cauchy problems of semilinear pseudo parabolic equations abstract this paper concerns with the cauchy problems of semilinear pseudo parabolic equations after establishing the necessary existence uniqueness and comparison principle for mild solutions which are also classical ones provided that the initial data are appropriately smooth we investigate large time behavior of solutions it is shown that there still exist the critical global existence exponent and the critical fujita exponent for pseudo parabolic equations and that these two critical exponents are consistent with the corresponding semilinear heat equations

auxiliary variable methods for markov chain monte carlo with applications abstract suppose that one wishes to sample from the density x using markov chain monte carlo mcmc an auxiliary variable u and its conditional distribution u x can be defined giving the joint distribution x u x u x a mcmc scheme that samples over this joint distribution can lead to substantial gains in efficiency compared to standard approaches the revolutionary algorithm of swendsen and wang is one such example besides reviewing the swendsen wang algorithm and its generalizations this article introduces a new auxiliary variable method called partial decoupling two applications in bayesian image analysis are considered a binary classification problem in which partial decoupling out performs swendsen wang and single site metropolis methods and a positron emission tomography pet reconstruction that uses the gray level prior of geman and mcclure a generalized swendsen wang algorithm is developed for this problem which reduces the computing time to the point where mcmc is a viab

extensions of fill s algorithm for perfect simulation fill s algorithm for perfect simulation for attractive finite state space models unbiased for user impatience is presented in terms of stochastic recursive sequences and extended in two ways repulsive discrete markov random fields with two coding sets like the auto poisson distribution on a lattice with 4 neighbourhood can be treated as monotone systems if a particular partial ordering and quasi maximal and quasi minimal states are used fill s algorithm then applies directly combining fill s rejection sampling with sandwiching leads to a version of the algorithm which works for general discrete conditionally specified repulsive models extensions to other types of models are briefly discussed

hidden markov models and disease mapping we present new methodology to extend hidden markov models to the spatial domain and use this class of models to analyze spatial heterogeneity of count data on a rare phenomenon this situation occurs commonly in many domains of application particularly in disease mapping we assume that the counts follow a poisson model at the lowest level of the hierarchy and introduce a finite mixture model for the poisson rates at the next level the novelty lies in the model for allocation to the mixture components which follows a spatially correlated process the potts model and in treating the number of components of the spatial mixture as unknown inference is performed in a bayesian framework using reversible jump markov chain monte carlo the model introduced can be viewed as a bayesian semiparametric approach to specifying flexible spatial distribution in hierarchical models performance of the model and comparison with an alternative well known markov random field specification for the poisson rates are de

fitting partial differential equations to space time dynamics the partial differential equations pdes governing the dynamics of reaction diffusion systems are reconstructed from data representing the spatially extended systems the fitted equations are validated by a comparison of their numerical solutions and the input data and by computation of the isoclines of the fitted pdes

estimation of parameters for macroparasite population evolution using approximate bayesian computation we estimate the parameters of a stochastic process model for a macroparasite population within a host using approximate bayesian computation abc the immunity of the host is an unobserved model variable and only mature macroparasites at sacrifice of the host are counted with very limited data process rates are inferred reasonably precisely modeling involves a three variable markov process for which the observed data likelihood is computationally intractable abc methods are particularly useful when the likelihood is analytically or computationally intractable the abc algorithm we present is based on sequential monte carlo is adaptive in nature and overcomes some drawbacks of previous approaches to abc the algorithm is validated on a test example involving simulated data from an autologistic model before being used to infer parameters of the markov process model for experimental data the fitted model explains the observed extra binomial variation in terms of a zero one immunity variable which has a short lived presence in the host

local autoencoding for parameter estimation in a hidden potts markov random field a local autoencoding lae method is proposed for the parameter estimation in a hidden potts markov random field model due to sampling cost markov chain monte carlo methods are rarely used in real time applications like other heuristic methods lae is based on a conditional independence assumption it adapts however the parameters in a block by block style with a simple hebbian learning rule experiments with given label fields show that the lae is able to converge in far less time than required for a scan it is also possible to derive an estimate for lae based on a cramer rao bound that is similar to the classical maximum pseudolikelihood method as a general algorithm lae can be used to estimate the parameters in anisotropic label fields furthermore lae is not limited to the classical potts model and can be applied to other types of potts models by simple label field transformations and straightforward learning rule extensions experimental results on image segmentations demonstrate the efficiency and generality of the lae algorithm

estimating varying coefficients for partial differential equation models partial differential equations pdes are used to model complex dynamical systems in multiple dimensions and their parameters often have important scientific interpretations in some applications pde parameters are not constant but can change depending on the values of covariates a feature that we call varying coefficients we propose a parameter cascading method to estimate varying coefficients in pde models from noisy data our estimates of the varying coefficients are shown to be consistent and asymptotically normally distributed the performance of our method is evaluated by a simulation study and by an empirical study estimating three varying coefficients in a pde model arising from lidar data

existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations in connection with the maximizing problem for the functional r u u lq u lp in w01 p 0 we consider the equation div u p 2 u x u q 2 u x x 1 p resp q p e has no nontrivial resp nonnegative nontrivial solution belonging to the class p u w01 p lq xi u q 2u lp p 1 i 1 2 n w01 p provided that is star shaped the crucial point of the proof of our result is to obtain an l estimate of weak solutions and to verify a certain pohozaev type inequality for weak solutions belonging to p

characterization of a two parameter mixed mode electrochemical behavior regime using neural networks abstract we use nonlinear signal processing techniques based on artificial neural networks to construct a continuous time model set of ordinary differential equations odes from experimental observations of mixed mode oscillations during the galvanostatic oxidation of hydrogen on platinum in the presence of bismuth and chloride ions the data was in the form of time series of the potential for different values of the applied current and chloride ion concentration we use the model to reconstruct the experimental dynamics and to explore the associated bifurcation structures in phase space using numerical bifurcation techniques we locate stable and unstable periodic solutions calculate eigenvalues and identify bifurcation points this approach constitutes a promising data post processing procedure for investigating phase space and parameter space of real experimental systems it allows us to infer phase space structures which the experiments can only probe with limited measurement precision and only at a discrete number of operating parameter settings for example the fitted model suggests the existence of a sub critical hopf bifurcation near the range of parameters probed in the experiments this might explain the experimental difficulty in locating small amplitude simple oscillations

ground state solution for a fourth order elliptic equation with logarithmic nonlinearity modeling epitaxial growth abstract this paper deals with a fourth order elliptic equation with logarithmic nonlinearity modeling epitaxial growth and the existence of a ground state solution is obtained

markov chains for exploring posterior distributions several markov chain methods are available for sampling from a posterior distribution two important examples are the gibbs sampler and the metropolis algorithm in addition several strategies are available for constructing hybrid algorithms this paper outlines some of the basic methods and strategies and discusses some related theoretical and practical issues on the theoretical side results from the theory of general state space markov chains can be used to obtain convergence rates laws of large numbers and central limit theorems for estimates obtained from markov chain methods these theoretical results can be used to guide the construction of more efficient algorithms for the practical use of markov chain methods standard simulation methodology provides several variance reduction techniques and also give guidance on the choice of sample size and allocation

efficient recursions for general factorisable models let n s valued categorical variables be jointly distributed according to a distribution known only up to an unknown normalising constant for an unnormalised joint likelihood expressible as a product of factors we give an algebraic recursion which can be used for computing the normalising constant and other summations a saving in computation is achieved when each factor contains a lagged subset of the components combining in the joint distribution with maximum computational efficiency as the subsets attain their minimum size if each subset contains at most r 1 of the n components in the joint distribution we term this a lag r model whose normalising constant can be computed using a forward recursion in o sr 1 computations as opposed to o sn for the direct computation we show how a lag r model represents a markov random field and allows a neighbourhood structure to be related to the unnormalised joint likelihood we illustrate the method by showing how the normalising constant of the ising or autologistic model can be computed

a monte carlo metropolis hastings algorithm for sampling from distributions with intractable normalizing constants simulating from distributions with intractable normalizing constants has been a long standing problem in machine learning in this letter we propose a new algorithm the monte carlo metropolis hastings mcmh algorithm for tackling this problem the mcmh algorithm is a monte carlo version of the metropolis hastings algorithm it replaces the unknown normalizing constant ratio by a monte carlo estimate in simulations while still converges as shown in the letter to the desired target distribution under mild conditions the mcmh algorithm is illustrated with spatial autologistic models and exponential random graph models unlike other auxiliary variable markov chain monte carlo mcmc algorithms such as the moller and exchange algorithms the mcmh algorithm avoids the requirement for perfect sampling and thus can be applied to many statistical models for which perfect sampling is not available or very expensive the mcmh algorithm can also be applied to bayesian inference for random effect models and missing data problems that involve simulations from a distribution with intractable integrals

identification of nonlinear spatiotemporal systems via partitioned filtering the problem of identifying continuous spatiotemporal nonlinear systems from noisy and indirect observations is determined by its computational complexity we propose a solution by means of nonlinear state space filtering along with a state partition technique the method is demonstrated to be computationally feasible for spatiotemporal data with properties that occur typically in experimental recordings it is applied to one component of the simulated chaotic data of a two component reaction diffusion system yielding estimates of both the unobserved state component and the diffusion constant

on periodic solutions of superlinear parabolic problems on etudie l existence des solutions periodiques non triviales positives des problemes paraboliques semilineaires on demontre des resultats de bifurcation

particle markov chain monte carlo methods summary markov chain monte carlo and sequential monte carlo methods have emerged as the two main tools to sample from high dimensional probability distributions although asymptotic convergence of markov chain monte carlo algorithms is ensured under weak assumptions the performance of these algorithms is unreliable when the proposal distributions that are used to explore the space are poorly chosen and or if highly correlated variables are updated independently we show here how it is possible to build efficient high dimensional proposal distributions by using sequential monte carlo methods this allows us not only to improve over standard markov chain monte carlo schemes but also to make bayesian inference feasible for a large class of statistical models where this was not previously so we demonstrate these algorithms on a non linear state space model and a levy driven stochastic volatility model

a recursive algorithm for markov random fields summary we propose a recursive algorithm as a more useful alternative to the brook expansion for the joint distribution of a vector of random variables when the original formulation is in terms of the corresponding full conditional distributions as occurs for markov random fields usually in practical applications the computational load will still be excessive but then the algorithm can be used to obtain the componentwise full conditionals of a system after marginalising over some variables or the joint distribution of subsets of the variables conditioned on values of the remainder which is required for block gibbs sampling as an illustrative example we apply the algorithm in the simplest nontrivial setting of hidden markov chains more important we demonstrate how it applies to markov random fields on regular lattices and to perfect block gibbs sampling for binary systems

global solution and blow up of a semilinear heat equation with logarithmic nonlinearity abstract we study the initial boundary value problem of a semilinear heat equation with logarithmic nonlinearity by using the logarithmic sobolev inequality and a family of potential wells we obtain the existence of global solution and blow up at under some suitable conditions on the other hand the results for decay estimates of the global solutions are also given our result in this paper means that the polynomial nonlinearity is a critical condition of blow up in finite time for the solutions of semilinear heat equations

efficient calculation of the normalizing constant of the autologistic and related models on the cylinder and lattice motivated by the autologistic model for the analysis of spatial binary data on the two dimensional lattice we develop efficient computational methods for calculating the normalizing constant for models for discrete data defined on the cylinder and lattice because the normalizing constant is generally unknown analytically statisticians have developed various ad hoc methods to overcome this difficulty our aim is to provide computationally and statistically efficient methods for calculating the normalizing constant so that efficient likelihood based statistical methods are then available for inference we extend the so called transition method to find a feasible computational method of obtaining the normalizing constant for the cylinder boundary condition to extend the result to the free boundary condition on the lattice we use an efficient path sampling markov chain monte carlo scheme the methods are generally applicable to association patterns other than spatial such as clustered binary data and to variables taking three or more values described by for example potts models

a bayesian reassessment of nearest neighbor classification the k nearest neighbor knn procedure is a well known deterministic method used in supervised classification this article proposes a reassessment of this approach as a statistical technique derived from a proper probabilistic model in particular we modify the assessment found in holmes and adams and evaluated by manocha and girolami where the underlying probabilistic model is not completely well defined once provided with a clear probabilistic basis for the knn procedure we derive computational tools for bayesian inference on the parameters of the corresponding model in particular we assess the difficulties inherent to both pseudo likelihood and path sampling approximations of an intractable normalizing constant we implement a correct mcmc sampler based on perfect sampling when perfect sampling is not available we use instead a gibbs sampling approximation illustrations of the performance of the corresponding bayesian classifier are provided for benchmark datasets demonstrating in particula

texture synthesis and pattern recognition for partially ordered markov models abstract the uses of texture in image analysis are widespread ranging from remotely sensed data to medical imaging to military applications image processing tasks that use texture characteristics include classification region segmentation and synthesis of data while there are several approaches available for texture modeling the research presented here is concerned with stochastic texture models stochastic approaches view a texture as the realization of a random field and are most useful when the texture appears noisy or when it lacks smooth geometric features the model introduced in this paper is a subclass of markov random fields mrfs called partially ordered markov models pomms markov random fields are a class of stochastic models that incorporate spatial dependency between data points one major disadvantage of mrfs is that in general an explicit form of the joint probability of the random variables describing the model is not obtainable however a popular subclass of mrfs called markov mesh models mmms allows the explicit description of the joint probability in terms of spatially local conditional probabilities we show how pomms are a generalization of mmms and demonstrate the versatility of pomms to texture synthesis and pattern recognition in imaging specifically we give a fast one pass algorithm for simulating textures using pomms and introduce examples of heterogeneous models that suggest potential applications for object recognition purposes then we address an inverse problem where we present results from a series of statistical experiments designed to estimate parameters of stochastic texture models for both binary and gray value data although the applications in this paper focus on imaging in their most general form pomms can be found in areas such as probabilistic expert systems bayesian hierarchical modeling influence diagrams and random graphs and networks

optimal scaling of discrete approximations to langevin diffusions we consider the optimal scaling problem for proposal distributions in hastings metropolis algorithms derived from langevin diffusions we prove an asymptotic diffusion limit theorem and show that the relative efficiency of the algorithm can be characterized by its overall acceptance rate independently of the target distribution the asymptotically optimal acceptance rate is 0 574 we show that as a function of dimension n the complexity of the algorithm is o n1 3 which compares favourably with the o n complexity of random walk metropolis algorithms we illustrate this comparison with some example simulations

on monte carlo methods for estimating ratios of normalizing constants recently estimating ratios of normalizing constants has played an important role in bayesian computations applications of estimating ratios of normalizing constants arise in many aspects of bayesian statistical inference in this article we present an overview and discuss the current monte carlo methods for estimating ratios of normalizing constants then we propose a new ratio importance sampling method and establish its theoretical framework we find that the ratio importance sampling method can be better than the current methods for example the bridge sampling method meng and wong and the path sampling method gelman and meng in the sense of minimizing asymptotic relative mean square errors of estimators an example is given for illustrative purposes finally we present two special applications and the general implementation issues for estimating ratios of normalizing constants

blow up phenomena for a pseudo parabolic equation with p laplacian and logarithmic nonlinearity terms abstract this paper deals with a pseudo parabolic equation with p laplacian and logarithmic nonlinearity terms under homogeneous dirichlet boundary condition in a smooth bounded domain which was studied in 12 and the global existence and finite time blow up of the weak solution were studied under different energy levels we generalize and extend those results by discussing the asymptotic behavior of the weak solution and proving that the weak solution converges to the corresponding stationary solution as time tends to infinity moreover a lower and an upper bound estimation for blow up time and rate are obtained for the blow up weak solution in different initial energy cases furthermore we establish the weak solution with high initial energy that is global bounded or blow up under some sufficient conditions as well as a nonblow up criterion and algebraic decay result in other assumptions

recovering coefficients of the complex ginzburg landau equation from experimental spatio temporal data two examples from hydrodynamics there are many examples where the description of the complexity of flows can only be achieved by the use of simple models these models obtained usually from phenomenological arguments need in general the knowledge of some parameters the challenge is then to determine the values of these parameters from experiments we will give two examples where we have been able to evaluate the coefficients of the complex ginzburg landau equation cgle from space time chaotic data applied to first a row of coupled cylinder wakes and then to wave propagation in the ekman layer of a rotating disk in the first case our analysis is based on a proper decomposition of experimental chaotic flow fields followed by a projection of the cgle onto the proper directions we show that our method is able to recover the parameters of the model which permits to reconstruct the spatio temporal chaos observed in the experiment the second physical system under consideration is the flow above a rotating disk and its cross flow instability our aim is to study the properties of the wavefield through a volterra series equation the kernels of the volterra expansion which contain relevant physical information about the system are estimated by fitting two point measurements via a nonlinear parametric model we then consider describing the wavefield with the cgle and derive analytical relations which express the coefficients of the ginzburg landau equation in terms of the kernels of the volterra expansion

an efficient markov chain monte carlo method for distributions with intractable normalising constants maximum likelihood parameter estimation and sampling from bayesian posterior distributions are problematic when the probability density for the parameter of interest involves an intractable normalising constant which is also a function of that parameter in this paper an auxiliary variable method is presented which requires only that independent samples can be drawn from the unnormalised density at any particular parameter value the proposal distribution is constructed so that the normalising constant cancels from the metropolis hastings ratio the method is illustrated by producing posterior samples for parameters of the ising model given a particular lattice realisation

estimating the granularity coefficient of a potts markov random field within a markov chain monte carlo algorithm this paper addresses the problem of estimating the potts parameter jointly with the unknown parameters of a bayesian model within a markov chain monte carlo mcmc algorithm standard mcmc methods cannot be applied to this problem because performing inference on requires computing the intractable normalizing constant of the potts model in the proposed mcmc method the estimation of is conducted using a likelihood free metropolis hastings algorithm experimental results obtained for synthetic data show that estimating jointly with the other unknown parameters leads to estimation results that are as good as those obtained with the actual value of on the other hand choosing an incorrect value of can degrade estimation performance significantly to illustrate the interest of this method the proposed algorithm is successfully applied to real bidimensional sar and tridimensional ultrasound images

fitting parameters in partial differential equations from partially observed noisy data abstract we describe a new method for parameter estimation in systems with nonlinear spatiotemporal dynamics the technique is able to reliably determine parameter estimates from noisy data and is even applicable in cases with unobserved components in a simulation study we investigate its performance in comparison to standard methods and show its superiority as numerical example we analyze in detail a complex ginzburg landau equation under realistic conditions

pseudolikelihood estimation for social networks abstract interest in log linear modeling for social network data has grown steadily since holland and leinhardt 1981 proposed their p 1 model that model was designed for a single binary relationship directed graph representing interactions between individuals it assumed that interactions between pairs of individuals are mutually independent subsequent work has extended the model in various ways including block modeling and the case of dependence between pairs of individuals in empirical work it would often be desirable to fit a wide variety of these models as the differences in predictions or goodness of fit are likely to provide insights into the data this has not been common practice however because estimation for some of the models has been difficult and the maximum likelihood schemes developed for others involve different computer programs not always available in standard packages the focus of this article is on a general estimation technique that maximizes the pseudolikelihood the pro

maximum likelihood estimation for spatial models by markov chain monte carlo stochastic approximation we propose a two stage algorithm for computing maximum likelihood estimates for a class of spatial models the algorithm combines markov chain monte carlo methods such as the metropolis hastings green algorithm and the gibbs sampler and stochastic approximation methods such as the off line average and adaptive search direction a new criterion is built into the algorithm so stopping is automatic once the desired precision has been set simulation studies and applications to some real data sets have been conducted with three spatial models we compared the algorithm proposed with a direct application of the classical robbins monro algorithm using wiebe s wheat data and found that our procedure is at least 15 times faster

second critical exponent and life span for pseudo parabolic equation this paper studies the second critical exponent and life span of solutions for the pseudo parabolic equation ut k ut u up in rn 0 t with p 1 k 0 it is proved that the second critical exponent i e the decay order of the initial data required by global solutions in the coexistence region of global and non global solutions is independent of the pseudo parabolic parameter k nevertheless it is revealed that the viscous term k ut relaxes restrictions on the amplitude of the initial data required by the global solutions moreover it is observed that the life span of the non global solutions will be delayed by the third order viscous term finally some numerical examples are given to illustrate all these results

perfect simulation of conditionally specified models we discuss how the ideas of producing perfect simulations based on coupling from the past for finite state space models naturally extend to multivariate distributions with infinite or uncountable state spaces such as auto gamma auto poisson and autonegative binomial models using gibbs sampling in combination with sandwiching methods originally introduced for perfect simulation of point processes

filtering via simulation auxiliary particle filters this article analyses the recently suggested particle approach to filtering time series we suggest that the algorithm is not robust to outliers for two reasons the design of the simulators and the use of the discrete support to represent the sequentially updating prior distribution here we tackle the first of these problems

kirchhoff type system with linear weak damping and logarithmic nonlinearities abstract for the nonlinear kirchhoff type wave system with logarithmic nonlinearities and weak dissipation the global well posedness of initial boundary value problem is analyzed focusing on the interplay between kirchhoff terms and logarithmic sources we investigate the kirchhoff system controlled by logarithmic forces thus amplifying the difficulties in blow up analysis which is the primary scenario of interest by employing potential well method and concavity method we obtain several results related to the sufficient conditions posed on subcritical initial energy and critical initial energy which is used to classify initial data for global existence and finite time blow up finally via careful analysis involving the unstable invariant set under supercritical initial energy we are able to show an affirmative result that the solution blows up in finite time when initial data satisfy some suitable assumptions

local dependence in random graph models characterization properties and statistical inference dependent phenomena such as relational spatial and temporal phenomena tend to be characterized by local dependence in the sense that units which are close in a well defined sense are dependent in contrast with spatial and temporal phenomena though relational phenomena tend to lack a natural neighbourhood structure in the sense that it is unknown which units are close and thus dependent owing to the challenge of characterizing local dependence and constructing random graph models with local dependence many conventional exponential family random graph models induce strong dependence and are not amenable to statistical inference we take first steps to characterize local dependence in random graph models inspired by the notion of finite neighbourhoods in spatial statistics and m dependence in time series and we show that local dependence endows random graph models with desirable properties which make them amenable to statistical inference we show that random graph models with local dependence satisfy a natural domain consistency condition which every model should satisfy but conventional exponential family random graph models do not satisfy in addition we establish a central limit theorem for random graph models with local dependence which suggests that random graph models with local dependence are amenable to statistical inference we discuss how random graph models with local dependence can be constructed by exploiting either observed or unobserved neighbourhood structure in the absence of observed neighbourhood structure we take a bayesian view and express the uncertainty about the neighbourhood structure by specifying a prior on a set of suitable neighbourhood structures we present simulation results and applications to two real world networks with ground truth

global existence and finite time blow up for a class of semilinear pseudo parabolic equations abstract in this paper we study a class of semilinear pseudo parabolic equations by introducing a family of potential wells we prove the invariance of some sets global existence nonexistence and asymptotic behavior of solutions with initial energy j u 0 d moreover we obtain finite time blow up with high initial energy j u 0 d by comparison principle

a bayesian nonparametric regression model with normalized weights a study of hippocampal atrophy in alzheimer s disease hippocampal volume is one of the best established biomarkers for alzheimer s disease however for appropriate use in clinical trials research the evolution of hippocampal volume needs to be well understood recent theoretical models propose a sigmoidal pattern for its evolution to support this theory the use of bayesian nonparametric regression mixture models seems particularly suitable due to the flexibility that models of this type can achieve and the unsatisfactory predictive properties of semiparametric methods in this article our aim is to develop an interpretable bayesian nonparametric regression model which allows inference with combinations of both continuous and discrete covariates as required for a full analysis of the dataset simple arguments regarding the interpretation of bayesian nonparametric regression mixtures lead naturally to regression weights based on normalized sums difficulty in working with the intractable normalizing constant is overcome thanks to recent advances in mcmc

blow up at infinity of solutions to a semilinear heat equation with logarithmic nonlinearity abstract in this short note the author establishes a blow up result for a semilinear heat equation with logarithmic nonlinearity by using the logarithmic sobolev inequality this improves a recent blow up result obtained in chen et al 2015 1

parameter estimation of partial differential equation models partial differential equation pde models are commonly used to model complex dynamic systems in applied sciences such as biology and finance the forms of these pde models are usually proposed by experts based on their prior knowledge and understanding of the dynamic system parameters in pde models often have interesting scientific interpretations but their values are often unknown and need to be estimated from the measurements of the dynamic system in the presence of measurement errors most pdes used in practice have no analytic solutions and can only be solved with numerical methods currently methods for estimating pde parameters require repeatedly solving pdes numerically under thousands of candidate parameter values and thus the computational load is high in this article we propose two methods to estimate parameters in pde models a parameter cascading method and a bayesian approach in both methods the underlying dynamic process modeled with the pde model is represented via basis function

constructing summary statistics for approximate bayesian computation semi automatic approximate bayesian computation summary many modern statistical applications involve inference for complex stochastic models where it is easy to simulate from the models but impossible to calculate likelihoods approximate bayesian computation abc is a method of inference for such models it replaces calculation of the likelihood by a step which involves simulating artificial data for different parameter values and comparing summary statistics of the simulated data with summary statistics of the observed data here we show how to construct appropriate summary statistics for abc in a semi automatic manner we aim for summary statistics which will enable inference about certain parameters of interest to be as accurate as possible theoretical results show that optimal summary statistics are the posterior means of the parameters although these cannot be calculated analytically we use an extra stage of simulation to estimate how the posterior means vary as a function of the data and we then use these estimates of our summary statistics within abc empirical results show that our approach is a robust method for choosing summary statistics that can result in substantially more accurate abc analyses than the ad hoc choices of summary statistics that have been proposed in the literature we also demonstrate advantages over two alternative methods of simulation based inference

finite time blow up for a class of parabolic or pseudo parabolic equations abstract in this paper we study the initial boundary value problem for a class of parabolic or pseudo parabolic equations u t a u t u b u k t u p 2 u x t 0 t where a 0 b l 1 with l 1 being the principal eigenvalue for on h 0 1 and k t 0 by using the potential well method levine s concavity method and some differential inequality techniques we obtain the finite time blow up results provided that the initial energy satisfies three conditions i j u 0 0 0 ii j u 0 0 d where d is a nonnegative constant iii 0 j u 0 0 c 0 where 0 involves the l 2 norm or h 0 1 norm of the initial data we also establish the lower and upper bounds for the blow up time in particular we obtain the existence of certain solutions blowing up in finite time with initial data at the nehari manifold or at arbitrary energy level

pulse bifurcations and instabilities in an excitable medium computations in finite ring domains we investigate the instabilities and bifurcations of traveling pulses in a model excitable medium in particular we discuss three different scenarios involving either the loss of stability or disappearance of stable pulses in numerical simulations beyond the instabilities we observe replication of pulses backfiring resulting in complex periodic or spatiotemporally chaotic dynamics as well as modulated traveling pulses we approximate the linear stability of traveling pulses through computations in a finite albeit large domain with periodic boundary conditions the critical eigenmodes at the onset of the instabilities are related to the resulting spatiotemporal dynamics and act upon the back of the pulses the first scenario has been analyzed earlier m g zimmermann et al physica d 110 92 1997 for high excitability low excitation threshold it involves the collision of a stable pulse branch with an unstable pulse branch in a so called t point in the framework of traveling wave ordinary differential equations pulses correspond to homoclinic orbits and the t point to a double heteroclinic loop we investigate this transition for a pulse in a domain with finite length and periodic boundary conditions numerical evidence of the proximity of the infinite domain t point in this setup appears in the form of two saddle node bifurcations alternatively for intermediate excitation threshold an entire cascade of saddle nodes causing a spiraling of the pulse branch appears near the parameter values corresponding to the infinite domain t point backfiring appears at the first saddle node bifurcation which limits the existence region of stable pulses the third case found in the model for large excitation threshold is an oscillatory instability giving rise to breathing traveling pulses that periodically vary in width and speed

initial boundary value problem for a class of semilinear pseudo parabolic equations with logarithmic nonlinearity abstract in this paper we study the initial boundary value problem for a class of semilinear pseudo parabolic equations with logarithmic nonlinearity by using the logarithmic sobolev inequality and a family of potential wells we obtain the existence of global solution blow up at and behavior of vacuum isolation of solutions on the other hand the asymptotic behavior of solutions is also discussed our result implies that the polynomial nonlinearity is important for the solutions of such kinds of semilinear pseudo parabolic equations to be blow up in finite time

multiple solutions for the semilinear elliptic equations with the sign changing logarithmic nonlinearity abstract in this paper we study the semilinear elliptic equations with the sign changing logarithmic nonlinearity under some conditions we obtain at least two nontrivial solutions by nehari manifold and logarithmic sobolev inequality here the results are quite different from these in the polynomial case

constrained monte carlo maximum likelihood for dependent data maximum likelihood estimates mles in autologistic models and other exponential family models for dependent data can be calculated with markov chain monte carlo methods the metropolis algorithm or the gibbs sampler which simulate ergodic markov chains having equilibrium distributions in the model from one realization of such a markov chain a monte carlo approximant to the whole likelihood function can be constructed the parameter value if any maximizing this function approximates the mle

instability of positive periodic solutions for semilinear pseudo parabolic equations with logarithmic nonlinearity abstract in this paper we consider the periodic problem for semilinear heat equation and pseudo parabolic equation with logarithmic source after establishing the existence of positive periodic solutions we discuss the instability of such solutions

instability sensitivity and degeneracy of discrete exponential families a number of discrete exponential family models for dependent data first and foremost relational data have turned out to be near degenerate and problematic in terms of markov chain monte carlo mcmc simulation and statistical inference i introduce the notion of instability with an eye to characterize detect and penalize discrete exponential family models that are near degenerate and problematic in terms of mcmc simulation and statistical inference i show that unstable discrete exponential family models are characterized by excessive sensitivity and near degeneracy in special cases the subset of the natural parameter space corresponding to nondegenerate distributions and mean value parameters far from the boundary of the mean value parameter space turns out to be a lower dimensional subspace of the natural parameter space these characteristics of unstable discrete exponential family models tend to obstruct mcmc simulation and statistical inference in applications to relational data i show that d

blow up and decay for a class of pseudo parabolic p laplacian equation with logarithmic nonlinearity abstract we consider the pseudo parabolic p laplacian equation u t u t p u u q 2 u log u in a bounded domain with homogeneous dirichlet boundary conditions when 2 p q p 1 2 n we obtain results of the decay and the finite time blow up for weak solutions

choice of dynamical variables for global reconstruction of model equations from time series the success of modeling from an experimental time series is determined to a significant extent by the choice of dynamical variables we propose a method for preliminary investigation of a time series whose purpose is to find out whether a global dynamical model with smooth functions can be constructed for the chosen variables the method consists in the estimation of single valuedness and continuity of relations between dynamical variables and variables to enter left hand sides of model equations the method is explained with numerical examples its efficiency is demonstrated by modeling a real nonlinear electric circuit

the calculation of posterior distributions by data augmentation abstract the idea of data augmentation arises naturally in missing value problems as exemplified by the standard ways of filling in missing cells in balanced two way tables thus data augmentation refers to a scheme of augmenting the observed data so as to make it more easy to analyze this device is used to great advantage by the em algorithm dempster laird and rubin 1977 in solving maximum likelihood problems in situations when the likelihood cannot be approximated closely by the normal likelihood maximum likelihood estimates and the associated standard errors cannot be relied upon to make valid inferential statements from the bayesian point of view one must now calculate the posterior distribution of parameters of interest if data augmentation can be used in the calculation of the maximum likelihood estimate then in the same cases one ought to be able to use it in the computation of the posterior distribution it is the purpose of this article to explain how this can be done the basic idea