a polynomial time approximation algorithm for the permanent of a matrix with nonnegative entries we present a polynomial time randomized algorithm for estimating the permanent of an arbitrary n n matrix with nonnegative entries this algorithm technically a fully polynomial randomized approximation scheme computes an approximation that is with high probability within arbitrarily small specified relative error of the true value of the permanent

a shape calculus based method for a transmission problem with a random interface the present work is devoted to an approximation of the statistical moments of the solution of a class of elliptic transmission problems in r 3 with uncertainly located transmission interfaces in this model the diffusion coefficient has a jump discontinuity across the random transmission interface which models linear diffusion in two different media separated by an uncertain surface we apply the shape calculus approach to approximate the solution perturbation by the so called shape derivative correspondingly statistical moments of the solution are approximated by the moments of the shape derivative we characterize the shape derivative as a solution of a related homogeneous transmission problem with nonzero jump conditions which is solved by the boundary integral equation method a rigorous theoretical framework is developed and the theoretical findings are supported by and illustrated in two particular examples

multilevel frames for sparse tensor product spaces for au f with an elliptic differential operator a mathcal h rightarrow mathcal h and stochastic data f the m point correlation function mathcal m m u of the random solution u satisfies a deterministic equation with the m fold tensor product operator a m of a sparse tensor products of hierarchic fe spaces in mathcal h are known to allow for approximations to mathcal m m u which converge at essentially the rate as in the case m 1 i e for the deterministic problem they can be realized by wavelet type fe bases von petersdorff and schwab in appl math 51 2 145 180 2006 schwab and todor in computing 71 43 63 2003 if wavelet bases are not available we show here how to achieve the fast computation of sparse approximations of mathcal m m u for galerkin discretizations of a by multilevel frames such as bpx or other multilevel preconditioners of any standard fem approximation for a numerical examples illustrate feasibility and scope of the method

sparse second moment analysis for elliptic problems in stochastic domains we consider the numerical solution of elliptic boundary value problems in domains with random boundary perturbations assuming normal perturbations with small amplitude and known mean field and two point correlation function we derive using a second order shape calculus deterministic equations for the mean field and the two point correlation function of the random solution for a model dirichlet problem which are 3rd order accurate in the boundary perturbation size using a variational boundary integral equation formulation on the unperturbed nominal boundary and a wavelet discretization we present and analyze an algorithm to approximate the random solution s mean and its two point correlation function at essentially optimal order in essentially mathcal o n work and memory where n denotes the number of unknowns required for consistent discretization of the boundary of the nominal domain

shape optimization for quadratic functionals and states with random right hand sides we consider shape optimization problems under uncertainties on the input parameters the presented theory applies to the minimization of the expectation of a quadratic objective for a state function that depends linearly on a random input parameter it covers important objectives such as tracking type functionals for elliptic second order partial differential equations and the compliance in linear elasticity we show that the robust objective and its gradient are completely determined by low order moments of the random input we derive a cheap deterministic algorithm to minimize this objective and present model cases in structural optimization

analytic regularity and collocation approximation for elliptic pdes with random domain deformations in this work we consider the problem of approximating the statistics of a given quantity of interest qoi that depends on the solution of a linear elliptic pde defined over a random domain parameterized by n random variables the elliptic problem is remapped onto a corresponding pde with a fixed deterministic domain we show that the solution can be analytically extended to a well defined region in c n with respect to the random variables a sparse grid stochastic collocation method is then used to compute the mean and variance of the qoi finally convergence rates for the mean and variance of the qoi are derived and compared to those obtained in numerical experiments

sparse finite elements for elliptic problems with stochastic loading we formulate elliptic boundary value problems with stochastic loading in a bounded domain d d we show well posedness of the problem in stochastic sobolev spaces and we derive a deterministic elliptic pde in d d for the spatial correlation of the random solution we show well posedness and regularity results for this pde in a scale of weighted sobolev spaces with mixed highest order derivatives discretization with sparse tensor products of any hierarchic finite element fe spaces in d yields optimal asymptotic rates of convergence for the spatial correlation even in the presence of singularities or for spatially completely uncorrelated data multilevel preconditioning in d d allows iterative solution of the discrete equation for the correlation kernel in essentially the same complexity as the solution of the mean field equation

dimer monomer model on the sierpinski gasket abstract we present the numbers of dimer monomers m d n on the sierpinski gasket s g d n at stage n with dimension d equal to two three and four the upper and lower bounds for the asymptotic growth constant defined as z s g d lim v ln m d n v where v is the number of vertices on s g d n are derived in terms of the results at a certain stage as the difference between these bounds converges quickly to zero as the calculated stage increases the numerical value of z s g d can be evaluated with more than a hundred significant figures accurate from the results for d 2 3 4 we conjecture the upper and lower bounds of z s g d for general dimension the corresponding results on the generalized sierpinski gasket s g d b n with d 2 and b 3 4 are also obtained

karhunen loeve approximation of random fields by generalized fast multipole methods kl approximation of a possibly instationary random field a x l2 dp l d subject to prescribed meanfield ea x a x dp and covariance va x x a x ea x a x ea x dp in a polyhedral domain d rd is analyzed we show how for stationary covariances va x x ga x x with ga z analytic outside of z 0 an m term approximate kl expansion am x of a x can be computed in log linear complexity the approach applies in arbitrary domains d and for nonseparable covariances ca it involves galerkin approximation of the kl eigenvalue problem by discontinuous finite elements of degree p 0 on a quasiuniform possibly unstructured mesh of width h in d plus a generalized fast multipole accelerated krylov eigensolver the approximate kl expansion am x of a x has accuracy o exp bm1 d if ga is analytic at z 0 and accuracy o m k d if ga is ck at zero it is obtained in o mn logn b operations where n o h d

on the absolutely continuous spectrum of a vector matrix dirac system a dirac system is considered which has a matrix valued long range short range and oscillatory potentials the system has one singular endpoint at infinity additional conditions on the potential are given which guarantee particular asymptotic behaviour of an energy functional associated with a certain set of solutions this asymptotic behaviour guarantees the existence of a purely absolutely continuous spectrum outside a gap containing the origin

covariance regularity and mathcal h h matrix approximation for rough random fields in an open bounded domain mathrm d subset mathbb r n d rn with smooth boundary partial mathrm d d or on a smooth closed and compact riemannian n manifold mathcal m subset mathbb r n 1 m rn 1 we consider the linear operator equation a u f au f where a is a boundedly invertible strongly elliptic pseudodifferential operator of order r in mathbb r r r with analytic coefficients covering all linear second order elliptic pdes as well as their boundary reductions here f in l 2 omega h t f l2 ht is an h t ht valued random field with finite second moments with h t ht denoting the isotropic sobolev space of not necessarily integer order t modelled on the domain mathrm d d or manifold mathcal m m respectively we prove that the random solution s covariance kernel k u a 1 otimes a 1 k f ku a 1 a 1 kf on mathrm d times mathrm d d d resp mathcal m times mathcal m m m is an asymptotically smooth function provided that the covariance function k f kf of the random data is a schwartz distributional kernel of an elliptic pseudodifferential operator as a consequence numerical mathcal h h matrix calculus allows the deterministic approximation of singular covariances k u ku of the random solution u a 1 f in l 2 omega h t r u a 1f l2 ht r in mathrm d times mathrm d d d text resp mathcal m times mathcal m resp m m with work versus accuracy essentially equal to that for the mean field approximation with splines of fixed order mathrm d d text resp mathcal m resp m overcoming the curse of dimensionality in this case

monomer dimer problem on random planar honeycomb lattice we consider the monomer dimer md problem on a random planar honeycomb lattice model namely the random multiple chain this is a lattice system with non periodic boundary condition whose generating process is inspired by the growth of single walled zigzag carbon nanotubes by applying algebraic and combinatorial techniques we establish a calculating expression of the md partition function for bipartite graphs which corresponds to the permanent of a matrix further by using the transfer matrix argument we show that the computing problem of the permanent of high order matrix can be converted into some lower order matrices for this family of lattices based on which we derive an explicit recurrence formula for evaluating the md partition function of multiple chains and random multiple chains finally we analyze the expectation of the number of monomer dimer arrangements on a random multiple chain and the asymptotic behavior of the annealed md entropy when the multiple chain becomes infinite in width and length respectively

hierarchical matrix approximation for the uncertainty quantification of potentials on random domains abstract computing statistical quantities of interest of the solution of partial differential equations on random domains is an important and challenging task in engineering we consider the computation of these quantities by the perturbation approach especially we discuss how third order accurate expansions of the mean and the correlation can numerically be computed these expansions become even fourth order accurate for certain types of boundary variations the correction terms are given by the solution of correlation equations in the tensor product domain which can efficiently be computed by means of h matrices they have recently been shown to be an efficient tool to solve correlation equations with rough data correlations that is with low sobolev smoothness or small correlation length in almost linear time numerical experiments in three dimensions for higher order ansatz spaces show the feasibility of the proposed algorithm the application to a non smooth domain is also included

packing dimers on 2p 1 x 2q 1 lattices we use computational method to investigate the number of ways to pack dimers on m x n lattices where both m and n are odd in this case there is always a single vacancy in the lattices we show that the dimer configuration numbers on 2k 1 x 2k 1 odd square lattices have some remarkable number theoretical properties in parallel to those of close packed dimers on 2k x 2k even square lattices for which an exact solution exists furthermore we demonstrate that there is an unambiguous logarithmic term in the finite size correction of free energy of odd by odd lattice strips with any width n or 1 this logarithmic term determines the distinct behavior of the free energy of odd square lattices these findings reveal a deep and previously unexplored connection between statistical physics models and number theory and indicate the possibility that the monomer dimer problem might be solvable

compression techniques for boundary integral equations asymptotically optimal complexity estimates matrix compression techniques in the context of wavelet galerkin schemes for boundary integral equations are developed and analyzed that exhibit optimal complexity in the following sense the fully discrete scheme produces approximate solutions within discretization error accuracy offered by the underlying galerkin method at a computational expense that is proven to stay proportional to the number of unknowns key issues are the second compression which reduces the near field complexity significantly and an additional a posteriori compression the latter is based on a general result concerning an optimal work balance that applies in particular to the quadrature used to compute the compressed stiffness matrix with sufficient accuracy in linear time

on the fast matrix multiplication in the boundary element method by panel clustering the boundary element method bem leads to a system of linear equations with a full matrix while fem yields sparse matrices this fact seems to require much computational work for the definition of the matrix for the solution of the system and in particular for the matrix vector multiplication which always occurs as an elementary in this paper a method for the approximate matrix vector multiplication is described which requires much less arithmetical work in addition the storage requirements are strongly reduced

instability of the absolutely continuous spectrum of ordinary differential operators under local perturbations abstract it is shown that a potential q exists such that a selfadjoint realization of lu u q x u has singular continuous spectrum in an interval i while all the selfadjoint realizations of l u u q x v x u where v is a continuous function with compact support have absolutely continuous spectrum in i

transfer matrices hyperbolic geometry and absolutely continuous spectrum for some discrete schrodinger operators on graphs abstract we prove the existence of absolutely continuous spectrum for a class of discrete schrodinger operators on tree like graphs we consider potentials whose radial behaviour is subject only to an l bound in the transverse direction the potential must satisfy a condition such as periodicity the graphs we consider include binary trees and graphs obtained from a binary tree by adding edges possibly with weights our methods are motivated by the one dimensional transfer matrix method interpreted as a discrete dynamical system on the hyperbolic plane this is extended to more general graphs leading to a formula for the green s function bounds on the green s function then follow from the contraction properties of the transformations that arise in this generalization the bounds imply the existence of absolutely continuous spectrum

computing monomer dimer systems through matrix permanent the monomer dimer model is fundamental in statistical mechanics however it is p complete in computation even for two dimensional problems a formulation for the partition function of the monomer dimer system is proposed in this paper by transforming the number of all matchings of a bipartite graph into the number of perfect matchings of an extended bipartite graph which can be given by a matrix permanent sequential importance sampling algorithm is applied to compute the permanents for two dimensional lattice with periodic condition the monomer dimer constant is known as h 2 0 662798972834 we obtain 0 6627 ifmmode pm else textpm fi 0 0002 for our approximation which shows the robustness and the efficiency of the algorithm for three dimensional problem our numerical result is 0 7847 ifmmode pm else textpm fi 0 0014 which agrees with the best known bounds

mathcal h matrix based second moment analysis for rough random fields and finite element discretizations we consider the efficient solution of strongly elliptic partial differential equations with random load based on the finite element method the solution s two point correlation can efficiently be approximated by means of an mathcal h matrix in particular if the correlation length is rather short or the correlation kernel is nonsmooth since the inverses of the finite element matrices which correspond to the differential operator under consideration can likewise efficiently be approximated in the mathcal h matrix format we can solve the correspondent mathcal h matrix equation in essentially linear time by using the mathcal h matrix arithmetic numerical experiments for three dimensional finite element discretizations for several correlation lengths and different smoothness are provided they validate the presented method and demonstrate that the computation times do not increase for nonsmooth or shortly correlated data

dense point spectrum for the one dimensional dirac operator with an electrostatic potential for the one dimensional dirac operator examples of electrostatic potentials with decay behaviour arbitrarily close to coulomb decay are constructed for which the operator has a prescribed set of eigenvalues dense in the whole or part of its essential spectrum a simple proof that the essential spectrum of one dimensional dirac operators with electrostatic potentials is never empty is given in the appendix

multi level monte carlo finite element method for elliptic pdes with stochastic coefficients in monte carlo methods quadrupling the sample size halves the error in simulations of stochastic partial differential equations spdes the total work is the sample size times the solution cost of an instance of the partial differential equation a multi level monte carlo method is introduced which allows in certain cases to reduce the overall work to that of the discretization of one instance of the deterministic pde the model problem is an elliptic equation with stochastic coefficients multi level monte carlo errors and work estimates are given both for the mean of the solutions and for higher moments the overall complexity of computing mean fields as well as k point correlations of the random solution is proved to be of log linear complexity in the number of unknowns of a single multi level solve of the deterministic elliptic problem numerical examples complete the theoretical analysis

majorization doubly stochastic matrices and comparison of eigenvalues abstract 1 basic properties of majorization 2 isotone maps and algebraic operations 3 double sub and superstochasticity 4 doubly stochastic matrices 5 doubly stochastic matrices with minimum permanent 6 comparison of eigenvalues 7 doubly stochastic maps

approximation of boundary element matrices this article considers the problem of approximating a general asymptotically smooth function in two variables typically arising in integral formulations of boundary value problems by a sum of products of two functions in one variable from these results an iterative algorithm for the low rank approximation of blocks of large unstructured matrices generated by asymptotically smooth functions is developed this algorithm uses only few entries from the original block and since it has a natural stopping criterion the approximative rank is not needed in advance

quasi monte carlo methods for elliptic pdes with random coefficients and applications we devise and implement quasi monte carlo methods for computing the expectations of nonlinear functionals of solutions of a class of elliptic partial differential equations with random coefficients our motivation comes from fluid flow in random porous media where relevant functionals include the fluid pressure velocity at any point in space or the breakthrough time of a pollution plume being transported by the velocity field our emphasis is on situations where a very large number of random variables is needed to model the coefficient field as an alternative to classical monte carlo we here employ quasi monte carlo methods which use deterministically chosen sample points in an appropriate usually high dimensional parameter space each realization of the pde solution requires a finite element fe approximation in space and this is done using a realization of the coefficient field restricted to a suitable regular spatial grid not necessarily the same as the fe grid in the statistically homogeneous case the corresponding covariance matrix can be diagonalized and the required coefficient realizations can be computed efficiently using fft in this way we avoid the use of a truncated karhunen loeve expansion but introduce high nominal dimension in parameter space numerical experiments with 2 dimensional rough random fields high variance and small length scale are reported showing that the quasi monte carlo method consistently outperforms the monte carlo method with a smaller error and a noticeably better than o n 1 2 convergence rate where n is the number of samples moreover the rate of convergence of the quasi monte carlo method does not appear to degrade as the nominal dimension increases examples with dimension as high as 10 6 are reported

fast wavelet transforms and numerical algorithms i a class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices in contrast the methods presented here are based on the recently developed theory of wavelets and are applicable to all calderon zygmund and pseudo differential operators the algorithms of this paper require order o n or o n log n operations to apply an n n matrix to a vector depending on the particular operator and the version of the algorithm being used and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here

approximating the permanent a randomised approximation scheme for the permanent of a 0 1s presented the task of estimating a permanent is reduced to that of almost uniformly generating perfect matchings in a graph the latter is accomplished by simulating a markov chain whose states are the matchings in the graph for a wide class of 0 1 matrices the approximation scheme is fully polynomial i e runs in time polynomial in the size of the matrix and a parameter that controls the accuracy of the output this class includes all dense matrices those that contain sufficiently many 1 s and almost all sparse matrices in some reasonable probabilistic model for 0 1 matrices of given density for the approach sketched above to be computationally efficient the markov chain must be rapidly mixing informally it must converge in a short time to its stationary distribution a major portion of the paper is devoted to demonstrating that the matchings chain is rapidly mixing apparently the first such result for a markov chain with genuinely c

the one dimensional schrodinger coulomb operator with definite parity as a model for one dimensional hydrogenic systems the schrodinger operator with the singular potential x 1 is studied by analyzing it in its reducing subspaces corresponding to even and odd parity we derive spectral properties common to all self adjoint extensions of the reduced operators and extract the boundary conditions at x 0 specific to the different self adjoint extensions a transcendental equation for the eigenvalues is derived which leads to a construct that we call the spectral helix for the distinguished self adjoint extension with dirichlet boundary conditions virial and supersymmetry relations are established

dimers on a simple quartic net with a vacancy a seminal milestone in lattice statistics is the exact solution of the enumeration of dimers on a simple quartic net obtained by fisher kasteleyn and temperley fkt in 1961 an outstanding related and yet unsolved problem is the enumeration of dimers on a net with vacant sites here we consider this vacant site problem with a single vacancy occurring at certain specific sites on the boundary of a simple quartic net first using a bijection between dimer and spanning tree configurations due to temperley kenyon propp and wilson we establish that the dimer generating function is independent of the location of the vacancy and deduce a closed form expression for the generating function we next carry out finite size analyses of this solution as well as that of the fkt solution our analyses lead to a logarithmic correction term in the large size expansion for the vacancy problem with free boundary conditions a concrete example exhibiting this difference is given we also find the central charge c 2 in the language of conformal field theory for the vacancy problem as versus the value c 1 when there is no vacancy

bounded solutions and absolute continuity of sturm liouville operators this work is a contribution to the theory of the absolutely continuous spectrum of one dimensional schriidinger operators i e sturm liouville operators generated by differential expressions of the form ru u qu in proofs of absolute continuity for such operators and also for the closely related dirac systems always asymptotic properties of solutions of zu lu for real or complex values of 1 have played an important role see for example 18 19 8 4 9 lo in these works and a number of others particular classes of potentials q were treated using properties of solutions in the course of the proof of absolute continuity somewhat more recently the connection between asymptotic properties of solutions and absolute continuity was given a theory i e results of the following form were proved let the solutions of zu lu satisfy some specific asymptotic property for a varying in some subset of r no or almost no a priori assumptions on q then operators generated by z are absolutely continuous in this set a first result of this type was given by carmona 3 who proved that local uniform boundedness of solutions together with their derivatives in some open a set implies pure absolute continuity in this set here we will mostly rely on the concept of non or racy of solutions which was shown by gilbert and pearson 7 12 61 to be a powerful tool in the investigation of the continuous spectrum of sturm liouville operators in section 2 we give some consequences of their results whi ch are useful in proofs of absolute continuity these consequences range from non existence of imbedded eigenvalues and the existence of absolutely continuous parts to pure absolute continuity and the existence of transient 210 0022 247x 92 5 00

two dimensional monomer dimer systems are computationally intractable the classic problem of counting monomer dimer arrangements on a two dimensional lattice is analyzed using techniques from theoretical computer science under a certain assumption made precise in the text it can be shown that the general problem is computationally intractable this negative result contrasts with the special case of a system with monomer density zero for which efficient solutions have been known for some time a second much easier result obtained under the same assumption is that the partition function of a three dimensional ising system is computationally intractable again the negative result contrasts with known efficient techniques for evaluating the partition function of a two dimensional system

logarithmic corrections in the free energy of monomer dimer model on plane lattices with free boundaries using exact computations we study the classical hard core monomer dimer models on m x n plane lattice strips with free boundaries for an arbitrary number v of monomers or vacancies we found a logarithmic correction term in the finite size correction of the free energy per lattice site the coefficient of the logarithmic correction term depends on the number of monomers present v and the parity of the width n of the lattice strip the coefficient equals to v when n is odd and v 2 when n is even the results are generalizations of the previous results for a single monomer in an otherwise fully packed lattice of dimers we also study the finite size correction in the low dimer density limit where the number of dimers d is fixed in this case the coefficient of the logarithmic correction term equals to d for both odd and even n

an interpolation based fast multipole method for higher order boundary elements on parametric surfaces in this article a black box higher order fast multipole method for solving boundary integral equations on parametric surfaces in three spatial dimensions is proposed such piecewise smooth surfaces are the topic of recent studies in isogeometric analysis due to the exact surface representation the rate of convergence of higher order methods is not limited by approximation errors of the surface an element wise clustering strategy yields a balanced cluster tree and an efficient numerical integration scheme for the underlying galerkin method by performing the interpolation for the fast multipole method directly on the reference domain the cost complexity in the polynomial degree is reduced by one order this gain is independent of the application of either mathcal h or mathcal h 2 matrices in fact several simplifications in the construction of mathcal h 2 matrices are pointed out which are a by product of the surface representation extensive numerical examples are provided in order to quantify and qualify the proposed method in this article a black box higher order fast multipole method for solving boundary integral equations on parametric surfaces in three spatial dimensions is proposed such piecewise smooth surfaces are the topic of recent studies in isogeometric analysis due to the exact surface representation the rate of convergence of higher order methods is not limited by approximation errors of the surface an element wise clustering strategy yields a balanced cluster tree and an efficient numerical integration scheme for the underlying galerkin method by performing the interpolation for the fast multipole method directly on the reference domain the cost complexity in the polynomial degree is reducedby one order this gain is independent of the application of either h or h 2 matrices in fact several simplificationsin the construction of h 2 matrices are pointed out which are a by product of the surface representation extensive numerical examples are provided in order to quantify and qualify the proposed method

numerical methods for differential equations in random domains physical phenomena in domains with rough boundaries play an important role in a variety of applications often the topology of such boundaries cannot be accurately described in all of its relevant detail due to either insufficient data or measurement errors or both this topological uncertainty can be efficiently handled by treating rough boundaries as random fields so that an underlying physical phenomenon is described by deterministic or stochastic differential equations in random domains to deal with this class of problems we propose a novel computational framework which is based on using stochastic mappings to transform the original deterministic stochastic problem in a random domain into a stochastic problem in a deterministic domain the latter problem has been studied more extensively and existing analytical numerical techniques can be readily applied in this paper we employ both a stochastic galerkin method and monte carlo simulations to solve the transformed stochastic problem we demonstrate our approach by applying it to an elliptic problem in single and double connected random domains and comment on the accuracy and convergence of the numerical methods

monomer dimer model on a scale free small world network abstract the explicit determination of the number of monomer dimer arrangements on a network is a theoretical challenge and exact solutions to monomer dimer problem are available only for few limiting graphs with a single monomer on the boundary e g rectangular lattice and quartic lattice however analytical research even numerical result for monomer dimer problem on scale free small world networks is still missing despite the fact that a vast variety of real systems display simultaneously scale free and small world structures in this paper we address the monomer dimer problem defined on a scale free small world network and obtain the exact formula for the number of all possible monomer dimer arrangements on the network based on which we also determine the asymptotic growth constant of the number of monomer dimer arrangements in the network we show that the obtained asymptotic growth constant is much less than its counterparts corresponding to two dimensional lattice and sierpinski fractal having the same average degree as the studied network which indicates from another aspect that scale free networks have a fundamentally distinct architecture as opposed to regular lattices and fractals without power law behavior

a sparse h matrix arithmetic general complexity estimates in a preceding paper hackbusch computing 62 1999 89 108 a class of matrices h matrices has been introduced which are data sparse and allow an approximate matrix arithmetic of almost linear complexity several types of h matrices have been analysed in hackbusch computing 62 1999 89 108 and hackbusch and khoromskij preprint mpi no 22 leipzig 1999 computing 64 2000 21 47 which are able to approximate integral nonlocal operators in fem and bem applications in the case of quasi uniform unstructured meshes in the present paper the general construction of h matrices on rectangular and triangular meshes is proposed and analysed first the reliability of h matrices in bem is discussed then we prove the optimal complexity of storage and matrix vector multiplication in the case of rather arbitrary admissibility parameters 1 and for nite elements up to the order 1 dened on quasi uniform rectangular triangular meshes in r d d 1 2 3 the almost linear complexity of the matrix addition multiplication and inversion of h matrices is also veried c 2000 elsevier science b v all rights reserved msc 65f05 65f30 65f50

analysis of the domain mapping method for elliptic diffusion problems on random domains in this article we provide a rigorous analysis of the solution to elliptic diffusion problems on random domains in particular based on the decay of the karhunen loeve expansion of the domain perturbation field we establish decay rates for the derivatives of the random solution that are independent of the stochastic dimension for the implementation of a related approximation scheme like quasi monte carlo quadrature stochastic collocation etc we propose parametric finite elements to compute the solution of the diffusion problem on each individual realization of the domain generated by the perturbation field this simplifies the implementation and yields a non intrusive approach having this machinery at hand we can easily transfer it to stochastic interface problems the theoretical findings are complemented by numerical examples for both stochastic interface problems and boundary value problems on random domains

approximating the permanent via importance sampling with application to the dimer covering problem we estimate the asymptotic growth rate of the number of dimer covers of a cubic lattice our estimate 3 0 4466 0 0006 is consistent with the lower bounds obtained by hammersley and later schrijver and the more recent improved upper bound obtained by ciucu obtaining this estimate is an important step toward approximating the partition function of the cubic monomer dimer system from the partition function all of the standard thermodynamic quantities can be evaluated it is well known that computing 3 is equivalent to computing the permanent of a certain 0 1 matrix we describe an extremely efficient monte carlo algorithm for approximating the permanent previous work on monte carlo approaches includes the pioneering results of jerrum and sinclair who use a rapidly mixing random walk our method is inspired by results of soules on convergence of sinkhorn balancing to obtain a maximum entropy doubly stochastic matrix we use the sinkhorn balanced matrix to generate an importance function that allows us to do direct random sampling rather than a random walk that converges to a limiting distribution

absolutely continuous spectrum of dirac operators for long range potentials abstract using the resolvent method and the technique of weighted l2 estimates we deduce the non existence of the singular continuous spectrum of the dirac operator p x on the interval 1 we assume that the hermitian matrix potential p x p1 x p2 x is divided into a long range part p 1 x o x e r r p 1 x o x e and a short range part p 2 x o x 1 e x with local singularities p x o x a j 1 of coulomb type for n nuclei a1 an r 3

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

capacity of finite state channels based on lyapunov exponents of random matrices the finite state markov channel fsmc is a time varying channel having states that are characterized by a finite state markov chain these channels have infinite memory which complicates their capacity analysis we develop a new method to characterize the capacity of these channels based on lyapunov exponents specifically we show that the input output and conditional entropies for this channel are equivalent to the largest lyapunov exponents for a particular class of random matrix products we then show that the lyapunov exponents can be expressed as expectations with respect to the stationary distributions of a class of continuous state space markov chains this class of markov chains which is closely related to the prediction filter in hidden markov models is shown to be nonirreducible hence much of the standard theory for continuous state space markov chains cannot be applied to establish the existence and uniqueness of stationary distributions nor do we have direct access to a central limit theorem clt in order to address these shortcomings we utilize several results from the theory of random matrix products and lyapunov exponents the stationary distributions for this class of markov chains are shown to be unique and continuous functions of the input symbol probabilities provided that the input sequence has finite memory these properties allow us to express mutual information and channel capacity in terms of lyapunov exponents we then leverage this connection between entropy and lyapunov exponents to develop a rigorous theory for computing or approximating entropy and mutual information for finite state channels with dependent inputs we develop a method for directly computing entropy of finite state channels that does not rely on simulation and establish its convergence we also obtain a new asymptotically tight lower bound for entropy based on norms of random matrix products in addition we prove a new functional clt for sample entropy and apply this theorem to characterize the error in simulated estimates of entropy finally we present numerical examples of mutual information computation for intersymbol interference isi channels and observe the capacity benefits of adding memory to the input sequence for such channels

approximating the number of monomer dimer coverings of a lattice we study the problem of counting the number of coverings of ad dimensional rectangular lattice by a specified number of monomers and dimers this problem arises in several models in statistical physics and has been widely studied a classical technique due to fisher kasteleyn and temperley solves the problem exactly in two dimensions when the number of monomers is zero the dimer covering problem but is not applicable in higher dimensions or in the presence of monomers this paper presents the first provably polynomial time approximation algorithms for computing the number of coverings with any specified number of monomers ind dimensional rectangular lattices with periodic boundaries for any fixed dimensiond and in two dimensional lattices with fixed boundaries the algorithms are based on monte carlo simulation of a suitable markov chain and in constrast to most monte carlo algorithms in statistical physics have rigorously derived performance guarantees that do not rely on any assumptions the method generalizes to counting coverings of any finite vertex transitive graph a class which includes most natural finite lattices with periodic boundary conditions

approximating the number of monomer dimer coverings in periodic lattices our starting point is an algorithm of kenyon randall and sinclair which is built upon the ideas of jerrum and sinclair giving an approximation to crucial parameters of the monomer dimer covering problem in polynomial time we make two key improvements to their algorithm we greatly reduce the number of simulations that must be run by estimating good values of the generating function parameter and we greatly reduce the number of steps that must be taken in each simulation by aggregating to a simulation with at most five states the result is an algorithm that is computationally feasible for modestly sized meshes we use our algorithm on two and three dimensional problems computing approximations to the coefficients of the generating function and some limiting values