spectral formulation and wkb approximation for rare event statistics in reaction systems we develop a spectral formulation and a stationary wkb approximation for calculating the probabilities of rare events large deviations from the mean in systems of reacting particles with infinite range interaction describable by a master equation we compare the stationary wkb approximation to a recent time dependent semiclassical approximation developed for the same class of problems by elgart and kamenev phys rev e 70 41106 2004 as a benchmark we use an exactly solvable problem of the binary annihilation reaction 2a 0

generalized bundle methods we study a class of generalized bundle methods for which the stabilizing term can be any closed convex function satisfying certain properties this setting covers several algorithms from the literature that have been so far regarded as distinct under a different hypothesis on the stabilizing term and or the function to be minimized we prove finite termination asymptotic convergence and finite convergence to an optimal point with or without limits on the number of serious steps and or requiring the proximal parameter to go to infinity the convergence proofs leave a high degree of freedom in the crucial implementative features of the algorithm i e the management of the bundle of subgradients beta strategy and of the proximal parameter t strategy we extensively exploit a dual view of bundle methods which are shown to be a dual ascent approach to one nonlinear problem in an appropriate dual space where nonlinear subproblems are approximately solved at each step with an inner linearization approach this allows us to precisely characterize the changes in the subproblems during the serious steps since the dual problem is not tied to the local concept of varepsilon subdifferential for some of the proofs a generalization of inf compactness called compactness is required this concept is related to that of asymptotically well behaved functions

a redistributed proximal bundle method for nonconvex optimization proximal bundle methods have been shown to be highly successful optimization methods for unconstrained convex problems with discontinuous first derivatives this naturally leads to the question of whether proximal variants of bundle methods can be extended to a nonconvex setting this work proposes an approach based on generating cutting planes models not of the objective function as most bundle methods do but of a local convexification of the objective function the corresponding convexification parameter is calculated on the fly in such a way that the algorithm can inform the user as to what proximal parameters are sufficiently large that the objective function is likely to have well defined proximal points this novel approach shown to be sound from both the objective function and subdifferential modelling perspectives opens the way to create workable nonconvex algorithms based on nonconvex mathcal vu theory both theoretical convergence analysis and some encouraging preliminary numerical experience are provided

convex proximal bundle methods in depth a unified analysis for inexact oracles the last few years have seen the advent of a new generation of bundle methods capable to handle inexact oracles polluted by noise proving convergence of a bundle method is never simple and coping with inexact oracles substantially increases the technicalities besides several variants exist to deal with noise each one needing an ad hoc proof to show convergence we state a synthetic convergence theory in which we highlight the main arguments and specify which assumption is used to establish each intermediate result the framework is comprehensive and generalizes in various ways a number of algorithms proposed in the literature based on the ingredients of our synthetic theory we consider various bundle methods adapted to oracles for which high accuracy is possible yet it is preferable not to make exact calculations often because they are too time consuming

a doubly stabilized bundle method for nonsmooth convex optimization we propose a bundle method for minimizing nonsmooth convex functions that combines both the level and the proximal stabilizations most bundle algorithms use a cutting plane model of the objective function to formulate a subproblem whose solution gives the next iterate proximal bundle methods employ the model in the objective function of the subproblem while level methods put the model in the subproblem s constraints the proposed algorithm defines new iterates by solving a subproblem that employs the model in both the objective function and in the constraints one advantage when compared to the proximal approach is that the level set constraint provides a certain lagrange multiplier which is used to update the proximal parameter in a novel manner we also show that in the case of inexact function and subgradient evaluations no additional procedure needs to be performed by our variant to deal with inexactness as opposed to the proximal bundle methods that require special modifications numerical experiments on almost 1 000 instances of different types of problems are presented our experiments show that the doubly stabilized bundle method inherits useful features of the level and the proximal versions and compares favorably to both of them

snakes in movement in this paper we propose a geometric partial differential equation pde for tracking one or several moving objects from a sequence of images which is based on a geometric model for active contours the active contour approach permits us to simultaneously handle both aspects finding the boundaries and tracking them we also describe a numerical scheme to solve the geometric equation and we present some numerical experiments

a method for globally minimizing concave functions over convex sets a method is described for globally minimizing concave functions over convex sets whose defining constraints may be nonlinear the algorithm generates linear programs whose solutions minimize the convex envelope of the original function over successively tighter polytopes enclosing the feasible region the algorithm does not involve cuts of the feasible region requires only simplex pivot operations and univariate search computations to be performed allows the objective function to be lower semicontinuous and nonseparable and is guaranteed to converge to the global solution computational aspects of the algorithm are discussed

demographic stochasticity and extinction in populations with allee effect we study simple stochastic scenarios based on birth and death markovian processes that describe populations with the allee effect to account for the role of demographic stochasticity in the mean field deterministic limit we recover well known deterministic evolution equations widely employed in population ecology the mean time to extinction is in general obtained by the wentzel kramers brillouin wkb approximation for populations with the strong and weak allee effects an exact solution for the mean time to extinction can be found via a recursive equation for special cases of the stochastic dynamics we study the conditions for the validity of the wkb solution and analyze the boundary between the weak and strong allee effect by comparing exact solutions with numerical simulations

a fast level set method for propagating interfaces a method is introduced to decrease the computational labor of the standard level set method for propagating interfaces the fast approach uses only points close to the curve at every time step we describe this new algorithm and compare its efficiency and accuracy with the standard level set approach

minimizing piecewise concave functions over polyhedra we introduce an iterative method for solving linearly constrained optimization problems whose nonsmooth nonconvex objective function is defined as the pointwise maximum of finitely many concave functions such problems often arise from reformulations of certain constraint structures e g binary constraints finite max min constraints in diverse areas of optimization we state a local optimization strategy which exploits piecewise concavity of the objective function giving rise to a linearized model corrected by a proximity term in addition we introduce an approximate line search strategy based on a curvilinear model which similarly to bundle methods can return either a satisfactory descent or a null step declaration termination at a point satisfying an approximate stationarity condition is proved we embed the local minimization algorithm into a variable neighborhood search scheme and a coordinate direction search heuristic whose aim is to improve the current estimate of the global minimizer

estimation of object motion parameters from noisy images an approach is presented for the estimation of object motion parameters based on a sequence of noisy images the problem considered is that of a rigid body undergoing unknown rotational and translational motion the measurement data consists of a sequence of noisy image coordinates of two or more object correspondence points by modeling the object dynamics as a function of time estimates of the model parameters including motion parameters can be extracted from the data using recursive and or batch techniques this permits a desired degree of smoothing to be achieved through the use of an arbitrarily large number of images some assumptions regarding object structure are presently made results are presented for a recursive estimation procedure the case considered here is that of a sequence of one dimensional images of a two dimensional object thus the object moves in one transverse dimension and in depth preserving the fundamental ambiguity of the central projection image model loss of depth information an iterated extended kalman filter is used for the recursive solution noise levels of 5 10 percent of the object image size are used approximate cramer rao lower bounds are derived for the model parameter estimates as a function of object trajectory and noise level this approach may be of use in situations where it is difficult to resolve large numbers of object match points but relatively long sequences of images 10 to 20 or more are available

variable neighborhood search principles and applications abstract systematic change of neighborhood within a possibly randomized local search algorithm yields a simple and effective metaheuristic for combinatorial and global optimization called variable neighborhood search vns we present a basic scheme for this purpose which can easily be implemented using any local search algorithm as a subroutine its effectiveness is illustrated by solving several classical combinatorial or global optimization problems moreover several extensions are proposed for solving large problem instances using vns within the successive approximation method yields a two level vns called variable neighborhood decomposition search vnds modifying the basic scheme to explore easily valleys far from the incumbent solution yields an efficient skewed vns svns heuristic finally we show how to stabilize column generation algorithms with help of vns and discuss various ways to use vns in graph theory i e to suggest disprove or give hints on how to prove conjectures an area where metaheuristics do not appear to have been applied before

piecewise convex maximization approach to multiknapsack we refine the reverse convex approach to binary programs into a piecewise convex maximization problem with only two pieces as a vital lead we address the multiknapsack problem the results of computational experiments are presented in contrast with the best known solutions found by heuristics and by the reverse convex approach

constraints on deformable models recovering 3d shape and nongrid motion abstract inferring the 3d structures of nonrigidly moving objects from images is a difficult yet basic problem in computational vision our approach makes use of dynamic elastically deformable object models that offer the geometric flexibility to satisfy a diversity of real world visual constraints we specialize these models to include intrinsic forces inducing a preference for axisymmetry image based constraints are applied as extrinsic forces that mold the symmetry seeking model into shapes consistent with image data we describe an extrinsic force that applies constraints derived from profiles of monocularly viewed objects we generalize this constraint force to incorporate profile information from multiple views and use it to exploit binocular image data for time varying images the force becomes dynamic and the model is able to infer not only depth but nonrigid motion as well we demonstrate the recovery of 3d shape and nonrigid motion from natural imagery

stochastic dynamics and logistic population growth the verhulst model is probably the best known macroscopic rate equation in population ecology it depends on two parameters the intrinsic growth rate and the carrying capacity these parameters can be estimated for different populations and are related to the reproductive fitness and the competition for limited resources respectively we investigate analytically and numerically the simplest possible microscopic scenarios that give rise to the logistic equation in the deterministic mean field limit we provide a definition of the two parameters of the verhulst equation in terms of microscopic parameters in addition we derive the conditions for extinction or persistence of the population by employing either the momentum space spectral theory or the real space wentzel kramers brillouin approximation to determine the probability distribution function and the mean time to extinction of the population our analytical results agree well with numerical simulations

proximal level bundle methods for convex nondifferentiable optimization saddle point problems and variational inequalities we study proximal level methods for convex optimization that use projections onto successive approximations of level sets of the objective corresponding to estimates of the optimal value we show that they enjoy almost optimal efficiency estimates we give extensions for solving convex constrained problems convex concave saddle point problems and variational inequalities with monotone operators we present several variants establish their efficiency estimates and discuss possible implementations in particular our methods require bounded storage in contrast to the original level methods of lemarechal nemirovskii and nesterov

a smooth method for the finite minimax problem we consider unconstrained minimax problems where the objective function is the maximum of a finite number of smooth functions we prove that under usual assumptions it is possible to construct a continuously differentiable function whose minimizers yield the minimizers of the max function and the corresponding minimum values on this basis we can define implementable algorithms for the solution of the minimax problem which are globally convergent at a superlinear convergence rate preliminary numerical results are reported

combined lp and quasi newton methods for minimax optimization we present an algorithm for minimax optimization that combines lp methods and quasi newton methods the quasi newton algorithm is used only if an irregular solution is detected in which case second order derivative information is needed in order to obtain a fast final rate of convergence we prove that the algorithm can converge only to a stationary point and that normally the final rate of convergence will be either quadratic or superlinear the performance is illustrated through some numerical examples

snakes active contour models a snake is an energy minimizing spline guided by external constraint forces and influenced by image forces that pull it toward features such as lines and edges snakes are active contour models they lock onto nearby edges localizing them accurately scale space continuation can be used to enlarge the capture region surrounding a feature snakes provide a unified account of a number of visual problems including detection of edges lines and subjective contours motion tracking and stereo matching we have used snakes successfully for interactive interpretation in which user imposed constraint forces guide the snake near features of interest

general variable neighborhood search for the continuous optimization we suggest a new heuristic for solving unconstrained continuous optimization problems it is based on a generalized version of the variable neighborhood search metaheuristic different neighborhoods and distributions induced from different metrics are ranked and used to get random points in the shaking step we also propose vns for solving constrained optimization problems the constraints are handled using exterior point penalty functions within an algorithm that combines sequential and exact penalty transformations the extensive computer analysis that includes the comparison with genetic algorithm and some other approaches on standard test functions are given with our approach we obtain encouraging results

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

colonization of a territory by a stochastic population under a strong allee effect and a low immigration pressure we study the dynamics of colonization of a territory by a stochastic population at low immigration pressure we assume a sufficiently strong allee effect that introduces in deterministic theory a large critical population size for colonization at low immigration rates the average precolonization population size is small thus invalidating the wkb approximation to the master equation we circumvent this difficulty by deriving an exact zero flux solution of the master equation and matching it with an approximate nonzero flux solution of the pertinent fokker planck equation in a small region around the critical population size this procedure provides an accurate evaluation of the quasistationary probability distribution of population sizes in the precolonization state and of the mean time to colonization for a wide range of immigration rates at sufficiently high immigration rates our results agree with wkb results obtained previously at low immigration rates the results can be very different

rare event statistics in reaction diffusion systems we present an efficient method to calculate probabilities of large deviations from the typical behavior rare events in reaction diffusion systems this method is based on a semiclassical treatment of an underlying quantum hamiltonian encoding the system s evolution to this end we formulate the corresponding canonical dynamical system and investigate its phase portrait this method is presented for a number of pedagogical examples

extinction in the lotka volterra model birth death processes often exhibit an oscillatory behavior we investigate a particular case where the oscillation cycles are marginally stable on the mean field level an iconic example of such a system is the lotka volterra model of predator prey interaction fluctuation effects due to discreteness of the populations destroy the mean field stability and eventually drive the system toward extinction of one or both species we show that the corresponding extinction time scales as a certain power law of the population sizes this behavior should be contrasted with the extinction of models stable in the mean field approximation in the latter case the extinction time scales exponentially with size

stochastic analysis of complex reaction networks using binomial moment equations the stochastic analysis of complex reaction networks is a difficult problem because the number of microscopic states in such systems increases exponentially with the number of reactive species direct integration of the master equation is thus infeasible and is most often replaced by monte carlo simulations while monte carlo simulations are a highly effective tool equation based formulations are more amenable to analytical treatment and may provide deeper insight into the dynamics of the network here we present a highly efficient equation based method for the analysis of stochastic reaction networks the method is based on the recently introduced binomial moment equations barzel and biham phys rev lett 106 150602 2011 the binomial moments are linear combinations of the ordinary moments of the probability distribution function of the population sizes of the interacting species they capture the essential combinatorics of the reaction processes reflecting their stoichiometric structure this leads to a simple and transparent form of the equations and allows a highly efficient and surprisingly simple truncation scheme unlike ordinary moment equations in which the inclusion of high order moments is prohibitively complicated the binomial moment equations can be easily constructed up to any desired order the result is a set of equations that enables the stochastic analysis of complex reaction networks under a broad range of conditions the number of equations is dramatically reduced from the exponential proliferation of the master equation to a polynomial and often quadratic dependence on the number of reactive species in the binomial moment equations the aim of this paper is twofold to present a complete derivation of the binomial moment equations to demonstrate the applicability of the moment equations for a representative set of example networks in which stochastic effects play an important role

extinction of an infectious disease a large fluctuation in a nonequilibrium system we develop a theory of first passage processes in stochastic nonequilibrium systems of birth death type using two closely related epidemiological models as examples our method employs the probability generating function technique in conjunction with the eikonal approximation in this way the problem is reduced to finding the optimal path to extinction a heteroclinic trajectory of an effective multidimensional classical hamiltonian system we compute this trajectory and mean extinction time of the disease numerically and uncover a nonmonotone spiral path to extinction of a disease we also obtain analytical results close to a bifurcation point where the problem is described by a hamiltonian previously identified in one species population models

population extinction risk in the aftermath of a catastrophic event we investigate how a catastrophic event modeled as a temporary fall of the reproduction rate increases the extinction probability of an isolated self regulated stochastic population using a variant of the verhulst logistic model as an example we combine the probability generating function technique with an eikonal approximation to evaluate the exponentially large increase in the extinction probability caused by the catastrophe this quantity is given by the eikonal action computed over the optimal path instanton of an effective classical hamiltonian system with a time dependent hamiltonian for a general catastrophe the eikonal equations can be solved numerically for simple models of catastrophic events analytic solutions can be obtained one such solution becomes quite simple close to the bifurcation point of the verhulst model the eikonal results for the increase in the extinction probability caused by a catastrophe agree well with numerical solutions of the master equation

a version of the bundle idea for minimizing a nonsmooth function conceptual idea convergence analysis numerical results during recent years various proposals for the minimization of a nonsmooth functional have been made amongst these the bundle concept turned out to be an especially fruitful idea based on this concept a number of authors have developed codes that can successfully deal with nonsmooth problems the aim of the paper is to show that by adding some features of the trust region philosophy to the bundle concept the end result is a distinguished member of the bundle family with a more stable behaviour than some other bundle versions the reliability and efficiency of this code is demonstrated on the standard academic test examples and on some real life problems

a proximal bundle method with approximate subgradient linearizations we give a proximal bundle method for minimizing a convex function f over a closed convex set it only requires evaluating f and its subgradients with an accuracy epsilon 0 which is fixed but possibly unknown it asymptotically finds points that are epsilon optimal when applied to lagrangian relaxation it allows for epsilon accurate solutions of lagrangian subproblems and finds epsilon optimal solutions of convex programs

stopping criteria for linesearch methods without derivatives in this paper acceptability criteria for the linesearch stepsize are introduced which require only function values simple algorithm models based on these criteria are presented some modifications of criteria based on the knowledge of the directional derivative are also illustrated

extinction of metastable stochastic populations we investigate the phenomenon of extinction of a long lived self regulating stochastic population caused by intrinsic demographic noise extinction typically occurs via one of two scenarios depending on whether the absorbing state n 0 is a repelling scenario a or attracting scenario b point of the deterministic rate equation in scenario a the metastable stochastic population resides in the vicinity of an attracting fixed point next to the repelling point n 0 in scenario b there is an intermediate repelling point n n 1 between the attracting point n 0 and another attracting point n n 2 in the vicinity of which the metastable population resides the crux of the theory is a dissipative variant of wkb wentzel kramers brillouin approximation which assumes that the typical population size in the metastable state is large starting from the master equation we calculate the quasistationary probability distribution of the population sizes and the exponentially long mean time to extinction for each of the two scenarios when necessary the wkb approximation is complemented i by a recursive solution of the quasistationary master equation at small n and ii by the van kampen system size expansion valid near the fixed points of the deterministic rate equation the theory yields both entropic barriers to extinction and pre exponential factors and holds for a general set of multistep processes when detailed balance is broken the results simplify considerably for single step processes and near the characteristic bifurcations of scenarios a and b

an introduction to stochastic modeling stochastic modeling probability review the major discrete distributions rtant continuous distributions some elementary exercises useful functions integrals and sums conditional probability and conditional expectation the discrete case the dice game craps random sums conditioning on a continuous random variable markov chains introduction definitions transition probability matrices of a markov chain some markov chain models first step analysis some special markov chains functionals of random walks and success runs another look at first step analysis the long run behavior of markov chains regular transition probability matrices examples the classification of states the basic limit theorem of markov chains reducible markov chains sequential decisions and markov chains poisson processes the poisson distribution and the poisson processes the law of rare events distributions associated with the poisson process the uniform distribution and poisson processes spatial poisson processes compound and marked poisson processes continuous time markov chains pure birth processes ptire death processes birth and death processes the limiting behavior of birth and death processes birth and death processes with absorbing states finite state continuous time markov chains set valued processes renewal phenomena definition of a renewal process and related concepts some examples of renewal processes the poisson process viewed as a renewal process the asymptotic 3ehavior as renewal process

geodesic active contours and level sets for the detection and tracking of moving objects this paper presents a new variational framework for detecting and tracking multiple moving objects in image sequences motion detection is performed using a statistical framework for which the observed interframe difference density function is approximated using a mixture model this model is composed of two components namely the static background and the mobile moving objects one both components are zero mean and obey laplacian or gaussian law this statistical framework is used to provide the motion detection boundaries additionally the original frame is used to provide the moving object boundaries then the detection and the tracking problem are addressed in a common framework that employs a geodesic active contour objective function this function is minimized using a gradient descent method a new approach named hermes is proposed which exploits aspects from the well known front propagation algorithms and compares favorably to them very promising experimental results are provided using real video sequences

on the mathematical foundations of nondifferentiable optimization in engineering design it is shown by example that a large class of engineering design problems can be transcribed into the form of a canonical optimization problem with inequality constraints involving mar functions such problems are commonly referred to as semi infinite optimization problems the bulk of this paper is devoted to the development of a mathematical theory for the construction of first order nondifferentiable optimization algorithms related to phase i phase ii methods of feasible directions which solve these semi infinite optimization problems the applicability of the theory is illustrated with examples that are relevant to engineering design

commitment versus persuasion in the three party constrained voter model in the framework of the three party constrained voter model where voters of two radical parties a and b interact with centrists c and c we study the competition between a persuasive majority and a committed minority in this model a s and b s are incompatible voters that can convince centrists or be swayed by them here radical voters are more persuasive than centrists whose sub population comprises susceptible agents c and a fraction of centrist zealots c whereas c s may adopt the opinions a and b with respective rates 1 a and 1 b with a b 0 c s are committed individuals that always remain centrists furthermore a and b voters can become susceptible centrists c with a rate 1 the resulting competition between commitment and persuasion is studied in the mean field limit and for a finite population on a complete graph at mean field level there is a continuous transition from a coexistence phase when b whereas all species coexist when a b when c and the initial density of centrists is low one finds lnn when n 1 our analytical findings are corroborated by stochastic simulations

population extinction in a time modulated environment the extinction time of an isolated population can be exponentially reduced by a periodic modulation of its environment we investigate this effect using as an example a stochastic branching annihilation process with a time dependent branching rate the population extinction is treated in eikonal approximation where it is described as an instanton trajectory of a proper reaction hamiltonian the modulation of the environment perturbs this trajectory and synchronizes it with the modulation phase we calculate the corresponding change in the action along the instanton using perturbation techniques supported by numerical calculations the techniques include a first order theory with respect to the modulation amplitude a second order theory in the spirit of the kapitsa pendulum effect and adiabatic theory valid for low modulation frequencies

piecewise linear approximations in nonconvex nonsmooth optimization we present a bundle type method for minimizing nonconvex nondifferentiable functions of several variables the algorithm is based on the construction of both a lower and an upper polyhedral approximation of the objective function in particular at each iteration a search direction is computed by solving a quadratic program aiming at maximizing the difference between the lower and the upper model a proximal approach is used to guarantee convergence to a stationary point under the hypothesis of weak semismoothness

an incremental method for solving convex finite min max problems we introduce a new approach to minimizing a function defined as the pointwise maximum over finitely many convex real functions next referred to as the component functions with the aim of working on the basis of incomplete knowledge of the objective function a descent algorithm is proposed which need not require at the current point the evaluation of the actual value of the objective function namely of all the component functions thus extending to min max problems the philosophy of the incremental approaches widely adopted in the nonlinear least squares literature given the nonsmooth nature of the problem we resort to the well established machinery of bundle methods we provide global convergence analysis of our method and in addition we study a subgradient aggregation scheme aimed at simplifying the problem of finding a tentative step this paper is completed by the numerical results obtained on a set of standard test problems

minimizing nonconvex nonsmooth functions via cutting planes and proximity control we describe an extension of the classical cutting plane algorithm to tackle the unconstrained minimization of a nonconvex not necessarily differentiable function of several variables the method is based on the construction of both a lower and an upper polyhedral approximation to the objective function and is related to the use of the concept of proximal trajectory convergence to a stationary point is proved for weakly semismooth functions

morphing active contours a method for deforming curves in a given image to a desired position in a second image is introduced the algorithm is based on deforming the first image toward the second one via a partial differential equation pde while tracking the deformation of the curves of interest in the first image with an additional coupled pde both the images and the curves on the frame slices of interest are used for tracking the technique can be applied to object tracking and sequential segmentation the topology of the deforming curve can change without any special topology handling procedures added to the scheme this permits for example the automatic tracking of scenes where due to occlusions the topology of the objects of interest changes from frame to frame in addition this work introduces the concept of projecting velocities to obtain systems of coupled pdes for image analysis applications we show examples for object tracking and segmentation of electronic microscopy

level set methods an overview and some recent results the level set method was devised by s osher and j a sethian 1988 j comput phys 79 12 49 as a simple and versatile method for computing and analyzing the motion of an interface in two or three dimensions bounds a possibly multiply connected region the goal is to compute and analyze the subsequent motion of under a velocity field v this velocity can depend on position time the geometry of the interface and the external physics the interface is captured for later time as the zero level set of a smooth at least lipschitz continuous function x t i e t x x t 0 is positive inside negative outside and is zero on t topological merging and breaking are well defined and easily performed in this review article we discuss recent variants and extensions including the motion of curves in three dimensions the dynamic surface extension method fast methods for steady state problems diffusion generated motion and the variational level set approach we also give a user s guide to the level set dictionary and technology and couple the method to a wide variety of problems involving external physics such as compressible and incompressible possibly reacting flow stefan problems kinetic crystal growth epitaxial growth of thin films vortex dominated flows and extensions to multiphase motion we conclude with a discussion of applications to computer vision and image processing

global optimization and multi knapsack a percolation algorithm abstract since the standard multi knapsack problem may be rewritten as a reverse convex problem we present a global optimization approach it is known from solving high dimensional nonconvex problems that pure cutting plane methods may fail and branch and bound is impractical due to a large duality gap on the other hand a strategy based on some sufficient optimality condition does not help much because it requires generating all level set points an intractable problem therefore we propose to combine both a cutting plane method and a sufficient optimality condition together with a random generation of level set points where the number of points is limited by a tabu list to prevent re examination of the same level set area experiments show that we end up with a small duality gap allowing a subsequent branch and bound approach to prove optimality

a bundle modification strategy for convex minimization we present a new bundle algorithm for minimizing convex not necessarily smooth functions the novelty of our approach is based on a bundle modification strategy that we apply whenever the stability center is updated and which is aimed at substituting the points of the bundle by new points characterized by possibly better values of the objective function convergence of the algorithm is proved and numerical results are presented

spectral theory of metastability and extinction in a branching annihilation reaction we apply the spectral method recently developed by the authors to calculate the statistics of a reaction limited multistep birth death process or chemical reaction that includes as elementary steps branching a 2a and annihilation 2a 0 the spectral method employs the generating function technique in conjunction with the sturm liouville theory of linear differential operators we focus on the limit when the branching rate is much higher than the annihilation rate and obtain accurate analytical results for the complete probability distribution including large deviations of the metastable long lived state and for the extinction time statistics the analytical results are in very good agreement with numerical calculations furthermore we use this example to settle the issue of the lacking boundary condition in the spectral formulation

the effect of deterministic noise in subgradient methods in this paper we study the influence of noise on subgradient methods for convex constrained optimization the noise may be due to various sources and is manifested in inexact computation of the subgradients and function values assuming that the noise is deterministic and bounded we discuss the convergence properties for two cases the case where the constraint set is compact and the case where this set need not be compact but the objective function has a sharp set of minima for example the function is polyhedral in both cases using several different stepsize rules we prove convergence to the optimal value within some tolerance that is given explicitly in terms of the errors in the first case the tolerance is nonzero but in the second case the optimal value can be obtained exactly provided the size of the error in the subgradient computation is below some threshold we then extend these results to objective functions that are the sum of a large number of convex functions in which case an incremental subgradient method can be used

noise driven unlimited population growth demographic noise causes unlimited population growth in a broad class of models which without noise would predict a stable finite population we study this effect on the example of a stochastic birth death model which includes immigration binary reproduction and death the unlimited population growth proceeds as an exponentially slow decay of a metastable probability distribution mpd of the population we develop a systematic wkb theory complemented by the van kampen system size expansion for the mpd and for the decay time important signatures of the mpd are a power law tail such that all the distribution moments except the zeroth one diverge and the presence in the solution of two different wkb modes

population extinction under bursty reproduction in a time modulated environment in recent years nondemographic variability has been shown to greatly affect dynamics of stochastic populations for example nondemographic noise in the form of a bursty reproduction process with an a priori unknown burst size or environmental variability in the form of time varying reaction rates have been separately found to dramatically impact the extinction risk of isolated populations in this work we investigate the extinction risk of an isolated population under the combined influence of these two types of nondemographic variation using the so called momentum space wentzel kramers brillouin wkb approach and accounting for the explicit time dependence in the reaction rates we arrive at a set of time dependent hamilton equations to this end we evaluate the population s extinction risk by finding the instanton of the time perturbed hamiltonian numerically whereas analytical expressions are presented in particular limits using various perturbation techniques we focus on two classes of time varying environments periodically varying rates corresponding to seasonal effects and a sudden decrease in the birth rate corresponding to a catastrophe all our theoretical results are tested against numerical monte carlo simulations with time dependent rates and also against a numerical solution of the corresponding time dependent hamilton equations

reducing the extinction risk of stochastic populations via nondemographic noise we consider nondemographic noise in the form of uncertainty in the reaction step size and reveal a dramatic effect this noise may have on the stability of self regulating populations employing the reaction scheme ma ka but allowing e g the product number k to be a priori unknown and sampled from a given distribution we show that such nondemographic noise can greatly reduce the population s extinction risk compared to the fixed k case our analysis is tested against numerical simulations and by using empirical data of different species we argue that certain distributions may be more evolutionary beneficial than others