally manageable and efficient moreover a solid foundation is thus provided for the adaptive error control in stochastic galerkin procedures different aspects of the proposed methodology are clarified through its application to an example problem from solid mechanics

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

kronecker product approximations for some function related matrices abstract a new approximation tool such as sums of kronecker products is recently found to provide a superb compression property on a series of numerical examples of quite a general nature the purpose of the paper is explanation of this phenomenon in the form of existence theorems for matrix approximations of low kronecker rank for some classes of function related matrices including important specimens from potential theory this lays the grounds for development of new approximation algorithms for example in the cases when a matrix is associated with a shift invariant function on the cartesian product of nonunform grids which is of great practical interest in the solution of integral equations on plates or screens

tensor structured galerkin approximation of parametric and stochastic elliptic pdes we investigate the convergence rate of approximations by finite sums of rank 1 tensors of solutions of multiparametric elliptic pdes such pdes arise for example in the parametric deterministic reformulation of elliptic pdes with random field inputs based for example on the m term truncated karhunen loeve expansion our approach could be regarded as either a class of compressed approximations of these solutions or as a new class of iterative elliptic problem solvers for high dimensional parametric elliptic pdes providing linear scaling complexity in the dimension m of the parameter space it is based on rank reduced tensor formatted separable approximations of the high dimensional tensors and matrices involved in the iterative process combined with the use of spectrally equivalent low rank tensor structured preconditioners to the parametric matrices resulting from a finite element discretization of the high dimensional parametric deterministic problems numerical illustrations for the m dimensional parametric elliptic pdes resulting from spdes on parameter spaces of dimensions m leq100 indicate the advantages of employing low rank tensor structured matrix formats in the numerical solution of such problems

pattern spectrum and multiscale shape representation the results of a study on multiscale shape description smoothing and representation are reported multiscale nonlinear smoothing filters are first developed using morphological opening and closings g matheron 1975 used openings and closings to obtain probabilistic size distributions of euclidean space sets continuous binary images these distributions are used to develop a concept of pattern spectrum a shape size descriptor a pattern spectrum is introduced for continuous graytone images and arbitrary multilevel signals as well as for discrete images by developing a discrete size family of patterns large jumps in the pattern spectrum at a certain scale indicate the existence of major protruding or intruding substructures of the signal at the scale an entropy like shape size complexity measure is also developed based on the pattern spectrum for shape representation a reduced morphological skeleton transform is introduced for discrete binary and graytone images this transform is a sequence of skeleton components sparse images which represent the original shape at various scales it is shown that the partially reconstructed images from the inverse transform on subsequences of skeleton components are the openings of the image at a scale determined by the number of eliminated components in addition two way correspondences are established among the degree of shape smoothing via multiscale openings or closings the pattern spectrum zero values and the elimination or nonexistence of skeleton components at certain scales

parallel tensor methods for high dimensional linear pdes abstract high dimensional partial differential equations pdes arise in a number of fields of science and engineering where they are used to describe the evolution of joint probability functions their examples include the boltzmann and fokker planck equations we develop new parallel algorithms to solve such high dimensional pdes the algorithms are based on canonical and hierarchical numerical tensor methods combined with alternating least squares and hierarchical singular value decomposition both implicit and explicit integration schemes are presented and discussed we demonstrate the accuracy and efficiency of the proposed new algorithms in computing the numerical solution to both an advection equation in six variables plus time and a linearized version of the boltzmann equation

blind parafac receivers for ds cdma systems this paper links the direct sequence code division multiple access ds cdma multiuser separation equalization detection problem to the parallel factor parafac model which is an analysis tool rooted in psychometrics and chemometrics exploiting this link it derives a deterministic blind parafac ds cdma receiver with performance close to non blind minimum mean squared error mmse the proposed parafac receiver capitalizes on code spatial and temporal diversity combining thereby supporting small sample sizes more users than sensors and or less spreading than users interestingly parafac does not require knowledge of spreading codes the specifics of multipath interchip interference doa calibration information finite alphabet constant modulus or statistical independence whiteness to recover the information bearing signals instead parafac relies on a fundamental result regarding the uniqueness of low rank three way array decomposition due to kruskal 1977 1988 and generalized herein to the complex valued case that guarantees identifiability of all relevant signals and propagation parameters these and other issues are also demonstrated in pertinent simulation experiments

ordered structural shape matching with primitive extraction by mathematical morphology abstract structural shape matching procedures begin by decomposing the shapes to be matched into primitive parts and extracting the properties of and relationships among these primitives the resultant structural descriptions are matched to structural models usually via a tree search whose time is exponential in the number of primitives in this paper we introduce a new fast method of structural shape matching called ordered structural matching which has polynomial time complexity instead of exponential the approach presented also uses mathematical morphology to rapidly decompose the shapes into primitives to be used in the matching the primitives so produced are pieces of the original shape but not necessarily those pieces that a human would select the method has been tested on the toy task of determining which plastic shapes match the shapes of the holes in a child s six sided toy box it is also directly applicable to the inspection and manipulation of certain airplane parts

multivariate regression and machine learning with sums of separable functions we present an algorithm for learning or estimating a function of many variables from scattered data the function is approximated by a sum of separable functions following the paradigm of separated representations the central fitting algorithm is linear in both the number of data points and the number of variables and thus is suitable for large data sets in high dimensions we present numerical evidence for the utility of these representations in particular we show that our method outperforms other methods on several benchmark data sets

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

an adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations in recent years there has been a growing interest in analyzing and quantifying the effects of random inputs in the solution of ordinary partial differential equations to this end the spectral stochastic finite element method ssfem is the most popular method due to its fast convergence rate recently the stochastic sparse grid collocation method has emerged as an attractive alternative to ssfem it approximates the solution in the stochastic space using lagrange polynomial interpolation the collocation method requires only repetitive calls to an existing deterministic solver similar to the monte carlo method however both the ssfem and current sparse grid collocation methods utilize global polynomials in the stochastic space thus when there are steep gradients or finite discontinuities in the stochastic space these methods converge very slowly or even fail to converge in this paper we develop an adaptive sparse grid collocation strategy using piecewise multi linear hierarchical basis functions hierarchical surplus is used as an error indicator to automatically detect the discontinuity region in the stochastic space and adaptively refine the collocation points in this region numerical examples especially for problems related to long term integration and stochastic discontinuity are presented comparisons with monte carlo and multi element based random domain decomposition methods are also given to show the efficiency and accuracy of the proposed method

uniform approximations for the natural modes and frequencies of spindle shaped acoustical resonators we wish to obtain uniform approxiations for the natural modes and frequencies of some elongated acoustic bodies or spindles these are bodies generated by plain curves of the type y h a a dxb also the end points are pointed finally we remark that when s o r ex r x is a body of infinite length with a nonzero cross section such a body is generated by a curve of the type y x i o kl x xx 1 2

asymptotic semiclassical equivalence for schrodinger equations with singular potentials and for related systems of two first order equations the asymptotic equivalence of systems of two ordinary first order linear differential equations with complex independent variable and a small parameter at the derivatives is analyzed in the case of arbitrary numbers and multiplicities of turning points and singular points the set of all the transformation matrices realizing the equivalence is described and a recursive procedure for constructing these matrices is developed by persistently using the determinant properties of the transformation matrices the number of integration operations at each step of this procedure is halved compared with the algorithms known before the theory is specialized to the case of time independent one dimensional schrodinger equations with singular potentials some generalizations to multichannel schrodinger equations are also presented

ingredients for a general purpose stochastic finite elements implementation abstract the stochastic finite element method has seen a number of incarnations over the past few years ranging from perturbation based methods and neumann expansion based methods to methods for evaluating bounds on the response variability and more recent functional expansions combined with error minimization techniques of these only the last class of methods provides an explicit expression to the solution of the problem as opposed to evaluating a global norm of this solution and can thus be viewed as a rational extension of concepts developed for the deterministic finite element method the main obstacle in developing a substantial user community of various stochastic finite element methods has been the lack of a general purpose formalism for addressing issues of general engineering interest this paper presents the main ingredients for developing a general purpose version of the spectral stochastic finite element method

global shape analysis by k syntactic similarity the k syntactic similarity approach is couched in graphical representation terms and its ability to provide global recognition capability while retaining a low time complexity is explored one potential application domain that of composite shape decomposition into approximately convex subshapes is described this is shown to be equivalent to finding cycles within a particular graph the approach yields valid decompositions in many cases of interest and is capable of identifying those cases where additional semantic considerations are necessary for proper analysis the permissible graph structures representing composite shapes given a reasonable set of relations are determined experimental results on nonideal data are given

description and discrimination of planar shapes using shape matrices a shape descriptor has been developed which can describe a shape independent of its translation rotation and scaling the description is in the form of a matrix and it is obtained by a polar quantization of a shape the quantization process takes into consideration not only a shape s outer geometry but its inner geometry as well the descriptor is information preserving and if the quantization parameters are selected properly it is possible to reconstruct an original shape from its description in this technique shape discrimination is possible by a simple exclusive or operation on shape descriptions

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

analysis of discrete ill posed problems by means of the l curve when discrete ill posed problems are analyzed and solved by various numerical regularization techniques a very convenient way to display information about the regularized solution is to plot the norm or seminorm of the solution versus the norm of the residual vector in particular the graph associated with tikhonov regularization plays a central role the main purpose of this paper is to advocate the use of this graph in the numerical treatment of discrete ill posed problems the graph is characterized quantitatively and several important relations between regularized solutions and the graph are derived it is also demonstrated that several methods for choosing the regularization parameter are related to locating a characteristic l shaped corner of the graph

statistical analysis and simulation of random shocks in stochastic burgers equation we study the statistical properties of random shock waves in stochastic burgers equation subject to random space time perturbations and random initial conditions by using the response excitation probability density function pdf method and the mori zwanzig mz formulation of irreversible statistical mechanics we derive exact reduced order equations for the one point and two point pdfs of the solution field in particular we compute the statistical properties of random shock waves in the inviscid limit by using an adaptive shock capturing discontinuous galerkin method in both physical and probability spaces we consider stochastic flows generated by high dimensional random initial conditions and random additive forcing terms yielding multiple interacting shock waves collapsing into clusters and settling down to a similarity state we also address the question of how random shock waves in space and time manifest themselves in probability space the proposed new mathematical framework can be applied to different conservation laws potentially leading to new insights into high dimensional stochastic dynamical systems and more efficient computational algorithms

partitioned treatment of uncertainty in coupled domain problems a separated representation approach abstract this work is concerned with the propagation of uncertainty across coupled domain problems with high dimensional random inputs a stochastic model reduction approach based on low rank separated representations is proposed for the partitioned treatment of the uncertainty space the construction of the coupled domain solution is achieved though a sequence of approximations with respect to the dimensionality of the random inputs associated with each individual sub domain and not the combined dimensionality hence drastically reducing the overall computational cost the coupling between the sub domain solutions is done via the classical finite element tearing and interconnecting feti method thus providing a well suited framework for parallel computing two high dimensional stochastic problems a 2d elliptic pde with random diffusion coefficient and a stochastic linear elasticity problem have been considered to study the performance and accuracy of the proposed stochastic coupling approach

low rank separated representation surrogates of high dimensional stochastic functions application in bayesian inference this study introduces a non intrusive approach in the context of low rank separated representation to construct a surrogate of high dimensional stochastic functions e g pdes odes in order to decrease the computational cost of markov chain monte carlo simulations in bayesian inference the surrogate model is constructed via a regularized alternative least square regression with tikhonov regularization using a roughening matrix computing the gradient of the solution in conjunction with a perturbation based error indicator to detect optimal model complexities the model approximates a vector of a continuous solution at discrete values of a physical variable the required number of random realizations to achieve a successful approximation linearly depends on the function dimensionality the computational cost of the model construction is quadratic in the number of random inputs which potentially tackles the curse of dimensionality in high dimensional stochastic functions furthermore this vector valued separated representation based model in comparison to the available scalar valued case leads to a significant reduction in the cost of approximation by an order of magnitude equal to the vector size the performance of the method is studied through its application to three numerical examples including a 41 dimensional elliptic pde and a 21 dimensional cavity flow

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

numerical operator calculus in higher dimensions when an algorithm in dimension one is extended to dimension d in nearly every case its computational cost is taken to the power d this fundamental difficulty is the single greatest impediment to solving many important problems and has been dubbed the curse of dimensionality for numerical analysis in dimension d we propose to use a representation for vectors and matrices that generalizes separation of variables while allowing controlled accuracy basic linear algebra operations can be performed in this representation using one dimensional operations thus bypassing the exponential scaling with respect to the dimension although not all operators and algorithms may be compatible with this representation we believe that many of the most important ones are we prove that the multiparticle schrodinger operator as well as the inverse laplacian can be represented very efficiently in this form we give numerical evidence to support the conjecture that eigenfunctions inherit this property by computing the ground state eigenfunction for a simplified schrodinger operator with 30 particles we conjecture and provide numerical evidence that functions of operators inherit this property in which case numerical operator calculus in higher dimensions becomes feasible

some comments on cp i study the typical configuration of a cp plot when the number of variables in the regression problem is large and there are many weak effects i show that a particular configuration that is very commonly seen can arise in a simple way i give a formula by means of which the risk incurred by the minimum cp rule can be estimated

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

to be or not to be intrusive the solution of parametric and stochastic equations the plain vanilla galerkin case in parametric equations stochastic equations are a special case one may want to approximate the solution such that it is easy to evaluate its dependence on the parameters interpolation in the parameters is an obvious possibility in this context often labeled as a collocation method in the frequent situation where one has a solver for a given fixed parameter value this may be used nonintrusively as a black box component to compute the solution at all the interpolation points independently of each other by extension all other methods and especially simple galerkin methods which produce some kind of coupled system are often classed as intrusive we show how for such plain vanilla galerkin formulations one may solve the coupled system in a nonintrusive way and even the simplest form of block solver has comparable efficiency this opens at least two avenues for possible speed up first to benefit from the coupling in the iteration by using more sophisticated block solvers and second

a continuous analogue of the tensor train decomposition abstract we develop new approximation algorithms and data structures for representing and computing with multivariate functions using the functional tensor train ft a continuous extension of the tensor train tt decomposition the ft represents functions using a tensor train ansatz by replacing the three dimensional tt cores with univariate matrix valued functions the main contribution of this paper is a framework to compute the ft that employs adaptive approximations of univariate fibers and that is not tied to any tensorized discretization the algorithm can be coupled with any univariate linear or nonlinear approximation procedure we demonstrate that this approach can generate multivariate function approximations that are several orders of magnitude more accurate for the same cost than those based on the conventional approach of compressing the coefficient tensor of a tensor product basis our approach is in the spirit of other continuous computation packages such as chebfun and yields an algorithm which requires the computation of continuous matrix factorizations such as the lu and qr decompositions of vector valued functions to support these developments we describe continuous versions of an approximate maximum volume cross approximation algorithm and of a rounding algorithm that re approximates an ft by one of lower ranks we demonstrate that our technique improves accuracy and robustness compared to tt and quantics tt approaches with fixed parameterizations of high dimensional integration differentiation and approximation of functions with local features such as discontinuities and other nonlinearities

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

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

a generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations we propose a new robust technique for solving stochastic partial differential equations the solution is approximated by a series of terms each of which being the product of a scalar stochastic function by a deterministic function none of these functions are fixed a priori but determined by solving a problem which can be interpreted as an extended eigenvalue problem this technique generalizes the classical spectral decomposition namely the karhunen loeve expansion ad hoc iterative techniques to build the approximation inspired by the power method for classical eigenproblems then transform the problem into the resolution of a few uncoupled deterministic problems and stochastic equations this method drastically reduces the calculation costs and memory requirements of classical resolution techniques used in the context of galerkin stochastic finite element methods finally this technique is particularly suitable to non linear and evolution problems since it enables the construction of a relevant reduced basis of deterministic functions which can be efficiently reused for subsequent resolutions

shape description by time series time series modeling techniques are adapted to represent or describe two dimensional closed contours both linear and nonlinear models are fitted it is found that to detect small changes in shape nonlinear modeling is necessary even though linear models may be sufficient to differentiate between shapes which differ widely a nonlinear model called the noncausal quadratic volterra model is developed for the purpose implementation is illustrated with shapes of aircraft

an anisotropic sparse grid stochastic collocation method for partial differential equations with random input data this work proposes and analyzes an anisotropic sparse grid stochastic collocation method for solving partial differential equations with random coefficients and forcing terms input data of the model the method consists of a galerkin approximation in the space variables and a collocation in probability space on sparse tensor product grids utilizing either clenshaw curtis or gaussian knots even in the presence of nonlinearities the collocation approach leads to the solution of uncoupled deterministic problems just as in the monte carlo method this work includes a priori and a posteriori procedures to adapt the anisotropy of the sparse grids to each given problem these procedures seem to be very effective for the problems under study the proposed method combines the advantages of isotropic sparse collocation with those of anisotropic full tensor product collocation the first approach is effective for problems depending on random variables which weigh approximately equally in the solution while the benefits of the latter approach become apparent when solving highly anisotropic problems depending on a relatively small number of random variables as in the case where input random variables are karhunen loeve truncations of smooth random fields this work also provides a rigorous convergence analysis of the fully discrete problem and demonstrates sub exponential convergence in the asymptotic regime and algebraic convergence in the preasymptotic regime with respect to the total number of collocation points it also shows that the anisotropic approximation breaks the curse of dimensionality for a wide set of problems numerical examples illustrate the theoretical results and are used to compare this approach with several others including the standard monte carlo in particular for moderately large dimensional problems the sparse grid approach with a properly chosen anisotropy seems to be very efficient and superior to all examined methods

sparse high order fem for elliptic spdes abstract we describe the analysis and the implementation of two finite element fe algorithms for the deterministic numerical solution of elliptic boundary value problems with stochastic coefficients they are based on separation of deterministic and stochastic parts of the input data by a karhunen loeve expansion truncated after m terms with a change of measure we convert the problem to a sequence of m dimensional parametric deterministic problems two sparse high order polynomial approximations of the random solution s joint pdf s parametrized in the input data s karhunen loeve expansion coordinates are analyzed a sparse stochastic galerkin fem sparse sgfem and a sparse stochastic collocation fem sparse scfem a priori and a posteriori error analysis is used to tailor the sparse polynomial approximations of the random solution s joint pdf s to the stochastic regularity of the input data scfem and sgfem yield deterministic approximations of the random solutions joint pdf s that converge spectrally in the number of deterministic problems to be solved numerical examples with random inputs of small correlation length in diffusion problems are presented high order gpc approximations of solutions with stochastic parameter spaces of dimension up to m 80 are computed on workstations

a sparse composite collocation finite element method for elliptic spdes this work presents a stochastic collocation method for solving elliptic pdes with random coefficients and forcing term which are assumed to depend on a finite number of random variables the method consists of a hierarchic wavelet discretization in space and a sequence of hierarchic collocation operators in the probability domain to approximate the solution s statistics the selection of collocation points is based on a smolyak construction of zeros of orthogonal polynomials with respect to the probability density function of each random input variable a sparse composition of levels of spatial refinements and stochastic collocation points is then proposed and analyzed resulting in a substantial reduction of overall degrees of freedom like in the monte carlo approach the algorithm results in solving a number of uncoupled purely deterministic elliptic problems which allows the integration of existing fast solvers for elliptic pdes numerical examples on two dimensional domains will then demonstrate the superiority of this sparse composite collocation finite element method compared to the full composite collocation finite element method and the monte carlo method

region extraction from complex shapes an algorithm is described which extracts primitive regions i e convex spiral shaped and biconcave lens from complex shapes the interior region bounded by the shape is decomposed by first slicing it into a set of convex subregions and then rotating and dissolving the various boundaries between subregions until a satisfactory decomposition is obtained the same algorithm also is used to decompose the exterior region between the shape and its convex hull the algorithm has been implemented as an algol w computer program for the univac 90 80 and results of running the program are presented for a wide variety of complex shapes these results compare favorably with the experience reported by previous programs

uniform asymptotic solutions of second order linear ordinary differential equations with singular points i formal theory abstract a formal uniform asymptotic solution of the differential equation d 2 u dz 2 2 r z u 0 z d for large when d contains any number and order of poles and turning points is constructed the development is a generalization of langer s method an asymptotic relation between the wronskian of any pair of linearly independent solutions of the differential equation and the wronskian of the pair of corresponding solutions of the comparison equation is derived a formal uniform asymptotic solution of a system of two first order ordinary differential equation is also obtained

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

morphological decomposition of 2 d binary shapes into convex polygons a heuristic algorithm in many morphological shape decomposition algorithms either a shape can only be decomposed into shape components of extremely simple forms or a time consuming search process is employed to determine a decomposition in this paper we present a morphological shape decomposition algorithm that decomposes a two dimensional 2 d binary shape into a collection of convex polygonal components a single convex polygonal approximation for a given image is first identified this first component is determined incrementally by selecting a sequence of basic shape primitives these shape primitives are chosen based on shape information extracted from the given shape at different scale levels additional shape components are identified recursively from the difference image between the given image and the first component simple operations are used to repair certain concavities caused by the set difference operation the resulting hierarchical structure provides descriptions for the given shape at different detail levels the experiments show that the decomposition results produced by the algorithm seem to be in good agreement with the natural structures of the given shapes the computational cost of the algorithm is significantly lower than that of an earlier search based convex decomposition algorithm compared to nonconvex decomposition algorithms our algorithm allows accurate approximations for the given shapes at low coding costs

uniform asymptotic solutions of second order linear ordinary differential equations with singular points ii some expansions abstract in this paper the general expansion produced in the companion paper j math anal appl 133 1988 is specialized to obtain expansions in terms of some special functions

structure seeking multilinear methods for the analysis of fmri data in comprehensive fmri studies of brain function the data structures often contain higher order ways such as trial task condition subject and group in addition to the intrinsic dimensions of time and space while multivariate bilinear methods such as principal component analysis pca have been used successfully for extracting information about spatial and temporal features in data from a single fmri run the need to unfold higher order data sets into bilinear arrays has led to decompositions that are nonunique and to the loss of multiway linkages and interactions present in the data these additional dimensions or ways can be retained in multilinear models to produce structures that are unique and which admit interpretations that are neurophysiologically meaningful multiway analysis of fmri data from multiple runs of a bilateral finger tapping paradigm was performed using the parallel factor parafac model a trilinear model was fitted to a data cube of dimensions voxels by time by run similarly a quadrilinear model was fitted to a higher way structure of dimensions voxels by time by trial by run the spatial and temporal response components were extracted and validated by comparison to results from traditional svd pca analyses based on scenarios of unfolding into lower order bilinear structures

optimization via separated representations and the canonical tensor decomposition abstract we introduce a new quadratically convergent algorithm for finding maximum absolute value entries of tensors represented in the canonical format the computational complexity of the algorithm is linear in the dimension of the tensor we show how to use this algorithm to find global maxima of non convex multivariate functions in separated form we demonstrate the performance of the new algorithms on several examples

calculating geometric properties from fourier representation abstract the area the location of the centroid and the second order moments of a region are determined and expressed in closed form in terms of the fourier coefficients of its boundary function the orientation of the central axes and the second order central moments can then be easily obtained bounds on the perimeter of the region are derived in terms of the fourier coefficients as well

some sequential algorithms for a generalized distance transformation based on minkowski operations a generalized distance transformation gdt of binary images and the related medial axis transformation mat are discussed these transformations are defined in a discrete space of arbitrary dimension and arbitrary grids the gdt is based on successive morphological operations using alternatively n arbitrary structuring elements n is called the period of the gdt the gdt differs from the classical distance transformations based on a point to point distance however the well known chessboard city block and hexagonal distance transformations are special cases of the one period gdt whereas the octagonal distance transformation is a special case of the two period gdt in this paper both one and two period gdts are discussed different sequential algorithms are proposed for computing such gdts these algorithms need a maximum of two scannings of the image the computation of the mat is also discussed

stochastic variational multiscale analysis of the advection diffusion equation advective diffusive regime and multi dimensional problems abstract we present the variational multiscale vms formulation for the stochastic advection diffusion equation in which the source of uncertainty may arise from random advective and diffusive fields or material parameters a random source term or a combination among them in this formulation a stabilization parameter arises that for the advection diffusion equation is a complicated function of the uncertain advection and diffusion parameters to efficiently incorporate the stochastic stabilization parameter in the numerical method we develop a projection based approach to obtain an approximation of the stabilization parameter in the form of a generalized polynomial chaos gpc expansion we demonstrate the current approach for problems which span the advective and diffusive regimes in the stochastic domain specifically four cases are considered with multi dimensional physical and stochastic domains these cases include an advection dominated case with an uncertain advection parameter a case spanning both the advective and diffusive regimes with uncertain advection and diffusion parameters a case with an uncertain source term and a case with scalar transport under an uncertain advection field in a channel with multiple branches for the latter case uniform and non uniform meshes are employed the expectation and variance from the stochastic galerkin and vms formulations are compared against their counterparts from analytical or monte carlo sampling methods where appropriate in summary an accurate and stable numerical solution is obtained using the vms formulation with projection based approximation of the stochastic stabilization parameter for multi dimensional problems including non trivial physical discretization

higher order principal component analysis for the approximation of tensors in tree based low rank formats this paper is concerned with the approximation of tensors using tree based tensor formats which are tensor networks whose graphs are dimension partition trees we consider hilbert tensor spaces of multivariate functions defined on a product set equipped with a probability measure this includes the case of multidimensional arrays corresponding to finite product sets we propose and analyse an algorithm for the construction of an approximation using only point evaluations of a multivariate function or evaluations of some entries of a multidimensional array the algorithm is a variant of higher order singular value decomposition which constructs a hierarchy of subspaces associated with the different nodes of the tree and a corresponding hierarchy of interpolation operators optimal subspaces are estimated using empirical principal component analysis of interpolations of partial random evaluations of the function the algorithm is able to provide an approximation in any tree based format with either a prescribed rank or a prescribed relative error with a number of evaluations of the order of the storage complexity of the approximation format under some assumptions on the estimation of principal components we prove that the algorithm provides either a quasi optimal approximation with a given rank or an approximation satisfying the prescribed relative error up to constants depending on the tree and the properties of interpolation operators the analysis takes into account the discretization errors for the approximation of infinite dimensional tensors for a tensor with finite and known rank in a tree based format the algorithm is able to recover the tensor in a stable way using a number of evaluations equal to the storage complexity of the representation of the tensor in this format several numerical examples illustrate the main results and the behavior of the algorithm for the approximation of high dimensional functions using hierarchical tucker or tensor train tensor formats and the approximation of univariate functions using tensorization

a method for shift rotation and scale invariant pattern recognition using the form and arrangement of pattern specific features abstract a new computer based method for visual pattern recognition is described the process extracts shape features from binary edge images represented in either real valued or discrete coordinates both separate and overlapping features are extracted maximizing their completeness and continuity without using templates or mathematical feature descriptions these features and their arrangement are encoded in a shift rotation and scale invariant manner and used to recognize incoming patterns from a set of previously encoded patterns stored in memory new patterns may be automatically classified and stored in memory allowing the system to learn and generalize without operator intervention testing shows the system to be completely shift rotation and scale invariant and insensitive to local image distortion

to be or not to be intrusive the solution of parametric and stochastic equations proper generalized decomposition a numerical method is proposed to compute a low rank galerkin approximation to the solution of a parametric or stochastic equation in a nonintrusive fashion the considered nonlinear problems are associated with the minimization of a parameterized differentiable convex functional we first introduce a bilinear parameterization of fixed rank tensors and employ an alternating minimization scheme for computing the low rank approximation in keeping with the idea of nonintrusiveness at each step of the algorithm the minimizations are carried out with a quasi newton method to avoid the computation of the hessian the algorithm is made nonintrusive through the use of numerical integration it only requires the evaluation of residuals at specific parameter values the algorithm is then applied to two numerical examples

binary shape recognition using the morphological skeleton transform abstract binary image representation by its morphological skeleton transform has been proposed in the past as an information preserving representation in this paper this skeleton representation is adopted as the start point for the development of a binary shape recognition scheme a skeleton matching algorithm is presented that efficiently characterizes the similarity between two skeletons as a distance measure this skeleton matching algorithm resembles the well known from speech processing elastic matching technique

a stochastic collocation method for elliptic partial differential equations with random input data in this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms input data of the model the input data are assumed to depend on a finite number of random variables the method consists in a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it can be seen as a generalization of the stochastic galerkin method proposed in i babuscka r tempone and g e zouraris siam j numer anal 42 2004 pp 800 825 and allows one to treat easily a wider range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction in the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method

generalized spectral decomposition method for solving stochastic finite element equations invariant subspace problem and dedicated algorithms stochastic galerkin methods have become a significant tool for the resolution of stochastic partial differential equations spde however they suffer from prohibitive computational times and memory requirements when dealing with large scale applications and high stochastic dimensionality some alternative techniques based on the construction of suitable reduced deterministic or stochastic bases have been proposed in order to reduce these computational costs recently a new approach based on the concept of generalized spectral decomposition gsd has been introduced for the definition and the automatic construction of reduced bases in this paper the concept of gsd initially introduced for a class of linear elliptic spde is extended to a wider class of stochastic problems the proposed definition of the gsd leads to the resolution of an invariant subspace problem which is interpreted as an eigen like problem this interpretation allows the construction of efficient numerical algorithms for building optimal reduced bases which are associated with dominant generalized eigenspaces the proposed algorithms by separating the resolution of reduced stochastic and deterministic problems lead to drastic computational savings their efficiency is illustrated on several examples where they are compared to classical resolution techniques

binary object representation and recognition using the hilbert morphological skeleton transform abstract a binary shape representation called the hilbert morphological skeleton transform hmst is introduced this representation combines the morphological skeleton transform mst with the clustering capabilities of the hilbert transform the hmst preserves the skeleton properties including information preservation progressive visualization and compact representation then an object recognition algorithm the hilbert skeleton matching algorithm hsma is introduced this algorithm performs a single sweep over the hmst s and renders the similarity between them as a distance measure testing the hsma against the skeleton matching algorithm sma and invariant moments revealed that the hsma algorithm achieves slightly better object recognition rates while substantially reducing the complexity in an experiment of 14 400 shape matches the hsma achieved a 90 36 recognition rate as opposed to 89 76 for the sma and 89 49 for invariant moments on the other hand the hsma improved the sma processing more than 40