ruin probability with claims modeled by a stationary ergodic stable process for a random walk with negative drift we study the exceedance probability ruin probability of a high threshold the steps of this walk claim sizes constitute a stationary ergodic stable process we study how ruin occurs in this situation and evaluate the asymptotic behavior of the ruin probability for a large variety of stationary ergodic stable processes our findings show that the order of magnitude of the ruin probability varies significantly from one model to another in particular ruin becomes much more likely when the claim sizes exhibit long range dependence the proofs exploit large deviation techniques for sums of dependent stable random variables and the series representation of a stable process as a function of a poisson process

using spanning graphs for efficient image registration we provide a detailed analysis of the use of minimal spanning graphs as an alignment method for registering multimodal images this yields an efficient graph theoretic algorithm that for the first time jointly estimates both an alignment measure and a viable descent direction with respect to a parameterized class of spatial transformations we also show how prior information about the interimage modality relationship from prealigned image pairs can be incorporated into the graph based algorithm a comparison of the graph theoretic alignment measure is provided with more traditional measures based on plug in entropy estimators this highlights previously unrecognized similarities between these two registration methods our analysis gives additional insight into the tradeoffs the graph based algorithm is making and how these will manifest themselves in the registration algorithm s performance

convergence of the nelder mead simplex method to a nonstationary point this paper analyzes the behavior of the nelder mead simplex method for a family of examples which cause the method to converge to a nonstationary point all the examples use continuous functions of two variables the family of functions contains strictly convex functions with up to three continuous derivatives in all the examples the method repeatedly applies the inside contraction step with the best vertex remaining fixed the simplices tend to a straight line which is orthogonal to the steepest descent direction it is shown that this behavior cannot occur for functions with more than three continuous derivatives the stability of the examples is analyzed

on the roots of certain polynomials arising from the analysis of the nelder mead simplex method abstract the study of the effect of dimensionality on the nelder mead simplex method for unconstrained optimization leads us to the study of a two parameter family of polynomials of the form p n z b az az n 1 z n we show that provided that a a b is real it is possible to use primarily the schur cohn criterion in order to determine the configuration of the roots of p n z with respect to the unit circle

fortified descent simplicial search method a general approach we propose a new simplex based direct search method for unconstrained minimization of a real valued function f of n variables as in other methods of this kind the intent is to iteratively improve an n dimensional simplex through certain reflection expansion contraction steps the method has three novel features first a user chosen integer bar m k specifies the number of good vertices to be retained in constructing the initial trial simplices reflected then either expanded or contracted at iteration k second a trial simplex is accepted only when it satisfies the criteria of fortified descent which are stronger than the criterion of strict descent used in most direct search methods third the number of additional function evaluations needed to check a trial reflected expanded simplex for fortified descent can be controlled if one of the initial trial simplices satisfies the fortified descent criteria it is accepted as the new simplex otherwise the simplex is shrunk a fraction of the way toward a best vertex and the process is restarted etc until either a trial simplex is accepted or the simplex effectively has shrunk to a single point we prove several theoretical properties of the new method if f is continuously differentiable bounded below and uniformly continuous on its lower level set and we choose bar m k with the same value at all iterations k then every cluster point of the generated sequence of iterates is a stationary point the same conclusion holds if the function is continuously differentiable bounded below and we choose bar m k 1 at all iterations k

on the convergence of pattern search algorithms we introduce an abstract definition of pattern search methods for solving nonlinear unconstrained optimization problems our definition unifies an important collection of optimization methods that neither compute nor explicitly approximate derivatives we exploit our characterization of pattern search methods to establish a global convergence theory that does not enforce a notion of sufficient decrease our analysis is possible because the iterates of a pattern search method lie on a scaled translated integer lattice this allows us to relax the classical requirements on the acceptance of the step at the expense of stronger conditions on the form of the step and still guarantee global convergence

precise large deviations for dependent regularly varying sequences we study a precise large deviation principle for a stationary regularly varying sequence of random variables this principle extends the classical results of nagaev theory probab appl 14 51 64 193 208 1969 and nagaev ann probab 7 745 789 1979 for iid regularly varying sequences the proof uses an idea of jakubowski stoch proc appl 44 291 327 1993 68 1 20 1997 in the context of central limit theorems with infinite variance stable limits we illustrate the principle for stochastic volatility models real valued functions of a markov chain satisfying a polynomial drift condition and solutions of linear and non linear stochastic recurrence equations

extremal properties of the determinant of the laplacian in the bergman metric on the moduli space of genus two riemann surfaces we study extremal properties of the determinant of the laplacian in the bergman metric on the moduli space of compact genus two riemann surfaces by a combination of analytical and numerical methods we identify four non degenerate critical points of this function and compute the signature of the hessian at these points the curve with the maximal number of automorphisms the burnside curve turns out to be the point of the absolute maximum our results agree with the mass formula for virtual euler characteristics of the moduli space a similar analysis is performed for bolza s strata of symmetric riemann surfaces of genus two

eigenvalue solution for the convected wave equation in a circular soft wall duct abstract a numerical approach to find the eigenvalues of the wave equation applied in a circular duct with convective flow and soft wall boundary conditions is proposed the characteristic equation is solved in the frequency domain as a function of a locally reacting acoustic impedance and the flow mach number in addition the presence of the convective flow couples the solution of the eigenvalues with the axial propagation constants the unknown eigenvalues are also related to these propagation constants by a quadratic expression that leads to two solutions these two solutions replaced into the characteristic equation generate two separate eigenvalue problems depending on the direction of propagation given that the resulting nonlinear complex valued equations do not provide the solution explicitly a numerical technique must be used the proposed approach is based on the minimization of the absolute value of the characteristic equation by the nelder mead simplex method the main advantage of this method is that it only uses function evaluations rather than derivatives and geometric reasoning the minimization is performed starting from very low frequencies and increasing by small steps to the particular frequency of interest the initial guess for the first frequency of calculation is provided as the hard wall eigenvalue solution then the solution from the previous step is used as the initial value for the next calculation this approach was specifically developed for applications with resonator type liners commonly used in the commercial aviation industry where the low frequency behavior resembles that of a hard wall and agrees with the first initial guess for the first frequency of calculation the numerical technique was found to be very robust in terms of convergence and stability also the method provides a physical meaning for each eigenvalue since the variation as a function of frequency can be clearly followed with respect to the values that are originally linked to the hard wall modes

on estimation of intermodulated frequencies for sinusoidal signals with unknown fundamental frequencies in this correspondence the problem of estimating the intermodulated frequencies of sinusoidal signals with unknown fundamental frequencies is investigated it is shown that by formulating the intermodulation structure as a constraint in the estimation of these frequencies a smaller cramer rao lower bound crlb can be obtained that is a more accurate estimate of these frequencies can be obtained the steps in the proposed estimation procedure are i obtaining initial estimate of the frequencies using existing methods ii establishing the intermodulation structure from the initial estimate of the frequencies and iii refining the initial estimate using the intermodulation structure the implementation and performance of the proposed estimation procedure is demonstrated by a simulation example

on designing an isotropic fiducial mark the crame spl acute r rao lower bound crlb of image registration error using an isotropic fiducial mark is derived the derived crlb is a function of the intensity profile of the fiducial mark following the development of the crlb a new method for designing an isotropic fiducial mark suitable for digital image registration is presented a parameterization method of the fiducial intensity profile is introduced which guarantees no aliasing effect when the fiducial mark is digitized with the proper sampling rate a method for computing the fiducial intensity profile based on minimization of the crlb registration error and subject to certain practical constraints is developed for imaging systems with a significant low pass effect it is proposed to pre emphasize the high frequency components of the fiducial mark by converting the designed gray scale fiducial marks into binary fiducial marks experimental results show that the designed fiducial mark can provide very accurate registration results and that the registration accuracy is independent of its location

large deviations for solutions to stochastic recurrence equations under kesten s condition in this paper we prove large deviations results for partial sums constructed from the solution to a stochastic recurrence equation we assume kesten s condition acta math 131 1973 207 248 under which the solution of the stochastic recurrence equation has a marginal distribution with power law tails while the noise sequence of the equations can have light tails the results of the paper are analogs to those obtained by a v nagaev theory probab appl 14 1969 51 64 193 208 and s v nagaev ann probab 7 1979 745 789 in the case of partial sums of i i d random variables in the latter case the large deviation probabilities of the partial sums are essentially determined by the largest step size of the partial sum for the solution to a stochastic recurrence equation the magnitude of the large deviation probabilities is again given by the tail of the maximum summand but the exact asymptotic tail behavior is also influenced by clusters of extreme values due to dependencies in the sequence we apply the large deviation results to study the asymptotic behavior of the ruin probabilities in the model

robust identification of harmonic oscillator parameters using the adjoint fokker planck equation we present a model based output only method for identifying from time series the parameters governing the dynamics of stochastically forced oscillators in this context suitable models of the oscillator s damping and stiffness properties are postulated guided by physical understanding of the oscillatory phenomena the temporal dynamics and the probability density function of the oscillation amplitude are described by a langevin equation and its associated fokker planck equation respectively one method consists in fitting the postulated analytical drift and diffusion coefficients with their estimated values obtained from data processing by taking the short time limit of the first two transition moments however this limit estimation loses robustness in some situations for instance when the data are band pass filtered to isolate the spectral contents of the oscillatory phenomena of interest in this paper we use a robust alternative where the adjoint fokker planck equation is solved to compute kramers moyal coefficients exactly and an iterative optimization yields the parameters that best fit the observed statistics simultaneously in a wide range of amplitudes and time scales the method is illustrated with a stochastic van der pol oscillator serving as a prototypical model of thermoacoustic instabilities in practical combustors where system identification is highly relevant to control

numerical study of blow up and dispersive shocks in solutions to generalized korteweg de vries equations abstract we present a detailed numerical study of solutions to general korteweg de vries equations with critical and supercritical nonlinearity both in the context of dispersive shocks and blow up we study the stability of solitons and show that they are unstable against being radiated away and blow up in the l 2 critical case the blow up mechanism by martel merle and raphael can be numerically identified in the limit of small dispersion it is shown that a dispersive shock always appears before an eventual blow up in the latter case always the first soliton to appear will blow up it is shown that the same type of blow up as for the perturbations of the soliton can be observed which indicates that the theory by martel merle and raphael is also applicable to initial data with a mass much larger than the soliton mass we study the scaling of the blow up time t in dependence of the small dispersion parameter and find an exponential dependence t and that there is a minimal blow up time t 0 greater than the critical time of the corresponding hopf solution for 0 to study the cases with blow up in detail we apply the first dynamic rescaling for generalized korteweg de vries equations this allows to identify the type of the singularity

an efficient technique to find semi analytical solutions for higher order multi point boundary value problems abstract a new semi analytical algorithm is presented to solve general multi point boundary value problems this method can be applied on nth order linear nonlinear singular and nonsingular multi point boundary value problems mathematical base of the method is presented convergence of the method is proved also the algorithm is applied to solve multi point boundary value problems including nonlinear sixth order nonlinear singular second order five point boundary value problems and a singularly perturbed boundary value problem comparison results show that the new method works more accurate than the other methods

on convergence toward an extreme value distribution in c 0 1 the structure of extreme value distributions in infinite dimensional space is well known we characterize the domain of attraction of such extreme value distributions in the framework of gine hahn and vatan we intend to use the result for statistical applications

extreme value theory ergodic theory and the boundary between short memory and long memory for stationary stable processes we study the partial maxima of stationary stable processes we relate their asymptotic behavior to the ergodic theoretical properties of the flow we observe a sharp change in the asymptotic behavior of the sequence of partial maxima as flow changes from being dissipative to being conservative and argue that this may indicate a change from a short memory process to a long memory process

large deviations for random walks under subexponentiality the big jump domain for a given one dimensional random walk sn with a subexponential step size distribution we present a unifying theory to study the sequences xn for which as n uniformly for x xn we also investigate the stronger local analogue our theory is self contained and fits well within classical results on domains of partial attraction and local limit theory when specialized to the most important subclasses of subexponential distributions that have been studied in the literature we reproduce known theorems and we supplement them with new results

extremal behavior of stochastic integrals driven by regularly varying levy processes we study the extremal behavior of a stochastic integral driven by a multivariate levy process that is regularly varying with index alpha 0 for predictable integrands with a finite alpha delta moment for some delta 0 we show that the extremal behavior of the stochastic integral is due to one big jump of the driving levy process and we determine its limit measure associated with regular variation on the space of cadlag functions

direct search methods on parallel machines direct search methods are methods designed to solve unconstrained minimization problems of the form mathop min limits x in mathbb r n f x where f n these methods are distinguished by the fact that they neither use nor require explicit derivative information the search for a local minimizer is driven solely by function information popular methods in this class include the factorial design algorithm of box 1 the pattern search algorithm of hooke and jeeves 4 and the simplex method of neider and mead 5

large deviations and ruin probabilities for solutions to stochastic recurrence equations with heavy tailed innovations in this paper we consider the stochastic recurrence equation y t a t y t 1 b t for an i i d sequence of pairs a t b t of nonnegative random variables where we assume that b t is regularly varying with index k 0 and ea k t x c 1 np y 1 x as x holds for some constant c 1 0 for k 1 we also study the large deviation probabilities p s n es n x x x n for some sequence x n whose growth depends on the heaviness of the tail of the distribution of y 1 we show that the relation p s n es n x c 2 np y 1 x holds uniformly for x x n and some constant c 2 0 then we apply the large deviation results to derive bounds for the ruin probability u p sup n 1 s n es n n u for any 0 we show that u c 3 up y 1 u 1 1 1 for some constant c 3 0 in contrast to the case of i i d regularly varying y t s when the above results hold with c 1 c 2 c 3 1 the constants c 1 c 2 and c 3 are different from 1

the effect of dropout on the efficiency of d optimal designs of linear mixed models dropout is often encountered in longitudinal data optimal designs will usually not remain optimal in the presence of dropout in this paper we study d optimal designs for linear mixed models where dropout is encountered moreover we estimate the efficiency loss in cases where a d optimal design for complete data is chosen instead of that for data with dropout two types of monotonically decreasing response probability functions are investigated to describe dropout our results show that the location of d optimal design points for the dropout case will shift with respect to that for the complete and uncorrelated data case owing to this shift the information collected at the d optimal design points for the complete data case does not correspond to the smallest variance we show that the size of the displacement of the time points depends on the linear mixed model and that the efficiency loss is moderate

on large delays in multi server queues with heavy tails we present upper and lower bounds for the tail distribution of the stationary waiting time d in the stable gi gi s first come first served fcfs queue these bounds depend on the value of the traffic load which is the ratio of mean service and mean interarrival times for service times with intermediate regularly varying tail distribution the bounds are exact up to a constant and we are able to establish a principle of s k big jumps in this case here k is the integer part of which gives the most probable way for the stationary waiting time to be large another corollary of the bounds obtained is to provide a new proof of necessity and sufficiency of conditions for the existence of moments of the stationary waiting time

convergence properties of the nelder mead simplex method in low dimensions the nelder mead simplex algorithm first published in 1965 is an enormously popular direct search method for multidimensional unconstrained minimization despite its widespread use essentially no theoretical results have been proved explicitly for the nelder mead algorithm this paper presents convergence properties of the nelder mead algorithm applied to strictly convex functions in dimensions 1 and 2 we prove convergence to a minimizer for dimension 1 and various limited convergence results for dimension 2 a counterexample of mckinnon gives a family of strictly convex functions in two dimensions and a set of initial conditions for which the nelder mead algorithm converges to a nonminimizer it is not yet known whether the nelder mead method can be proved to converge to a minimizer for a more specialized class of convex functions in two dimensions

detection and remediation of stagnation in the nelder mead algorithm using a sufficient decrease condition the nelder mead algorithm can stagnate and converge to a nonoptimal point even for very simple problems in this note we propose a test for sufficient decrease which if passed for all iterations will guarantee convergence of the nelder mead iteration to a stationary point if the objective function is smooth and the diameters of the nelder mead simplices converge to zero failure of this condition is an indicator of potential stagnation as a remedy we propose a new step which we call an oriented restart that reinitializes the simplex to a smaller one with orthogonal edges whose orientation is determined by an approximate descent direction from the current best point we also give results that apply when the objective function is a low amplitude perturbation of a smooth function we illustrate our results with some numerical examples

intrinsic graph structure estimation using graph laplacian a graph is a mathematical representation of a set of variables where some pairs of the variables are connected by edges common examples of graphs are railroads the internet and neural networks it is both theoretically and practically important to estimate the intensity of direct connections between variables in this study a problem of estimating the intrinsic graph structure from observed data is considered the observed data in this study are a matrix with elements representing dependency between nodes in the graph the dependency represents more than direct connections because it includes influences of various paths for example each element of the observed matrix represents a co occurrence of events at two nodes or a correlation of variables corresponding to two nodes in this setting spurious correlations make the estimation of direct connection difficult to alleviate this difficulty a digraph laplacian is used for characterizing a graph a generative model of this observed matrix is proposed and a parameter estimation algorithm for the model is also introduced the notable advantage of the proposed method is its ability to deal with directed graphs while conventional graph structure estimation methods such as covariance selections are applicable only to undirected graphs the algorithm is experimentally shown to be able to identify the intrinsic graph structure

a global multilevel coordinate search procedure for estimating the unsaturated soil hydraulic properties we present a new inverse modeling procedure to characterize the hydraulic properties of partially saturated soils from soil moisture measurements during a natural transient flow experiment the inversion of the governing one dimensional richards equation is carried out using the global multilevel coordinate search optimization algorithm in sequential combination with the local nelder mead simplex algorithm gmcs nms we introduce this optimization method in the area of unsaturated zone hydrology since it is adapted for solving accurately and efficiently complex nonlinear problems several numerical experiments have been conducted to evaluate the proposed inversion method using synthetic error free and error contaminated data for different textured soils inversion of the simulated error free data and examination of the related response surfaces demonstrated the uniqueness of the inverse solution and the suitability of the gmcs nms strategy when identifying four key parameters of the hydraulic functions described by the mualem van genuchten model inversion of the error contaminated data proved further the good stability of the inverse solution that is consistent with the needs required by real experiments

video denoising deblocking and enhancement through separable 4 d nonlocal spatiotemporal transforms we propose a powerful video filtering algorithm that exploits temporal and spatial redundancy characterizing natural video sequences the algorithm implements the paradigm of nonlocal grouping and collaborative filtering where a higher dimensional transform domain representation of the observations is leveraged to enforce sparsity and thus regularize the data 3 d spatiotemporal volumes are constructed by tracking blocks along trajectories defined by the motion vectors mutually similar volumes are then grouped together by stacking them along an additional fourth dimension thus producing a 4 d structure termed group where different types of data correlation exist along the different dimensions local correlation along the two dimensions of the blocks temporal correlation along the motion trajectories and nonlocal spatial correlation i e self similarity along the fourth dimension of the group collaborative filtering is then realized by transforming each group through a decorrelating 4 d separable transform and then by shrinkage and inverse transformation in this way the collaborative filtering provides estimates for each volume stacked in the group which are then returned and adaptively aggregated to their original positions in the video the proposed filtering procedure addresses several video processing applications such as denoising deblocking and enhancement of both grayscale and color data experimental results prove the effectiveness of our method in terms of both subjective and objective visual quality and show that it outperforms the state of the art in video denoising

exploring the 3d geometry of the diffusion kurtosis tensor impact on the development of robust tractography procedures and novel biomarkers diffusion kurtosis imaging dki is a diffusion weighted technique which overcomes limitations of the commonly used diffusion tensor imaging approach this technique models non gaussian behaviour of water diffusion by the diffusion kurtosis tensor kt which can be used to provide indices of tissue heterogeneity and a better characterisation of the spatial architecture of tissue microstructure in this study the geometry of the kt is elucidated using synthetic data generated from multi compartmental models where diffusion heterogeneity between intra and extra cellular media is taken into account as well as the sensitivity of the results to each model parameter and to synthetic noise furthermore based on the assumption that the maxima of the kt are distributed perpendicularly to the direction of well aligned fibres a novel algorithm for estimating fibre direction directly from the kt is proposed and compared to the fibre directions extracted from dki based orientation distribution function odf estimates previously proposed in the literature synthetic data results showed that for fibres crossing at high intersection angles direction estimates extracted directly from the kt have smaller errors than the dki based odf estimation approaches dki odf nevertheless the proposed method showed smaller angular resolution and lower stability to changes of the simulation parameters on real data tractography performed on these kt fibre estimates suggests a higher sensitivity than the dki based odf in resolving lateral corpus callosum fibres reaching the pre central cortex when diffusion acquisition is performed with five b values using faster acquisition schemes kt based tractography did not show improved performance over the dki odf procedures nevertheless it is shown that direct kt fibre estimates are more adequate for computing a generalised version of radial kurtosis maps

blume emery griffiths model on the square lattice with repulsive biquadratic coupling using a real space renormalization group procedure with no adjustable parameters we investigate the blume emery griffiths model on the square lattice the formalism respects sublattice symmetry allowing the study of both signs of k the biquadratic exchange coupling our results for k 0 are compared with other renormalization group calculations and with exact results in order to assess the magnitude of the errors introduced by our approximate calculation the quantitative agreement is excellent values for critical parameters differ in some cases by less than 1 from exact ones for k 0 our results lead to a rich phase diagram with antiquadrupolar and ferromagnetic ordered phases contrarily to monte carlo simulations these two phases meet only at zero temperature both antiquadrupolar disordered and ferromagnetic disordered transitions are found to be continuous and no ferrimagnetic phase is found

wiener system modelling by exponentially weighted aggregation abstractin the paper the aggregation algorithm for wiener system modelling is proposed the method uses exponential weights to combine aggregate models from a given collection of d linear dynamic and dm nonlinear static models of the genuine system blocks that can be a priori hypotheses other system identification outcomes etc the resulting model has a noteworthy property it is almost as accurate as the best hypothetical wiener model built from a given collection this is caused by the fact that the error introduced by the aggregation routine decreases in the mean squared sense with growing number of measurements n as fast as cqn 1ln dmd where c and q are known constants simulations confirm theoretical findings and demonstrate applicability of the algorithm in engineering problems for small and moderate measurement data sets

a large deviations approach to limit theory for heavy tailed time series in this paper we propagate a large deviations approach for proving limit theory for generally multivariate time series with heavy tails we make this notion precise by introducing regularly varying time series we provide general large deviation results for functionals acting on a sample path and vanishing in some neighborhood of the origin we study a variety of such functionals including large deviations of random walks their suprema the ruin functional and further derive weak limit theory for maxima point processes cluster functionals and the tail empirical process one of the main results of this paper concerns bounds for the ruin probability in various heavy tailed models including garch stochastic volatility models and solutions to stochastic recurrence equations

the maximum of a branching random walk with semiexponential increments we consider an infinite galton watson tree and label the vertices with a collection of i i d random variables y in the case where the upper tail of the distribution of y is semiexponential we then determine the speed of the corresponding tree indexed random walk in contrast to the classical case where the random variables y have finite exponential moments the normalization in the definition of the speed depends on the distribution of y interpreting the random variables y as displacements of the offspring from the parent y describes a branching random walk the result on the speed gives a limit theorem for the maximum of the branching random walk that is for the position of the rightmost particle in our case this maximum grows faster than linear in time

deterministic impulse control problems we prove that the optimal cost function of a deterministic impulse control problem is the unique viscosity solution of a first order hamilton jacobi quasi variational inequality in mathbb r n

automatic parameter estimation for the discrete algebraic reconstruction technique dart computed tomography ct is a technique for noninvasive imaging of physical objects in the discrete algebraic reconstruction technique dart prior knowledge about the material s densities is exploited to obtain high quality reconstructed images from a limited number of its projections in practice this prior knowledge is typically not readily available here a fully automatic method called projection distance minimization dart pdm dart is proposed in which the optimal grey level parameters are adaptively estimated during the reconstruction process to apply pdm dart only the number of different grey levels should be known in advance simulation as well as real ct experiments show that pdm dart is capable of computing reconstructed images of which the quality is similar to reconstructions computed by conventional dart based on exact prior knowledge thereby eliminating the need for tedious and error prone user interaction