variable transformation to obtain geometric ergodicity in the random walk metropolis algorithm a random walk metropolis sampler is geometrically ergodic if its equilibrium density is super exponentially light and satisfies a curvature condition stochastic process appl 85 2000 341 361 many applications including bayesian analysis with conjugate priors of logistic and poisson regression and of log linear models for categorical data result in posterior distributions that are not super exponentially light we show how to apply the change of variable formula for diffeomorphisms to obtain new densities that do satisfy the conditions for geometric ergodicity sampling the new variable and mapping the results back to the old gives a geometrically ergodic sampler for the original variable this method of obtaining geometric ergodicity has very wide applicability

nonregular languages in the kicked rotor a symbolic dynamics of the markov chain model for trajectories of the kicked rotor standard map in sticky regions near resonant orbits is developed two sets of trajectories are shown to correspond to a context free and a context sensitive language respectively

on distribution of number of trades in different time windows in the stock market abstract properties of distributions of the number of trades in different intraday time intervals for five stocks traded in micex are studied the dependence of the mean number of trades on the capital turnover is analyzed correlation analysis using factorial and h q moments demonstrates the multifractal nature of these distributions as well as some peculiar changes in the correlation pattern guided by the analogy with the analysis of particle multiplicity distributions in multiparticle production at high energies an evolution equation relating changes in capital turnover and a number of trades is proposed we argue that such equation can describe the observed features of the distribution of the number of trades in the stock market

reachability analysis of dynamical systems having piecewise constant derivatives abstract in this paper we consider a class of hybrid systems namely dynamical systems with piecewise constant derivatives pcd systems such systems consist of a partition of the euclidean space into a finite set of polyhedral sets regions within each region the dynamics is defined by a constant vector field hence discrete transitions occur only on the boundaries between regions where the trajectories change their direction with respect to such systems we investigate the reachability question given an effective description of the systems and of two polyhedral subsets p and q of the state space is there a trajectory starting at some x p and reaching some point in q our main results are a decision procedure for two dimensional systems and an undecidability result for three or more dimensions

a stochastic newton mcmc method for large scale statistical inverse problems with application to seismic inversion we address the solution of large scale statistical inverse problems in the framework of bayesian inference the markov chain monte carlo mcmc method is the most popular approach for sampling the posterior probability distribution that describes the solution of the statistical inverse problem mcmc methods face two central difficulties when applied to large scale inverse problems first the forward models typically in the form of partial differential equations that map uncertain parameters to observable quantities make the evaluation of the probability density at any point in parameter space very expensive and second the high dimensional parameter spaces that arise upon discretization of infinite dimensional parameter fields make the exploration of the probability density function prohibitive the challenge for mcmc methods is to construct proposal functions that simultaneously provide a good approximation of the target density while being inexpensive to manipulate here we present a so called stoch

inverse scattering exact solution of the gel fand levitan equation one formulation of inverse scattering theory involves the gel fand levitan equation we present a procedure for finding exact solutions to this equation this procedure can be applied whenever the reflection coefficient is a rational function of the wave number k with an arbitrary number of poles we present graphs of the potential as a function of distance for several cases with 3 4 5 and 6 poles prior to this paper no 4 5 or 6 pole case had ever been treated successfully

existence and uniqueness of monotone measure preserving maps introduction given a pair of borel probability measures and v on rd it is natural to ask whether v can be obtained from by redistributing its mass in a canonical way in the case of the line d 1 the answer is clear as long as both measures are free from atoms x v x 0mthere is a map y x of the line to itself for which uniquely determined almost everywhere this map may be taken to be nondecreasing by a suitable choice of y x e rw o at the remaining points interpreting and v as the initial and final distribution of a one dimensional fluid the transformation y x gives a rearrangement of fluid particles yielding final state v from the initial state this rearrangement is characterized by the fact that it preserves particle ordering obviating any need for two particles to cross although the generalization of this construction to higher dimensions is the focus of this paper the one dimensional case will be pursued slightly further when the measures are absolutely continuous with respect to lebesgue d x f x dx and dr y g y dymthen formally at least neglecting regularity issues the fundamental theorem of calculus yields

quasi linear geostatistical theory for inversing a quasi linear theory is presented for the geostatistical solution to the inverse problem the archetypal problem is to estimate the log transmissivity function from observations of head and log transmissivity at selected locations the unknown is parameterized as a realization of a random field and the estimation problem is solved in two phases structural analysis where the random field is characterized followed by estimation of the log transmissivity conditional on all observations the proposed method generalizes the linear approach of kitanidis and vomvoris 1983 the generalized method is superior to the linear method in cases of large contrast in formation properties but informative measurements i e there are enough observations that the variance of estimation error of the log transmissivity is small the methodology deals rigorously with unknown drift coefficients and yields estimates of covariance parameters that are unbiased and grid independent the applicability of the methodology is demonstrated through an example that includes structural analysis determination of best estimates and conditional simulations

bayesian inverse problems with l_1 priors a randomize then optimize approach prior distributions for bayesian inference that rely on the l 1 norm of the parameters are of considerable interest in part because they promote parameter fields with less regularity than gaussian priors e g discontinuities and blockiness these l 1 type priors include the total variation tv prior and the besov space b 1 1 s prior and in general yield non gaussian posterior distributions sampling from these posteriors is challenging particularly in the inverse problem setting where the parameter space is high dimensional and the forward problem may be nonlinear this paper extends the randomize then optimize rto method an optimization based sampling algorithm developed for bayesian inverse problems with gaussian priors to inverse problems with l 1 type priors we use a variable transformation to convert an l 1 type prior to a standard gaussian prior such that the posterior distribution of the transformed parameters is amenable to metropolized sampling via rto we demonstrate thi

a numerical resolution of the gelfand levitan equation abstract the gelfand levitan equation is the main equation in inverse scatterring theory a new numerical method for solving this equation is presented based on a predictor corrector scheme

a kalman filter solution of the inverse scattering problem with a rational reflection coefficient abstract this paper presents a new inverse scattering method for reconstructing the reflectivity function of symmetric two component wave equations or the potential of a schrodinger equation when the reflection coefficient is rational this method relies on the so called chandrasekhar equations which implement the kalman filter associated to a stationary state space model these equations are derived by using first a general layer stripping principle to obtain some differential equations for reconstructing a general scattering medium and by specializing these recursions to the case when the probing waves have a state space model

financial multifractality and its subtleties an example of dax abstract detailed study of multifractal characteristics of the financial time series of asset values and of its returns is performed using a collection of the high frequency deutsche aktienindex data the tail index the renyi exponents based on the box counting algorithm for the graph d q and the generalized hurst exponents h q are computed in parallel for short and daily return times the results indicate a more complicated nature of the stock market dynamics than just consistent multifractal

small universal turing machines abstract let utm m n be the class of universal turing machine with m states and n symbols universal turing machines are proved to exist in the following classes utm 24 2 utm 10 3 utm 7 4 utm 5 5 utm 4 6 utm 3 10 and utm 2 18

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

dimension independent likelihood informed mcmc many bayesian inference problems require exploring the posterior distribution of high dimensional parameters that represent the discretization of an underlying function this work introduces a family of markov chain monte carlo mcmc samplers that can adapt to the particular structure of a posterior distribution over functions two distinct lines of research intersect in the methods developed here first we introduce a general class of operator weighted proposal distributions that are well defined on function space such that the performance of the resulting mcmc samplers is independent of the discretization of the function second by exploiting local hessian information and any associated low dimensional structure in the change from prior to posterior distributions we develop an inhomogeneous discretization scheme for the langevin stochastic differential equation that yields operator weighted proposals adapted to the non gaussian structure of the posterior the resulting dimension independent and likelihood informed dili mcmc samplers may be useful for a large class of high dimensional problems where the target probability measure has a density with respect to a gaussian reference measure two nonlinear inverse problems are used to demonstrate the efficiency of these dili samplers an elliptic pde coefficient inverse problem and path reconstruction in a conditioned diffusion

inverse scattering for optical couplers exact solution of marchenko equations an exact solution of marchenko equations for rational scattering data of arbitrary order is developed with reference to the case of two coupled waves propagating in a lossless optical waveguide some numerical examples are presented for the synthesis of coupling structures having a prescribed frequency response

neo classical theory of competition or adam smith s hand as mathematized ideology abstract orthodox economic theory utility maximization rational agents efficient markets in equilibrium is based on arbitrarily postulated nonempiric notions the disagreement between economic reality and a key feature of neo classical economic theory was criticized empirically by osborne i show that the orthodox theory is internally self inconsistent for the very reason suggested by osborne lack of invertibility of demand and supply as functions of price to obtain price as functions of supply and demand the reason for the noninvertibililty arises from nonintegrable excess demand dynamics a feature of their theory completely ignored by economists

physical constraints on hypercomputation many attempts to transcend the fundamental limitations to computability implied by the halting problem for turing machines depend on the use of forms of hypercomputation that draw on notions of infinite or continuous as opposed to bounded or discrete computation thus such schemes may include the deployment of actualised rather than potential infinities of physical resources or of physical representations of real numbers to arbitrary precision here we argue that such bases for hypercomputation are not materially realisable and so cannot constitute new forms of effective calculability

eigenvalues and eigenfunctions associated with the gel fand levitan equation it is shown here that the solutions of the gel fand levitan equation for inverse potential scattering on the line may be expressed in terms of the eigenvalues and eigenfunctions of certain associated operators of trace class the details are sketched for the case of rational reflection coefficients and carried out for the simplest class of examples

closed form analytic maps in one and two dimensions can simulate universal turing machines we show closed form analytic functions consisting of a finite number of trigonometric terms can simulate turing machines with exponential slowdown in one dimension or in real time in two or more

randomize then optimize a method for sampling from posterior distributions in nonlinear inverse problems high dimensional inverse problems present a challenge for markov chain monte carlo mcmc type sampling schemes typically they rely on finding an efficient proposal distribution which can be dif

rates of convergence of the hastings and metropolis algorithms we apply recent results in markov chain theory to hastings and metropolis algorithms with either independent or symmetric candidate distributions and provide necessary and sufficient conditions for the algorithms to converge at a geometric rate to a prescribed distribution pi in the independence case in mathbb r k these indicate that geometric convergence essentially occurs if and only if the candidate density is bounded below by a multiple of pi in the symmetric case in mathbb r only we show geometric convergence essentially occurs if and only if pi has geometric tails we also evaluate recently developed computable bounds on the rates of convergence in this context examples show that these theoretical bounds can be inherently extremely conservative although when the chain is stochastically monotone the bounds may well be effective

bayesian inference with optimal maps we present a new approach to bayesian inference that entirely avoids markov chain simulation by constructing a map that pushes forward the prior measure to the posterior measure existence and uniqueness of a suitable measure preserving map is established by formulating the problem in the context of optimal transport theory we discuss various means of explicitly parameterizing the map and computing it efficiently through solution of an optimization problem exploiting gradient information from the forward model when possible the resulting algorithm overcomes many of the computational bottlenecks associated with markov chain monte carlo advantages of a map based representation of the posterior include analytical expressions for posterior moments and the ability to generate arbitrary numbers of independent posterior samples without additional likelihood evaluations or forward solves the optimization approach also provides clear convergence criteria for posterior approximation and facilitates model selection through automatic evaluation of the marginal likelihood we demonstrate the accuracy and efficiency of the approach on nonlinear inverse problems of varying dimension involving the inference of parameters appearing in ordinary and partial differential equations

on the solutions of the gel fand levitan equation the solutions of the gel fand levitan equation are obtained in a closed form for general rational reflection coefficients and as a convergent limit of closed forms for reflection coefficients sufficiently close to being rational

riemann manifold langevin and hamiltonian monte carlo methods the paper proposes metropolis adjusted langevin and hamiltonian monte carlo sampling methods defined on the riemann manifold to resolve the shortcomings of existing monte carlo algorithms when sampling from target densities that may be high dimensional and exhibit strong correlations the methods provide fully automated adaptation mechanisms that circumvent the costly pilot runs that are required to tune proposal densities for metropolis hastings or indeed hamiltonian monte carlo and metropolis adjusted langevin algorithms this allows for highly efficient sampling even in very high dimensions where different scalings may be required for the transient and stationary phases of the markov chain the methodology proposed exploits the riemann geometry of the parameter space of statistical models and thus automatically adapts to the local structure when simulating paths across this manifold providing highly efficient convergence and exploration of the target density the performance of these riemann manifold monte carlo methods is rigorously assessed by performing inference on logistic regression models log gaussian cox point processes stochastic volatility models and bayesian estimation of dynamic systems described by non linear differential equations substantial improvements in the time normalized effective sample size are reported when compared with alternative sampling approaches matlab code that is available from http www ucl ac uk statistics research rmhmc allows replication of all the results reported

reversible jump markov chain monte carlo computation and bayesian model determination markov chain monte carlo methods for bayesian computation have until recently been restricted to problems where the joint distribution of all variables has a density with respect to some fixed standard underlying measure they have therefore not been available for application to bayesian model determination where the dimensionality of the parameter vector is typically not fixed this paper proposes a new framework for the construction of reversible markov chain samplers that jump between parameter subspaces of differing dimensionality which is flexible and entirely constructive it should therefore have wide applicability in model determination problems the methodology is illustrated with applications to multiple change point analysis in one and two dimensions and to a bayesian comparison of binomial experiments

implicit sampling for hierarchical bayesian inversion and applications in fractional multiscale diffusion models abstract this paper presents an implicit sampling method for hierarchical bayesian inverse problems a widely used approach for sampling posterior distribution is markov chain monte carlo mcmc however the samples generated by mcmc are usually strongly correlated which may lead to a small size of effective samples the implicit sampling method proposed in chorin and tu 2009 can generate independent samples and capture inherent non gaussian features of the posterior in implicit sampling method the posterior samples are generated by a map and distribution around the maximum a posterior map point for practical applications some parameters in prior density are often unknown and the hierarchical bayesian formulation is used to estimate the map point effectively it is applied to the bayesian inverse problems of multi term time fractional diffusion models in heterogeneous media to effectively capture the heterogeneity effect we present a mixed generalized multiscale finite element method mixed gmsfem to substantially speed up the bayesian inversion finally we carry out a few numerical examples to effectively identify various unknown inputs

low dimensional hybrid systems decidable undecidable don t know even though many attempts have been made to define the boundary between decidable and undecidable hybrid systems the affair is far from being resolved more and more low dimensional systems are being shown to be undecidable with respect to reachability and many open problems in between are being discovered in this paper we present various two dimensional hybrid systems for which the reachability problem is undecidable we show their undecidability by simulating minsky machines their proximity to the decidability frontier is understood by inspecting the most parsimonious constraints necessary to make reachability over these automata decidable we also show that for other two dimensional systems the reachability question remains unanswered by proving that it is as hard as the reachability problem for piecewise affine maps on the real line which is a well known open problem

polar factorization and monotone rearrangement of vector valued functions given a probability space x p and a bounded domain r in rd equipped with the lebesgue measure 1 i normalized so that 10 i i it is shown under additional technical assumptions on x and q that for every vector valued function u e lp x p rd there is a unique polar factorization u v s where is a convex function defined on r and s is a measure preserving mapping from x p into q i i provided that u is nondegenerate in the sense that p u e 0 for each lebesgue negligible subset e of rd through this result the concepts of polar factorization of real matrices helmholtz decomposition of vector fields and nondecreasing rearrangements of real valued functions are unified the monge amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate monge kantorovich problem

exact solutions of the one dimensional acoustic wave equations for several new velocity profiles transmission and reflection coefficients after reviewing the relationship of the acoustic equation to the schrodinger equation solutions of the latter are used to construct new indices of refraction or equivalently sound velocity profiles svp for which the solutions of the acoustic equation are expressed in terms of elementary functions the new indices of refraction can be used to form layers to model more general inhomogeneous media just as piecewise constant slabs or other indices have been used in the past four of the new indices of refraction also provide especially simple closed form examples which can be used to describe how waves are scattered in media with such indices more general scattering problems are discussed in particular the notion of outgoing wave is gone into the five indices given in the present paper are the first of several which have been obtained from schrodinger equations with potentials for which the eigenfunctions can be found in closed form others will be given in later papers