cramer rao induced bounds for candecomp parafac tensor decomposition this paper presents a cramer rao lower bound crlb on the variance of unbiased estimates of factor matrices in canonical polyadic cp or candecomp parafac cp decompositions of a tensor from noisy observations i e the tensor plus a random gaussian i i d tensor a novel expression is derived for a bound on the mean square angular error of factors along a selected dimension of a tensor of an arbitrary dimension the expression needs less operations for computing the bound o nr6 than the best existing state of the art algorithm o n3r6 operations where n and r are the tensor order and the tensor rank insightful expressions are derived for tensors of rank 1 and rank 2 of arbitrary dimension and for tensors of arbitrary dimension and rank where two factor matrices have orthogonal columns the results can be used as a gauge of performance of different approximate cp decomposition algorithms prediction of their accuracy and for checking stability of a given decomposition of a tensor condition whether the crlb is finite or not a novel expression is derived for a hessian matrix needed in popular damped gauss newton method for solving the cp decomposition of tensors with missing elements beside computing the crlb for these tensors the expression may serve for design of damped gauss newton algorithm for the decomposition

bayesian factorizations of big sparse tensors it has become routine to collect data that are structured as multiway arrays tensors there is an enormous literature on low rank and sparse matrix factorizations but limited consideration of extensions to the tensor case in statistics the most common low rank tensor factorization relies on parallel factor analysis parafac which expresses a rank k tensor as a sum of rank one tensors in contingency table applications in which the sample size is massively less than the number of cells in the table the low rank assumption is not sufficient and parafac has poor performance we induce an additional layer of dimension reduction by allowing the effective rank to vary across dimensions of the table taking a bayesian approach we place priors on terms in the factorization and develop an efficient gibbs sampler for posterior computation theory is provided showing posterior concentration rates in high dimensional settings and the methods are shown to have excellent performance in simulations and several

induction for secant varieties of segre varieties this paper studies the dimension of secant varieties to segre varieties the problem is cast both in the setting of tensor algebra and in the setting of algebraic geometry an inductive procedure is built around the ideas of successive specializations of points and projections this reduces the calculation of the dimension of the secant variety in a high dimensional case to a sequence of calculations of partial secant varieties in low dimensional cases as applications of the technique we give a complete classification of defective p secant varieties to segre varieties for p 6 we generalize a theorem of catalisano geramita gimigliano on non defectivity of tensor powers of pn we determine the set of p for which unbalanced segre varieties have defective p secant varieties in addition we completely describe the dimensions of the secant varieties to the deficient segre varieties p1 x p1 x pn x pn and p2 x p3 x p3 in the final section we propose a series of conjectures about defective segre varieties

typical tensorial rank abstract upper bounds on the typical rank r n m l of tensors maximal border rank rank of almost all tensors of a given shape n m l are presented these improve previous results by a tkinson and lloyd for cubic shape tensors the typical rank is determined exactly r n n n n3 3n 2 n 3

exponential splitting for nonautonomous linear discrete time systems in banach spaces in this paper we consider some concepts of exponential splitting for nonautonomous linear discrete time systems these concepts are generalizations of some well known concepts of uniform and nonuniform exponential dichotomies connections between these concepts are presented and some illustrating examples prove that these are distinct

fast nonnegative matrix factorization an active set like method and comparisons nonnegative matrix factorization nmf is a dimension reduction method that has been widely used for numerous applications including text mining computer vision pattern discovery and bioinformatics a mathematical formulation for nmf appears as a nonconvex optimization problem and various types of algorithms have been devised to solve the problem the alternating nonnegative least squares anls framework is a block coordinate descent approach for solving nmf which was recently shown to be theoretically sound and empirically efficient in this paper we present a novel algorithm for nmf based on the anls framework our new algorithm builds upon the block principal pivoting method for the nonnegativity constrained least squares problem that overcomes a limitation of the active set method we introduce ideas that efficiently extend the block principal pivoting method within the context of nmf computation our algorithm inherits the convergence property of the anls framework and can easily be extended to other constrained nmf formulations extensive computational comparisons using data sets that are from real life applications as well as those artificially generated show that the proposed algorithm provides state of the art performance in terms of computational speed

existence and roughness of the exponential dichotomy for skew product semiflow in banach spaces abstract in this paper we introduce a concept of exponential dichotomy for skew product semiflow in infinite dimensional banach spaces which is an extension of the classic concept for evolution operators this concept is used to study the roughness property of the skew product semiflow also we introduce the concept of discrete skew product and give a necessary and sufficient condition for this discrete skew product to have a discrete dichotomy after that we give necessary and sufficient conditions for the existence of exponential dichotomy for skew product semiflow moreover we prove that the exponential dichotomy for skew product semiflow is not destroyed by small perturbation finally we apply these results to parabolic partial differential equations and functional differential equations

effects of error covariance structure on estimation of model averaging weights and predictive performance 1 when conducting model averaging for assessing groundwater conceptual model uncertainty the averaging weights are often evaluated using model selection criteria such as aic aicc bic and kic akaike information criterion corrected akaike information criterion bayesian information criterion and kashyap information criterion respectively however this method often leads to an unrealistic situation in which the best model receives overwhelmingly large averaging weight close to 100 which cannot be justified by available data and knowledge it was found in this study that this problem was caused by using the covariance matrix c of measurement errors for estimating the negative log likelihood function common to all the model selection criteria this problem can be resolved by using the covariance matrix cek of total errors including model errors and measurement errors to account for the correlation between the total errors an iterative two stage method was developed in the context of maximum likelihood inverse modeling to iteratively infer the unknown cek from the residuals during model calibration the inferred cek was then used in the evaluation of model selection criteria and model averaging weights while this method was limited to serial data using time series techniques in this study it can be extended to spatial data using geostatistical techniques the method was first evaluated in a synthetic study and then applied to an experimental study in which alternative surface complexation models were developed to simulate column experiments of uranium reactive transport it was found that the total errors of the alternative models were temporally correlated due to the model errors the iterative two stage method using cek resolved the problem that the best model receives 100 model averaging weight and the resulting model averaging weights were supported by the calibration results and physical understanding of the alternative models using cek obtained from the iterative two stage method also improved predictive performance of the individual models and model averaging in both synthetic and experimental studies

parameter identification of groundwater aquifer models a generalized least squares approach this paper concerns the methods of estimating aquifer transmissivities on the basis of unsteady state hydraulic head data traditionally the criterion of minimizing the sum of the squares of errors has been used to match the observed data with the model response the data used for optimization usually contain noise that is not necessarily uncorrelated it is well understood that the results of identification methods are very sensitive to measurement errors in data in this study the ordinary least squares ols technique is carried out along with a generalized least squares gls technique specifically designed to reduce the effect of correlated errors the trace of the covariance matrix is used as a measure of overall accuracy and reliability of the estimated parameters the effectiveness of the ols and gls techniques in dealing with noisy data is demonstrated by using a hypothetical example the results of numerical experiments suggest that gls offers a promising approach in efficiently improving the reliability of the estimated parameters

incorporation of prior information on parameters into nonlinear regression groundwater flow models 1 theory prior information on the parameters of a groundwater flow model can be used to improve parameter estimates obtained from nonlinear regression solution of a modeling problem two scales of prior information can be available 1 prior information having known reliability that is bias and random error structure and 2 prior information consisting of best available estimates of unknown reliability a regression method that incorporates the second scale of prior information assumes the prior information to be fixed for any particular analysis to produce improved although biased parameter estimates approximate optimization of two auxiliary parameters of the formulation is used to help minimize the bias which is almost always much smaller than that resulting from standard ridge regression it is shown that if both scales of prior information are available then a combined regression analysis may be made

robust nonuniform dichotomies and parameter dependence abstract we establish the robustness of linear cocycles with an exponential dichotomy under sufficiently small lipschitz perturbations in the sense that the existence of an exponential dichotomy for a given cocycle persists under these perturbations we consider cocycles in banach spaces as well as the general case of nonuniform exponential dichotomies and also the general case of an exponential behavior e c n given by an arbitrary sequence n including the usual exponential behavior n n as a very special case moreover we show that the projections of the exponential dichotomies obtained from the perturbation vary continuously with the parameter and in fact that they are locally lipschitz on finite dimensional parameters

on the complexity of nonnegative matrix factorization nonnegative matrix factorization nmf has become a prominent technique for the analysis of image databases text databases and other information retrieval and clustering applications the problem is most naturally posed as continuous optimization in this report we define an exact version of nmf then we establish several results about exact nmf i that it is equivalent to a problem in polyhedral combinatorics ii that it is np hard and iii that a polynomial time local search heuristic exists

tensor svd statistical and computational limits in this paper we propose a general framework for tensor singular value decomposition tensor singular value decomposition svd which focuses on the methodology and theory for extracting the hidden low rank structure from high dimensional tensor data comprehensive results are developed on both the statistical and computational limits for tensor svd this problem exhibits three different phases according to the signal to noise ratio snr in particular with strong snr we show that the classical higher order orthogonal iteration achieves the minimax optimal rate of convergence in estimation with weak snr the information theoretical lower bound implies that it is impossible to have consistent estimation in general with moderate snr we show that the non convex maximum likelihood estimation provides optimal solution but with np hard computational cost moreover under the hardness hypothesis of hypergraphic planted clique detection there are no polynomial time algorithms performing consistently in general

the inverse problem for confined aquifer flow identification and estimation with extensions the contributions of this work are twofold first a methodology for estimating the elements of parameter matrices in the governing equation of flow in a confined aquifer is developed the estimation techniques for the distributed parameter inverse problem pertain to linear least squares and generalized least squares methods the linear relationship among the known heads and unknown parameters of the flow equation provides the background for developing criteria for determining the identifiability status of unknown parameters under conditions of exact or overidentification it is possible to develop statistically consistent parameter estimators and their asymptotic distributions the estimation techniques namely two stage least squares and three stage least squares are applied to a specific groundwater inverse problem and compared between themselves and with an ordinary least squares estimator the three stage estimator provides the closer approximation to the actual parameter values but it also shows relatively large standard errors as compared to the ordinary and two stage estimators the estimation techniques provide the parameter matrices required to simulate the unsteady groundwater flow equation second a nonlinear maximum likelihood estimation approach to the inverse problem is presented the statistical properties of maximum likelihood estimators are derived and a procedure to construct confidence intervals and do hypothesis testing is given the relative merits of the linear and maximum likelihood estimators are analyzed other topics relevant to the identification and estimation methodologies i e a continuous time solution to the flow equation coping with noise corrupted head measurements and extension of the developed theory to nonlinear cases are also discussed a simulation study is used to evaluate the methods developed in this study

tensor decomposition for signal processing and machine learning tensors or multiway arrays are functions of three or more indices i j k ldots similar to matrices two way arrays which are functions of two indices r c for row column tensors have a rich history stretching over almost a century and touching upon numerous disciplines but they have only recently become ubiquitous in signal and data analytics at the confluence of signal processing statistics data mining and machine learning this overview article aims to provide a good starting point for researchers and practitioners interested in learning about and working with tensors as such it focuses on fundamentals and motivation using various application examples aiming to strike an appropriate balance of breadth and depth that will enable someone having taken first graduate courses in matrix algebra and probability to get started doing research and or developing tensor algorithms and software some background in applied optimization is useful but not strictly required the material covered includes tensor rank and rank decomposition basic tensor factorization models and their relationships and properties including fairly good coverage of identifiability broad coverage of algorithms ranging from alternating optimization to stochastic gradient statistical performance analysis and applications ranging from source separation to collaborative filtering mixture and topic modeling classification and multilinear subspace learning

estimation of aquifer parameters under transient and steady state conditions 1 maximum likelihood method incorporating prior information in this series of three papers a method is presented to estimate the parameters of groundwater flow models under steady and nonsteady state conditions the parameters include values and directions of principal hydraulic conductivities or transmissivities in anisotropic media specific storage or storativity interior and boundary recharge or leakage rates coefficients of head dependent interior and boundary sources and boundary heads in transient situations the initial head distribution can also be estimated if the system is originally at a steady state paper 1 of the series discusses some of the advantage in treating the inverse problem statistically and in regularizing its solution by means of penalty criteria based on prior estimates of the parameters the inverse problem is posed in the framework of maximum likelihood theory cast in a manner that accounts for prior information about the parameters since not all the factors which contribute to the prior errors can be quantified statistically at the outset the covariance matrices of these errors are expressed in terms of several parameters which if unknown can be estimated jointly with the hydraulic parameters by a stagewise optimization process when transient head data are separated by a fixed time interval the temporal structure of these data is approximated by a lag one autoregressive model with a correlation coefficient that can be treated as another unknown parameter estimation errors are analyzed by examining the lower bound of their covariance matrix in the eigenspace paper 1 concludes by suggesting that certain model identification criteria developed in the time series literature all of which are based on the maximum likelihood concept might be useful for selecting the best groundwater model or the best method of parameterizing a particular model among a number of given alternatives

perron frobenius theorem for nonnegative multilinear forms and extensions we prove an analog of perron frobenius theorem for multilinear forms with nonnegative coefficients and more generally for polynomial maps with nonnegative coefficients we determine the geometric convergence rate of the power algorithm to the unique normalized eigenvector

uniqueness of nonnegative tensor approximations we show that for a nonnegative tensor a best nonnegative rank r approximation is almost always unique its best rank one approximation may always be chosen to be a best nonnegative rank one approximation and the set of nonnegative tensors with nonunique best rank one approximations forms an algebraic hypersurface we show that the last part holds true more generally for real tensors and thereby determine a polynomial equation so that a real or nonnegative tensor that does not satisfy this equation is guaranteed to have a unique best rank one approximation we also establish an analogue for real or nonnegative symmetric tensors in addition we prove a singular vector variant of the perron frobenius theorem for positive tensors and apply it to show that a best nonnegative rank r approximation of a positive tensor can never be obtained by deflation as an aside we verify that the euclidean distance ed discriminants of the segre variety and the veronese variety are hypersurfaces and give defining equations of these ed discriminants

a composite l1 parameter estimator for model fitting in groundwater flow and solute transport simulation this paper proposes a composite l1 parameter estimator to solve the inverse problems in groundwater flow and solute transport modeling the estimator is formulated using a weighted l1 norm as the error measure between the vector of the observed hydraulic heads and solute concentrations and the corresponding vector of the heads and concentrations computed from a finite element model for steady state groundwater flow and solute transport in a two dimensional areal confined aquifer system the parameters currently considered are the hydraulic conductivities the dispersivities the porosities of the aquifer system and the solute source concentration s the gradients of the state variables with respect to the parameters are computed analytically using a formulation stemming from the finite element model for the state variables the solutions to two hypothetical groundwater parameter estimation problems are presented to illustrate that the proposed estimator does have the property of being fairly accurate and more significantly being robust in handling observation data containing certain outliers

aquifer parameter identification with kriging and optimum parameterization a new parameter identification method is developed for a two dimensional unsteady state groundwater flow model the method is a direct approach as it requires no iterative searching of parameters the unknown functional coefficients parameters are optimally determined by a generalized matrix inversion scheme kriging an unbiased minimum variance estimator is used as a presampling filter to reconstruct the head distribution for the entire flow domain the ill posedness of the inverse problem as characterized by instability and nonuniqueness is controlled by reparameterization utilizing the technique of finite elements a set of statistics are developed to quantify explicitly the tradeoff between the system modeling error and the error associated with the parameter uncertainty these two types of errors are shown to be a function of the parameter dimension residual mean square and trace of the covariance matrix of the estimated parameters serve as good indicators for the determination of the optimum parameter dimension a set of numerical experiments are conducted to illustrate the methodology

existence and robustness of nonuniform h k μ ν dichotomies for nonautonomous impulsive differential equations abstract for the linear nonautonomous impulsive differential equation we propose a new notion called the nonuniform h k dichotomy which extends the notion of dichotomy of uniform and nonuniform types and is related to the theory of nonuniform hyperbolicity we establish a sufficient criterion for the existence of the nonuniform h k dichotomy in terms of appropriate lyapunov exponents for the linear perturbation we develop the robustness or roughness of nonuniform h k dichotomies in the sense that a nonuniform h k dichotomy persists under a sufficiently small linear perturbation

input output admissibility and exponential trichotomy of difference equations abstract we propose a new method for the study of the asymptotic behavior of difference equations in infinite dimensional spaces providing characterizations for the property of uniform exponential trichotomy we deduce the structure of the stable unstable and bounded subspace and prove the uniqueness of the projection families we introduce a new admissibility concept with respect to a discrete input output system and prove that this is a necessary and sufficient condition for the existence of uniform exponential trichotomy throughout the paper there is no assumption on the coefficients and the obtained results are applicable to any class of difference equations

aquifer parameter identification with optimum dimension in parameterization this paper presents a systematic procedure whereby the functional coefficients imbedded in a two dimensional partial differential equation which governs unsteady groundwater flow are optimally identified the coefficients to be identified are transmissivities which vary spatially finite elements are used to represent the unknown transmissivity function parametrically in terms of nodal values over a suitable discretization of a flow region a modified gauss newton algorithm is used for parameter optimization covariance analysis is used to estimate the reliability of the estimated parameters as the dimension of the unknown parameter increases the modeling error represented by a least squares criterion will generally decrease but errors in data would be propagated to a greater degree into the estimated parameters thus reducing the reliability of estimation the reliability of the estimated parameters is characterized by a norm of the covariance matrix this information is used for the determination of the optimum dimension in parameterization

robustness of general dichotomies abstract for a linear equation v a t v we consider general dichotomies that may exhibit stable and unstable behaviors with respect to arbitrary asymptotic rates e c t for some function t this includes as a special case the usual exponential behavior when t t we also consider the general case of nonuniform exponential dichotomies we establish the robustness of the exponential dichotomies in banach spaces in the sense that the existence of an exponential dichotomy for a given linear equation persists under sufficiently small linear perturbations we also establish the continuous dependence with the perturbation of the constants in the notion of dichotomy

exponential dichotomy roughness on banach spaces in the present paper we extend existing results on exponential dichotomy roughness for linear ode systems to infinite dimensional banach space we give new conditions for the existence of exponential dichotomy roughness in infinite dimensional space and in the finite interval case we also improve previous results by indicating the exact values of the dichotomic constants of the perturbed equation

exponential dichotomy and dichotomy radius for difference equations abstract the aim of this paper is to provide a new approach concerning the characterization of exponential dichotomy of difference equations by means of admissible pair of sequence spaces we classify the classes of input and output spaces respectively and deduce necessary and sufficient conditions for exponential dichotomy applicable for a large variety of systems by an example we show that the obtained results are the most general in this topic as an application we deduce a general lower bound for the dichotomy radius of difference equations in terms of input output operators acting on sequence spaces which are invariant under translations

optimum experimental design for parameter identification in groundwater hydrology the experimental design for parameter identification of a confined groundwater system is formulated as a nonlinear mixed integer programming problem the decision variables considered are the number and locations of the pumping and observation wells and the pumping rates the constraints include the allowable drawdown the restricted locations of pumping wells and observation wells the permissible pumping rates and the required information the amount of the required information is determined by the prescribed reliability of the response for the future operation of the groundwater system the galerkin finite element crank nicolson approach is used to develop the simulation models for solving the groundwater governing equation for the system responses the sensitivity coefficients are computed by the influence coefficient method the principle of superposition is used to combine the simulation models with the optimization model the characteristics of the optimization problem have been investigated and are clarified and its difficulties are pointed out an efficient heuristic solution approach is proposed for solving this problem a test problem is constructed to demonstrate the usefulness of the proposed methodology observations generated from the optimal design are used to identify the parameters a covariance analysis is then carried out to verify the validity of the assumptions made in the formulation of the optimization problem the results obtained indicate that the proposed methodology is feasible and can be used to solve complicated experimental design problems associated with a confined groundwater system

smoothness of spectral subbundles and reducibility of quasi periodic linear differential systems abstract in this paper we study linear differential systems 1 x a t x where a is an n n matrix valued function defined on the k torus t k and t t is a given irrational twist flow on t k first we show that if a c n t k where n 0 1 2 then the spectral subbundles are of class c n on t k next we assume that a is sufficiently smooth on t k and satisfies a suitable small divisors inequality we show that if 1 satisfies the full spectrum assumption then there is a quasi periodic linear change of variables x p t y that transforms 1 to a constant coefficient system y by finally we study the case where the matrix a t in 1 is the jacobian matrix of a nonlinear vector field x evaluated along a quasi periodic solution x t of 2 x x we give sufficient conditions in terms of smoothness and small divisors inequalities in order that there is a coordinate system z defined in the vicinity of h the hull of so that the linearized system 1 can be represented in the form z dz where d is a constant matrix our results represent substantial improvements over known methods because we do not require that a be close to a constant coefficient system

review of parameter identification procedures in groundwater hydrology the inverse problem the purpose of this survey is to review parameter identification procedures in groundwater hydrology and to examine computational techniques which have been developed to solve the inverse problem parameter identification methods are classified under the error criterion used in the formulation of the inverse problem the problem of ill posedness in connection with the inverse problem is addressed typical inverse solution techniques are highlighted the review also includes the evaluation of methods used for computing the sensitivity matrix statistics which can be used to estimate the parameter uncertainty are outlined attempts have been made to compare and contrast representative inverse procedures and direction for future research is suggested

low dimensional polytope approximation and its applications to nonnegative matrix factorization in this study nonnegative matrix factorization is recast as the problem of approximating a polytope on the probability simplex by another polytope with fewer facets working on the probability simplex has the advantage that data are limited to a compact set with a known boundary making it easier to trace the approximation procedure in particular the supporting hyperplane that separates a point from a disjoint polytope a fact asserted by the hahn banach theorem can be calculated in finitely many steps this approach leads to a convenient way of computing the proximity map which in contrast to most existing algorithms where only an approximate map is used finds the unique and global minimum per iteration this paper sets up a theoretical framework outlines a numerical algorithm and suggests an effective implementation testing results strongly evidence that this approach obtains a better low rank nonnegative matrix approximation in fewer steps than conventional methods

eigenvalues and invariants of tensors abstract a tensor is represented by a supermatrix under a co ordinate system in this paper we define e eigenvalues and e eigenvectors for tensors and supermatrices by the resultant theory we define the e characteristic polynomial of a tensor an e eigenvalue of a tensor is a root of the e characteristic polynomial in the regular case a complex number is an e eigenvalue if and only if it is a root of the e characteristic polynomial we convert the e characteristic polynomial of a tensor to a monic polynomial and show that the coefficients of that monic polynomial are invariants of that tensor i e they are invariant under co ordinate system changes we call them principal invariants of that tensor the maximum number of principal invariants of mth order n dimensional tensors is a function of m and n we denote it by d m n and show that d 1 n 1 d 2 n n d m 2 m for m 3 and d m n m n 1 m for m n 3 we also define the rank of a tensor all real eigenvectors associated with nonzero e eigenvalues are in a subspace with dimension equal to its rank

tensor deflation for candecomp parafac part i alternating subspace update algorithm candecomp parafac cp approximates multiway data by sum of rank 1 tensors unlike matrix decomposition the procedure which estimates the best rank r tensor approximation through r sequential best rank 1 approximations does not work for tensors because the deflation does not always reduce the tensor rank in this paper we propose a novel deflation method for the problem when one factor matrix of a rank r cp decomposition is of full column rank the decomposition can be performed through r 1 rank 1 reductions at each deflation stage the residue tensor is constrained to have a reduced multilinear rank for decomposition of order 3 tensors of size r times r times r and rank r estimation of one rank 1 tensor has a computational cost of cal o r 3 per iteration which is lower than the cost cal o r 4 of the als algorithm for the overall cp decomposition the method can be extended to tracking one or a few rank one tensors of slow changes or inspect variations of common patterns in individual datasets

fast nonnegative matrix tensor factorization based on low rank approximation nonnegative matrix factorization nmf algorithms often suffer from slow convergence speed due to the nonnegativity constraints especially for large scale problems low rank approximation methods such as principle component analysis pca are widely used in matrix factorizations to suppress noise reduce computational complexity and memory requirements however they cannot be applied to nmf directly so far as they result in factors with mixed signs in this paper low rank approximation is introduced to nmf named lranmf which is not only able to reduce the computational complexity of nmf algorithms significantly but also suppress bipolar noise in fact the new update rules are typically about times faster than traditional ones of nmf here is the number of observations and is the low rank of latent factors therefore lranmf is particularly efficient in the case where which is the general case in nmf the proposed update rules can also be incorporated into most existing nmf algorithms straightforwardly as long as they are based on euclidean distance then the concept of lranmf is generalized to the tensor field to perform a fast sequential nonnegative tucker decomposition ntd by applying the proposed methods the practicability of nmf ntd is significantly improved simulations on synthetic and real data show the validity and efficiency of the proposed approaches

inverse modeling for locating dense nonaqueous pools in groundwater under steady flow conditions in this work we develop an inverse modeling procedure to identify the location and the dimensions of a single component dense nonaqueous phase liquid dnapl pool in a saturated porous medium under steady flow conditions the inverse problem is formulated as a least squares minimization problem and solved by a search procedure based on the levenberg marquardt method model output is calculated by an existing three dimensional analytical model describing the transport of solute from a dissolving distributed noise upon the forward model generated concentration field we further test the algorithm s ability to predict the location and size of a dnapl pool placed in a controlled three dimensional bench scale experiment in this case we apply the levenberg marquardt algorithm to the minimization of three types of residuals ordinary residuals weighted residuals with weights equal to the square of the inverse of the observations and weighted residuals with weights obtained by adding a constant term to the observed concentrations the results are sensitive to the location of the observation wells and to the type of residuals minimized in general better results in terms of pool location and dimensions were obtained by the minimization of weighted residuals with weights obtained by adding a constant term to the observed concentrations the results also indicate that the inverse problem is nonunique and nonconvex even in the absence of observation errors finally the sensitivity of the inverse modeling scheme to transport parameter uncertainty was addressed the inverse solution was found to be extremely sensitive to errors in the dispersion coefficients and relatively insensitive to errors in the mass transfer coefficient

the number of eigenvalues of a tensor abstract eigenvectors of tensors as studied recently in numerical multilinear algebra correspond to fixed points of self maps of a projective space we determine the number of eigenvectors and eigenvalues of a generic tensor and we show that the number of normalized eigenvalues of a symmetric tensor is always finite we also examine the characteristic polynomial and how its coefficients are related to discriminants and resultants

subtracting a best rank 1 approximation may increase tensor rank it has been shown that a best rank r approximation of an order k tensor may not exist when r 2 and k 3 this poses a serious problem to data analysts using tensor decompositions it has been observed numerically that generally this issue cannot be solved by consecutively computing and subtracting best rank 1 approximations the reason for this is that subtracting a best rank 1 approximation generally does not decrease tensor rank in this paper we provide a mathematical treatment of this property for real valued 2 2 2 tensors with symmetric tensors as a special case regardless of the symmetry we show that for generic 2 2 2 tensors which have rank 2 or 3 subtracting a best rank 1 approximation results in a tensor that has rank 3 and lies on the boundary between the rank 2 and rank 3 sets hence for a typical tensor of rank 2 subtracting a best rank 1 approximation increases the tensor rank

linearization functors on real convex sets we prove that linearizing multilinear optimization problems leads to new functorial operations on real convex sets these operations are convex analogues of hom functors tensor products symmetric powers exterior powers and general schur functors on vector spaces and lead to novel constructions even for polyhedra we discuss their general theory and introduce mechanisms to compute them or approximate them in ways amenable to efficient computation

rank and optimal computation of generic tensors abstract the typical rank maximal border rank of tensors of a given size and the set of optimal bilinear computations of typical tensors of a given rank are investigated for the size n n 3 with n odd the complement of the set of tensors of maximal border rank is a hypersurface its equation is given

tensor deflation for candecomp parafac part ii initialization and error analysis in part i of the study of the tensor deflation for candecomp parafac we have shown that the rank 1 tensor deflation is applicable under some conditions part ii of the study presents several initialization algorithms suitable for the algorithm proposed in part i in addition part ii contains an algorithm for the case when one or more factor matrices in the estimated model is constrained to be orthogonal finally part ii provides an error analysis of the tensor deflation algorithm which shows that there is a marginal loss of accuracy of the deflation algorithm compared to the ordinary cp decomposition

generic and typical ranks of multi way arrays abstract the concept of tensor rank was introduced in the 20s in the 70s when methods of component analysis on arrays with more than two indices became popular tensor rank became a much studied topic the generic rank may be seen as an upper bound to the number of factors that are needed to construct a random tensor we explain in this paper how to obtain numerically in the complex field the generic rank of tensors of arbitrary dimensions based on terracini s lemma and compare it with the algebraic results already known in the real or complex fields in particular we examine the cases of symmetric tensors tensors with symmetric matrix slices complex tensors enjoying hermitian symmetries or merely tensors with free entries

projected gradient methods for nonnegative matrix factorization nonnegative matrix factorization nmf can be formulated as a minimization problem with bound constraints although bound constrained optimization has been studied extensively in both theory and practice so far no study has formally applied its techniques to nmf in this letter we propose two projected gradient methods for nmf both of which exhibit strong optimization properties we discuss efficient implementations and demonstrate that one of the proposed methods converges faster than the popular multiplicative update approach a simple matlab code is also provided