effect of symmetry to the structure of positive solutions in nonlinear elliptic problems ii abstract we consider the problem u hu f u 0 in r u 0 on r u 0 in r where r x r n r 1

algebraic methods for interactive proof systems a new algebraic technique for the construction of interactive proof systems is presented our technique is used to prove that every language in the polynomial time hierarchy has an interactive proof system this technique played a pivotal role in the recent proofs that ip pspace 28 and that mip nexp 4

probabilistic algorithms for deciding equivalence of straight line programs let q be any algebraic structure and the set of all total programs over q using the instruction set z 1 z x y z x y z x y z x y a program is total if no division by zero occurs during any computation let the equivalence problem for be the problem of deciding for two given programs in whether or not they compute the same funcuon the following results are proved 1 if q is an inftmte field e g the rauonal numbers or the complex numbers then the equwalence problem for is probabilistlcally decidable in polynomml time the result also holds for programs with no dwlslon instructions and q an infimte integral domain e g the integers 2 if q is a finite field or if q is a fimte set of integers of cardmahty 2 then the equivalence problem is np hard the case when the field q is finite but its cardinality is a funcuon of the size of the instance to the eqmvalence problem is also considered an example is shown for which a sharp boundary between the classes np hard and probabihsticauy decidable exists provided they are not identical classes

existence of nontrivial solutions for a semilinear elliptic problem with supercritical exponent let n 3 p 2 2n n 2 and rn be a bounded domain with a boundary our purpose in this paper is to consider the existence of solutions of problem p u up 1on u 0on u 0on

entire nodal solutions to the pure critical exponent problem arising from concentration abstract we obtain new sign changing solutions to the problem u u 2 2 u u d 1 2 r n for n 4 where 2 2 n n 2 is the critical sobolev exponent these solutions arise as asymptotic profiles of sign changing solutions to the problem p u u p 2 u in u 0 on in some bounded smooth domains in r n for p 2 2 as p 2 we exhibit solutions u p to p which blow up at a single point as p 2 developing a peak whose asymptotic profile is a rescaling of a nonradial sign changing solution to problem we also obtain existence and multiplicity of sign changing nonradial solutions to the bahri coron problem 2 in annuli

radial solutions for the brezis nirenberg problem involving large nonlinearities abstract let us consider the problem 0 1 u a x u u p in b 1 u 0 in b 1 u 0 on b 1 where b 1 is the unit ball in r n n 3 and a x 0 is a smooth radial function under some suitable assumptions on the regular part of the green function of the operator u n 1 r u a r u we prove the existence of a radial solution to 0 1 for p large enough

asymptotic behavior of positive solution in an annulus abstract this paper concerns the positive solutions of superlinear elliptic equation with hardy potential u x 2 u u p 2 u in u 0 in u 0 on where is an annulus of r n n 2 we obtain qualitative properties of solutions as p

pp is as hard as the polynomial time hierarchy in this paper two interesting complexity classes pp and oplus text p are compared with ph the polynomial time hierarchy it is shown that every set in ph is polynomial time turing reducible to a set in pp and ph is included in text bp cdot oplus text p as a consequence of the results it follows that text pp subseteq text ph or oplus text p subseteq text ph implies a collapse of ph a stronger result is also shown every set in pp ph is polynomial time turing reducible to a set in pp

the complexity of partial derivatives abstract let l denote the nonscalar complexity in k x 1 x n we prove l x 1 x n 3l using this we determine the complexity of single power sums single elementary symmetric functions the resultant and the discriminant as root functions up to order of magnitude also we linearly reduce matrix inversion to computing the determinant

emulation of higher order tensors in manifold monte carlo methods for bayesian inverse problems the bayesian approach to inverse problems relies predominantly on markov chain monte carlo methods for posterior inference the typical nonlinear concentration of posterior measure observed in many such inverse problems presents severe challenges to existing simulation based inference methods motivated by these challenges the exploitation of local geometric information in the form of covariant gradients metric tensors levi civita connections and local geodesic flows have been introduced to more effectively locally explore the configuration space of the posterior measure however obtaining such geometric quantities usually requires extensive computational effort and despite their effectiveness affects the applicability of these geometrically based monte carlo methods in this paper we explore one way to address this issue by the construction of an emulator of the model from which all geometric objects can be obtained in a much more computationally feasible manner the main concept is to approximate the geometric quantities using a gaussian process emulator which is conditioned on a carefully chosen design set of configuration points which also determines the quality of the emulator to this end we propose the use of statistical experiment design methods to refine a potentially arbitrarily initialized design online without destroying the convergence of the resulting markov chain to the desired invariant measure the practical examples considered in this paper provide a demonstration of the significant improvement possible in terms of computational loading suggesting this is a promising avenue of further development

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

global bifurcations of solutions of emden fowler type equation δu x λf u x on an annulus in rn n 3 abstract in this article we study global and symmetry breaking bifurcations of solutions of emden fowler type elliptic differential equations as the main tool we use a topological invariant which is suitable for study of global bifurcations and continuations of critical points of equivariant functionals namely we will apply degree for s 1 equivariant orthogonal maps defined in rybicki 1994 nonlinear anal tma 23 83 102

convergence of adaptive mixtures of importance sampling schemes in the design of ecient simulation algorithms one is often beset with a poor choice of proposal distributions although the performances of a given kernel can clarify how adequate it is for the problem at hand a permanent on line modification of kernels causes concerns about the validity of the resulting algorithm while the issue is quite complex and most often intractable for mcmc algorithms the equivalent version for importance sampling algorithms can be validated quite precisely we derive sucient convergence conditions for a wide class of population monte carlo algorithms and show that rao blackwellized versions asymptotically achieve an optimum in terms of a kullback divergence criterion while more rudimentary versions simply do not benefit from repeated updating

on the applicability of regenerative simulation in markov chain monte carlo we consider the central limit theorem and the calculation of asymptotic standard errors for the ergodic averages constructed in markov chain monte carlo chan geyer 1994 established a central limit theorem for ergodic averages by assuming that the underlying markov chain is geometrically ergodic and that a simple moment condition is satisfied while it is relatively straightforward to check chan geyer s conditions their theorem does not lead to a consistent and easily computed estimate of the variance of the asymptotic normal distribution conversely mykland et al 1995 discuss the use of regeneration to establish an alternative central limit theorem with the advantage that a simple consistent estimator of the asymptotic variance is readily available however their result assumes a pair of unwieldy moment conditions whose verification is difficult in practice in this paper we show that the conditions of chan geyer s theorem are sufficient to establish the central limit theorem of mykland et al this result in conjunction with other recent developments should pave the way for more widespread use of the regenerative method in markov chain monte carlo our results are illustrated in the context of the slice sampler

fixed width output analysis for markov chain monte carlo markov chain monte carlo is a method of producing a correlated sample to estimate features of a target distribution through ergodic averages a fundamental question is when sampling should stop that is at what point the ergodic averages are good estimates of the desired quantities we consider a method that stops the simulation when the width of a confidence interval based on an ergodic average is less than a user specified value hence calculating a monte carlo standard error is a critical step in assessing the simulation output we consider the regenerative simulation and batch means methods of estimating the variance of the asymptotic normal distribution we give sufficient conditions for the strong consistency of both methods and investigate their finite sample properties in various examples

existence of many positive solutions of semilinear elliptic equations on annulus abstract we study the existence of many nonradial positive solutions in an annulus of rn it is an improvement of results in c v coffman a nonlinear boundary value problem with many positive solutions j differential equations 54 1984 429 437 an annulus is invariant under many group actions and these group actions act naturaly on the functional and leave the functional invariant therefore when we look for critical points of the functional we can restrict the functional to the manifold consisting of all fixed points under the group action we are going to use different group actions to obtain critical points and furthermore we can distinguish them

positive solutions of elliptic equations involving supercritical growth positive radial solutions of elliptic equation involving supercritical growth are analysed as their supremum norm tends to infinity it is shown that they converge uniformly away from the origin as well as in h 1 to the unique singular solution

nonexistence results for elliptic problems with supercritical nonlinearity in nontrivial domains let be a bounded domain in rn with n 3 and assume p 2n n 2 we study problems of the type u u p 2u 0 in u 0 on we prove that for suitable p 2n n 2 there exists some bounded domain with nontrivial topology in the sense of bahri and coron such that the above problem has only the solution u 0

compact embedding from w01 2 ω to lq x ω and its application to nonlinear elliptic boundary value problem with variable critical exponent we study the compact embedding from w01 2 to lq x with a variable critical exponent 1 q x 2n n 2 n 3 if there exist a point x0 a small 0 0 0 such that q x0 2n n 2 and q x 2n n 2 c0 log 1 x x0 l for x x0 as an application we show an existence of a positive solution to the nonlinear elliptic boundary value problem u uq x 1 in u x 0 on

stability of non isolated asymptotic profiles for fast diffusion the stability of asymptotic profiles of solutions to the cauchy dirichlet problem for fast diffusion equation fde for short is discussed the main result of the present paper is the stability of any asymptotic profiles of least energy it is noteworthy that this result can cover non isolated profiles e g those for thin annular domain cases the method of proof is based on the ojasiewicz simon inequality which is usually used to prove the convergence of solutions to prescribed limits as well as a uniform extinction estimate for solutions to fde besides local minimizers of an energy functional associated with this issue are characterized furthermore the instability of positive radial asymptotic profiles in thin annular domains is also proved by applying the ojasiewicz simon inequality in a different way

multi peak solutions for super critical elliptic problems in domains with small holes abstract this paper deals with the slightly super critical elliptic problem u u n 2 n 2 e in u 0 in u 0 on 1 where e 0 is a small parameter and r n is a bounded domain with smooth boundary assuming that the domain exhibits k sufficiently small holes multiple solutions are constructed by gluing double spike patterns located near each of the holes

sampling based approaches to calculating marginal densities abstract stochastic substitution the gibbs sampler and the sampling importance resampling algorithm can be viewed as three alternative sampling or monte carlo based approaches to the calculation of numerical estimates of marginal probability distributions the three approaches will be reviewed compared and contrasted in relation to various joint probability structures frequently encountered in applications in particular the relevance of the approaches to calculating bayesian posterior densities for a variety of structured models will be discussed and illustrated

generalized darting monte carlo one of the main shortcomings of markov chain monte carlo samplers is their inability to mix between modes of the target distribution in this paper we show that advance knowledge of the location of these modes can be incorporated into the mcmc sampler by introducing mode hopping moves that satisfy detailed balance the proposed sampling algorithm explores local mode structure through local mcmc moves e g diffusion or hybrid monte carlo but in addition also represents the relative strengths of the different modes correctly using a set of global moves this mode hopping mcmc sampler can be viewed as a generalization of the darting method 1 we illustrate the method on learning markov random fields and evaluate it against the spherical darting algorithm on a real world vision application of inferring 3d human body pose distributions from 2d image information

existence of clustering high dimensional bump solutions of superlinear elliptic problems on expanding annuli abstract we consider the nonlinear elliptic problem u u p in r u 0 in r u 0 in r where p 1 and r x r n r x r 1 with n 3 it is known that as r the number of nonequivalent solutions of the above problem goes to when p 1 n 2 n 2 n 3 here we prove the same phenomenon for any p 1 by finding o n 1 symmetric clustering bump solutions which concentrate near the set x 1 x n r x n 0 for large r 0

multipoint metropolis method with application to hybrid monte carlo abstract we propose the multipoint metropolis algorithm as an extension of the orientational bias monte carlo of frenkel and smit a ratio statistics similar to that in the metropolis algorithm is introduced to maintain the detailed balance the multipoint idea can be applied to improve the efficiency of a general markov chain based monte carlo algorithm to illustrate we describe two variations of the idea the random grid metropolis and the multipoint hybrid monte carlo and apply them to a number of examples

discretization of continuous markov chains and markov chain monte carlo convergence assessment abstract we show that continuous state space markov chains can be rigorously discretized into finite markov chains the idea is to subsample the continuous chain at renewal times related to small sets that control the discretization once a finite markov chain is derived from the markov chain monte carlo output general convergence properties on finite state spaces can be exploited for convergence assessment in several directions our choice is based on a divergence criterion derived from kemeny and snell which is first evaluated on parallel chains with a stopping time and then implemented more efficiently on two parallel chains only using birkhoff s pointwise ergodic theorem for stopping rules the performance of this criterion is illustrated on three standard examples

logarithmic advice classes abstract karp and lipton 1980 introduced the notion of nonuniform complexity classes where a certain amount of additional information the advice is given for free the advice only depends on the length of the input karp and lipton initiated the study of classes with either logarithmic or polynomial advice however later yap 1983 schoning 1984 balcazar 1987 and ko and schoning 1985 concentrated on the study of classes of the form c poly where c is p np or pspace and poly denotes a polynomial size advice this paper considers classes of the form c log as a main result it is shown that in the context of an np log computation log bounded advice is equivalent to a sparse oracle in np in contrast it has been shown that a poly bounded advice to an arbitrary sparse oracle set furthermore a general theorem is presented that generalizes karp and lipton s round robin tournament method

geometric convergence and central limit theorems for multidimensional hastings and metropolis algorithms we develop results on geometric ergodicity of markov chains and apply these and other recent results in markov chain theory to multidimensional hastings and metropolis algorithms for those based on random walk candidate distributions we find sufficient conditions for moments and moment generating functions to converge at a geometric rate to a prescribed distribution by phrasing the conditions in terms of the curvature of the densities we show that the results apply to all distributions with positive densities in a large class which encompasses many commonly used statistical forms from these results we develop central limit theorems for the metropolis algorithm converse results showing non geometric convergence rates for chains where the rejection rate is not bounded away from unity are also given these show that the negative definiteness property is not redundant

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

a nonlinear elliptic equation with critical exponents effect of geometry and topology of the domain let n 3 and rn be a bounded domain with a smooth boundary and consider the semilinear boundary value problem of the form u u u 2 2uin u 0in u 0on p where r and 2 2n n 2 we show the effect of topology and geometry of the domain on the existence and multiplicity of solutions of this problem

infinitely many non radial solutions to a critical equation on annulus abstract in this paper we build infinitely many non radial sign changing solutions to the critical problem p u u 4 n 2 u in u 0 on on the annulus x r n a x b n 3 in particular for any integer k large enough we build a non radial solution which look like the unique positive solution u 0 to p crowned by k negative bubbles arranged on a regular polygon with radius r 0 such that r 0 n 2 2 u 0 r 0 max a r b r n 2 2 u 0 r

multiple existence of solutions for a nonlinear elliptic problem on a riemannian manifold abstract let m g be a compact connected orientable smooth n dimensional riemannian manifold without boundary let p 2 2 where 2 2 n n 2 we consider the multiple existence of solutions of the problem p e 2 g u u u p 2 u on m where g stands for the laplacian in m

a shuffled complex evolution metropolis algorithm for optimization and uncertainty assessment of hydrologic model parameters author s vrugt ja gupta hv bouten w sorooshian s abstract markov chain monte carlo mcmc methods have become increasingly popular for estimating the posterior probability distribution of parameters in hydrologic models however mcmc methods require the a priori definition of a proposal or sampling distribution which determines the explorative capabilities and efficiency of the sampler and therefore the statistical properties of the markov chain and its rate of convergence in this paper we present an mcmc sampler entitled the shuffled complex evolution metropolis algorithm scem ua which is well suited to infer the posterior distribution of hydrologic model parameters the scem ua algorithm is a modified version of the original sce ua global optimization algorithm developed by duan et al 1992 the scem ua algorithm operates by merging the strengths of the metropolis algorithm controlled random search competitive evolution and complex shuffling in order to continuously update the proposal distribution and evolve the sampler to the posterior target distribution three case studies demonstrate that the adaptive capability of the scem ua algorithm significantly reduces the number of model simulations needed to infer the posterior distribution of the parameters when compared with the traditional metropolis hastings samplers

positive stationary solutions and threshold results for the non homogeneous semilinear parabolic equation with robin boundary conditions abstract let be a smooth bounded domain in r n considering the following robin problem for a semilinear parabolic equation 0 1 u t u u p f x x t 0 t u u 0 x t 0 t u x 0 u 0 x x we show that for any function f x satisfying f which will be given in the introduction there exists a positive number f such that problem 0 1 has no stationary solution if 0 f and has at least two stationary solutions when f moreover among all stationary solutions of problem 0 1 there is a minimal one we prove further that the minimal stationary solution of problem 0 1 is stable whereas any other stationary solutions of problem 0 1 are initial datum thresholds for the existence and non existence of a global solution to problem 0 1

learn from thy neighbor parallel chain and regional adaptive mcmc starting with the seminal paper of haario saksman and tamminen haario saksman and tamminen 2001 a substantial amount of work has been done to validate adaptive markov chain monte carlo algorithms in this paper we focus on two practical aspects of adaptive metropolis samplers first we draw attention to the deficient performance of standard adaptation when the target distribution is multimodal we propose a parallel chain adaptation strategy that incorporates multiple markov chains which are run in parallel second we note that the current adaptive mcmc paradigm implicitly assumes that the adaptation is uniformly efficient on all regions of the state space however in many practical instances different optimal kernels are needed in different regions of the state space we propose here a regional adaptation algorithm in which we account for possible errors made in defining the adaptation regions this corresponds to the more realistic case in which one does not know exactly the optimal regions

nonexistence of bounded energy solutions for a critical exponent problem abstract in this paper we discuss positive solutions for certain weighted elliptic equations with critical sobolev exponent in r n the weights depend on a positive parameter which is allowed to increase to infinity while for small values of solutions are completely classified an attempt to such a classification is much more difficult for large values of the parameter in the present work we prove the nonexistence of solutions with bounded energy as increases to infinity we also prove a multiplicity result for high energy solutions

adaptive markov chain monte carlo through regeneration abstract markov chain monte carlo mcmc is used for evaluating expectations of functions of interest under a target distribution this is done by calculating averages over the sample path of a markov chain having as its stationary distribution for computational efficiency the markov chain should be rapidly mixing this sometimes can be achieved only by careful design of the transition kernel of the chain on the basis of a detailed preliminary exploratory analysis of an alternative approach might be to allow the transition kernel to adapt whenever new features of are encountered during the mcmc run however if such adaptation occurs infinitely often then the stationary distribution of the chain may be disturbed we describe a framework based on the concept of markov chain regeneration which allows adaptation to occur infinitely often but does not disturb the stationary distribution of the chain or the consistency of sample path averages

a minimax method for finding multiple critical points and its applications to semilinear pdes most minimax theorems in critical point theory require one to solve a two level global optimization problem and therefore are not for algorithm implementation the objective of this research is to develop numerical algorithms and corresponding mathematical theory for finding multiple saddle points in a stable way in this paper inspired by the numerical works of choi mckenna and ding costa chen and the idea to define a solution submanifold some local minimax theorems are established which require us to solve only a two level local optimization problem based on the local theory a new local numerical minimax method for finding multiple saddle points is developed the local theory is applied and the numerical method is implemented successfully to solve a class of semilinear elliptic boundary value problems for multiple solutions on some nonconvex non star shaped and multiconnected domains numerical solutions are illustrated by their graphics for visualization in a subsequent paper y li and j zhou convergence results of a minimax method for finding critical points in review we establish some convergence results for the algorithm

approximate bayesian inference with the weighted likelihood bootstrap we introduce the weighted likelihood bootstrap wlb as a way to simulate approximately from a posterior distribution this method is often easy to implement requiring only an algorithm for calculating the maximum likelihood estimator such as iteratively reweighted least squares in the generic weighting scheme the wlb is first order correct under quite general conditions inaccuracies can be removed by using the wlb as a source of samples in the sampling importance resampling sir algorithm which also allows incorporation of particular prior information the sir adjusted wlb can be a competitive alternative to other integration methods in certain models asymptotic expansions elucidate the second order properties of the wlb which is a generalization of rubin s bayesian bootstrap the calculation of approximate bayes factors for model comparison is also considered we note that given a sample simulated from the posterior distribution the required marginal likelihood may be simulation consistently estimated by the harmonic mean of the associated likelihood values a modification of this estimator that avoids instability is also noted these methods provide simple ways of calculating approximate bayes factors and posterior model probabilities for a very wide class of models

asymptotic behaviour of positive solutions of elliptic equations with critical and supercritical growth ii the nonradial case abstract we consider positive solutions ue of a family of elliptic boundary value problems ie as e 0 they approach in some sense the problem i0 which has a nonlinearity with critical or supercritical growth it is shown how the solution ue concentrates at a point how this happens asymptotically and where in the domain it happens

on the use of local optimizations within metropolis hastings updates summary we propose new metropolis hastings algorithms for sampling from multimodal dis tributions on n tjelmeland and hegstad have obtained direct mode jumping proposals by optimization within metropolis hastings updates and different proposals for forward and backward steps we generalize their scheme by allowing the probability distribution for forward and backward kernels to depend on the current state we use the new setting to combine mode jumping proposals and proposals from a prior approximation we obtain that the frequency of proposals from the different proposal kernels is automatically adjusted to their quality mode jumping proposals include local optimizations when combining this with a prior approximation it is tempting to use local optimization results not only for mode jumping proposals but also to improve the prior approximation we show how this idea can be implemented the resulting algorithm is adaptive but has a markov structure we evaluate the effectiveness of the proposed algorithms in two simulation examples

batch means and spectral variance estimators in markov chain monte carlo calculating a monte carlo standard error mcse is an important step in the statistical analysis of the simulation output obtained from a markov chain monte carlo experiment an mcse is usually based on an estimate of the variance of the asymptotic normal distribution we consider spectral and batch means methods for estimating this variance in particular we establish conditions which guarantee that these estimators are strongly consistent as the simulation effort increases in addition for the batch means and overlapping batch means methods we establish conditions ensuring consistency in the mean square sense which in turn allows us to calculate the optimal batch size up to a constant of proportionality finally we examine the empirical finite sample properties of spectral variance and batch means estimators and provide recommendations for practitioners

symmetry properties of positive solutions for fully nonlinear elliptic systems abstract we investigate symmetry properties of positive solutions for fully nonlinear uniformly elliptic systems such as f i x d u i d 2 u i f i x u 1 u n d u i 0 1 i n in a bounded domain in r n with dirichlet boundary condition u 1 u n 0 on here f i s are nonincreasing with the radius r x and satisfy a cooperativity assumption in addition each f i is the sum of a locally lipschitz with a nondecreasing function in the variable u i and may have superlinear gradient growth we show that symmetry occurs for systems with nondifferentiable f i s by developing a unified treatment of the classical moving planes method in the spirit of gidas ni nirenberg we also present different applications of our results including uniqueness of positive solutions for lane emden systems in the subcritical case in a ball and symmetry for a class of systems with natural growth in the gradient

asymptotically radial solutions in expanding annular domains in this paper we consider the problem left begin array ll delta u u p quad rm in omega r u 0 quad quad quad rm on partial omega r quad quad quad 0 1 end array right where p 1 and r is a smooth bounded domain with a hole which is diffeomorphic to an annulus and expands as r longrightarrow infty the main goal of the paper is to prove for large r the existence of a positive solution to 0 1 which is close to the positive radial solution in the corresponding diffeomorphic annulus the proof relies on a careful analysis of the spectrum of the linearized operator at the radial solution as well as on a delicate analysis of the nondegeneracy of suitable approximating solutions

importance sampling for simulation of markovian physical processes transient markov chains are sometimes simulated to estimate rare event probabilities for illustration chains defined by an airborne particle dispersion model are used to estimate the probabilities that released particles reach various locations such estimated probabilities are needed for many purposes including exposure calculations for affected populations and optimization of detector placement by using experimental designs for simulation runs and embedding fitted regression models of output data in importance sampling transition kernels convergence is improved by factors of tens to hundreds

abundance of entire solutions to nonlinear elliptic equations by the variational method abstract we study entire bounded solutions to the equation u u u 3 0 in r 2 our approach is purely variational and is based on concentration arguments and symmetry considerations this method allows us to construct in an unified way several types of solutions with various symmetries radial breather type rectangular triangular hexagonal etc both positive and sign changing it is also applicable for more general equations in any dimension

adaptive sampling for bayesian variable selection our paper proposes adaptive monte carlo sampling schemes for bayesian variable selection in linear regression that improve on standard markov chain methods we do so by considering metropolis hastings proposals that make use of accumulated information about the posterior distribution obtained during sampling adaptation needs to be done carefully to ensure that sampling is from the correct ergodic distribution we give conditions for the validity of an adaptive sampling scheme in this problem and for simulating from a distribution on a finite state space in general and suggest a class of adaptive proposal densities which uses best linear prediction to approximate the gibbs sampler our sampling scheme is computationally much faster per iteration than the gibbs sampler and when this is taken into account the efficiency gains when using our sampling scheme compared to alternative approaches are substantial in terms of precision of estimation of posterior quantities of interest for a given amount of computation time we compare our method with other sampling schemes for examples involving both real and simulated data the methodology developed in the paper can be extended to variable selection in more general problems

blow up and asymptotic behavior of solutions for reaction diffusion equations with free boundaries abstract we study the blow up phenomena and the asymptotic behavior of time global solutions for reaction diffusion equations u t u x x a u u p a r and p 1 with free boundaries and initial data x we give a sharp threshold value a p 0 such that the solution blows up in finite time when vanishes i e u 0 as t when and when it converges as t to 0 or to an evenly decreasing positive stationary solution depending on whether a 0 or a 0 moreover when blow up happens we show that the blow up set is compact in the occupying domain of initial data and the free boundaries keep bounded

the multiple try method and local optimization in metropolis sampling abstract this article describes a new metropolis like transition rule the multiple try metropolis for markov chain monte carlo mcmc simulations by using this transition rule together with adaptive direction sampling we propose a novel method for incorporating local optimization steps into a mcmc sampler in continuous state space numerical studies show that the new method performs significantly better than the traditional metropolis hastings m h sampler with minor tailoring in using the rule the multiple try method can also be exploited to achieve the effect of a griddy gibbs sampler without having to bear with griddy approximations and the effect of a hit and run algorithm without having to figure out the required conditional distribution in a random direction

variational method for estimating the rate of convergence of markov chain monte carlo algorithms we demonstrate the use of a variational method to determine a quantitative lower bound on the rate of convergence of markov chain monte carlo mcmc algorithms as a function of the target density and proposal density the bound relies on approximating the second largest eigenvalue in the spectrum of the mcmc operator using a variational principle and the approach is applicable to problems with continuous state spaces we apply the method to one dimensional examples with gaussian and quartic target densities and we contrast the performance of the random walk metropolis hastings algorithm with a smart variant that incorporates gradient information into the trial moves a generalization of the metropolis adjusted langevin algorithm we find that the variational method agrees quite closely with numerical simulations we also see that the smart mcmc algorithm often fails to converge geometrically in the tails of the target density except in the simplest case we examine and even then care must be taken to choose the appropriate scaling of the deterministic and random parts of the proposed moves again this calls into question the utility of smart mcmc in more complex problems finally we apply the same method to approximate the rate of convergence in multidimensional gaussian problems with and without importance sampling there we demonstrate the necessity of importance sampling for target densities which depend on variables with a wide range of scales

existence of positive solutions for a class of semilinear and quasilinear elliptic equations with supercritical case abstract in this paper we consider a class of semilinear elliptic dirichlet problems in a bounded regular domain with cylindrical symmetry involving concave convex nonlinearities with supercritical growth using a new sobolev embedding theorem and variational method we show the existence of two positive solutions of the problem additionally we study the quasilinear elliptic equation and obtain a similar result

sobolev spaces of symmetric functions and applications abstract we prove sharp pointwise estimates for functions in the sobolev spaces of radial functions defined in a ball as a consequence we obtain some imbeddings of such sobolev spaces in weighted l q spaces we also prove similar imbeddings for sobolev spaces of functions with partial symmetry our techniques lead to new hardy type inequalities it is important to observe that we do not require any vanishing condition on the boundary to obtain all our estimates we apply these imbeddings to obtain radial solutions and partially symmetric solutions for a biharmonic equation of the henon type under both dirichlet and navier boundary conditions the delicate question of the regularity of these solutions is also established

nodal solutions for an elliptic problem involving large nonlinearities abstract let us consider the problem 0 1 u a x u u p 1 u in b 1 u 0 on b 1 where b 1 is the unit ball in r n n 3 and a x 0 is a smooth radial function under some suitable assumptions on the regular part of the green function of the operator u n 1 r u a r u we prove the existence of a changing sign solution to 0 1 for p large enough

improving samc using smoothing methods theory and applications to bayesian model selection problems stochastic approximation monte carlo samc has recently been proposed by liang liu and carroll j amer statist assoc 102 2007 305 320 as a general simulation and optimization algorithm in this paper we propose to improve its convergence using smoothing methods and discuss the application of the new algorithm to bayesian model selection problems the new algorithm is tested through a change point identification example the numerical results indicate that the new algorithm can outperform samc and reversible jump mcmc significantly for the model selection problems the new algorithm represents a general form of the stochastic approximation markov chain monte carlo algorithm it allows multiple samples to be generated at each iteration and a bias term to be included in the parameter updating step a rigorous proof for the convergence of the general algorithm is established under verifiable conditions this paper also provides a framework on how to improve efficiency of monte carlo simulations by incorporating some nonparametric techniques

stochastic approximation in monte carlo computation the wang landau wl algorithm is an adaptive markov chain monte carlo algorithm used to calculate the spectral density for a physical system a remarkable feature of the wl algorithm is that it is not trapped by local energy minima which is very important for systems with rugged energy landscapes this feature has led to many successful applications of the algorithm in statistical physics and biophysics however there does not exist rigorous theory to support its convergence and the estimates produced by the algorithm can reach only a limited statistical accuracy in this article we propose the stochastic approximation monte carlo samc algorithm which overcomes the shortcomings of the wl algorithm we establish a theorem concerning its convergence the estimates produced by samc can be improved continuously as the simulation proceeds samc also extends applications of the wl algorithm to continuum systems the potential uses of samc in statistics are discussed through two classes of applications i

fixed point methods and accretivity for perturbed nonlinear equations in banach spaces abstract in this paper we use fixed point theorems to guarantee the existence of solutions for inclusions of the form a u u f u g where a is a quasi m accretive operator defined in a banach space 0 and the nonlinear perturbation f satisfies some suitable conditions we apply the obtained results among other things to guarantee the existence of solutions of boundary value problems of the type u x u x f u x g x x and u 0 on where the laplace operator should be understood in the sense of distributions over and to study the existence and uniqueness of solution for a nonlinear integro differential equation posed in l 1

instability analysis of saddle points by a local minimax method the objective of this work is to develop some tools for local instability analysis of multiple critical points which can be computationally carried out the morse index can be used to measure local instability of a nondegenerate saddle point however it is very expensive to compute numerically and is ineffective for degenerate critical points a local weak linking index can also be defined to measure local instability of a degenerate saddle point but it is still too difficult to compute in this paper a local instability index called a local minimax index is defined by using a local minimax method this new instability index is known beforehand and can help in finding a saddle point numerically relations between the local minimax index and other local instability indices are established those relations also provide ways to numerically compute the morse local linking indices in particular the local minimax index can be used to define a local instability index of a saddle point relative to a reference trivial critical point even in a banach space while others failed to do so

existence of radial solutions for nonlinear elliptic equations with gradient terms in annular domains abstract this paper is concerned with the problem of elliptic equations in annular domains involving derivative terms using nonlinear analysis methods some results regarding existence of solutions are established more precisely at least one radial solution is obtained basing on schauder s fixed point theorem and contraction mapping theorem respectively some iterative technique is also applied which allows us to overcome difficulties that the nonlinearity is related to derivative terms

efficient parametrisations for normal linear mixed models summary the generality and easy programmability of modern sampling based methods for maximisation of likelihoods and summarisation of posterior distributions have led to a tremendous increase in the complexity and dimensionality of the statistical models used in practice however these methods can often be extremely slow to converge due to high correlations between or weak identifiability of certain model parameters we present simple hierarchical centring reparametrisations that often give improved convergence for a broad class of normal linear mixed models in particular we study the two stage hierarchical normal linear model the laird ware model for longitudinal data and a general structure for hierarchically nested linear models using analytical arguments simulation studies and an example involving clinical markers of acquired immune deficiency syndrome aids we indicate when reparametrisation is likely to provide substantial gains in efficiency

regeneration in markov chain samplers abstract markov chain sampling has recently received considerable attention in particular in the context of bayesian computation and maximum likelihood estimation this article discusses the use of markov chain splitting originally developed for the theoretical analysis of general state space markov chains to introduce regeneration into markov chain samplers this allows the use of regenerative methods for analyzing the output of these samplers and can provide a useful diagnostic of sampler performance the approach is applied to several samplers including certain metropolis samplers that can be used on their own or in hybrid samplers and is illustrated in several examples

permanent does not have succinct polynomial size arithmetic circuits of constant depth we show that over fields of characteristic zero there does not exist a polynomial p n and a constant free succinct arithmetic circuit family f n using division by constants where f n has size at most p n and depth o 1 such that f n computes the nxn permanent a circuit family f n is succinct if there exists a nonuniform boolean circuit family c n with o logn many inputs and size n o 1 such that c n can correctly answer direct connection language queries about f n succinctness is a relaxation of uniformity to obtain this result we develop a novel technique that further strengthens the connection between black box derandomization of polynomial identity testing and lower bounds for arithmetic circuits from this we obtain the lower bound by giving an explicit construction computable in the polynomial hierarchy of a hitting set for arithmetic circuits

a high linking algorithm for sign changing solutions of semilinear elliptic equations where n is a bounded domain with regular boundary the operator can be replaced by a more general second order uniformly elliptical operator in divergence form the solutions we are interested in nding are of minimax type and sign changing nonlinear elliptic equations of type 1 1 arise naturally in physics engineering and mathematical biology it is known that many such equations have multiple solutions their study has attracted the attention of many pure and applied mathematicians during the past two decades let us assume that f satis es the following regularity and growth conditions a1 f x t is locally lipschitz continuous in