analysis of the parareal time parallel time integration method the parareal algorithm is a method to solve time dependent problems parallel in time it approximates parts of the solution later in time simultaneously to parts of the solution earlier in time in this paper the relation of the parareal algorithm to space time multigrid and multiple shooting methods is first briefly discussed the focus of the paper is on new convergence results that show superlinear convergence of the algorithm when used on bounded time intervals and linear convergence for unbounded intervals

gibbs sampling methods for stick breaking priors a rich and flexible class of random probability measures which we call stick breaking priors can be constructed using a sequence of independent beta random variables examples of random measures that have this characterization include the dirichlet process its two parameter extension the two parameter poisson dirichlet process finite dimensional dirichlet priors and beta two parameter processes the rich nature of stick breaking priors offers bayesians a useful class of priors for nonparametric problems while the similar construction used in each prior can be exploited to develop a general computational procedure for fitting them in this article we present two general types of gibbs samplers that can be used to fit posteriors of bayesian hierarchical models based on stick breaking priors the first type of gibbs sampler referred to as a polya urn gibbs sampler is a generalized version of a widely used gibbs sampling method currently employed for dirichlet process computing this method applies t

asymptotic behaviour of the posterior distribution in overfitted mixture models abstract summary we study the asymptotic behaviour of the posterior distribution in a mixture model when the number of components in the mixture is larger than the true number of components a situation which is commonly referred to as an overfitted mixture we prove in particular that quite generally the posterior distribution has a stable and interesting behaviour since it tends to empty the extra components this stability is achieved under some restriction on the prior which can be used as a guideline for choosing the prior some simulations are presented to illustrate this behaviour

distribution spaces and a new construction of stochastic processes associated with the grassmann algebra we associate with the grassmann algebra a topological algebra of distributions which allows the study of processes analogous to the corresponding free stochastic processes with stationary increments as well as their derivatives

regularized estimation of large covariance matrices this paper considers estimating a covariance matrix of p variables from n observations by either banding the sample covariance matrix or estimating a banded version of the inverse of the covariance we show that these estimates are consistent in the operator norm as long as logp 2 n 0 and obtain explicit rates the results are uniform over some fairly natural well conditioned families of covariance matrices we also introduce an analogue of the gaussian white noise model and show that if the population covariance is embeddable in that model and well conditioned then the banded approximations produce consistent estimates of the eigenvalues and associated eigenvectors of the covariance matrix the results can be extended to smooth versions of banding and to non gaussian distributions with su ciently short tails a resampling approach is proposed for choosing the banding parameter in practice this approach is illustrated numerically on both simulated and real data

parallel high order integrators in this work we discuss a class of defect correction methods which is easily adapted to create parallel time integrators for multicore architectures and is ideally suited for developing methods which can be order adaptive in time the method is based on integral deferred correction idc which was itself motivated by spectral deferred correction by dutt greengard and rokhlin bit 40 2000 pp 241 266 the method presented here is a revised formulation of explicit idc dubbed revisionist idc ridc which can achieve p th order accuracy in wall clock time comparable to a single forward euler simulation on problems where the time to evaluate the right hand side of a system of differential equations is greater than latency costs of interprocessor communication such as in the case of the n body problem the key idea is to rewrite the defect correction framework so that after initial start up costs each correction loop can be lagged behind the previous correction loop in a manner that facilitates running the predictor and m p 1 correctors in parallel on an interval which has k steps where k gg p we prove that given an r th order runge kutta method in both the prediction and m correction loops of ridc then the method is order r times m 1 the parallelization in ridc uses a small number of cores the number of processors used is limited by the order one wants to achieve using a four core cpu it is natural to think about fourth order ridc built with forward euler or eighth order ridc constructed with second order runge kutta numerical tests on an n body simulation show that ridc methods can be significantly faster than popular runge kutta methods such as the classical fourth order runge kutta scheme in a pde setting one can imagine coupling ridc time integrators with parallel spatial evaluators thereby increasing the parallelization the ideas behind ridc extend to implicit and semi implicit idc and have high potential in this area

a numerical method for the accurate solution of the fokker planck landau equation in the nonhomogeneous case abstract a new approach for the accurate numerical solution of the fokker planck landau fpl equation in the nonhomogeneous case is presented the method couples through a time splitting algorithm a finite volume scheme for the transport with a fast spectral solver for the efficient solution of the collision operator recently introduced the scheme allows the use of different grids in the velocity space for the transport and the collision phases the use of a suitable explicit runge kutta solver for the time integration of the collision phase permits avoid once of excessive small time steps induced by the stiffness of the diffusive collision operator numerical results for both space homogeneous and space nonhomogeneous situations show the efficiency and the accuracy of the present method

a high order boris integrator this work introduces the high order boris sdc method for integrating the equations of motion for electrically charged particles in electric and magnetic fields boris sdc relies on a combination of the boris integrator with spectral deferred corrections sdc sdc can be considered as preconditioned picard iteration to compute the stages of a collocation method in this interpretation inverting the preconditioner corresponds to a sweep with a low order method in boris sdc the boris method a second order lorentz force integrator based on velocity verlet is used as a sweeper preconditioner the presented method provides a generic way to extend the classical boris integrator which is widely used in essentially all particle based plasma physics simulations involving magnetic fields to a high order method stability convergence order and conservation properties of the method are demonstrated for different simulation setups boris sdc reproduces the expected high order of convergence for a single particle and for the center of mass of a particle cloud in a penning trap and shows good long term energy stability introduces boris sdc an arbitrary order extension of the boris integrator boris sdc is based on merging the boris method with spectral deferred corrections sdc for a 2nd order problem using interpretation as preconditioned iteration analyzes properties of high order boris sdc for particles in a penning trap

accelerating the convergence of spectral deferred correction methods in the recent paper by dutt greengard and rokhlin a variant of deferred or defect correction methods is presented which couples gaussian quadrature with the picard integral equation formulation of the initial value ordinary differential equation the resulting spectral deferred correction sdc methods have been shown to possess favorable accuracy and stability properties even for versions with very high order of accuracy in this paper we show that for linear problems the iterations in the sdc algorithm are equivalent to constructing a preconditioned neumann series expansion for the solution of the standard collocation discretization of the ode this observation is used to accelerate the convergence of sdc using the gmres krylov subspace method for nonlinear problems the gmres acceleration is coupled with a linear implicit approach stability and accuracy analyses show the accelerated scheme provides an improvement in the accuracy efficiency and stability of the original sdc approach furthermore preliminary numerical experiments show that accelerating the convergence of sdc methods can effectively eliminate the order reduction previously observed for stiff ode systems

spike and slab variable selection frequentist and bayesian strategies variable selection in the linear regression model takes many apparent faces from both frequentist and bayesian standpoints in this paper we introduce a variable selection method referred to as a rescaled spike and slab model we study the importance of prior hierarchical specifications and draw connections to frequentist generalized ridge regression estimation specifically we study the usefulness of continuous bimodal priors to model hypervariance parameters and the effect scaling has on the posterior mean through its relationship to penalization several model selection strategies some frequentist and some bayesian in nature are developed and studied theoretically we demonstrate the importance of selective shrinkage for effective variable selection in terms of risk misclassification and show this is achieved using the posterior from a rescaled spike and slab model we also show how to verify a procedure s ability to reduce model uncertainty in finite samples using a specialized forward selection strategy using this tool we illustrate the effectiveness of rescaled spike and slab models in reducing model uncertainty

on a class of quaternionic positive definite functions and their derivatives in this paper we start the study of stochastic processes over the skew field of quaternions we discuss the relation between positive definite functions and the covariance of centered gaussian processes and the construction of stochastic processes and their derivatives the use of perfect spaces and strong algebras and the notion of fock space are crucial in this framework

high dimensional sparse factor modeling applications in gene expression genomics we describe studies in molecular profiling and biological pathway analysis that use sparse latent factor and regression models for microarray gene expression data we discuss breast cancer applications and key aspects of the modeling and computational methodology our case studies aim to investigate and characterize heterogeneity of structure related to specific oncogenic pathways as well as links between aggregate patterns in gene expression profiles and clinical biomarkers based on the metaphor of statistically derived factors as representing biological subpathway structure we explore the decomposition of fitted sparse factor models into pathway subcomponents and investigate how these components overlay multiple aspects of known biological activity our methodology is based on sparsity modeling of multivariate regression anova and latent factor models as well as a class of models that combines all components hierarchical sparsity priors address questions of dimension reduction and multiple co

gleason s problem and schur multipliers in the multivariable quaternionic setting abstract we define and study the counterparts of gleason s problem of the drury arveson s space and of schur multipliers when the unit ball of c n is replaced by the unit ball of h n where h denotes the algebra of real quaternions schur multipliers are characterized in terms of coisometric operator matrices in quaternionic spaces finally we define the counterpart of blaschke factors in this setting and study a one point interpolation problem

sparse bayesian infinite factor models we focus on sparse modelling of high dimensional covariance matrices using bayesian latent factor models we propose a multiplicative gamma process shrinkage prior on the factor loadings which allows introduction of infinitely many factors with the loadings increasingly shrunk towards zero as the column index increases we use our prior on a parameter expanded loading matrix to avoid the order dependence typical in factor analysis models and develop an efficient gibbs sampler that scales well as data dimensionality increases the gain in efficiency is achieved by the joint conjugacy property of the proposed prior which allows block updating of the loadings matrix we propose an adaptive gibbs sampler for automatically truncating the infinite loading matrix through selection of the number of important factors theoretical results are provided on the support of the prior and truncation approximation bounds a fast algorithm is proposed to produce approximate bayes estimates latent factor regression methods are developed for prediction and variable selection in applications with high dimensional correlated predictors operating characteristics are assessed through simulation studies and the approach is applied to predict survival times from gene expression data

generalized fock spaces and the stirling numbers the bargmann fock segal space plays an important role in mathematical physics and has been extended into a number of directions in the present paper we imbed this space into a gelfand triple the spaces forming the frechet part i e the space of test functions of the triple are characterized both in a geometric way and in terms of the adjoint of multiplication by the complex variable using the stirling numbers of the second kind the dual of the space of test functions has a topological algebra structure of the kind introduced and studied by the first named author and salomon

quaternion valued positive definite functions on locally compact abelian groups and nuclear spaces in this paper we study quaternion valued positive definite functions on locally compact abelian groups real countably hilbertian nuclear spaces and on the space r n x 1 x 2 x d r endowed with the tychonoff topology in particular we prove a quaternionic version of the bochner minlos theorem a tool for proving this result is a classical matricial analogue of the bochner minlos theorem which we believe is new we will see that in all these various settings the integral representation is with respect to a quaternion valued measure which has certain symmetry properties

topological convolution algebras abstract in this paper we introduce a dual of reflexive frechet counterpart of banach algebras of the form p n p where the p are dual of banach spaces with associated norms p which carry inequalities of the form a b p a p q a q b p and b a p a p q a q b p for p q d where d is preassigned and a p q is a constant we study the functional calculus and the spectrum of the elements of these algebras we then focus on the particular case p l 2 s p where s is a borel semi group in a locally compact group g and multiplication is convolution we give a sufficient condition on the measures p for such inequalities to hold finally we present three examples one is the algebra of germs of holomorphic functions in zero the second related to dirichlet series and the third in the setting of non commutative stochastic distributions

the horseshoe estimator for sparse signals this paper proposes a new approach to sparsity called the horseshoe estimator the horseshoe is a close cousin of other widely used bayes rules arising from for example double exponential and cauchy priors in that it is a member of the same family of multivariate scale mixtures of normals but the horseshoe enjoys a number of advantages over existing approaches including its robustness its adaptivity to dierent sparsity patterns and its analytical tractability we prove two theorems that formally characterize both the horseshoe s adeptness at large outlying signals and its super ecient rate of convergence to the correct estimate of the sampling density in sparse situations finally using a combination of real and simulated data we show that the horseshoe estimator corresponds quite closely to the answers one would get by pursuing a full bayesian model averaging approach using a discrete mixture prior to model signals and noise

integral deferred correction methods constructed with high order runge kutta integrators spectral deferred correction sdc methods for solving ordinary differential equations odes were introduced by dutt greengard and rokhlin 2000 it was shown in that paper that sdc methods can achieve arbitrary high order accuracy and possess nice stability properties their sdc methods are constructed with low order integrators such as forward euler or backward euler and are able to handle stiff and non stiff terms in the odes in this paper we use high order runge kutta rk integrators to construct a family of related methods which we refer to as integral deferred correction idc methods the distribution of quadrature nodes is assumed to be uniform and the corresponding local error analysis is given the smoothness of the error vector associated with an idc method measured by the discrete sobolev norm is a crucial tool in our analysis the expected order of accuracy is demonstrated through several numerical examples superior numerical stability and accuracy regions are observed when high order rk integrators are used to construct idc methods

bayesian cumulative shrinkage for infinite factorizations the dimension of the parameter space is typically unknown in a variety of models that rely on factorizations for example in factor analysis the number of latent factors is not known and has to be inferred from the data although classical shrinkage priors are useful in such contexts increasing shrinkage priors can provide a more effective approach that progressively penalizes expansions with growing complexity in this article we propose a novel increasing shrinkage prior called the cumulative shrinkage process for the parameters that control the dimension in overcomplete formulations our construction has broad applicability and is based on an interpretable sequence of spike and slab distributions which assign increasing mass to the spike as the model complexity grows using factor analysis as an illustrative example we show that this formulation has theoretical and practical advantages relative to current competitors including an improved ability to recover the model dimension an adaptive markov chain monte carlo algorithm is proposed and the performance gains are outlined in simulations and in an application to personality data

uniqueness of the scalar product in the tensor product of quaternion hilbert modules in the construction of a tensor product of quaternion hilbert modules given in a previous work real complex and quaternionic inner products were defined in the vector spaces formed from the tensor product of quaternion algebras h modulo an appropriate left ideal in each case under conditions that are necessary for the definition of a scalar product in the quaternionic hilbert modules and a natural condition on the algebraic structure it is proven that the scalar products which are defined are unique

the spectral theorem for quaternionic unbounded normal operators based on the s spectrum in this paper we prove the spectral theorem for quaternionic unbounded normal operators using the notion of s spectrum the proof technique consists of first establishing a spectral theorem for quaternionic bounded normal operators and then using a transformation which maps a quaternionic unbounded normal operator to a quaternionic bounded normal operator with this paper we complete the foundation of spectral analysis of quaternionic operators the s spectrum has been introduced to define the quaternionic functional calculus but it turns out to be the correct object also for the spectral theorem for quaternionic normal operators the lack of a suitable notion of spectrum was a major obstruction to fully understand the spectral theorem for quaternionic normal operators a prime motivation for studying the spectral theorem for quaternionic unbounded normal operators is given by the subclass of unbounded anti self adjoint quaternionic operators which play a crucial role in the quaternionic quantum mechanics