noncausal gauss markov random fields parameter structure and estimation the parameter structure of noncausal homogeneous gauss markov random fields gmrf defined on finite lattices is studied for first order nearest neighbor and a special class of second order fields a complete characterization of the parameter space and a fast implementation of the maximum likelihood estimator of the field parameters are provided for general higher order fields tight bounds for the parameter space are presented and an efficient procedure for ml estimation is described experimental results illustrate the application of the approach presented and the viability of the present method in fitting noncausal models to 2 d data

ergodic properties of levy flights coexisting with subdiffusion and related models abstract we analyze ergodic properties of two different stationary processes resulting from a combination of levy flights and subdiffusion the first one comes from the lamperti transformation of subordinated stable process the second one is a sequence of increments of subordinated levy process we prove that both processes are mixing and ergodic we also study increments and the lamperti transformation of time changed fractional brownian motion it appears that these models of anomalous diffusion are also ergodic and mixing we believe that the derived results will help to verify ergodicity in single particle tracking experiments

what is the fractional laplacian a comparative review with new results abstract the fractional laplacian in r d which we write as 2 with 0 2 has multiple equivalent characterizations moreover in bounded domains boundary conditions must be incorporated in these characterizations in mathematically distinct ways and there is currently no consensus in the literature as to which definition of the fractional laplacian in bounded domains is most appropriate for a given application the riesz or integral definition for example admits a nonlocal boundary condition where the value of a function must be prescribed on the entire exterior of the domain in order to compute its fractional laplacian in contrast the spectral definition requires only the standard local boundary condition these differences among others lead us to ask the question what is the fractional laplacian beginning from first principles we compare several commonly used definitions of the fractional laplacian theoretically through their stochastic interpretations as well as their analytical properties then we present quantitative comparisons using a sample of state of the art methods we discuss recent advances on nonzero boundary conditions and present new methods to discretize such boundary value problems radial basis function collocation for the riesz fractional laplacian and nonharmonic lifting for the spectral fractional laplacian in our numerical studies we aim to compare different definitions on bounded domains using a collection of benchmark problems we consider the fractional poisson equation with both zero and nonzero boundary conditions where the fractional laplacian is defined according to the riesz definition the spectral definition the directional definition and the horizon based nonlocal definition we verify the accuracy of the numerical methods used in the approximations for each operator and we focus on identifying differences in the boundary behaviors of solutions to equations posed with these different definitions through our efforts we aim to further engage the research community in open problems and assist practitioners in identifying the most appropriate definition and computational approach to use for their mathematical models in addressing anomalous transport in diverse applications

efficient geo graph contiguity and hole algorithms for geographic zoning and dynamic plane graph partitioning graph partitioning is an intractable problem that arises in many practical applications heuristics such as local search generate good though suboptimal solutions in limited time such heuristics must be able to explore the solution space quickly and when the solution space is constrained differentiate feasible solutions from infeasible ones geographic zoning problems allocate some resource e g political representation school enrollment police patrols to contiguous zones modeled by partitions of an embedded planar graph each vertex corresponds to an area of the plane e g census block town county and local search moves one area from its current zone to a different zone in each iteration enforcing contiguity constraints may require significant computation when the graph is large while existing algorithms require linear or polylogarithmic time in the number of vertices to assess contiguity in each local search iteration the geo graph paradigm shows how contiguity can be verified by examining only the set of vertices that border the transferred area i e those areas whose boundaries share at least a single point with the boundary of the transferred area this paper develops efficient algorithms that examine these vertices more quickly than traditional search based methods allowing practitioners to more fully consider their zoning options when creating zones with local search

upper and lower bounds for the sales force deployment problem with explicit contiguity constraints abstract the sales force deployment problem arises in many selling organizations this complex planning problem involves the concurrent resolution of four interrelated subproblems sizing of the sales force sales representatives locations sales territory alignment and sales resource allocation the objective is to maximize the total profit for this a well known and accepted concave sales response function is used unfortunately literature is lacking approaches that provide valid upper bounds therefore we propose a model formulation with an infinite number of binary variables the linear relaxation is solved by column generation where the variables with maximum reduced costs are obtained analytically for the optimal objective function value of the linear relaxation an upper bound is provided to obtain a very tight gap for the objective function value of the optimal integer solution we introduce a branch and price approach moreover we propose explicit contiguity constraints based on flow variables in a series of computational studies we consider instances which may occur in the pharmaceutical industry the largest instance comprises 50 potential locations and more than 500 sales coverage units we are able to solve this instance in 1273 seconds with a gap of less than 0 01 a comparison with drexl and haase 1999 shows that we are able to halve the solution gap due to tight upper bounds provided by the column generation procedure

transfer matrix renormalization group method for general markov random fields abstract we seek the numerical calculation of partition functions of general markov random fields mrfs on an infinitely long twisted cylindrical lattice by using the transfer matrix renormalization group tmrg method the tmrg is a variant of the density matrix renormalization group dmrg which automatically truncates the hilbert space so that the properties of large systems can be precisely calculated while the dimension of the renormalized hilbert space remains constant we apply the tmrg to the decimation of the fundamental transfer matrix that we have proposed previously for general mrfs instead of the standard s e scheme for tmrg we propose a new e s e scheme and propose an alternative method for selecting the renormalized basis specifically the new e s e scheme keeps those singular value decomposition svd components of the fundamental transfer matrix that are relevant for the perron state and truncates the other irrelevant ones results for the ising model show that our method exhibits very impressive accuracy under a rather restricted computational resource simulations for another four general mrfs demonstrate that our tmrg method is superior to the classical monte carlo method in accuracy computational speed and in the possibility of treating a much larger system

algebraic signal processing theory 1 d space in our paper titled ldquoalgebraic signal processing theory foundation and 1 d timerdquo appearing in this issue of the ieee transactions on signal processing we presented the algebraic signal processing theory an axiomatic and general framework for linear signal processing the basic concept in this theory is the signal model defined as the triple a m phi where a is a chosen algebra of filters m an associated a module of signals and phi is a generalization of the z transform each signal model has its own associated set of basic sp concepts including filtering spectrum and fourier transform examples include infinite and finite discrete time where these notions take their well known forms in this paper we use the algebraic theory to develop infinite and finite space signal models these models are based on a symmetric space shift operator which is distinct from the standard time shift we present the space signal processing concepts of filtering or convolution ldquoz transform rdquo spectrum and fourier transform for finite length space signals we obtain 16 variants of space models which have the 16 discrete cosine and sine transforms dcts dsts as fourier transforms using this novel derivation we provide missing signal processing concepts associated with the dcts dsts establish them as precise analogs to the dft get deep insight into their origin and enable the easy derivation of many of their properties including their fast algorithms

mutually compatible gibbs random fields a unified theory for the mathematical description of gibbs random fields that answers some important theoretical and practical questions about their statistical behavior is presented the local transfer function is introduced and the joint probability measure of the general gibbs random field is derived in terms of this function the resulting probability structure is required to satisfy the property of mutual compatibility a necessary and sufficient condition for a gibbs random field to be mutually compatible is developed and used to prove that a mutually compatible gibbs random field is a unilateral markov random field the existence of some special nontrivial cases of gibbs random fields that are mutually compatible is demonstrated conditions on the translation invariance and isotropy of the general gibbs random field with a free boundary are studied the class of gibbs random fields with a homogeneous local transfer function and the class of horizontally and vertically translation invariant gibbs random fields are introduced and treated the concept of a translation invariant gibbs random field is also explored the problem of the statistical inference of mutually compatible gibbs random fields is discussed

markov modeling for bayesian restoration of two dimensional layered structures bayesian estimation of two dimensional stratified structures is described the major point addressed is the derivation of a statistical prior model that adequately describes such layered media this problem is of interest in applications in which the data are generally processed in one dimension only in order to take local interactions into account a markovian description is used the model is derived so as to fulfill a set of constraints that summarize physical and geometrical characteristics of the problem as well as practical requirements the resulting class of markov random fields presents a unilateral structure on a nonrectangular lattice and a hierarchical organization which involves a line process in addition it is shown to be an extension of one dimensional models already in use the properties of the model are investigated and its practicality is demonstrated by an application to seismic deconvolution simulation results show significant improvements with respect to the usual one dimensional methods

recovering the historical distribution for nonlinear space fractional diffusion equation with temporally dependent thermal conductivity in higher dimensional space abstract in this paper we investigate the problem of recovering the historical distribution for a nonlinear space fractional diffusion equation with temporally dependent thermal conductivity in higher dimensional space this problem is obtained from the classical diffusion equation by replacing the second order space derivative with a fractional laplacian of order 1 2 1 which is usually used to model the anomalous diffusion the problem is severely ill posed to regularize the problem we propose a modified version of the tikhonov regularization method a stability estimate of holder type is established finally several numerical examples based on the finite difference approximation and the discrete fourier transform are presented to illustrate the theoretical results

attraction repulsion expectation maximization algorithm for image reconstruction and sensor field estimation in this paper we propose an attraction repulsion expectation maximization arem algorithm for image reconstruction and sensor field estimation we rely on a new method for density estimation to address the problems of image reconstruction from limited samples and sensor field estimation from randomly scattered sensors density estimation methods often suffer from undesirable phenomena such as over fitting and over smoothing specifically various density estimation techniques based on a gaussian mixture model gmm tend to cluster the gaussian functions together thus resulting in over fitting on the other hand other approaches repel the gaussian functions and yield over smooth density estimates we propose a method that seeks an equilibrium between over fitting and over smoothing in density estimation by incorporating attraction and repulsion forces among the gaussian functions and determining the optimal balance between the competing forces experimentally we model the attractive and repulsive forces by introducing the gibbs and inverse gibbs distributions respectively the maximization of the likelihood function augmented by the gibbs density mixture is solved under the expectation maximization em method computer simulation results are provided to demonstrate the effectiveness of the proposed arem algorithm in image reconstruction and sensor field estimation

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

model parameter estimation for reciprocal gaussian random processes the problem of estimating the model parameters of a discrete index reciprocal gaussian random process from a limited number of noisy observations is addressed the general case of a first order multivariate process is analyzed stating its basic properties and deriving a linear equation set that relates the model parameters including the unknown variance of the observation noise to the generally nonstationary autocorrelation function of the observed process it generalizes to the reciprocal processes the so called high order yule walker equations for ar processes based on these results a practical estimation algorithm is proposed

algebraic signal processing theory 1 d nearest neighbor models we present a signal processing framework for the analysis of discrete signals represented as linear combinations of orthogonal polynomials we demonstrate that this representation implicitly changes the associated shift operation from the standard time shift to the nearest neighbor shift introduced in this paper using the algebraic signal processing theory we construct signal models based on this shift and derive their corresponding signal processing concepts including the proper notions of signal and filter spaces z transform convolution spectrum and fourier transform the presented results extend the algebraic signal processing theory and provide a general theoretical framework for signal analysis using orthogonal polynomials

fractional diffusions with time varying coefficients this paper is concerned with the fractionalized diffusion equations governing the law of the fractional brownian motion bh t we obtain solutions of these equations which are probability laws extending that of bh t our analysis is based on mcbride fractional operators generalizing the hyper bessel operators l and converting their fractional power l into erdelyi kober fractional integrals we study also probabilistic properties of the random variables whose distributions satisfy space time fractional equations involving caputo and riesz fractional derivatives some results emerging from the analysis of fractional equations with time varying coefficients have the form of distributions of time changed random variables

stochastic solution of space time fractional diffusion equations classical and anomalous diffusion equations employ integer derivatives fractional derivatives and other pseudodifferential operators in space in this paper we show that replacing the integer time derivative by a fractional derivative subordinates the original stochastic solution to an inverse stable subordinator process whose probability distributions are mittag leffler type this leads to explicit solutions for space time fractional diffusion equations with multiscaling space fractional derivatives and additional insight into the meaning of these equations

stochastic relaxation on partitions with connected components and its application to image segmentation we present a new method of segmentation in which images are segmented in partitions with connected components we give computationally inexpensive algorithms for probability simulation and simulated annealing on the space of partitions with connected components of a general graph in particular hastings algorithms 1970 and generalized metropolis algorithms are defined to avoid heavy computation to achieve segmentation we propose a hierarchical approach which at each step minimizes a cost function on the space of partitions with connected components of a graph the algorithm is applied to segment gray level color and textured images

absolute phase image reconstruction a stochastic nonlinear filtering approach this paper formulates and proposes solutions to the problem of estimating reconstructing the absolute not simply modulo 2 spl pi phase of a complex random field from noisy observations of its real and imaginary parts this problem is representative of a class of important imaging techniques such as interferometric synthetic aperture radar optical interferometry magnetic resonance imaging and diffraction tomography we follow a bayesian approach then not only a probabilistic model of the observation mechanism but also prior knowledge concerning the phase image to be reconstructed are needed we take as prior a nonsymmetrical half plane autoregressive nshp ar gauss markov random field gmrf based on a reduced order state space formulation of the linear nshp ar model and on the nonlinear observation mechanism a recursive stochastic nonlinear filter is derived the corresponding estimates are compared with those obtained by the extended kalman bucy filter a classical linearizing approach to the same problem a set of examples illustrate the effectiveness of the proposed approach

detection of gauss markov random fields with nearest neighbor dependency the problem of hypothesis testing against independence for a gauss markov random field gmrf is analyzed assuming an acyclic dependency graph an expression for the log likelihood ratio of detection is derived assuming random placement of nodes over a large region according to the poisson or uniform distribution and nearest neighbor dependency graph the error exponent of the neyman pearson detector is derived using large deviations theory the error exponent is expressed as a dependency graph functional and the limit is evaluated through a special law of large numbers for stabilizing graph functionals the exponent is analyzed for different values of the variance ratio and correlation it is found that a more correlated gmrf has a higher exponent at low values of the variance ratio whereas the situation is reversed at high values of the variance ratio

a fast algorithm for solving the space time fractional diffusion equation abstract in this paper we propose a fast algorithm for efficient and accurate solution of the space time fractional diffusion equations defined in a rectangular domain the spatial discretization is done by using the central finite difference scheme and matrix transfer technique due to its nonlocality numerical discretization of the spectral fractional laplacian s 2 results in a large dense matrix this causes considerable challenges not only for storing the matrix but also for computing matrix vector products in practice by utilizing the compact structure of the discrete system and the discrete sine transform our algorithm avoids to store the large matrix from discretizing the nonlocal operator and also significantly reduces the computational costs we then use the laplace transform method for time integration of the semi discretized system and a weighted trapezoidal method to numerically compute the convolutions needed in the resulting scheme various experiments are presented to demonstrate the efficiency and accuracy of our method

sequential and parallel image restoration neural network implementations sequential and parallel image restoration algorithms and their implementations on neural networks are proposed for images degraded by linear blur and contaminated by additive white gaussian noise maximum a posteriori map estimation and regularization theory lead to the same high dimension convex optimization problem the commonly adopted strategy in using neural networks for image restoration is to map the objective function of the optimization problem into the energy of a predefined network taking advantage of its energy minimization properties departing from this approach we propose neural implementations of iterative minimization algorithms which are first proved to converge the developed schemes are based on modified hopfield 1985 networks of graded elements with both sequential and parallel updating schedules an algorithm supported on a fully standard hopfield network binary elements and zero autoconnections is also considered robustness with respect to finite numerical precision is studied and examples with real images are presented

optimal node density for detection in energy constrained random networks the problem of optimal node density maximizing the neyman pearson detection error exponent subject to a constraint on average per node energy consumption is analyzed the spatial correlation among the sensor measurements is incorporated through a gauss markov random field gmrf model with euclidean nearest neighbor dependency graph a constant density deployment of sensors under the uniform or poisson distribution is assumed it is shown that the optimal node density crucially depends on the ratio between the measurement variances under the two hypotheses and displays a threshold behavior below the threshold value of the variance ratio the optimal node density tends to infinity under any feasible average energy constraint on the other hand when the variance ratio is above the threshold the optimal node density is the minimum value at which it is feasible to process and deliver the likelihood ratio sufficient statistic of the sensor measurements to the fusion center in this regime of the variance ratio an upper bound on the optimal node density based on a proposed 2 approximation fusion scheme and a lower bound based on the minimum spanning tree are established under an alternative formulation where the energy consumption per unit area is constrained the optimal node density is shown to be strictly finite for all values of the variance ratio and bounds on this optimal node density are provided

digital image restoration using spatial interaction models this paper is concerned with developing fast nonrecursive algorithms for the minimum mean squared error restoration of degraded images the degradation is assumed to be due to a space invariant periodic nonseparable known point spread function and additive white noise our basic approach is to represent the images by a class of spatial interaction models namely the simultaneous autoregressve models and the conditional markov models defined on toroidal lattices and develop minimum mean squared error restoration algorithms using these models the restoration algorithms are optimal if the parameters characterizing the interaction models are exactly known however in practice the parameters are estimated from the images by using spatial interaction models we develop restoration algorithms that do not require the availability of the original image or its prototype the specific structure of the underlying lattice enables the implementation of the filters using fast fourier transform fft computations several restoration examples are given

two families of functions related to the fractional powers of generators of strongly continuous contraction semigroups abstract two families of functions on 0 are related to the theory of fractional powers of generators of strongly continuous semigroups namely the family t t 0 of density functions of the one sided stable semigroup of order 0 1 and a family e 0 of mittag leffler type functions for the latter family we make this relation visible we collect some important properties of these functions and we improve a known result on the laplace transform of the functions e and their derivatives in the sense that we enlarge the domains of the laplace variable we furthermore find an expression for the laplace transform of the functions t x t x 0 in terms of the derivative e

computationally tractable stochastic image modeling based on symmetric markov mesh random fields in this paper the properties of a new class of causal markov random fields named symmetric markov mesh random field are initially discussed it is shown that the symmetric markov mesh random fields from the upper corners are equivalent to the symmetric markov mesh random fields from the lower corners based on this new random field a symmetric corner independent and isotropic image model is then derived which incorporates the dependency of a pixel on all its neighbors the introduced image model comprises the product of several local 1d density and 2d joint density functions of pixels in an image thus making it computationally tractable and practically feasible by allowing the use of histogram and joint histogram approximations to estimate the model parameters an image restoration application is also presented to confirm the effectiveness of the model developed the experimental results demonstrate that this new model provides an improved tool for image modeling purposes compared to the conventional markov random field models

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

geo graphs an efficient model for enforcing contiguity and hole constraints in planar graph partitioning political districting is an intractable problem with significant ramifications for political representation districts often are required to satisfy some legal constraints but these typically are not very restrictive allowing decision makers to influence the composition of these districts without violating relevant laws for example while districts must often comprise a single contiguous area a vast collection of acceptable solutions i e sets of districts remains choosing the best set of districts from this collection can be treated as a planar graph partitioning problem when districts must be contiguous successfully solving this problem requires an efficient computational method for evaluating contiguity constraints common methods for assessing contiguity can require significant computation as the problem size grows this paper introduces the geo graph a new graph model that ameliorates the computational burdens associated with enforcing contiguity constraints in planar graph partitioning when each vertex corresponds to a particular region of the plane through planar graph duality the geo graph provides a scale invariant method for enforcing contiguity constraints in local search furthermore geo graphs allow district holes which typically are considered undesirable to be rigorously and efficiently integrated into the partitioning process

classification of textures using gaussian markov random fields the problem of texture classification arises in several disciplines such as remote sensing computer vision and image analysis in this paper we present two feature extraction methods for the classification of textures using two dimensional 2 d markov random field mrf models it is assumed that the given m m texture is generated by a gaussian mrf model in the first method the least square ls estimates of model parameters are used as features in the second method using the notion of sufficient statistics it is shown that the sample correlations over a symmetric window including the origin are optimal features for classification simple minimum distance classifiers using these two feature sets yield good classification accuracies for a seven class problem

partition function estimation of gibbs random field images using monte carlo simulations a monte carlo simulation technique for estimating the partition function of a general gibbs random field image is proposed by expressing the partition function as an expectation an importance sampling approach for estimating it using monte carlo simulations is developed as expected the resulting estimators are unbiased and consistent computations can be performed iteratively by using simple monte carlo algorithms with remarkable success as demonstrated by simulations the work concentrates on binary second order gibbs random fields defined on a rectangular lattice however the proposed methods can be easily extended to more general gibbs random fields their potential contribution to optimal parameter estimation and hypothesis testing problems for general gibbs random field images using a likelihood approach is anticipated

stochastic approximation algorithms for partition function estimation of gibbs random fields we present an analysis of previously proposed monte carlo algorithms for estimating the partition function of a gibbs random field we show that this problem reduces to estimating one or more expectations of suitable functionals of the gibbs states with respect to properly chosen gibbs distributions as expected the resulting estimators are consistent certain generalizations are also provided we study computational complexity with respect to grid size and show that monte carlo partition function estimation algorithms can be classified into two categories e type algorithms that are of exponential complexity and p type algorithms that are of polynomial complexity turing reducible to the problem of sampling from the gibbs distribution e type algorithms require estimating a single expectation whereas p type algorithms require estimating a number of expectations with respect to gibbs distributions which are chosen to be sufficiently close to each other in the latter case the required number of expectations is of polynomial order with respect to grid size we compare computational complexity by using both theoretical results and simulation experiments we determine the most efficient e type and p type algorithms and conclude that p type algorithms are more appropriate for partition function estimation we finally suggest a practical and efficient p type algorithm for this task

bayes smoothing algorithms for segmentation of binary images modeled by markov random fields a new image segmentation algorithm is presented based on recursive bayes smoothing of images modeled by markov random fields and corrupted by independent additive noise the bayes smoothing algorithm yields the a posteriori distribution of the scene value at each pixel given the total noisy image in a recursive way the a posteriori distribution together with a criterion of optimality then determine a bayes estimate of the scene the algorithm presented is an extension of a 1 d bayes smoothing algorithm to 2 d and it gives the optimum bayes estimate for the scene value at each pixel computational concerns in 2 d however necessitate certain simplifying assumptions on the model and approximations on the implementation of the algorithm in particular the scene noiseless image is modeled as a markov mesh random field a special class of markov random fields and the bayes smoothing algorithm is applied on overlapping strips horizontal vertical of the image consisting of several rows columns it is assumed that the signal scene values vector sequence along the strip is a vector markov chain since signal correlation in one of the dimensions is not fully used along the edges of the strip estimates are generated only along the middle sections of the strips the overlapping strips are chosen such that the union of the middle sections of the strips gives the whole image the bayes smoothing algorithm presented here is valid for scene random fields consisting of multilevel discrete or continuous random variables

distributed order fractional diffusions on bounded domains fractional derivatives can be used to model time delays in a diffusion process when the order of the fractional derivative is distributed over the unit interval it is useful for modeling a mixture of delay sources this paper provides explicit strong solutions and stochastic analogues for distributed order time fractional diffusion equations on bounded domains with dirichlet boundary conditions

identification of 2 d noncausal gauss markov random fields parameter identification of multidimensional noncausal markov random fields is an important paradigm in multidimensional signal processing and modeling and the solutions to this problem are employed in many areas of image processing an original procedure for estimating the model parameters of discrete index 2 d noncausal gauss markov random fields gmrfs from noisy observations is proposed valid for both finite and infinite lattices and for any kind of boundary conditions starting from a suitable local representation of the gmrf and taking into account the symmetry property of so called field potentials a linear equation set relating the model parameters to the 2 d autocorrelation function known or estimated of the observed field is derived its solution gives the parameter estimates of the gmrf together with the estimate of the possibly unknown variance of the observation noise

fundamental kernel based method for backward space time fractional diffusion problem based on kernel based approximation technique we devise in this paper an efficient and accurate numerical scheme for solving a backward space time fractional diffusion problem bstfdp the kernels used in the approximation are the fundamental solutions of the space time fractional diffusion equation expressed in terms of inverse fourier transform of mittag leffler functions the use of inverse fast fourier transform ifft technique enables an accurate and efficient evaluation of the fundamental solutions and gives a robust numerical algorithm for the solution of the bstfdp since the bstfdp is intrinsic ill posed we apply the standard tikhonov regularization technique to obtain a stable solution to the highly ill conditioned resultant system of linear equations for choosing optimal regularization parameter we combine the regularization technique with the generalized cross validation gcv method for an optimal placement of the source points in the use of fundamental solutions meanwhile the proposed algorithm also speeds up the previous method given in dou and hon 2014 several numerical examples are constructed to verify the accuracy and efficiency of the proposed method

noniterative manipulation of discrete energy based models for image analysis abstract with emphasis on the graph structure of energy based models devoted to image analysis we investigate efficient procedures for sampling and inferring we show that triangulated graphs whom trees are simple instances of always support causal models for which noniterative procedures can be devised to minimize the energy to extract probabilistic descriptions to sample from corresponding prior and posterior distributions or to infer from local marginals the relevance and efficiency of these procedures are illustrated for classification problems

finding non overlapping clusters for generalized inference over graphical models graphical models use graphs to compactly capture stochastic dependencies amongst a collection of random variables inference over graphical models corresponds to finding marginal probability distributions given joint probability distributions in general this is computationally intractable which has led to a quest for finding efficient approximate inference algorithms we propose a framework for generalized inference over graphical models that can be used as a wrapper for improving the estimates of approximate inference algorithms instead of applying an inference algorithm to the original graph we apply the inference algorithm to a block graph defined as a graph in which the nodes are non overlapping clusters of nodes from the original graph this results in marginal estimates of a cluster of nodes which we further marginalize to get the marginal estimates of each node our proposed block graph construction algorithm is simple efficient and motivated by the observation that approximate inference is more accurate on graphs with longer cycles we present extensive numerical simulations that illustrate our block graph framework with a variety of inference algorithms e g those in the libdai software package these simulations show the improvements provided by our framework

on the nonexistence of completely transitive codes completely transitive codes were introduced by p sole 1990 as a special case of binary linear completely regular codes the existence of such codes is closely related to the existence of certain permutation groups the nonexistence of highly transitive permutation groups allows us to prove the nonexistence of completely transitive codes with error correcting capability greater than 4

systems of equations driven by stable processes let zjt j 1 d be independent one dimensional symmetric stable processes of index 0 2 we consider the system of stochastic differential equations

bayesian methods for multiaspect target tracking in image sequences in this paper we introduce new algorithms for automatic tracking of multiaspect targets in cluttered image sequences we depart from the conventional correlation filter kalman filter association approach to target tracking and propose instead a nonlinear bayesian methodology that enables direct tracking from the image sequence incorporating the statistical models for the background clutter target motion and target aspect change proposed algorithms include 1 a batch hidden markov model hmm smoother and a sequential hmm filter for joint multiframe target detection and tracking and 2 two mixed state sequential importance sampling trackers based on the sampling importance resampling sir and the auxiliary particle filtering apf techniques performance studies show that the proposed algorithms outperform the association of a bank of template correlators and a kalman filter in adverse scenarios of low target to clutter ratio and uncertainty in the true target aspect

multidimensional advection and fractional dispersion extension of the fractional diffusion equation to two or three dimensions is not as simple as extension of the second order equation this is revealed by the solutions of the equations unlike the gaussian the most general stable vector cannot be generated with an atomistic measure on the coordinate axes a random combination of maximally skewed stable variables on the unit sphere generates a stable vector that is a general model of a diffusing particle subsets are symmetric stable vectors that have previously appeared in the literature and the well known multidimensional brownian motion a multidimensional fractional differential operator is defined in the process

martingale and weak solutions for a stochastic nonlocal burgers equation on finite intervals abstract this work is about the existence of martingale solutions and weak solutions for a stochastic nonlocal burgers equation on a one dimensional bounded domain the existence of a martingale solution is shown by using a galerkin approximation prokhorov s theorem and skorokhod s embedding theorem the same galerkin approximation also leads to the existence of weak solution for the corresponding deterministic nonlocal burgers equation on a bounded domain

on the foundations of relaxation labeling processes a large class of problems can be formulated in terms of the assignment of labels to objects frequently processes are needed which reduce ambiguity and noise and select the best label among several possible choices relaxation labeling processes are just such a class of algorithms they are based on the parallel use of local constraints between labels this paper develops a theory to characterize the goal of relaxation labeling the theory is founded on a definition of con sistency in labelings extending the notion of constraint satisfaction in certain restricted circumstances an explicit functional exists that can be maximized to guide the search for consistent labelings this functional is used to derive a new relaxation labeling operator when the restrictions are not satisfied the theory relies on variational cal culus it is shown that the problem of finding consistent labelings is equivalent to solving a variational inequality a procedure nearly identical to the relaxation operator derived under restricted circum stances serves in the more general setting further a local convergence result is established for this operator the standard relaxation labeling formulas are shown to approximate our new operator which leads us to conjecture that successful applications of the standard methods are explainable by the theory developed here observations about con vergence and generalizations to higher order compatibility relations are described

markov random field texture models we consider a texture to be a stochastic possibly periodic two dimensional image field a texture model is a mathematical procedure capable of producing and describing a textured image we explore the use of markov random fields as texture models the binomial model where each point in the texture has a binomial distribution with parameter controlled by its neighbors and number of tries equal to the number of gray levels was taken to be the basic model for the analysis a method of generating samples from the binomial model is given followed by a theoretical and practical analysis of the method s convergence examples show how the parameters of the markov random field control the strength and direction of the clustering in the image the power of the binomial model to produce blurry sharp line like and blob like textures is demonstrated natural texture samples were digitized and their parameters were estimated under the markov random field model a hypothesis test was used for an objective assessment of goodness of fit under the markov random field model overall microtextures fit the model well the estimated parameters of the natural textures were used as input to the generation procedure the synthetic microtextures closely resembled their real counterparts while the regular and inhomogeneous textures did not

discrete markov image modeling and inference on the quadtree noncasual markov or energy based models are widely used in early vision applications for the representation of images in high dimensional inverse problems due to their noncausal nature these models generally lead to iterative inference algorithms that are computationally demanding in this paper we consider a special class of nonlinear markov models which allow one to circumvent this drawback these models are defined as discrete markov random fields mrf attached to the nodes of a quadtree the quadtree induces causality properties which enable the design of exact noniterative inference algorithms similar to those used in the context of markov chain models we first introduce an extension of the viterbi algorithm which enables exact maximum a posteriori map estimation on the quadtree two other algorithms related to the mpm criterion and to bouman and shapiro s 1994 sequential map smap estimator are derived on the same hierarchical structure the estimation of the model hyper parameters is also addressed two expectation maximization em type algorithms allowing unsupervised inference with these models are defined the practical relevance of the different models and inference algorithms is investigated in the context of image classification problem on both synthetic and natural images

local search algorithms for political districting abstract electoral district planning plays an important role in a political election especially when a majority voting rule is adopted because it interferes in the translation of votes into seats the practice of gerrymandering can easily take place if the shape of electoral districts is not controlled in this paper we consider the following formulation of the political districting problem given a connected graph territory with n nodes territorial units partition its set of nodes into k classes such that the subgraph induced by each class district is connected and a given vector of functions of the partition is minimized the nonlinearity of such functions and the connectivity constraints make this network optimization problem a very hard one thus the use of local search heuristics is justified experimentation on a sample of medium large real life instances has been carried out in order to compare the performance of four local search metaheuristics i e descent tabu search simulated annealing and old bachelor acceptance our experiments with italian political districting provided strong evidence in favor of the use of automatic procedures actually except for descent all local search algorithms showed a very good performance for this problem in particular in our sample of regions old bachelor acceptance produced the best results in the majority of the cases especially when the objective function was compactness moreover the district maps generated by this heuristic dominate the institutional district plan with respect to all the districting criteria under consideration when properly designed automatic procedures tend to be impartial and yield good districting alternatives moreover they are remarkably fast and thus they allow for the exploration of a large number of scenarios

modeling and estimation of discrete time gaussian reciprocal processes discrete time gaussian reciprocal processes are characterized in terms of a second order two point boundary value nearest neighbor model driven by a locally correlated noise whose correlation is specified by the model dynamics this second order model is the analog for reciprocal processes of the standard first order state space models for markov processes the model is used to obtain a solution to the smoothing problem for reciprocal processes the resulting smoother obeys second order equations whose structure is similar to that of the kalman filter for gauss markov processes it is shown that the smoothing error is itself a reciprocal process

noncausal predictive image codec the paper describes a lossy image codec that uses a noncausal or bilateral prediction model coupled with vector quantization the noncausal prediction model is an alternative to the causal or unilateral model that is commonly used in differential pulse code modulation dpcm and other codecs with a predictive component we show how to obtain a recursive implementation of the noncausal image model without compromising its optimality and how to apply this in coding in much the same way as a causal predictor we report experimental compression results that demonstrate the superiority of using a noncausal model based predictor over using traditional causal predictors the codec is shown to produce high quality compressed images at low bit rates such as 0 375 b pixel this quality is contrasted with the degraded images that are produced at the same bit rates by codecs using causal predictors or standard discrete cosine transform joint photographic experts group based dct jpeg based algorithms

dct dst and gauss markov fields conditions for equivalence the correspondence addresses the intriguing question of which random models are equivalent to the discrete cosine transform dct and discrete sine transform dst common knowledge states that these transforms are asymptotically equivalent to first order gauss causal markov random processes we establish that the dct and the dst are exactly equivalent to homogeneous one dimensional 1 d and two dimensional 2 d gauss noncausal markov random fields defined on finite lattices with appropriate boundary conditions

the viterbi algorithm and markov noise memory this work designs sequence detectors for channels with intersymbol interference isi and correlated and or signal dependent noise we describe three major contributions i first by modeling the noise as a finite order markov process we derive the optimal maximum likelihood sequence detector mlsd and the optimal maximum a posteriori map sequence detector extending to the correlated noise case the viterbi algorithm we show that when the signal dependent noise is conditionally gauss markov the branch metrics in the mlsd are computed from the conditional second order noise statistics we evaluate the branch metrics using a bank of finite impulse response fir filters ii second we characterize the error performance of the mlsd and map sequence detector the error analysis of these detectors is complicated by the correlation asymmetry of the channel noise we derive upper and lower bounds and computationally efficient approximations to these bounds based on the banded structure of the inverses of gauss markov covariance matrices an experimental study shows the tightness of these bounds iii finally we derive several classes of suboptimal sequence detectors and demonstrate how these and others available in the literature relate to the mlsd we compare their error rate performance and their relative computational complexity and show how the structure of the mlsd and the performance evaluation guide us in choosing a best compromise between several types of suboptimal sequence detectors

unsupervised image restoration and edge location using compound gauss markov random fields and the mdl principle discontinuity preserving bayesian image restoration typically involves two markov random fields one representing the image intensities gray levels to be recovered and another one signaling discontinuities edges to be preserved the usual strategy is to perform joint maximum a posterori map estimation of the image and its edges which requires the specification of priors for both fields instead of taking an edge prior we interpret discontinuities in fact their locations as deterministic unknown parameters of the compound gauss markov random field cgmrf which is assumed to model the intensities this strategy should allow inferring the discontinuity locations directly from the image with no further assumptions however an additional problem emerges the number of parameters edges is unknown to deal with it we invoke the minimum description length mdl principle according to mdl the best edge configuration is the one that allows the shortest description of the image and its edges taking the other model parameters noise and cgmrf variances also as unknown we propose a new unsupervised discontinuity preserving image restoration criterion implementation is carried out by a continuation type iterative algorithm which provides estimates of the number of discontinuities their locations the noise variance the original image variance and the original image itself restored image experimental results with real and synthetic images are reported

computation of image spatial entropy using quadrilateral markov random field shannon entropy is a powerful tool in image analysis but its reliable computation from image data faces an inherent dimensionality problem that calls for a low dimensional and closed form model for the pixel value distributions the most promising such models are markovian however the conventional markov random field is hampered by noncausality and its causal versions are also not free of difficulties for example the markov mesh random field has its own limitations due to the strong diagonal dependency in its local neighboring system a new model named quadrilateral markov random field qmrf is introduced in this paper in order to overcome these limitations a property of qmrf with neighboring size of 2 is then used to decompose an image prior into a product of 2 d joint pdfs in which they are estimated using a joint histogram under the homogeneity assumption in addition the paper includes an extension of the introduced method to the computation of image spatial mutual information comparisons on synthesized images as well as two applications with real images are presented to motivate the developments in this paper and demonstrate the advantages in the performance of the introduced method over the existing ones

inference for discrete markov fields the simplest nontrivial case abstract markov fields provide a general context for describing the strength and structure of spatial interactions the gibbs markov equivalence theorem preston 1974 parameterizes markov fields via their neighborhood structures yielding exponential families in canonical form likelihood inference is therefore apparently straightforward the requisite normalizing constants however are obstreperous even when asymptotic characterizations can be obtained substantial location errors arise during implementation moreover markov fields can exhibit phase transitions and long range interactions thereby creating identifiability problems these issues are illustrated in the simplest nontrivial case the classical ising model of ferromagnetism

bayesian reconstruction of signals invariant under a space group symmetry from fourier transform magnitudes a bayesian signal reconstruction problem motivated by x ray crystallography which includes magnitude of fourier transform measurements and a markov random field a priori model was approximately solved by doerschuk see j opt soc amer a vol 8 no 8 p 1222 1232 and p 1207 1221 1991 these ideas are extended to signals that obey a space group symmetry which is a crucial extension for x ray crystallography application performance statistics based on simulated data are presented

estimation of markov random field prior parameters using markov chain monte carlo maximum likelihood developments in statistics now allow maximum likelihood estimators for the parameters of markov random fields mrfs to be constructed we detail the theory required and present an algorithm that is easily implemented and practical in terms of computation time we demonstrate this algorithm on three mrf models the standard potts model an inhomogeneous variation of the potts model and a long range interaction model better adapted to modeling real world images we estimate the parameters from a synthetic and a real image and then resynthesize the models to demonstrate which features of the image have been captured by the model segmentations are computed based on the estimated parameters and conclusions drawn

asymptotics of the partition function of a general markov random field on an infinite rectangular lattice abstract we investigate theoretically and numerically the asymptotics of the partition function of a general markov random field mrf on an infinite rectangular lattice we first propose the general local energy function lef parameterized mrf then we prove that the thermodynamic limit of the free energy of the mrf can be exactly characterized by the perron root of the fundamental transfer matrix of a particular markov additive process map this matrix possesses a special structure and many interesting properties that enable parallel computation of the perron root and may be beneficial for deriving an analytical form of the free energy we also develop another transfer matrix for numerical computation of the desired perron root specifically the former is a site to site transfer matrix on a twisted cylindrical lattice while the latter is the one associated with a row to row transition on a vertical strip numerical results show that our methods exhibit consistent finite size scaling behavior even for small values of the lattice width this study reveals that the fundamental transfer matrix is an alternative direction of research on the analysis of the partition function of general mrfs within the scope of matrix algebra

matrices with banded inverses inversion algorithms and factorization of gauss markov processes the paper considers the inversion of full matrices whose inverses are l banded we derive a nested inversion algorithm for such matrices applied to a tridiagonal matrix the algorithm provides its explicit inverse as an element wise product hadamard product of three matrices when related to gauss markov random processes gmrp this result provides a closed form factored expression for the covariance matrix of a first order gmrp this factored form leads to the interpretation of a first order gmrp as the product of three independent processes a forward independent increments process a backward independent increments process and a variance stationary process we explore the nonuniqueness of the factorization and design it so that the forward and backward factor processes have minimum energy we then consider the issue of approximating general nonstationary gaussian processes by gauss markov processes under two optimality criteria the kullback leibler distance and maximum entropy the problem reduces to approximating general covariances by covariance matrices whose inverses are banded our inversion result is an efficient algorithmic solution to this problem we evaluate the information loss between the original process and its gauss markov approximation

gauss markov random fields cmrf with continuous indices gauss markov random fields gmrfs play an important role in the modeling of physical phenomena the paper addresses the second order characterization and the sample path description of gmrf s when the indexing parameters take values in bounded subsets of spl rfr sup d d spl ges 1 using results of pitt 1994 we give conditions for the covariance of a gmrf to be the green s function of a partial differential operator and conversely for the green s function of an operator to be the covariance of a gmrf we then develop a minimum mean square error representation for the field in terms of a partial differential equation driven by correlated noise the paper establishes for gmrf s on spl rfr sup d second order characterizations that parallel the corresponding results for gmrf s on finite lattices

green s formula for integro differential operators abstract in this report we establish green s formula for an integro differential operator and apply it to describe a class of self adjoint fractional order differential operators a found symmetric fractional order caputo riemann liouville type operator can be considered as a fractional analogue of the classical sturm liouville operator in some sense

telescoping recursive representations and estimation of gauss markov random fields we present telescoping recursive representations for both continuous and discrete indexed noncausal gauss markov random fields our recursions start at the boundary a hypersurface in and telescope inwards for example for images the telescoping representation reduce recursions from to i e to recursions on a single dimension under appropriate conditions the recursions for the random field are linear stochastic differential difference equations driven by white noise for which we derive recursive estimation algorithms that extend standard algorithms like the kalman bucy filter and the rauch tung striebel smoother to noncausal markov random fields

θ schemes for finite element discretization of the space time fractional diffusion equations the numerical solution of a space time fractional diffusion equation used to model the anomalous diffusion is considered spatial discretization is effected using a finite element method whereas the scheme is used for temporal discretization the fully discrete scheme is analyzed for all 0 1 to determine conditional and unconditional stability regimes for the scheme and also to obtain error estimates for the approximate solution the analysis is facilitated by making use of a variational formulation of the equations that is based on a recently developed nonlocal calculus one dimensional numerical examples are provided that illustrate the theoretical stability and convergence results

stationary markov random fields on a finite rectangular lattice this paper provides a complete characterization of stationary markov random fields on a finite rectangular non toroidal lattice in the basic case of a second order neighborhood system equivalently it characterizes stationary markov fields on z sup 2 whose restrictions to finite rectangular subsets are still markovian i e even on the boundaries until now pickard random fields formed the only known class of such fields first we derive a necessary and sufficient condition for markov random fields on a finite lattice to be stationary it is shown that their joint distribution factors in terms of the marginal distribution on a generic 2 spl times 2 cell which must fulfil some consistency constraints second we solve the consistency constraints and provide a complete characterization of such measures in three cases symmetric measures and gaussian measures are shown to necessarily belong to the pickard class whereas binary measures belong either to the pickard class or to a new nontrivial class which is further studied in particular the corresponding fields admit a simple parameterization and may be simulated in a simple although nonunilateral manner

modeling diffusive transport with a fractional derivative without singular kernel abstract in this paper we present an alternative representation of the diffusion equation and the diffusion advection equation using the fractional calculus approach the spatial time derivatives are approximated using the fractional definition recently introduced by caputo and fabrizio in the range 0 2 for the space and time domain respectively in this representation two auxiliary parameters x and t are introduced these parameters related to equation results in a fractal space time geometry provide an entire new family of solutions for the diffusion processes the numerical results showed different behaviors when compared with classical model solutions in the range 0 1 the concentration exhibits the non markovian levy flights and the subdiffusion phenomena when 1 the classical case is recovered when 1 2 the concentration exhibits the markovian levy flights and the superdiffusion phenomena finally when 2 the concentration is anomalous dispersive and we found ballistic diffusion

space time fractional diffusion on bounded domains abstract fractional diffusion equations replace the integer order derivatives in space and time by their fractional order analogues they are used in physics to model anomalous diffusion this paper develops strong solutions of space time fractional diffusion equations on bounded domains as well as probabilistic representations of these solutions which are useful for particle tracking codes

an inverse time dependent source problem for a time space fractional diffusion equation abstract this paper is devoted to identify a time dependent source term in a time space fractional diffusion equation by using the usual initial and boundary data and an additional measurement data at an inner point the existence and uniqueness of a weak solution for the corresponding direct problem with homogeneous dirichlet boundary condition are proved we provide the uniqueness and a stability estimate for the inverse time dependent source problem based on the separation of variables we transform the inverse source problem into a first kind volterra integral equation with the source term as the unknown function and then show the ill posedness of the problem further we use a boundary element method combined with a generalized tikhonov regularization to solve the volterra integral equation of the fist kind the generalized cross validation rule for the choice of regularization parameter is applied to obtain a stable numerical approximation to the time dependent source term numerical experiments for six examples in one dimensional and two dimensional cases show that our proposed method is effective and stable

stochastic classical solutions for space time fractional evolution equations on a bounded domain abstract space time fractional evolution equations are a powerful tool to model diffusion displaying space time heterogeneity we prove existence uniqueness and stochastic representation of classical solutions for an extension of caputo evolution equations featuring time nonlocal initial conditions we discuss the interpretation of the new stochastic representation as part of the proof a new result about inhomogeneous caputo evolution equations is proven

estimation and choice of neighbors in spatial interaction models of images some aspects of statistical inference for a class of spatial interaction models for finite images are presented primarily the simultaneous autoregressive sar models and conditional markov cm models each of these models is characterized by a set of neighbors a set of coefficients and a noise sequence of specified characteristics we are concerned with two problems the estimation of the unknown parameters in both sar and cm models and the choice of an appropriate model from a class of such competing models assuming gaussian distributed variables we discuss maximum likelihood ml estimation methods in general the ml scheme leads to nonlinear optimization problems to avoid excessive computation an iterative scheme is given for sar models which gives approximate ml estimates in the gaussian case and reasonably good estimates in some non gaussian situations as well likewise for cm models an easily computable consistent estimate is given the asymptotic mean squared error mse of this estimate for a four neighbor cm model is shown tn be substantially less than the mse of the popular coding estimate asymptotically consistent decision rules are given for choosing an appropriate sar or cm model the usefulness of the estimation scheme and the decision rule for the choice of neighbors is illustrated by using synthetic patterns synthetic patterns obeying known sar and cm models are generated and the models corresponding to true and several competing neighbor sets are fitted the estimation scheme yields estimates close to the parameters of the true models and the decision rule for the choice of neighbors picks up the true model from the class of competing models

a tabu search heuristic and adaptive memory procedure for political districting abstract in political districting problems the aim is to partition a territory into electoral constituencies subject to some side constraints the most common side constraints include contiguity population equality compactness and socio economic homogeneity we propose a formulation in which the various constraints are integrated into a single multicriteria function we solve the problem by means of a tabu search and adaptive memory heuristic the procedure is illustrated on real data from the city of edmonton

recursive filtering and smoothing for reciprocal gaussian processes pinned boundary case the least square estimation problem for pinned to zero discrete index reciprocal gaussian processes in additive white noise is solved thus completing and extending some previous results available in the literature in particular following the innovations approach a finite set of recursive equations is obtained for the filter and for the three standard classes of smoothers fixed point fixed interval fixed lag recursive expressions for the mean square performance of the proposed estimators are also given

intrinsic ultracontractivity and conditional gauge for symmetric stable processes abstract it is shown in this paper that the conditional gauge theorem holds for symmetric stable processes on bounded c 1 1 domains in r n where 0 n 2 two of the major tools used to prove this conditional gauge theorem are logarithmic sobolev inequality and intrinsic ultracontractivity

a theoretical analysis of monte carlo algorithms for the simulation of gibbs random field images various theoretical and computational issues about the algorithms behavior are addressed the concept of relative entropy is introduced as the primary analytical tool and convergence of the simulation algorithms is discussed in terms of the relative entropy this approach allows a view of the simulation of gibbs random field images as a constrained convex optimization problem and it results in a systematic study of various monte carlo simulation algorithms under a common analytical framework the problems of proper initialization of maximizing the rate of convergence at each iteration and of minimizing the rejection rate are discussed a computational comparison of various monte carlo simulation algorithms is also presented

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

recursive structure of noncausal gauss markov random fields an approach is developed for noncausal gauss markov random fields gmrfs that enables the use of recursive procedures while retaining the noncausality of the field recursive representations are established that are equivalent to the original field this is achieved by first presenting a canonical representation for gmrfs that is based on the inverse of the covariance matrix which is called the potential matrix it is this matrix rather than the field covariance that reflects in a natural way the mrf structure from its properties two equivalent one sided representations are derived each of which is obtained as the successive iterates of a riccati type equation for homogeneous fields these unilateral descriptions are symmetrized versions of each other the study of only one riccati equation being required it is proven that this riccati equation converges at a geometric rate therefore the one sided representations are asymptotically invariant these unilateral representations make it possible to process the fields with well known recursive techniques such as kalman bucy filters and two point smoothers

block matrices with l block banded inverse inversion algorithms block banded matrices generalize banded matrices we study the properties of positive definite full matrices p whose inverses a are l block banded we show that for such matrices the blocks in the l block band of p completely determine p namely all blocks of p outside its l block band are computed from the blocks in the l block band of p we derive fast inversion algorithms for p and its inverse a that when compared to direct inversion are faster by two orders of magnitude of the linear dimension of the constituent blocks we apply these inversion algorithms to successfully develop fast approximations to kalman bucy filters in applications with high dimensional states where the direct inversion of the covariance matrix is computationally unfeasible

on a riesz feller space fractional backward diffusion problem with a nonlinear source abstract in this paper a backward diffusion problem for a space fractional diffusion equation with a nonlinear source in a strip is investigated this problem is obtained from the classical diffusion equation by replacing the second order space derivative with a riesz feller derivative of order 0 2 a nonlinear problem is severely ill posed therefore we propose two new modified regularization solutions to solve it we further show that the approximated problems are well posed and their solutions converge if the original problem has a classical solution in addition the convergence estimates are presented under a priori bounded assumption of the exact solution for estimating the error of the proposed method a numerical example has been implemented