random walks on fractals higher moments the authors investigate by numerical simulation the higher moments for sn the number of distinct sites visited in an n step random walk on fractal structures the sierpinski gasket the sierpinski web and the percolation clusters of the square planar and the simple cubic lattices they find that these moments scale similarly e g the ratio of the standard deviation to sn the average sn is constant in time

random walk on fractals numerical studies in two dimensions monte carlo calculations are used to investigate some statistical properties of random walks on fractal structures two kinds of lattices are used the sierpinski gasket and the infinite percolation cluster in two dimensions among other problems the authors study i the range rn of the walker number of distinct visited sites during n steps average value sn variance sigma n and asymptotic distribution ii renewal theory return to the original site probability of return p0 n mean number of returns nu n the probability distribution of the walker position p n r after n steps is discussed the asymptotic behaviour n 1 of these quantities exhibits power laws with associated exponents the numerical values of these exponents are in good agreement with recent theoretical predictions alexander and orbach 1982 rammal and toulouse 1982

to what class of fractals does the alexander orbach conjecture apply discussion de la decouverte numerique recente d alexander et orbach selon laquelle pour un agregat infini naissant en percolation le rapport d f d w dimension fractale de l agregat dimension fractale d une marche aleatoire dans l agregat est approximativement independant de d pour d 1

accelerating ordered subsets image reconstruction for x ray ct using spatially nonuniform optimization transfer statistical image reconstruction algorithms in x ray computed tomography ct provide improved image quality for reduced dose levels but require substantial computation time iterative algorithms that converge in few iterations and that are amenable to massive parallelization are favorable in multiprocessor implementations the separable quadratic surrogate sqs algorithm is desirable as it is simple and updates all voxels simultaneously however the standard sqs algorithm requires many iterations to converge this paper proposes an extension of the sqs algorithm that leads to spatially nonuniform updates the nonuniform nu sqs encourages larger step sizes for the voxels that are expected to change more between the current and the final image accelerating convergence while the derivation of nu sqs guarantees monotonic descent ordered subsets os algorithms can also accelerate sqs provided suitable subset balance conditions hold these conditions can fail in 3 d helical cone beam ct due to incomplete sampling outside the axial region of interest roi this paper proposes a modified os algorithm that is more stable outside the roi in helical ct we use ct scans to demonstrate that the proposed nu os sqs algorithm handles the helical geometry better than the conventional os methods and converges in less than half the time of ordinary os sqs

a splitting based iterative algorithm for accelerated statistical x ray ct reconstruction statistical image reconstruction using penalized weighted least squares pwls criteria can improve image quality in x ray computed tomography ct however the huge dynamic range of the statistical weights leads to a highly shift variant inverse problem making it difficult to precondition and accelerate existing iterative algorithms that attack the statistical model directly we propose to alleviate the problem by using a variable splitting scheme that separates the shift variant and nearly invariant components of the statistical data model and also decouples the regularization term this leads to an equivalent constrained problem that we tackle using the classical method of multipliers framework with alternating minimization the specific form of our splitting yields an alternating direction method of multipliers admm algorithm with an inner step involving a nearly shift invariant linear system that is suitable for fft based preconditioning using cone type filters the proposed method can efficiently handle a variety of convex regularization criteria including smooth edge preserving regularizers and non smooth sparsity promoting ones based on the l1 norm and total variation numerical experiments with synthetic and real in vivo human data illustrate that cone filter preconditioners accelerate the proposed admm resulting in fast convergence of admm compared to conventional nonlinear conjugate gradient ordered subsets and state of the art mfista split bregman algorithms that are applicable for ct

3d forward and back projection for x ray ct using separable footprints iterative methods for 3d image reconstruction have the potential to improve image quality over conventional filtered back projection fbp in x ray computed tomography ct however the computation burden of 3d cone beam forward and back projectors is one of the greatest challenges facing practical adoption of iterative methods for x ray ct moreover projector accuracy is also important for iterative methods this paper describes two new separable footprint sf projector methods that approximate the voxel footprint functions as 2d separable functions because of the separability of these footprint functions calculating their integrals over a detector cell is greatly simplified and can be implemented efficiently the sf tr projector uses trapezoid functions in the transaxial direction and rectangular functions in the axial direction whereas the sf tt projector uses trapezoid functions in both directions simulations and experiments showed that both sf projector methods are more accurate than the distance driven dd projector which is a current state of the art method in the field the sf tt projector is more accurate than the sf tr projector for rays associated with large cone angles the sf tr projector has similar computation speed with the dd projector and the sf tt projector is about two times slower

growth model for fractal scale free networks generated by a random walk abstract the diversity of networked systems with fractal structures suggests that knowing the underlying mechanism that generates the fractality is necessary for building a model of the development of complex networks in the present paper we propose a growth model of a network generated by a random walk and show that the evolving graph forms a fractal structure with various properties including the scale free property if the graph which provides a space where a random walk occurs by itself is formed by the random walk the proposed model is regulated by two parameters p v and p e which define the probability of either a roundabout path via a new vertex or a shortcut being formed by the random walk respectively the power law exponent describing the vertex degree distribution is determined by the ratio p e p v and is related to an internal factor f i via the relation 1 f i 1 where f i is a parameter that describes the local structure generated by the random walk a sufficiently small p v provides the small world property to the model network the small world property is usually considered to be incompatible with the fractal scaling property open m c l c d c where open m c is the average number of vertices which can be reached from a randomly chosen vertex in at most l c steps however we demonstrate that fractality can be reconciled with the small world property by introducing a size dependent fractal cluster dimension d c

scaling behaviour for excitation trapping on fractals the authors study the decay behaviour of a nearest neighbour random walker which gets trapped at the first encounter of a sink as underlying structures they consider fractals sierpinski gaskets and regular lattices and establish the scaling behaviour of the decay law with respect to x nd2 ln 1 p where n is the number of steps p the sink concentration and d the spectral dimension the decay scales well for x large for small x scaling is reasonable for the gaskets and the linear chain and rather poor for the square lattice

combining ordered subsets and momentum for accelerated x ray ct image reconstruction statistical x ray computed tomography ct reconstruction can improve image quality from reduced dose scans but requires very long computation time ordered subsets os methods have been widely used for research in x ray ct statistical image reconstruction and are used in clinical pet and spect reconstruction in particular os methods based on separable quadratic surrogates os sqs are massively parallelizable and are well suited to modern computing architectures but the number of iterations required for convergence should be reduced for better practical use this paper introduces os sqs momentum algorithms that combine nesterov s momentum techniques with os sqs methods greatly improving convergence speed in early iterations if the number of subsets is too large the os sqs momentum methods can be unstable so we propose diminishing step sizes that stabilize the method while preserving the very fast convergence behavior experiments with simulated and real 3d ct scan data illustrate the performance of the proposed algorithms

block iterative methods for image reconstruction from projections the simultaneous mart algorithm smart and the expectation maximization method for likelihood maximization emml are extended to block iterative versions bi smart and bi emml that converge to a solution in the feasible case for any choice of subsets the bi emml reduces to the ordered subset emml of hudson et al 1992 1994 when their subset balanced property holds

globally convergent image reconstruction for emission tomography using relaxed ordered subsets algorithms we present two types of globally convergent relaxed ordered subsets os algorithms for penalized likelihood image reconstruction in emission tomography modified block sequential regularized expectation maximization bsrem and relaxed os separable paraboloidal surrogates os sps the global convergence proof of the existing bsrem de pierro and yamagishi 2001 required a few a posteriori assumptions by modifying the scaling functions of bsrem we are able to prove the convergence of the modified bsrem under realistic assumptions our modification also makes stepsize selection more convenient in addition we introduce relaxation into the os sps algorithm erdogan and fessler 1999 that otherwise would converge to a limit cycle we prove the global convergence of diagonally scaled incremental gradient methods of which the relaxed os sps is a special case main results of the proofs are from nedic and bertsekas 2001 and correa and lemarechal 1993 simulation results showed that both new algorithms achieve global convergence yet retain the fast initial convergence speed of conventional unrelaxed ordered subsets algorithms

mean first passage time for diffusion on fractal lattices with imposed boundary conditions abstract relation between the mean first passage time and the chemical distance for diffusion on fractal lattices with imposed periodic boundary conditions is analysed heuristically it is argued that this relation is completely controlled by the chemical and the fractal dimensions and two parameters depending on the number of lattice sites according to the power law equations extensive computer simulations have been performed in order to confirm the analytical formulae

rate processes on fractals theory simulations and experiments heterogeneous kinetics are shown to differ drastically from homogeneous kinetics for the elementary reaction a a products we show that the diffusion limited reaction rate is proportional tot h a 2 or to a x whereh 1 d s 2 x 1 2 d s h 2 h 1 andd s is the effective spectral dimension we note that ford d s 1 h 1 2 andx 3 for percolating clustersd s 4 3 h 1 3 andx 5 2 while for dust ds h 1 2 and x 3 scaling arguments supercomputer simulations and experiments give a consistent picture the interplay of energetic and geometric heterogeneity results in fractal like kinetics and is relevant to excitation fusion experiments in porous membranes films and polymeric glasses however in isotopic mixed crystals the geometric fractal nature percolation clusters dominates

a new class of incremental gradient methods for least squares problems the least mean squares lms method for linear least squares problems differs from the steepest descent method in that it processes data blocks one by one with intermediate adjustment of the parameter vector under optimization this mode of operation often leads to faster convergence when far from the eventual limit and to slower sublinear convergence when close to the optimal solution we embed both lms and steepest descent as well as other intermediate methods within a one parameter class of algorithms and we propose a hybrid class of methods that combine the faster early convergence rate of lms with the faster ultimate linear convergence rate of steepest descent these methods are well suited for neural network training problems with large data sets furthermore these methods allow the effective use of scaling based for example on diagonal or other approximations of the hessian matrix

variance of random walks on cayley trees application to the trapping problem the authors investigate the trapping of particles on regularly multifurcating cayley trees by using random walk methods in order to find a good approximation for the survival probability of the walker they use a cumulant expansion and obtain an exact expression for the variance of the range of nearest neighbour random walks the expressions are corroborated to high accuracy by simulation calculations the method may turn out to be useful for more complex problems such as walks on ultrametric spaces

grouped coordinate ascent algorithms for penalized likelihood transmission image reconstruction presents a new class of algorithms for penalized likelihood reconstruction of attenuation maps from low count transmission scans we derive the algorithms by applying to the transmission log likelihood a version of the convexity technique developed by de pierro for emission tomography the new class includes the single coordinate ascent sca algorithm and lange s convex algorithm for transmission tomography as special cases the new grouped coordinate ascent gca algorithms in the class overcome several limitations associated with previous algorithms 1 fewer exponentiations are required than in the transmission maximum likelihood expectation maximization ml em algorithm or in the sca algorithm 2 the algorithms intrinsically accommodate nonnegativity constraints unlike many gradient based methods 3 the algorithms are easily parallelizable unlike the sca algorithm and perhaps line search algorithms we show that the gca algorithms converge faster than the sca algorithm even on conventional workstations an example from a low count positron emission tomography pet transmission scan illustrates the method

critical behavior of the ideal gas bose einstein condensation in the apollonian network we show that the ideal boson gas displays a finite temperature bose einstein condensation transition in the complex apollonian network exhibiting scale free small world and hierarchical properties the single particle tight binding hamiltonian with properly rescaled hopping amplitudes has a fractal like energy spectrum the energy spectrum is analytically demonstrated to be generated by a nonlinear mapping transformation a finite size scaling analysis over several orders of magnitudes of network sizes is shown to provide precise estimates for the exponents characterizing the condensed fraction correlation size and specific heat the critical exponents as well as the power law behavior of the density of states at the bottom of the band are similar to those of the ideal boson gas in lattices with spectral dimension d s 2ln 3 ln 9 5 3 74

maple code for the gamma algorithm for global optimization of uncertain functions in economy and finance problems with uncertainties are ubiquitous in many areas of life and the economy due to a lack of information as regards the economy and in finance problems with uncertainties stock prices marketing problems inflation unemployment are usually formulated by giving bounds on maximum and minimum values of certain parameters i e box constraints in such situations it is necessary to make a choice of better parameters that produce finite intervals of possible values for a given uncertain function at each point of the parameter space the gamma algorithm presents a method for making that choice a variant of the gamma algorithm based on the cubic algorithm is considered for global optimization of uncertain functions with box constraints in r n the set monotonic algorithm contains a block for problems with equality constraints and operates within the unit cube 0 1 n for all problems on this basis a maple code of modular structure is developed for full global optimization of uncertain functions in n variables the code does not create ill conditioned situations graphics are included and the solution set can be visualized in plane projections and sections an example related to minsky s financial instability hypothesis is presented with a graph to illustrate the use of the code

biased and greedy random walks on two dimensional lattices with quenched randomness the greedy ant within a disordered environment the main characteristics of biased greedy random walks bgrws on two dimensional lattices with real valued quenched disorder on the lattice edges are studied here the disorder allows for negative edge weights in previous studies considering the negative weight percolation nwp problem this was shown to change the universality class of the existing static percolation transition in the presented study four different types of bgrws and an algorithm based on the ant colony optimization heuristic were considered regarding the bgrws the precise configurations of the lattice walks constructed during the numerical simulations were influenced by two parameters a disorder parameter ensuremath rho that controls the amount of negative edge weights on the lattice and a bias strength b that governs the drift of the walkers along a certain lattice direction the random walks are greedy in the sense that the local optimal choice of the walker is to preferentially traverse edges with a negative weight associated with a net gain of energy for the walker here the pivotal observable is the probability that after termination a lattice walk exhibits a total negative weight which is here considered as percolating the behavior of this observable as function of ensuremath rho for different bias strengths b is put under scrutiny upon tuning ensuremath rho the probability to find such a feasible lattice walk increases from zero to 1 this is the key feature of the percolation transition in the nwp model here we address the question how well the transition point ensuremath rho mathrm c resulting from numerically exact and static simulations in terms of the nwp model can be resolved using simple dynamic algorithms that have only local information available one of the basic questions in the physics of glassy systems

statistics of the true self avoiding walk in one dimension numerical results of a monte carlo study for the true self avoiding walk in one dimension are presented for any positive value of the strength parameter 0 g infinity the root mean square displacement rn and the range sn of the walk are characterised by two universal exponents nu 2 3 or 0 003 and s 2 3 or 0 01 respectively for negative g self attracting walk both rn and sn exhibit a saturation effect limn to infinity rn rinfinity g and limn to infinity sn sinfinity g with rinfinity g approximately g 1 and sinfinity g approximately g 1 at g or approximately 0 a simple scaling analysis in n and g is proposed and found to be consistent with the monte carlo results

a generating function for the second moment of the distinct number of sites visited by an n step lattice random walk we derive a generating function for the second moment of the distinct number of sites visited by an n step lattice random walk the formalism allows us to find asymptotic forms for the second moment as previously given in the mathematical literature using much more complicated techniques the general technique can in principle be utilized in a derivation of higher moments

accelerated image reconstruction using ordered subsets of projection data the authors define ordered subset processing for standard algorithms such as expectation maximization em for image restoration from projections ordered subsets methods group projection data into an ordered sequence of subsets or blocks an iteration of ordered subsets em is defined as a single pass through all the subsets in each subset using the current estimate to initialize application of em with that data subset this approach is similar in concept to block kaczmarz methods introduced by eggermont et al 1981 for iterative reconstruction simultaneous iterative reconstruction sirt and multiplicative algebraic reconstruction mart techniques are well known special cases ordered subsets em os em provides a restoration imposing a natural positivity condition and with close links to the em algorithm os em is applicable in both single photon spect and positron emission tomography pet in simulation studies in spect the os em algorithm provides an order of magnitude acceleration over em with restoration quality maintained

random walks on ultrametric spaces mean and variance of the range the authors study thermally activated hopping processes on regularly multifurcating ultrametric spaces ums and investigate the first two moments of the range rn where n is the number of steps paralleling the recent findings for cayley trees the knowledge of exact generating functions for these moments permits to draw conclusions about their behaviour for large n for the first moment the mean number of distinct sites visited sn they derive analytically the cross over from a linear behaviour in the transient regime gamma 1 to a sublinear increase sn approximately ngamma in the recurrent regime gamma 1 here as usual for ums the parameter gamma depends linearly on the temperature the second moment of rn the variance vn displays different behaviour corresponding to strongly transient gamma 2 not strongly transient 1 gamma 2 and recurrent gamma 1 random walks all results are verified by comparison to computer data obtained through simulations and through numerical series expansions

statistics of the one dimensional riemann walk the riemann walk is the lattice version of the levy flight for the one dimensional riemann walk of levy exponent 0 2 we study the statistics of the support i e set of visited sites after t steps we consider a wide class of support related observables m t including the number s t of visited sites and the number i t of sequences of adjacent visited sites for t we obtain the asymptotic power laws for the averages variances and correlations of these observables logarithmic correction factors appear for 2 3 and 1 bulk and surface observables have different power laws for 1 2 fluctuations are shown to be universal for 2 3 2 this means that in the limit t the deviations from average m t m t m 0304 t 0304 are fully described either by a single m independent stochastic process when 2 3 1 or by two such processes one for the bulk and one for the surface observables when 1 2

incremental subgradient methods for nondifferentiable optimization we consider a class of subgradient methods for minimizing a convex function that consists of the sum of a large number of component functions this type of minimization arises in a dual context from lagrangian relaxation of the coupling constraints of large scale separable problems the idea is to perform the subgradient iteration incrementally by sequentially taking steps along the subgradients of the component functions with intermediate adjustment of the variables after processing each component function this incremental approach has been very successful in solving large differentiable least squares problems such as those arising in the training of neural networks and it has resulted in a much better practical rate of convergence than the steepest descent method in this paper we establish the convergence properties of a number of variants of incremental subgradient methods including some that are stochastic based on the analysis and computational experiments the methods appear very promising and effective for important classes of large problems a particularly interesting discovery is that by randomizing the order of selection of component functions for iteration the convergence rate is substantially improved

region of interest reconstruction from truncated data in circular cone beam ct the circular scanning trajectory is one of the most widely adopted data acquisition configurations in computed tomography ct the feldkamp davis kress fdk algorithm and its various modifications have been developed for reconstructing approximately three dimensional images from circular cone beam data when data contain transverse truncations however these algorithms may reconstruct images with significant truncation artifacts it is of practical significance to develop algorithms that can reconstruct region of interest roi images from truncated circular cone beam data that are free of truncation artifacts and that have an accuracy comparable to that obtained from nontruncated cone beam data in this work we have investigated and developed a backprojection filtration bpf based algorithm for roi image reconstruction from circular cone beam data containing transverse truncations furthermore we have developed a weighted bpf algorithm to exploit redundant information in data for improving image quality in an effort to validate and evaluate the proposed bpf algorithms for circular cone beam ct we have performed numerical studies by using both computer simulation data and experimental data acquired with a radiotherapy cone beam ct system quantitative results in these studies demonstrate that the proposed bpf algorithms for circular cone beam ct can reconstruct roi images free of truncation artifacts

on the convergence of the lms algorithm with adaptive learning rate for linear feedforward networks we consider the problem of training a linear feedforward neural network by using a gradient descent like lms learning algorithm the objective is to find a weight matrix for the network by repeatedly presenting to it a finite set of examples so that the sum of the squares of the errors is minimized kohonen showed that with a small but fixed learning rate or stepsize some subsequences of the weight matrices generated by the algorithm will converge to certain matrices close to the optimal weight matrix in this paper we show that by dynamically decreasing the learning rate during each training cycle the sequence of matrices generated by the algorithm will converge to the optimal weight matrix we also show that for any given 0 the lms algorithm with decreasing learning rates will generate an optimal weight matrix i e a matrix of distance at most away from the optimal matrix after o 1 training cycles this is in contrast to 1 log 1 training cycles needed to generate an optima

fast model based x ray ct reconstruction using spatially nonhomogeneous icd optimization recent applications of model based iterative reconstruction mbir algorithms to multislice helical ct reconstructions have shown that mbir can greatly improve image quality by increasing resolution as well as reducing noise and some artifacts however high computational cost and long reconstruction times remain as a barrier to the use of mbir in practical applications among the various iterative methods that have been studied for mbir iterative coordinate descent icd has been found to have relatively low overall computational requirements due to its fast convergence this paper presents a fast model based iterative reconstruction algorithm using spatially nonhomogeneous icd nh icd optimization the nh icd algorithm speeds up convergence by focusing computation where it is most needed the nh icd algorithm has a mechanism that adaptively selects voxels for update first a voxel selection criterion vsc determines the voxels in greatest need of update then a voxel selection algorithm vsa selects the order of successive voxel updates based upon the need for repeated updates of some locations while retaining characteristics for global convergence in order to speed up each voxel update we also propose a fast 1 d optimization algorithm that uses a quadratic substitute function to upper bound the local 1 d objective function so that a closed form solution can be obtained rather than using a computationally expensive line search algorithm we examine the performance of the proposed algorithm using several clinical data sets of various anatomy the experimental results show that the proposed method accelerates the reconstructions by roughly a factor of three on average for typical 3 d multislice geometries

an expanded theoretical treatment of iteration dependent majorize minimize algorithms the majorize minimize mm optimization technique has received considerable attention in signal and image processing applications as well as in statistics literature at each iteration of an mm algorithm one constructs a tangent majorant function that majorizes the given cost function and is equal to it at the current iterate the next iterate is obtained by minimizing this tangent majorant function resulting in a sequence of iterates that reduces the cost function monotonically a well known special case of mm methods are expectation maximization algorithms in this paper we expand on previous analyses of mm due to fessler and hero that allowed the tangent majorants to be constructed in iteration dependent ways also this paper overcomes an error in one of those earlier analyses there are three main aspects in which our analysis builds upon previous work first our treatment relaxes many assumptions related to the structure of the cost function feasible set and tangent majorants for example the cost function can be nonconvex and the feasible set for the problem can be any convex set second we propose convergence conditions based on upper curvature bounds that can be easier to verify than more standard continuity conditions furthermore these conditions allow for considerable design freedom in the iteration dependent behavior of the algorithm finally we give an original characterization of the local region of convergence of mm algorithms based on connected e g convex tangent majorants for such algorithms cost function minimizers will locally attract the iterates over larger neighborhoods than typically is guaranteed with other methods this expanded treatment widens the scope of the mm algorithm designs that can be considered for signal and image processing applications allows us to verify the convergent behavior of previously published algorithms and gives a fuller understanding overall of how these algorithms behave

wiener sausage volume moments the statistical characteristics of a spatial region visited by a spherical brownian particle during timet wiener sausage are investigated the expectation value and dispersion of this quantity are obtained for a space of arbitrary dimension in the one dimensional case the distribution of probability density and the moments of any order are determined for this quantity

variance of the range of a random walk tn the expectation of the square of the number of distinct sites occupied by a random walk in steps 1 throughn is obtained from its relation to the dual first occupancy probabilityfij x x and the latter quantity is obtained from a recursion with the first occupancy probabilityf k x the variancevn of the number of distinct sites occupied is calculated directly from tn the procedure is illustrated by the calculation ofvn 4096 n and the derivation of asymptotic expansions forvn for a particular random walk in dimensions 1 through 3

image recovery using partitioned separable paraboloidal surrogate coordinate ascent algorithms iterative coordinate ascent algorithms have been shown to be useful for image recovery but are poorly suited to parallel computing due to their sequential nature this paper presents a new fast converging parallelizable algorithm for image recovery that can be applied to a very broad class of objective functions this method is based on paraboloidal surrogate functions and a concavity technique the paraboloidal surrogates simplify the optimization problem the idea of the concavity technique is to partition pixels into subsets that can be updated in parallel to reduce the computation time for fast convergence pixels within each subset are updated sequentially using a coordinate ascent algorithm the proposed algorithm is guaranteed to monotonically increase the objective function and intrinsically accommodates nonnegativity constraints a global convergence proof is summarized simulation results show that the proposed algorithm requires less elapsed time for convergence than iterative coordinate ascent algorithms with four parallel processors the proposed algorithm yields a speedup factor of 3 77 relative to single processor coordinate ascent algorithms for a three dimensional 3 d confocal image restoration problem

monotonic algorithms for transmission tomography we present a framework for designing fast and monotonic algorithms for transmission tomography penalized likelihood image reconstruction the new algorithms are based on paraboloidal surrogate functions for the log likelihood due to the form of the log likelihood function it is possible to find low curvature surrogate functions that guarantee monotonicity unlike previous methods the proposed surrogate functions lead to monotonic algorithms even for the nonconvex log likelihood that arises due to background events such as scatter and random coincidences the gradient and the curvature of the likelihood terms are evaluated only once per iteration since the problem is simplified at each iteration the cpu time is less than that of current algorithms which directly minimize the objective yet the convergence rate is comparable the simplicity monotonicity and speed of the new algorithms are quite attractive the convergence rates of the algorithms are demonstrated using real and simulated pet transmission scans

random walks on percolating clusters with energetic disorder single random walker simulations on energetically disordered percolation clusters in two dimensions are presented exponential gaussian and uniform site energy distributions are investigated the superposition of spatial and energetic disorder leads to reduced random walk ranges with decreasing temperature an analogue subordination rule is derived random walk on an energetically disordered fractal is equivalent to that on a geometrical fractal with a lower spectral dimension this rule is strictly followed for the exponential distribution but only approximately for the gaussian and uniform distributions the last two distributions and especially the uniform one show a crossover behaviour analogous to that of random walks on percolation clusters away from criticality

metric properties of fractal lattices the authors study the connectivity dimension d of fractal lattices viewed as networks graphs of sites and constant length bond two examples are investigated in detail the 2d sierpinski gasket and 2d infinite percolation clusters on square and triangular lattices in the first case d is shown to coincide with the fractal dimension whereas d 1 72 or 0 02 appears as a universal exponent for percolation clusters in two dimensions considered as an intrinsic parameter the connectivity dimension d is compared with other intrinsic and extrinsic characteristic parameters of fractal lattices in particular the authors argue that d or d or d holds on fractal lattices in general d fractal dimension d spectral dimension

globally convergent algorithms for maximum a posteriori transmission tomography this paper reviews and compares three maximum likelihood algorithms for transmission tomography one of these algorithms is the em algorithm one is based on a convexity argument devised by de pierro see ieee trans med imaging vol 12 p 328 333 1993 in the context of emission tomography and one is an ad hoc gradient algorithm the algorithms enjoy desirable local and global convergence properties and combine gracefully with bayesian smoothing priors preliminary numerical testing of the algorithms on simulated data suggest that the convex algorithm and the ad hoc gradient algorithm are computationally superior to the em algorithm this superiority stems from the larger number of exponentiations required by the em algorithm the convex and gradient algorithms are well adapted to parallel computing

global first passage times of fractal lattices the global first passage time density of a network is the probability that a random walker released at a random site arrives at an absorbing trap at time t we find simple expressions for the mean global first passage time for five fractals the d dimensional sierpinski gasket t fractal hierarchical percolation model mandelbrot given curve and a deterministic tree we also find an exact expression for the second moment and show that the variance of the first passage time var t scales with the number of nodes within the fractal n such that var t similar to n 4 d where d is the spectral dimension

statistical properties of the set of sites visited by the two dimensional random walk we study the support i e the set of visited sites of a t step random walk on a two dimensional square lattice in the large t limit a broad class of global properties m t of the support is considered including for example the number s t of its sites the length of its boundary the number of islands of unvisited sites that it encloses the number of such islands of given shape size and orientation and the number of occurrences in space of specific local patterns of visited and unvisited sites on a finite lattice we determine the scaling functions that describe the averages on appropriate lattice size dependent time scales on an infinite lattice we first observe that the all increase with t as where k is an m dependent positive integer we then consider the class of random processes constituted by the fluctuations around average we show that to leading order as t gets large these fluctuations are all proportional to a single universal random process normalized to for the probability law of tends to that of varadhan s renormalized local time of self intersections an implication is that in the long time limit all are proportional to

convergent incremental optimization transfer algorithms application to tomography no convergent ordered subsets os type image reconstruction algorithms for transmission tomography have been proposed to date in contrast in emission tomography there are two known families of convergent os algorithms methods that use relaxation parameters and methods based on the incremental expectation maximization em approach this paper generalizes the incremental em approach by introducing a general framework incremental optimization transfer the proposed algorithms accelerate convergence speeds and ensure global convergence without requiring relaxation parameters the general optimization transfer framework allows the use of a very broad family of surrogate functions enabling the development of new algorithms this paper provides the first convergent os type algorithm for nonconcave penalized likelihood pl transmission image reconstruction by using separable paraboloidal surrogates sps which yield closed form maximization steps we found it is very effective to achieve fast convergence rates by starting with an os algorithm with a large number of subsets and switching to the new transmission incremental optimization transfer triot algorithm results show that triot is faster in increasing the pl objective than nonincremental ordinary sps and even os sps yet is convergent

quantum field theory models on fractal spacetime i introduction and overview deux types de modeles de champs scalaires sont consideres l un a son propagateur donne par un temps propre ou une representation chaleur noyau l autre a un propagateur hierarchique la version du reseau fractal du propagateur a temps propre est une reflexion positive

on the relation between the isra and the em algorithm for positron emission tomography the image space reconstruction algorithm isra was proposed as a modification of the expectation maximization em algorithm based on physical considerations for application in volume emission computered tomography as a consequence of this modification isra searches for least squares solutions instead of maximizing poisson likelihoods as the em algorithm it is shown that both algorithms may be obtained from a common mathematical framework this fact is used to extend isra for penalized likelihood estimates

diffusion on percolation clusters at criticality the concept of fractal dimensionality is used to study the problem of diffusion on percolation clusters the authors find from monte carlo simulations that the fractal dimensionality of a random walk on a critical percolation cluster in three dimensional space is d 3 3 or 0 1 where the size of the cluster is restricted to be larger than the span of the walk and is d 3 9 or 0 1 for a walk on clusters not subject to this restriction for two dimensional space they find d approximately d approximately 2 7 or p0 1 the exponent d and d is related to the scaling of the average length r of n steps via rd varies as n the fracton dimensionality which is related to the density of states was found to be 1 26 or 0 1 these results are in good agreement with the predictions of alexander and orbach 1982

fractal chemical kinetics reacting random walkers computer simulations on binary reactions of random walkers a a a on fractal spaces bear out a recent conjecture p po i oct s where p is the instantaneous walker density and p0 the initial one andf d j2 where d s is the spectral dimension for the sierpinski gaskets d 2 2f 1 38 d 1 365 d 3 2f 1 56 d s 1 547 biased initial random distributions are compared to unbiased ones for site percolation d 2 p 0 60 2f 1 35 d 1 35 d 3 p 0 32 2f 1 37 d s 1 4 fractal to euclidean crossovers are also observed for energetically disordered lattices the effective 2f from reacting walkers and d s from single walkers are in good agreement in both two and three dimensions

single random walker on disordered lattices random walks on square lattice percolating clusters were followed for up to 2 10 steps the mean number of distinct sites visited sn gives a spectral dimension of d s 1 30 5 0 03 consistent with superuniversality d s 4 3 but closer to the alternative ds 182 139 based on the low dimensionality correction simulations are also given for walkers on an energetically disordered lattice with a jump probability that depends on the local energy mismatch and the temperature an apparent fractal behavior is observed for a low enough reduced temperature above this temperature the walker exhibits a crossover from fractal to euclidean behavior walks on two and three dimensional lattices are similar except that those in three dimensions are more efficient

globally convergent edge preserving regularized reconstruction an application to limited angle tomography we introduce a generalization of a deterministic relaxation algorithm for edge preserving regularization in linear inverse problems this algorithm transforms the original possibly nonconvex optimization problem into a sequence of quadratic optimization problems and has been shown to converge under certain conditions when the original cost functional being minimized is strictly convex we prove that our more general algorithm is globally convergent i e converges to a local minimum from any initialization under less restrictive conditions even when the original cost functional is nonconvex we apply this algorithm to tomographic reconstruction from limited angle data by formulating the problem as one of regularized least squares optimization the results demonstrate that the constraint of piecewise smoothness applied through the use of edge preserving regularization can provide excellent limited angle tomographic reconstructions two edge preserving regularizers one convex the other nonconvex are used in numerous simulations to demonstrate the effectiveness of the algorithm under various limited angle scenarios and to explore how factors such as the choice of error norm angular sampling rate and amount of noise affect the reconstruction quality and algorithm performance these simulation results show that for this application the nonconvex regularizer produces consistently superior results

a local update strategy for iterative reconstruction from projections a method for bayesian reconstruction which relies on updates of single pixel values rather than the entire image at each iteration is presented the technique is similar to gauss seidel gs iteration for the solution of differential equations on finite grids the computational cost per iteration of the gs approach is found to be approximately equal to that of gradient methods for continuously valued images gs is found to have significantly better convergence at modes representing high spatial frequencies in addition gs is well suited to segmentation when the image is constrained to be discretely valued it is shown that bayesian segmentation using gs iteration produces useful estimates at much lower signal to noise ratios than required for continuously valued reconstruction the convergence properties of gradient ascent and gs for reconstruction from integral projections are analyzed and simulations of both maximum likelihood and maximum a posteriori cases are included

on the span of brownian motion in a field in one dimension abstract we study the effect of a field on the span of a particle diffusing on a line i e the length covered by a brownian particle which moves on a line for time t in the presence of a constant field this is the one dimensional analog of the wiener sausage volume exact expressions are found for the probability density for the span together with the first two moments our results indicate that at very short times the dominant effect is diffusion while at very long times the field plays the dominant role

exact fractals with adjustable fractal and fracton dimensionalities a family of exact fractals is presented by adjusting two external parameters a wide range of fractal and fracton dimensionalities can be achieved this includes the case of fracton dimensionality of 2 which is critical for diffusion

global optimization over general compact sets by the beta algorithm a maple code a variant of the beta algorithm based on the cubic algorithm 1 2 is presented for global optimization of holder continuous functions over general compact sets the set monotonic algorithm contains a block for problems with equality constraints and operates within the unit cube 0 1 n universal for all problems on this basis a maple code of modular structure is developed for full global optimization of functions of n variables the code does not create ill conditioned situations graphics are included and the solution set can be visualized in plane projections and sections the code is ready for engineering applications results of numerical experiments are presented with graphs to illustrate the use of the code

spatial resolution properties of penalized likelihood image reconstruction space invariant tomographs this paper examines the spatial resolution properties of penalized likelihood image reconstruction methods by analyzing the local impulse response the analysis shows that standard regularization penalties induce space variant local impulse response functions even for space invariant tomographic systems paradoxically for emission image reconstruction the local resolution is generally poorest in high count regions we show that the linearized local impulse response induced by quadratic roughness penalties depends on the object only through its projections this analysis leads naturally to a modified regularization penalty that yields reconstructed images with nearly uniform resolution the modified penalty also provides a very practical method for choosing the regularization parameter to obtain a specified resolution in images reconstructed by penalized likelihood methods

true self avoiding walks on fractal lattices above the upper marginal dimension we study by monte carlo simulations the critical behaviour and the cross over scaling of the true self avoiding walks tsaw on fractal lattices above the upper marginal dimension we estimate the flory exponent nu which characterizes the rms end to end distances of saw on a percolation cluster at percolation thresholds both in three and four dimensions and on a dla cluster in three dimensions results were in good agreement with the predictions of the known flory formulae we also discuss the fractal to euclidean and the rw to tsaw cross over scaling monte carlo data appear to collapse in the scaling regions for both cases however we found that the scaling function for the latter is different from that on the regular lattices

true self avoiding walks on a percolation cluster the authors investigate by monte carlo simulations the critical behaviour of true self avoiding walks tsaws on a percolation cluster performed very close to the percolation threshold specifically they generate tsaws on a site percolated incipient infinite cluster for various values of the self avoidance parameter g 0 they found that such walks exhibit critical behaviour different from that of ordinary self avoiding walks and also from that of random walks of no constraint the flory exponent obtained was v 0432 or 0 005 for all g 0 which agrees well with the flory type formula suggested by rammal 1984

random walk statistics on fractal structures we consider some statistical properties of simple random walks on fractal structures viewed as networks of sites and bonds range renewal theory mean first passage time etc asymptotic behaviors are shown to be controlled by the fractal d and spectral d dimensionalities of the considered structure a simple decimation procedure giving the value of d is outlined and illustrated in the case of the sierpinski gaskets recent results for the trapping problem the self avoiding walk and the true self avoiding walk are briefly reviewed new numerical results for diffusion on percolation clusters are also presented

diffusion and fracton dimensionality on fractals and on percolation clusters the authors present simulations of diffusion on an exact fractal and on percolation clusters at criticality for two and three dimensions the results for the fractal support the rammal and toulouse proposition 1983 that ds n dn varies as b n s n the results for percolation are in excellent agreement with the alexander and orbach conjecture 1982 that the fracton dimensionality d 4 3

a modified expectation maximization algorithm for penalized likelihood estimation in emission tomography the maximum likelihood ml expectation maximization em approach in emission tomography has been very popular in medical imaging for several years in spite of this no satisfactory convergent modifications have been proposed for the regularized approach here a modification of the em algorithm is presented the new method is a natural extension of the em for maximizing likelihood with concave priors convergence proofs are given

multi material decomposition using statistical image reconstruction for spectral ct spectral computed tomography ct provides information on material characterization and quantification because of its ability to separate different basis materials dual energy de ct provides two sets of measurements at two different source energies in principle two materials can be accurately decomposed from dect measurements however many clinical and industrial applications require three or more material images for triple material decomposition a third constraint such as volume conservation mass conservation or both is required to solve three sets of unknowns from two sets of measurements the recently proposed flexible image domain id multi material decomposition method assumes each pixel contains at most three materials out of several possible materials and decomposes a mixture pixel by pixel we propose a penalized likelihood pl method with edge preserving regularizers for each material to reconstruct multi material images using a similar constraint from sinogram data we develop an optimization transfer method with a series of pixel wise separable quadratic surrogate pwsqs functions to monotonically decrease the complicated pl cost function the pwsqs algorithm separates pixels to allow simultaneous update of all pixels but keeps the basis materials coupled to allow faster convergence rate than our previous proposed material and pixel wise sqs algorithms comparing with the id method using 2 d fan beam simulations the pl method greatly reduced noise streak and cross talk artifacts in the reconstructed basis component images and achieved much smaller root mean square errors

true self avoiding walk on fractals the authors investigate the true self avoiding walk on fractal spaces the number sn of sites visited during an n step walk and the root mean square displacement rn are calculated via extensive monte carlo simulations on the 2d sierpinski gasket for all positive values of the repulsion parameter g they found the same exponents s 0 815 or 0 001 and nu 0 510 or 0 005 respectively both rn and sn are shown to exhibit a simple scaling behaviour in n and g the authors compare the results with a simple flory like prediction for the exponent nu

global optimization in problems with uncertainties the gamma algorithm abstract problems with uncertainties can be viewed and formalized making use of multifunctions or general set valued functions a new concept of global optimality is proposed which allows us to solve global optimization problems with uncertainties in natural setting without imposing artificial constraints on uncertainties nor introducing a kind of partial ordering in order to apply conventional optimality concepts and optimization techniques nor considering solution in probability with the new concept deterministic optimization requires two optimization procedures a study of the subject is presented with many illustrative examples then a monotonic iterative algorithm is developed which renders approximate solutions with precision specified in advance a notion of piecewise continuous function of several variables is proposed and the method is then generalized for uncertain functions defined by a closed set valued function with piecewise continuous upper and lower boundaries the max2 f reduction precision and decomposition lemmas are proved to facilitate practical applications deferred deletions of sets with discontinuities are introduced and convergence theorem is proved for the modified algorithm

a field theoretic approach to the wiener sausage the wiener sausage the volume traced out by a sphere attached to a brownian particle is a classical problem in statistics and mathematical physics initially motivated by a range of field theoretic technical questions we present a single loop renormalised perturbation theory of a stochastic process closely related to the wiener sausage which however proves to be exact for the exponents and some amplitudes the field theoretic approach is particularly elegant and very enjoyable to see at work on such a classic problem while we recover a number of known classical results the field theoretic techniques deployed provide a particularly versatile framework which allows easy calculation with different boundary conditions even of higher momenta and more complicated correlation functions at the same time we provide a highly instructive non trivial example for some of the technical particularities of the field theoretic description of stochastic processes such as excluded volume lack of translational invariance and immobile particles the aim of the present work is not to improve upon the well established results for the wiener sausage but to provide a field theoretic approach to it in order to gain a better understanding of the field theoretic obstacles to overcome