on non archimedean recurrence equations and their applications in the present paper we study stability of recurrence equations which in particular case contain dynamics of rational functions generated by contractive functions defined on an arbitrary non archimedean algebra moreover multirecurrence equations are considered we also investigate reverse recurrence equations which have application in the study of p adic gibbs measures note that our results also provide the existence of unique solutions of nonlinear functional equations we should stress that the non archimedeanity of the algebra is essentially used therefore the methods applied in the present paper are not valid in the archimedean setting

level spacing distributions and the bessel kernel scaling models of randomn n hermitian matrices and passing to the limitn leads to integral operators whose fredholm determinants describe the statistics of the spacing of the eigenvalues of hermitian matrices of large order for the gaussian unitary ensemble and for many others as well the kernel one obtains by scaling in the bulk of the spectrum is the sine kernel frac sin pi x y pi x y rescaling the gue at the edge of the spectrum leads to the kernel frac ai x ai y ai x ai y x y where ai is the airy function in previous work we found several analogies between properties of this airy kernel and known properties of the sine kernel a system of partial differential equations associated with the logarithmic differential of the fredholm determinant when the underlying domain is a union of intervals a representation of the fredholm determinant in terms of a painleve transcendent in the case of a single interval and also in this case asymptotic expansions for these determinants and related quantities achieved with the help of a differential operator which commutes with the integral operator in this paper we show that there are completely analogous properties for a class of kernels which arise when one rescales the laguerre or jacobi ensembles at the edge of the spectrum namely frac j alpha sqrt x sqrt y j alpha sqrt y sqrt x j alpha sqrt x j alpha sqrt y 2 x y wherej z is the bessel function of order in the cases 1 2 these become after a variable change the kernels which arise when taking scaling limits in the bulk of the spectrum for the gaussian orthogonal and symplectic ensembles in particular an asymptotic expansion we derive will generalize ones found by dyson for the fredholm determinants of these kernels

on a class of rational p adic dynamical systems in this paper we investigate the behavior of trajectories of one class of rational p adic dynamical systems in complex p adic field cp we studied siegel disks and attractors of such dynamical systems we found the basin of the attractor of the system it is proved that such dynamical systems are not ergodic on a unit sphere with respect to the haar measure

spiking the random matrix hard edge we characterize the limiting smallest eigenvalue distributions or hard edge laws for sample covariance type matrices drawn from a spiked population in the case of a single spike the results are valid in the context of the general beta ensembles for multiple spikes the necessary construction restricts matters to real complex or quaternion beta 1 2 or 4 ensembles the limit laws are described in terms of random integral operators and partial differential equations satisfied by the corresponding distribution functions are derived as corollaries we also show that under a natural limit all spiked hard edge laws derived here degenerate to the critically spiked soft edge laws or deformed tracy widom laws the latter were first described at beta 2 by baik ben arous and peche ann probab 33 1643 1697 2005 and from a unified beta random operator point of view by bloemendal and virag probab theory relat fields 156 795 825 2013 ann probab arxiv 1109 3704 2011

a feti preconditioner for two dimensional edge element approximations of maxwell s equations on nonmatching grids a class of feti methods for the mortar approximation of a vector field problem in two dimensions is proposed edge element discretizations of lowest degree are considered the method proposed can be employed with geometrically conforming and nonconforming partitions our numerical results show that its condition number increases only with the number of unknowns in each subdomain and is independent of the number of subdomains and the size of the problem

a method of finite element tearing and interconnecting and its parallel solution algorithm a novel domain decomposition approach for the parallel finite element solution of equilibrium equations is presented the spatial domain is partitioned into a set of totally disconnected subdomains each assigned to an individual processor lagrange multipliers are introduced to enforce compatibility at the interface nodes in the static case each floating subdomain induces a local singularity that is resolved in two phases first the rigid body modes are eliminated in parallel from each local problem and a direct scheme is applied concurrently to all subdomains in order to recover each partial local solution next the contributions of these modes are related to the lagrange multipliers through an orthogonality condition a parallel conjugate projected gradient algorithm is developed for the solution of the coupled system of local rigid modes components and lagrange multipliers which completes the solution of the problem when implemented on local memory multiprocessors this proposed method of tearing and interconnecting requires less interprocessor communications than the classical method of substructuring it is also suitable for parallel vector computers with shared memory moreover unlike parallel direct solvers it exhibits a degree of parallelism that is not limited by the bandwidth of the finite element system of equations

geometric location of periodic points of 2 ramified power series abstract in this paper we study the geometric location of periodic points of power series defined over fields of positive characteristic p we find a lower bound for the norm of all nonzero periodic points in the open unit disk of 2 ramified power series we prove that this bound is optimal for a large class of power series our main technical result is a computation of the first significant terms of p power iterates of 2 ramified power series as a by product we obtain a self contained proof of the characterization of 2 ramified power series

diffusing particles with electrostatic repulsion we study a diffusion model of an interacting particles system with general drift and diffusion coefficients and electrostatic inter particles repulsion more precisely the finite particle system is shown to be well defined thanks to recent results on multivalued stochastic differential equations see 2 and then we consider the behaviour of this system when the number of particles n goes to infinity through the empirical measure process in the particular case of affine drift and constant diffusion coefficient we prove that a limiting measure valued process exists and is the unique solution of a deterministic pde our treatment of the convergence problem as n uparrow infty is partly similar to that of t chan 3 and l c g rogers z shi 5 except we consider here a more general case allowing collisions between particles which leads to a second order limiting pde

a preconditioner for substructuring based on constrained energy minimization a preconditioner for substructuring based on constrained energy minimization concepts is presented the preconditioner is applicable to both structured and unstructured meshes and offers a straightforward approach for the iterative solution of second and fourth order structural mechanics problems the approach involves constraints associated with disjoint sets of nodes on substructure boundaries these constraints provide the means for preconditioning at both the substructure and global levels numerical examples are presented that demonstrate the good performance of the method in terms of iterations compute time and condition numbers of the preconditioned equations

the largest sample eigenvalue distribution in the rank 1 quaternionic spiked model of wishart ensemble we solve the largest sample eigenvalue distribution problem in the rank 1 spiked model of the quaternionic wishart ensemble which is the first case of a statistical generalization of the laguerre symplectic ensemble lse on the soft edge we observe a phase change phenomenon similar to that in the complex case and prove that the new distribution at the phase change point is the goe tracy widom distribution

domain decomposition methods for large linearly elliptic three dimensional problems abstract the idea of solving large problems using domain decomposition technique appears particularly attractive on present day large scale parallel computers but the performance of such techniques used on a parallel computer depends on both the numerical efficiency of the proposed algorithm and the efficiency of its parallel implementation the approach proposed herein splits the computational domain in unstructured subdomains of arbitrary shape and solves for unknown on the interface using the associated trace operator the steklov poincare operator on the continuous level or the schur complement matrix after a finite element discretization and a preconditioned conjugate gradient method this algorithm involves the solution of dirichlet and of neumann problems defined on each subdomain and which can be solved in parallel this method has been implemented on a cray 2 computer using multitasking and on an intel hypercube it was tested on a large scale industrial ill conditioned three dimensional linear elasticity problem which gives a fair indication of its performance in a real life environment in such situations the proposed method appears operational and competitive on both machines compared to standard techniques it yields faster results with far less memory requirements

uncertainty propagation using wiener haar expansions an uncertainty quantification scheme is constructed based on generalized polynomial chaos pc representations two such representations are considered based on the orthogonal projection of uncertain data and solution variables using either a haar or a legendre basis governing equations for the unknown coefficients in the resulting representations are derived using a galerkin procedure and then integrated in order to determine the behavior of the stochastic process the schemes are applied to a model problem involving a simplified dynamical system and to the classical problem of rayleigh benard instability for situations involving random parameters close to a critical point the computational implementations show that the wiener haar wha representation provides more robust predictions that those based on a wiener legendre wle decomposition however when the solution depends smoothly on the random data the wle scheme exhibits superior convergence suggestions regarding future extensions are finally drawn based on these experiences

brownian motions of ellipsoids on donne un traitement elementaire des mouvements browniens d ellipsoides et en particulier des resultats remarquables obtenus par dynkin

self normalized processes exponential inequalities moment bounds and iterated logarithm laws self normalized processes arise naturally in statistical applications being unit free they are not affected by scale changes moreover self normalization often eliminates or weakens moment assumptions in this paper we present several exponential and moment inequalities particularly those related to laws of the iterated logarithm for self normalized random variables including martingales tail probability bounds are also derived for random variables bt 0 and at let yt exp at 2bt2 2 we develop inequalities for the moments of at bt or supt 0at bt log logbt 1 2 and variants thereof when eyt 1 or when yt is a supermartingale for all belonging to some interval our results are valid for a wide class of random processes including continuous martingales with at mt and b t sqrt langle m rangle t and sums of conditionally symmetric variables di with at i 1tdi and b t sqrt sum i 1 t d i 2 a sharp maximal inequality for conditionally symmetric random variables and for continuous local martingales with values in rm m 1 is also established another development in this paper is a bounded law of the iterated logarithm for general adapted sequences that are centered at certain truncated conditional expectations and self normalized by the square root of the sum of squares the key ingredient in this development is a new exponential supermartingale involving i 1tdi and i 1tdi2 a compact law of the iterated logarithm for self normalized martingales is also derived in this connection

sparse tensor discretization of elliptic spdes we propose and analyze sparse deterministic stochastic tensor galerkin finite element methods sparse sgfems for the numerical solution of elliptic partial differential equations pdes with random coefficients in a physical domain d subset mathbb r d in tensor product sgfems the variational solution to the boundary value problem is approximated in tensor product finite element spaces v gamma otimes v d where v gamma and v d denote suitable finite dimensional subspaces of the stochastic and deterministic function spaces respectively these approaches lead to sgfem algorithms of complexity o n gamma n d where n gamma dim v gamma and n d dim v d in this work we use hierarchic sequences v gamma 1 subset v gamma 2 subset ldots and v d 1 subset v d 2 subset ldots of finite dimensional spaces to approximate the law of the random solution the hierarchies of approximation spaces allow us to define sparse tensor product spaces v gamma ell hat otimes v d ell ell 1 2 dots yielding algorithms of o n gamma log n d n d log n gamma work and memory we estimate the convergence rate of sgfem for algebraic decay of the input random field karhunen loeve coefficients we give an algorithm for an input adapted a priori selection of deterministic and stochastic discretization spaces the convergence rate in terms of the total number of degrees of freedom of the proposed method is superior to monte carlo approximations numerical examples illustrate the theoretical results and demonstrate superiority of the sparse tensor product discretization proposed here versus the full tensor product approach

p adic 2 1 rational dynamical systems abstract we study the dynamics of an arbitrary 2 1 rational function f x x 2 a x b c x d on the field c p of complex p adic numbers we show that the p adic dynamical system generated by f has a very rich behavior siegel disks may either coincide or be disjoint for different fixed points of the dynamical system also we find the basin of the attractor of the system varying the parameters it is proven that there are periodic trajectories for some values of the parameters there are trajectories which go arbitrary far from fixed points

stochastic model reduction for chaos representations abstract this paper addresses issues of model reduction of stochastic representations and computational efficiency of spectral stochastic galerkin schemes for the solution of partial differential equations with stochastic coefficients in particular an algorithm is developed for the efficient characterization of a lower dimensional manifold occupied by the solution to a stochastic partial differential equation spde in the hilbert space spanned by wiener chaos a description of the stochastic aspect of the problem on two well separated scales is developed to enable the stochastic characterization on the fine scale using algebraic operations on the coarse scale with such algorithms at hand the solution of spde s becomes both computationally manageable and efficient moreover a solid foundation is thus provided for the adaptive error control in stochastic galerkin procedures different aspects of the proposed methodology are clarified through its application to an example problem from solid mechanics

kronecker product approximations for some function related matrices abstract a new approximation tool such as sums of kronecker products is recently found to provide a superb compression property on a series of numerical examples of quite a general nature the purpose of the paper is explanation of this phenomenon in the form of existence theorems for matrix approximations of low kronecker rank for some classes of function related matrices including important specimens from potential theory this lays the grounds for development of new approximation algorithms for example in the cases when a matrix is associated with a shift invariant function on the cartesian product of nonunform grids which is of great practical interest in the solution of integral equations on plates or screens

tensor structured galerkin approximation of parametric and stochastic elliptic pdes we investigate the convergence rate of approximations by finite sums of rank 1 tensors of solutions of multiparametric elliptic pdes such pdes arise for example in the parametric deterministic reformulation of elliptic pdes with random field inputs based for example on the m term truncated karhunen loeve expansion our approach could be regarded as either a class of compressed approximations of these solutions or as a new class of iterative elliptic problem solvers for high dimensional parametric elliptic pdes providing linear scaling complexity in the dimension m of the parameter space it is based on rank reduced tensor formatted separable approximations of the high dimensional tensors and matrices involved in the iterative process combined with the use of spectrally equivalent low rank tensor structured preconditioners to the parametric matrices resulting from a finite element discretization of the high dimensional parametric deterministic problems numerical illustrations for the m dimensional parametric elliptic pdes resulting from spdes on parameter spaces of dimensions m leq100 indicate the advantages of employing low rank tensor structured matrix formats in the numerical solution of such problems

rank 1 real wishart spiked model in this paper we consider n dimensional real wishart matrices y in the class input amssym w bbb r sigma m in which all but one eigenvalue of is 1 let the nontrivial eigenvalue of be 1 then as n m with m n 2 finite and nonzero the eigenvalue distribution of y will converge into the marchenko pastur distribution inside a bulk region when increases from 0 one starts to see a stray eigenvalue of y outside of the support of the marchenko pastur density as this stray eigenvalue leaves the bulk region a phase transition will occur in the largest eigenvalue distribution of the wishart matrix in this paper we will compute the asymptotics of the largest eigenvalue distribution when the phase transition occurs we will first establish the results that are valid for all n and m and will use them to carry out the asymptotic analysis in particular we have derived a contour integral formula for the harish chandra itzykson zuber integral when x and y are real symmetric and y is a rank 1 matrix this allows us to write down a fredholm determinant formula for the largest eigenvalue distribution and analyze it using orthogonal polynomial techniques as a result we obtain an integral formula for the largest eigenvalue distribution in the large n limit characterized by painleve transcendents the approach used in this paper is very different from a recent paper by bloemenal and virag in which the largest eigenvalue distribution was obtained using a stochastic operator method in particular the painleve formula for the largest eigenvalue distribution obtained in this paper is new 2012 wiley periodicals inc

a stochastic deterministic coupling method for continuum mechanics in this paper we present a novel approach that allows to couple a deterministic continuum model with a stochastic continuum one the coupling strategy is performed in the arlequin framework which is based on a volume coupling and a partition of the energy a suitable functional space is chosen for the weak enforcement of the continuity between the two models the choice of this space ensures that the mean of the stochastic solution equals the deterministic solution point wise and enforces appropriate boundary conditions on the stochastic dimension the proof of the existence of the solution of the mixed problem is provided the numerical strategy is also reviewed in particular with a view at the monte carlo method finally examples show the interest of the method and possible strategies for use in adaptive modeling

painleve representation of tracy widom _ beta distribution for beta 6 in rumanov j math phys 56 013508 2015 we found explicit lax pairs for the soft edge of beta ensembles with even integer values of beta using this general result the case beta 6 is further considered here this is the smallest even beta when the corresponding lax pair and its relation to painleve ii pii have not been known before unlike cases beta 2 and 4 it turns out that again everything can be expressed in terms of the hastings mcleod solution of pii in particular a second order nonlinear ordinary differential equation ode for the logarithmic derivative of tracy widom distribution for beta 6 involving the pii function in the coefficients is found which allows one to compute asymptotics for the distribution function the ode is a consequence of a linear system of three odes for which the local singularity analysis yields series solutions with exponents in the set 4 3 1 3 and 2 3

the distribution of lyapunov exponents exact results for random matrices simple exact expressions are derived for all the lyapunov exponents of certainn dimensional stochastic linear dynamical systems in the case of the product of independent random matrices each of which has independent gaussian entries with mean zero and variance 1 n the exponents have an exponential distribution asn in the case of the time ordered product integral of exp n 1 2dw where the entries of then n matrixw t are independent standard wiener processes the exponents are equally spaced for fixedn and thus have a uniform distribution as n

tensor decompositions and applications this survey provides an overview of higher order tensor decompositions their applications and available software a tensor is a multidimensional or n way array decompositions of higher order tensors i e n way arrays with n geq 3 have applications in psycho metrics chemometrics signal processing numerical linear algebra computer vision numerical analysis data mining neuroscience graph analysis and elsewhere two particular tensor decompositions can be considered to be higher order extensions of the matrix singular value decomposition candecomp parafac cp decomposes a tensor as a sum of rank one tensors and the tucker decomposition is a higher order form of principal component analysis there are many other tensor decompositions including indscal parafac2 candelinc dedicom and paratuck2 as well as nonnegative variants of all of the above the n way toolbox tensor toolbox and multilinear engine are examples of software packages for working with tensors

an adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations in recent years there has been a growing interest in analyzing and quantifying the effects of random inputs in the solution of ordinary partial differential equations to this end the spectral stochastic finite element method ssfem is the most popular method due to its fast convergence rate recently the stochastic sparse grid collocation method has emerged as an attractive alternative to ssfem it approximates the solution in the stochastic space using lagrange polynomial interpolation the collocation method requires only repetitive calls to an existing deterministic solver similar to the monte carlo method however both the ssfem and current sparse grid collocation methods utilize global polynomials in the stochastic space thus when there are steep gradients or finite discontinuities in the stochastic space these methods converge very slowly or even fail to converge in this paper we develop an adaptive sparse grid collocation strategy using piecewise multi linear hierarchical basis functions hierarchical surplus is used as an error indicator to automatically detect the discontinuity region in the stochastic space and adaptively refine the collocation points in this region numerical examples especially for problems related to long term integration and stochastic discontinuity are presented comparisons with monte carlo and multi element based random domain decomposition methods are also given to show the efficiency and accuracy of the proposed method

feti and neumann neumann iterative substructuring methods connections and new results the feti and neumann neumann families of algorithms are among the best know and most severely tested domain decomposition methods for elliptic partial differential equations they are iterative substructuring methods and have many algorithmic components in common but there are also differences the purpose of this paper is to further unify the theory for these two families of methods and to introduce a new family of feti algorithms bounds on the rate of convergence which are uniform with respect to the coefficients of a family of elliptic problems with heterogeneous coefficients are established for these new algorithms the theory for a variant of the neumann neumann algorithm is also redeveloped stressing similarities to that for the feti methods

two level finite element method and its application to the helmholtz equation a two level finite element method is introduced and its application to the helmholtz equation is considered the method retains the desirable features of the galerkin method enriched with residual free bubbles while it is not limited to discretizations using elements with simple geometry the method can be applied to other equations and to irregular shaped domains

limits of spiked random matrices ii given a large high dimensional sample from a spiked population the top sample covariance eigenvalue is known to exhibit a phase transition we show that the largest eigenvalues have asymptotic distributions near the phase transition in the rank one spiked real wishart setting and its general analogue proving a conjecture of baik et al ann probab 33 1643 1697 2005 we also treat shifted mean gaussian orthogonal and ensembles such results are entirely new in the real case in the complex case we strengthen existing results by providing optimal scaling assumptions one obtains the known limiting random schrodinger operator on the half line but the boundary condition now depends on the perturbation we derive several characterizations of the limit laws in which appears as a parameter including a simple linear boundary value problem this pde description recovers known explicit formulas at 2 4 yielding in particular a new and simple proof of the painleve representations for these tracy widom distributions

partitioned treatment of uncertainty in coupled domain problems a separated representation approach abstract this work is concerned with the propagation of uncertainty across coupled domain problems with high dimensional random inputs a stochastic model reduction approach based on low rank separated representations is proposed for the partitioned treatment of the uncertainty space the construction of the coupled domain solution is achieved though a sequence of approximations with respect to the dimensionality of the random inputs associated with each individual sub domain and not the combined dimensionality hence drastically reducing the overall computational cost the coupling between the sub domain solutions is done via the classical finite element tearing and interconnecting feti method thus providing a well suited framework for parallel computing two high dimensional stochastic problems a 2d elliptic pde with random diffusion coefficient and a stochastic linear elasticity problem have been considered to study the performance and accuracy of the proposed stochastic coupling approach

on p adic λ model on the cayley tree we consider a nearest neighbor p adic model with spin values 1 on the cayley tree of order k 1 we prove that a p adic gibbs measure is unique for p 3 if p 2 then we find a condition which guarantees uniqueness of p adic gibbs measure besides the results are applied to the p adic ising model

matrix models for beta ensembles this paper constructs tridiagonal random matrix models for general 0 hermite gaussian and laguerre wishart ensembles these generalize the well known gaussian and wishart models for 1 2 4 furthermore in the cases of the laguerre ensembles we eliminate the exponent quantization present in the previously known models we further discuss applications for the new matrix models and present some open problems

numerical operator calculus in higher dimensions when an algorithm in dimension one is extended to dimension d in nearly every case its computational cost is taken to the power d this fundamental difficulty is the single greatest impediment to solving many important problems and has been dubbed the curse of dimensionality for numerical analysis in dimension d we propose to use a representation for vectors and matrices that generalizes separation of variables while allowing controlled accuracy basic linear algebra operations can be performed in this representation using one dimensional operations thus bypassing the exponential scaling with respect to the dimension although not all operators and algorithms may be compatible with this representation we believe that many of the most important ones are we prove that the multiparticle schrodinger operator as well as the inverse laplacian can be represented very efficiently in this form we give numerical evidence to support the conjecture that eigenfunctions inherit this property by computing the ground state eigenfunction for a simplified schrodinger operator with 30 particles we conjecture and provide numerical evidence that functions of operators inherit this property in which case numerical operator calculus in higher dimensions becomes feasible

universality of the stochastic bessel operator we establish universality at the hard edge for general beta ensembles assuming that the background potential v is a polynomial such that x mapsto v x 2 is strongly convex beta ge 1 and the dimension difference parameter a ge 0 the method rests on the corresponding tridiagonal matrix models showing that their appropriate continuum scaling limit is given by the stochastic bessel operator as conjectured in edelman and sutton j stat phys 127 1121 1165 2007 and rigorously established in ramirez and rider commun math phys 288 887 906 2009 the latter characterizes the hard edge in the case of linear potential and all beta the classical beta laguerre ensembles

beta ensembles stochastic airy spectrum and a diffusion we prove that the largest eigenvalues of the beta ensembles of random matrix theory converge in distribution to the low lying eigenvalues of the random schrodinger operator d2 dx2 x 2 bx restricted to the positive half line where b x is white noise in doing so we extend the definition of the tracy widom distributions to all 0 and also analyze their tails last in a parallel development we provide a second characterization of these laws in terms of a one dimensional diffusion the proofs rely on the associated tridiagonal matrix models and a universality result showing that the spectrum of such models converge to that of their continuum operator limit in particular we show how tracy widom laws arise from a functional central limit theorem

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

proper generalized decomposition for nonlinear convex problems in tensor banach spaces tensor based methods are receiving a growing interest in scientific computing for the numerical solution of problems defined in high dimensional tensor product spaces a family of methods called proper generalized decompositions pgd methods have been recently introduced for the a priori construction of tensor approximations of the solution of such problems in this paper we give a mathematical analysis of a family of progressive and updated pgds for a particular class of problems associated with the minimization of a convex functional over a reflexive tensor banach space

the two level feti method for static and dynamic plate problems part i an optimal iterative solver for biharmonic systems abstract we present a lagrange multiplier based substructuring method for solving iteratively large scale systems of equations arising from the finite element discretization of static and dynamic plate bending problems the proposed method is essentially an extension of the feti domain decomposition algorithm to fourth order problems the main idea is to enforce exactly the continuity of the transverse displacement field at the substructure corners throughout the preconditioned conjugate projected gradient iterations this results in a two level feti substructuring method where the condition number of the preconditioned interface problem does not grow with the number of substructures and grows at most polylogarithmically with the number of elements per substructure these theoretically proven optimal convergence properties of the new feti method are numerically demonstrated for several finite element plate bending static and transient problems the two level iterative solver presented in this paper is applicable to a large family of biharmonic time independent as well as time dependent systems it is also extendible to shell problems

a generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations we propose a new robust technique for solving stochastic partial differential equations the solution is approximated by a series of terms each of which being the product of a scalar stochastic function by a deterministic function none of these functions are fixed a priori but determined by solving a problem which can be interpreted as an extended eigenvalue problem this technique generalizes the classical spectral decomposition namely the karhunen loeve expansion ad hoc iterative techniques to build the approximation inspired by the power method for classical eigenproblems then transform the problem into the resolution of a few uncoupled deterministic problems and stochastic equations this method drastically reduces the calculation costs and memory requirements of classical resolution techniques used in the context of galerkin stochastic finite element methods finally this technique is particularly suitable to non linear and evolution problems since it enables the construction of a relevant reduced basis of deterministic functions which can be efficiently reused for subsequent resolutions

sparse high order fem for elliptic spdes abstract we describe the analysis and the implementation of two finite element fe algorithms for the deterministic numerical solution of elliptic boundary value problems with stochastic coefficients they are based on separation of deterministic and stochastic parts of the input data by a karhunen loeve expansion truncated after m terms with a change of measure we convert the problem to a sequence of m dimensional parametric deterministic problems two sparse high order polynomial approximations of the random solution s joint pdf s parametrized in the input data s karhunen loeve expansion coordinates are analyzed a sparse stochastic galerkin fem sparse sgfem and a sparse stochastic collocation fem sparse scfem a priori and a posteriori error analysis is used to tailor the sparse polynomial approximations of the random solution s joint pdf s to the stochastic regularity of the input data scfem and sgfem yield deterministic approximations of the random solutions joint pdf s that converge spectrally in the number of deterministic problems to be solved numerical examples with random inputs of small correlation length in diffusion problems are presented high order gpc approximations of solutions with stochastic parameter spaces of dimension up to m 80 are computed on workstations

feti dp a dual primal unified feti method part i a faster alternative to the two level feti method the feti method and its two level extension feti 2 are two numerically scalable domain decomposition methods with lagrange multipliers for the iterative solution of second order solid mechanics and fourth order beam plate and shell structural problems respectively the feti 2 method distinguishes itself from the basic or one level feti method by a second set of lagrange multipliers that are introduced at the subdomain cross points to enforce at each iteration the exact continuity of a subset of the displacement field at these specific locations in this paper we present a dual primal formulation of the feti 2 concept that eliminates the need for that second set of lagrange multipliers and unifies all previously developed one level and two level feti algorithms into a single dual primal feti dp method we show that this new feti dp method is numerically scalable for both second order and fourth order problems we also show that it is more robust and more computationally efficient than existing feti solvers particularly when the number of subdomains and or processors is very large copyright 2001 john wiley sons ltd

from random matrices to stochastic operators we propose that classical random matrix models are properly viewed as finite difference schemes for stochastic differential operators three particular stochastic operators commonly arise each associated with a familiar class of local eigenvalue behavior the stochastic airy operator displays soft edge behavior associated with the airy kernel the stochastic bessel operator displays hard edge behavior associated with the bessel kernel the article concludes with suggestions for a stochastic sine operator which would display bulk behavior associated with the sine kernel

algorithms for numerical analysis in high dimensions nearly every numerical analysis algorithm has computational complexity that scales exponentially in the underlying physical dimension the separated representation introduced previously allows many operations to be performed with scaling that is formally linear in the dimension in this paper we further develop this representation by i discussing the variety of mechanisms that allow it to be surprisingly efficient ii addressing the issue of conditioning iii presenting algorithms for solving linear systems within this framework and iv demonstrating methods for dealing with antisymmetric functions as arise in the multiparticle schrodinger equation in quantum mechanics numerical examples are given

multi element probabilistic collocation method in high dimensions we combine multi element polynomial chaos with analysis of variance anova functional decomposition to enhance the convergence rate of polynomial chaos in high dimensions and in problems with low stochastic regularity specifically we employ the multi element probabilistic collocation method mepcm 1 and so we refer to the new method as mepcm a we investigate the dependence of the convergence of mepcm a on two decomposition parameters the polynomial order m and the effective dimension n with n n and n the nominal dimension numerical tests for multi dimensional integration and for stochastic elliptic problems suggest that n m for monotonic convergence of the method we also employ mepcm a to obtain error bars for the piezometric head at the hanford nuclear waste site under stochastic hydraulic conductivity conditions finally we compare the cost of mepcm a against monte carlo in several hundred dimensions and we find mepcm a to be more efficient for up to 600 dimensions for a specific multi dimensional integration problem involving a discontinuous function

increasing subsequences and the hard to soft edge transition in matrix ensembles our interest is in the cumulative probabilities pr l t l for the maximum length of increasing subsequences in poissonized ensembles of random permutations random fixed point free involutions and reversed random fixed point free involutions it is shown that these probabilities are equal to the hard edge gap probability for matrix ensembles with unitary orthogonal and symplectic symmetry respectively the gap probabilities can be written as a sum over correlations for certain determinantal point processes from these expressions a proof can be given that the limiting form of pr l t l in the three cases is equal to the soft edge gap probability for matrix ensembles with unitary orthogonal and symplectic symmetry respectively thereby reclaiming theorems due to baik deift johansson and baik rains

a scalable lagrange multiplier based domain decomposition method for time dependent problems we present a new efficient and scalable domain decomposition method for solving implicitly linear and non linear time dependent problems in computational mechanics the method is derived by adding a coarse problem to the recently proposed transient feti substructuring algorithm in order to propagate the error globally and accelerate convergence it is proved that in the limit for large time steps the new method converges toward the feti algorithm for time independent problems computational results confirm that the optimal convergence properties of the time independent feti method are preserved in the time dependent case we employ an iterative scheme for solving efficiently the coarse problem on massively parallel processors and demonstrate the effective scalability of the new transient feti method with the large scale finite element dynamic analysis on the paragon xp s and ibm sp2 systems of several diffraction grating finite element structural models we also show that this new domain decomposition method outperforms the popular direct skyline solver the coarse problem presented herein is applicable and beneficial to a large class of lagrange multiplier based substructuring algorithms for time dependent problems including the fictitious domain decomposition method

a multiscale method with patch for the solution of stochastic partial differential equations with localized uncertainties abstract we here propose a multiscale numerical method for the solution of stochastic parametric partial differential equations with localized uncertainties described with a finite number of random variables it is based on a multiscale domain decomposition method that exploits the localized side of uncertainties and incidentally improves the conditioning of the problem by operating a separation of scales an efficient iterative algorithm is proposed that requires the solution of a sequence of simple global problems at a macro scale involving a deterministic operator and local problems at a micro scale for which we have the possibility to use fine approximation spaces global and local problems are solved using tensor approximation methods allowing the representation of high dimensional stochastic parametric solutions convergence properties of these tensor based methods which are closely related to spectral decompositions benefit from the separation of scales different types of uncertainties are considered at the micro level they may be associated with some variability in the operator or source terms or even with some geometrical variability in the latter case specific reformulations of local problems using fictitious domain methods are introduced

functional equations in a p adic context abstract recently in the complex context several results were obtained concerning functional equations of the form p f q g where p and q are polynomials of only two or three terms whose coefficients are small functions in certain cases the equation does not admit any pair of admissible solutions and in other cases it only admits pairs of solutions that are of a very particular type here we consider similar questions when the ground field is a p adic complete algebraically closed field of characteristic 0 and we derive results that are often analogous for instance if f n a 1 f n m b 1 c g n a 2 g n m b 2 with a i b j small functions with regard to f g and a 2 b 2 non identically 0 then c b 1 b 2 and f h g with h m a 1 a 2 however contrary to the complex context here results apply not only to meromorphic functions defined in the whole field but also to unbounded meromorphic functions defined inside an open disc the main tool is the p adic nevanlinna theory

some nonoverlapping domain decomposition methods the purpose of this paper is to give a unified investigation of a class of nonoverlapping domain decomposition methods for solving second order elliptic problems in two and three dimensions the methods under scrutiny fall into two major categories the substructuring type methods and the neumann neumann type methods the basic framework used for analysis is the parallel subspace correction method or additive schwarz method and other technical tools include local global and global local techniques the analyses for both two and three dimensional cases are carried out simultaneously some internal relationships between various algorithms are observed and several new variants of the algorithms are also derived

on metric properties of limit sets of contractive analytic non archimedean dynamical systems abstract let k be an algebraically closed field which is complete with respect to a non trivial non archimedean absolute value we study metric properties of the limit set of a semigroup g generated by a finite set of contractive analytic functions on o z k z 1 we prove that the limit set of g is uniformly perfect if the derivative of each generating function of g does not vanish on o furthermore we show that if each coefficient of the generating functions is in the field q p of p adic numbers or the limit set satisfies the strong open set condition then has the doubling property this yields that the limit set is quasisymmetrically equivalent to the space z 2 of 2 adic integers we also give a counterexample to show that not all limit sets have the doubling property the berkovich space is introduced to study the limit set and we prove that the limit set has a positive capacity in the berkovich space which yields that there exists an equilibrium measure whose support is contained in the limit set we also show that if the semigroup is generated by a countable set of contractive analytic functions then its limit set can be non compact however if coefficients of the generating functions lie in q p then the limit set is compact

on uniqueness of gibbs measure for p adic countable state potts model on the cayley tree abstract in the present paper we consider countable state of p adic potts model on the tree under some condition on weights we establish uniqueness of gibbs measures for the model note that this condition does not depend on values of the prime p an analogous fact is not true when the number of spins is finite

a stochastic collocation method for elliptic partial differential equations with random input data in this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms input data of the model the input data are assumed to depend on a finite number of random variables the method consists in a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it can be seen as a generalization of the stochastic galerkin method proposed in i babuscka r tempone and g e zouraris siam j numer anal 42 2004 pp 800 825 and allows one to treat easily a wider range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction in the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method

a feti preconditioned conjugate gradient method for large scale stochastic finite element problems in the spectral stochastic finite element method for analyzing an uncertain system the uncertainty is represented by a set of random variables and a quantity of interest such as the system response is considered as a function of these random variables consequently the underlying galerkin projection yields a block system of deterministic equations where the blocks are sparse but coupled the solution of this algebraic system of equations becomes rapidly challenging when the size of the physical system and or the level of uncertainty is increased this paper addresses this challenge by presenting a preconditioned conjugate gradient method for such block systems where the preconditioning step is based on the dual primal finite element tearing and interconnecting method equipped with a krylov subspace reusage technique for accelerating the iterative solution of systems with multiple and repeated right hand sides preliminary performance results on a linux cluster suggest that the proposed solution method is numerically scalable and demonstrate its potential for making the uncertainty quantification of realistic systems tractable

reduced chaos decomposition with random coefficients of vector valued random variables and random fields we develop a stochastic functional representation that is adapted to problems involving various forms of epistemic uncertainties including modeling error and data paucity the new representation builds on the polynomial chaos decomposition and eventually yields a karhunen loeve expansion with random multiplicative coefficients in this expansion one set of uncertainty is captured in the usual manner as uncorrelated scalar random variables another component of the uncertainty statistically independent from the first is captured by constructing the usually deterministic functions in the kl expansion as random functions we think of the first set of uncertainties as associated with a coarse scale model and of the second set as associated with subscale fluctuations not captured in the coarse scale description

reduced chaos expansions with random coefficients in reduced dimensional stochastic modeling of coupled problems summary we address the curse of dimensionality in methods for solving stochastic coupled problems with an emphasis on stochastic expansion methods such as those involving polynomial chaos expansions the proposed method entails a partitioned iterative solution algorithm that relies on a reduced dimensional representation of information exchanged between subproblems to allow each subproblem to be solved within its own stochastic dimension while interacting with a reduced projection of the other subproblems the proposed method extends previous work by the authors by introducing a reduced chaos expansion with random coefficients the representation of the exchanged information by using this reduced chaos expansion with random coefficients enables an expeditious construction of doubly stochastic polynomial chaos expansions that separate the effect of uncertainty local to a subproblem from the effect of statistically independent uncertainty coming from other subproblems through the coupling after laying out the theoretical framework we apply the proposed method to a multiphysics problem from nuclear engineering copyright 2013 john wiley sons ltd