an inventory model with limited production capacity and uncertain demands ii the discounted cost criterion this paper considers a single item periodic review inventory model with uncertain demands we assume a finite production capacity in each period with stationary data a convex one period cost function and a continuous demand distribution we show under a few additional unrestrictive assumptions that a modified basic stock policy is optimal under the discounted cost criterion both for finite and infinite planning horizons in addition we characterize the optimal base stock levels in several ways

uniqueness and stability of optimal policies of finite state markov decision processes in this paper we consider infinite horizon discrete time optimal control of markov decision processes mdps with finite state spaces and compact action sets we restrict attention to unicost mdps which form a class that contains all the weakly communicating mdps the unicost mdps are characterized as those mdps for which there exists a solution to the single optimality equation we address the problem of uniqueness and stability of minimizing markov actions our main result asserts that when we endow the set of unicost mdps with a certain natural metric under which it is complete then the class of mdps with essentially unique and stable minimizing markov actions contains the intersection of countably many open dense sets hence is itself dense thus the property of having essentially unique and stable minimizing markov actions is generic for unicost mdps

the collocation points of the fundamental solution method for the potential problem we propose an algorithm for computing harmonic functions in a two dimensional domain with prescribed dirichlet data our algorithm is a variant of what is called a fundamental solution method 1 3 this method requires us to select 2n points in the two dimensional plane n of which are called collocation points and the remaining n are called charge points representing the position of the singularities of the fundamental solution it is known that there exists a set of 2n points by which the error is exponentially small 1 3 6 however these papers are concerned mainly with existence and as far as the authors know few fast and reliable algorithms are known for good position of the points in this paper we propose a new rule for the position of the points and examine its efficiency by numerical experiments the new rule uses fft effectively

on average reward semi markov decision processes with a general multichain structure in this paper we investigate average reward semi markov decision processes with a general multichain structure using a data transformation method by solving the transformed discrete time average markov decision processes we can obtain significant and interesting information on the original average semi markov decision processes if the original semi markov decision processes satisfy some appropriate conditions then stationary optimal policies in the transformed discrete time models are also optimal in the original semi markov decision processes

stability and convergence of the method of fundamental solutions for helmholtz problems on analytic domains the method of fundamental solutions mfs is a popular tool to solve laplace and helmholtz boundary value problems its main drawback is that it often leads to ill conditioned systems of equations in this paper we investigate for the interior helmholtz problem on analytic domains how the singularities charge points of the mfs basis functions have to be chosen such that approximate solutions can be represented by the mfs basis in a numerically stable way for helmholtz problems on the unit disc we give a full analysis which includes the high frequency short wavelength limit for more difficult and nonconvex domains such as crescents we demonstrate how the right choice of charge points is connected to how far into the complex plane the solution of the boundary value problem can be analytically continued which in turn depends on both domain shape and boundary data using this we develop a recipe for locating charge points which allows us to reach error norms of typically 10 1 1 on a wide variety of analytic domains at high frequencies of order only 3 points per wavelength are needed which compares very favorably to boundary integral methods

interface delocalization in the three dimensional ising model the interface delocalization in the three dimensional ising model is studied by real space renormalization group methods the first order cumulant expansion approximation is used defect free energies for a boundary plane of defects and an internal plane of defects are calculated in the whole temperature region the phase diagrams are also obtained the method and the model analyzed may give a correct phase diagram only in the regime of continuous interface delocalization the interface delocalization is obtained for the boundary defect and also for the internal defect if the systems on two sides of the internal defect plane have a different degree of order the delocalization transition does not occur in the case of the internal defect plane between two equally ordered systems

domain decomposition methods with fundamental solutions for helmholtz problems with discontinuous source terms abstract the direct application of the classical method of fundamental solutions mfs is restricted to homogeneous linear partial differential equations pdes the use of fundamental solutions with different frequencies allowed the extension of the mfs to non homogeneous pdes in particular for poisson or helmholtz equations and for elastostatic or elastodynamic problems this method has been called method of fundamental solutions for domains mfs d but it faces an approximation problem when the non homogeneous term presents discontinuities because the fundamental solutions are analytic functions outside the source point set in this paper we analyze two domain decomposition techniques for overcoming this approximation problem the problem is set in the context of the modified helmholtz equation and we also establish the missing density results that justify both the mfs and the mfs d approximations numerical results are presented comparing a direct and an iterative domain decomposition technique with simulations in non trivial domains

on the range of the first two dirichlet and neumann eigenvalues of the laplacian in this paper we study the set of points in the plane defined by x y lambda 1 omega lambda 2 omega omega 1 where lambda 1 omega lambda 2 omega are either the two first eigenvalues of the dirichlet laplacian or the two first non trivial eigenvalues of the neumann laplacian we consider the case of general open sets together with the case of convex open domains we give some qualitative properties of these sets show some pictures obtained through numerical computations and state several open problems

perturbative analysis of the method of particular solutions for improved inclusion of high lying dirichlet eigenvalues the dirichlet eigenvalue or drum problem in a domain omega subset mathbb r 2 becomes numerically challenging at high eigenvalue frequency e in this regime the method of particular solutions mps gives spectral accuracy for many domain shapes it requires a number of degrees of freedom scaling as sqrt e the number of wavelengths on the boundary in contrast to direct discretization for which this scaling is e our main result is an inclusion bound on eigenvalues that is a factor o sqrt e tighter than the classical bound of moler payne and that is optimal in that it reflects the true slopes of curves appearing in the mps we also present an mps variant that cures a normalization problem in the original method while evaluating basis functions only on the boundary this method is efficient at high frequencies where we show that in practice our inclusion bound can give three extra digits of eigenvalue accuracy with no extra effort

qualitative and numerical analysis of a spectral problem with perimeter constraint we consider the problem of optimizing the k text th eigenvalue of the dirichlet laplace operator under perimeter constraint we provide a new method based on a gamma convergence result for approximating the corresponding optimal shapes we also give new optimality conditions in the case of multiple eigenvalues we deduce from previous conditions the fact that optimal shapes never contain flat parts in their boundaries

an exponentially convergent nonpolynomial finite element method for time harmonic scattering from polygons in recent years nonpolynomial finite element methods have received increasing attention for the efficient solution of wave problems as with their close cousin the method of particular solutions high efficiency comes from using solutions to the helmholtz equation as basis functions we present and analyze such a method for the scattering of two dimensional scalar waves from a polygonal domain that achieves exponential convergence purely by increasing the number of basis functions in each element key ingredients are the use of basis functions that capture the singularities at corners and the representation of the scattered field towards infinity by a combination of fundamental solutions the solution is obtained by minimizing a least squares functional which we discretize in such a way that a matrix least squares problem is obtained we give computable exponential bounds on the rate of convergence of the least squares functional that are in very good agreement with the observed numerical convergence challenging numerical examples including a nonconvex polygon with several corner singularities and a cavity domain are solved to around 10 digits of accuracy with a few seconds of cpu time the examples are implemented concisely with mpspack a matlab toolbox for wave computations with nonpolynomial basis functions developed by the authors a code example is included

an algorithm to identify and compute average optimal policies in multichain markov decision processes this paper concerns discrete time finite state multichain mdps with compact action sets the optimality criterion is long run average cost simple examples illustrate that optimal stationary markov policies do not always exist we establish the existence of e optimal policies that are stationary markovian and develop an algorithm that computes these approximate optimal policies we establish a necessary and sufficient condition for the existence of an optimal policy that is stationary markovian and in case that such an optimal policy exists the algorithm computes it

evaluation of abramowitz functions in the right half of the complex plane a numerical scheme is developed for the evaluation of abramowitz functions jn in the right half of the complex plane for n 1 2 the scheme utilizes series expansions for z r with r determined by the required precision and least squares laurent polynomial approximations on each sub region in the intermediate region 1 z r for n 2 jn is evaluated via a forward recurrence relation the scheme achieves nearly machine precision for n 1 2 at a cost that is competitive as compared with software packages for the evaluation of other special functions in the complex domain

denumerable undiscounted semi markov decision processes with unbounded rewards this paper establishes the existence of a solution to the optimality equations in undis counted semi markov decision models with countable state space under conditions generalizing the hitherto obtained results in particular we merely require the existence of a finite set of states in which every pair of states can reach each other via some stationary policy instead of the traditional and restrictive assumption that every stationary policy has a single irreducible set of states a replacement model and an inventory model illustrate why this extension is essential our approach differs fundamentally from classical approaches we convert the optimality equations into a form suitable for the application of a fixed point theorem

numerical treatment of acoustic problems with boundary singularities by the singular boundary method abstract the singular boundary method sbm is a novel boundary type meshless method based on the fundamental solution of the given governing equation the sbm employs the origin intensity factors to circumvent the singularities resulting from the fundamental solutions in this paper we investigate the acoustic problems with boundary singularities using the sbm this is achieved by combining the sbm with the singularity subtraction techniques where the solution is decomposed into the singular solution and the regular solution the singular solution is derived analytically which satisfies the governing equation and the corresponding boundary conditions containing the singularities then the regular solution is obtained by the sbm numerical examples show the excellent performance of the proposed technique

an efficient wave based method for 2d acoustic problems containing corner singularities abstract the wave based method is an efficient alternative prediction technique based on a trefftz approach which as compared to element based methods can be applied in a wider frequency range at a reasonable cost this paper discusses the use of this method for the particular case of 2d acoustic problems where singularities are present in the corners of the acoustic domain the conventional set of expansion functions used to approximate the dynamic pressure field is extended with special purpose functions that incorporate the singular behaviour in the region around the singular corners this enrichment comes with a beneficial convergence rate while only a small increase of computational effort is incurred this is illustrated for various problem settings including different combinations of both homogeneous and non homogeneous boundary conditions and bounded as well as unbounded problems are considered

efficient implementation of the mfs the three scenarios in this study we investigate the approximation of the solutions of harmonic problems subject to dirichlet boundary conditions by the method of fundamental solutions mfs in particular we study the application of the mfs to dirichlet problems in a disk the mfs discretization yields systems which possess special features which can be exploited by using fast fourier transform fft based techniques we describe three possible formulations related to the ratio of boundary points to sources namely when the number of boundary points is equal larger and smaller than the number of sources we also present some numerical experiments and provide an efficient matlab implementation of the resulting algorithms

optimal threshold probability and expectation in semi markov decision processes we consider undiscounted semi markov decision process with a target set and our main concern is a problem minimizing threshold probability we formulate the problem as an infinite horizon case with a recurrent class we show that an optimal value function is a unique solution to an optimality equation and there exists a stationary optimal policy also several value iteration methods and a policy improvement method are given in our model furthermore we investigate a relationship between threshold probabilities and expectations for total rewards

a meshfree method for elasticity problems with interfaces in this work we develop a meshfree method based on fundamental solutions basis functions for a transmission problem in linear elasticity the addressed problem consists in given the displacement field on the boundary compute the corresponding displacement field of an elastic object which has piecewise constant lame coefficients the lame coefficients are assumed to be constant in non overlapping subdomains and on the corresponding interface interior boundaries non homogeneous jump conditions on the displacement and on the traction vectors are considered the main properties of the method are analyzed and illustrated with several numerical simulations in 2d domains

comparable upper and lower bounds for boundary values of neumann eigenfunctions and tight inclusion of eigenvalues for smooth bounded domains in rn we prove upper and lower l2 bounds on the boundary data of neumann eigenfunctions and we prove quasiorthogonality of this boundary data in a spectral window the bounds are tight in the sense that both are independent of the eigenvalues this is achieved by working with an appropriate norm for boundary functions which includes a spectral weight that is a function of the boundary laplacian this spectral weight is chosen to cancel concentration at the boundary that can happen for whispering gallery type eigenfunctions these bounds are closely related to wave equation estimates due to tataru using this we bound the distance from an arbitrary helmholtz parameter e 0 to the nearest neumann eigenvalue in terms of boundary normal derivative data of a trial function u solving the helmholtz equation e u 0 this inclusion bound improves over previously known bounds by a factor of e5 6 analogously to a recently improved inclusion bound in the dirichlet case due to the first two authors finally we apply our theory to present an improved numerical implementation of the method of particular solutions for computation of neumann eigenpairs on smooth planar domains we show that the new inclusion bound improves the relative accuracy in a computed neumann eigenvalue around the 42000th from nine to fourteen digits with negligible extra numerical effort

average cost optimal stationary policies in infinite state markov decision processes with unbounded costs we deal with infinite state markov decision processes with unbounded costs three simple conditions based on the optimal discounted value function guarantee the existence of an expected average cost optimal stationary policy sufficient conditions are the existence of a distinguished state of smallest discounted value and a single stationary policy inducing an irreducible ergodic markov chain for which the average cost of a first passage from any state to the distinguished state is finite a result to verify this is also given two examples illustrate the ease of applying the criteria

conditions for existence of average and blackwell optimal stationary policies in denumerable markov decision processes abstract sufficiency conditions are given for the existence of stationary blackwell optimal policies these conditions concern only the one step transition probability matrices and their resolvents and therefore appear to be more natural than those recently introduced by dekker and hordijk we use the same operator theoretic approach and standard results about spectral theory of bounded linear operators

local magnetization of the two dimensional ising model near the phase boundary abstract we study the two dimensional ising model on a square lattice with bonds which differ on the two sides of the column of the defect vertical couplings we compute exactly the local magnetization near the defect column we also show explicitly that near the critical point the magnetization has the scaling form we calculate the asymptotic behaviour of the scaling function in the different special cases

on the second optimality equation for semi markov decision models for a semi markov decision model with average return the validity of the second optimality equation is shown in the nonmodified form where the actions run through the set of all admissible actions rather than through the set of maximum points conserving actions for the first optimal equation as a consequence the existence of a strongly optimal stationary policy is shown the results seem to be known only for finite state finite action models whereas here countable state compact action models with unbounded rewards are considered

asymptotic stability of the fundamental solution method abstract in a previous paper 1988 we studied the numerical stability of the fundamental solution method applied to the dirichlet problem of laplace s equation and presented a practical scheme for examining the stability of the method using the singular value decomposition in the present paper we give an asymptotic estimate of the stability with respect to the number of collocation points

a meshfree numerical method for acoustic wave propagation problems in planar domains with corners and cracks the numerical solution of acoustic wave propagation problems in planar domains with corners and cracks is considered since the exact solution of such problems is singular in the neighborhood of the geometric singularities the standard meshfree methods based on global interpolation by analytic functions show low accuracy in order to circumvent this issue a meshfree modification of the method of fundamental solutions is developed where the approximation basis is enriched by an extra span of corner adapted non smooth shape functions the high accuracy of the new method is illustrated by solving several boundary value problems for the helmholtz equation modelling physical phenomena from the fields of room acoustics and acoustic resonance

interpolation in the limit of increasingly flat radial basis functions abstract many types of radial basis functions such as multiquadrics contain a free parameter in the limit where the basis functions become increasingly flat the linear system to solve becomes highly ill conditioned and the expansion coefficients diverge nevertheless we find in this study that limiting interpolants often exist and take the form of polynomials in the 1 d case we prove that with simple conditions on the basis function the interpolants converge to the lagrange interpolating polynomial hence differentiation of this limit is equivalent to the standard finite difference method we also summarize some preliminary observations regarding the limit in 2 d

statistical properties of many particle eigenfunctions wavefunction correlations and density matrices for few or many particles are derived from the properties of semiclassical energy green functions universal features of fixed energy microcanonical random wavefunction correlation functions appear which reflect the emergence of the canonical ensemble as n this arises through a little known asymptotic limit of bessel functions constraints due to symmetries boundaries and collisions between particles can be included

method of moments analysis of resonant circular arrays of cylindrical dipoles we perform a method of moments mom analysis of a circular array of cylindrical dipoles the array is known from earlier theoretical and experimental studies to possess very narrow resonances the earlier theoretical studies were carried out using the two term theory the present paper is a direct continuation of a recent work showing that the problem possesses unique and particular difficulties the main difficulties are overcome herein using a set of improved kernels in the usual hallen type integral equations these kernels had been developed in previous works and were successfully incorporated into the aforementioned two term theory analyses we make a detailed comparison of our mom results to two term theory results and also to the earlier experimental results

on the use of quasi equidistant source points over the sphere surface for the method of fundamental solutions abstract the method of fundamental solutions is broadly used in science and engineering to numerically solve the direct time harmonic scattering problem in 2d the choice of source points is usually made by considering an inner pseudo boundary over which equidistant source points are placed in 3d however this problem is much more challenging since in general n equidistant points over a closed surface do not exist in this paper we discuss a method to obtain a quasi equidistant point distribution over the unit sphere surface giving rise to a delaunay triangulation that might also be used for other boundary element methods we give theoretical estimates for the expected distance between points and the expect area of each triangle we illustrate the feasibility of the proposed method in terms of the comparison with the expected values for distance and area we also provide numerical evidence that this point distribution leads to a good conditioning of the linear system associated with the direct scattering problem being therefore an adequated choice of source points for the method of fundamental solutions

a forecast horizon and a stopping rule for general markov decision processes in this paper we are concerned with stochastic optimal control problems over an infinite horizon when the reward is discounted there is only one optimality criterion and standard dynamic programming can be applied now in the undiscounted case the total reward might be unbounded for any feasible policy and there is not a unique optimahty criterion any more greatest average reward overtaking optimality average overtaking optimality and i optimality are such criteria 4 lo the most used of them is the greatest average reward and several technics have been proposed to get an average optimal policy at least for finite state space markov decision processes mdp i 1 these results are all based on the existence of a fixed point solution to the optimality equations for a relative value function in the case of denumerable state space weak conditions have been recently exhibited under which a fixed point solution exists 3 in the case of finite state space non homogeneous mdp the relative value function is also shown to converge under so called weak ergodicity con ditions s while the latter assumptions are stronger one can easily check them by inspection of the one step transition probability matrices and also

high order finite point method for the solution to the sound propagation problems abstract in this paper we present an accuracy improvement of the meshfree finite point method this high order method has been used to solve the sound propagation problems which can be modelled by linearized euler equations high accuracy has been obtained using polynomial reconstruction of variables involved in the riemann solver the order of the meshfree method will be verified on 2d acoustic pulse problem which serves as a benchmark problem with known analytical solution

efficient numerical solution of acoustic scattering from doubly periodic arrays of axisymmetric objects we present a high order accurate boundary based solver for three dimensional 3d frequency domain scattering from a doubly periodic grating of smooth axisymmetric sound hard or transmission obstacles we build the one obstacle solution operator using separation into p azimuthal modes via the fft the method of fundamental solutions with n proxy points lying on a curve and dense direct least squares solves the effort is o n 3 p with a small constant periodizing then combines fast multipole summation of nearest neighbors with an auxiliary global helmholtz basis expansion to represent the distant contributions and enforcing quasiperiodicity and radiation conditions on the unit cell walls eliminating the auxiliary coefficients and preconditioning with the one obstacle solution operator leaves a well conditioned square linear system that is solved iteratively the solution time per incident wave is then o n p at fixed frequency our scheme avoids singular quadratures periodic green s functions and lattice sums and its convergence rate is unaffected by resonances within obstacles we include numerical examples such as scattering from a grating of period 13 13 comprising highly resonant sound hard cups each needing n p 64800 surface unknowns to 10 digit accuracy in half an hour on a desktop

numerical algorithms based on analytic function values at roots of unity let f z be an analytic or meromorphic function in the closed unit disk sampled at the n th roots of unity based on these data how can we approximately evaluate f z or f m z at a point z in the disk how can we calculate the zeros or poles of f in the disk these questions exhibit in the purest form certain algorithmic issues that arise across computational science in areas including integral equations partial differential equations and large scale linear algebra we analyze some of the possibilities and emphasize the distinction between algorithms based on polynomial or rational interpolation and those based on trapezoidal rule approximations of cauchy integrals we then show how these developments apply to the problem of computing the eigenvalues in the disk of a matrix of large dimension finally we highlight the power of rational in comparison with polynomial approximations for some of these problems

recurrence conditions for average and blackwell optimality in denumerable state markov decision chains in a previous paper dekker and hordijk 1988 presented an operator theoretical approach for multichain markov decision processes with a countable state space compact action sets and unbounded rewards conditions were presented guaranteeing the existence of a laurent series expansion for the discounted rewards the existence of average and blackwell optimal policies and the existence of solutions for the average and blackwell optimality equations while these assumptions were operator oriented and formulated as conditions for the deviation matrix we will show in this paper that the same approach can also be carried out under recurrence conditions these new conditions seem easier to check in general and are especially suited for applications in queueing models

average sensitive and blackwell optimal policies in denumerable markov decisionchains with unbounded rewards in this paper we consider a discrete time markov decision chain with a denumerable state space and compact action sets and we assume that for all states the rewards and transition probabilities depend continuously on the actions the first objective of this paper is to develop an analysis for average optimality without assuming a special markov chain structure in doing so we present a set of conditions guaranteeing average optimality which are automatically fulfilled in the finite state and action model the second objective is to study simultaneously average and discount optimality as veinott veinott a f jr 1969 on discrete dynamic programming with sensitive discount optimality criteria ann math statist 40 1635 1660 did for the finite state and action model we investigate the concepts of n discount and blackwell optimality in the denumerable state space using a laurent series expansion for the discounted rewards under the same condition as for average optimality we establish solutions to the n discount optimality equations for every n

an augmented rbf method for solving fractional sturm liouville eigenvalue problems in this work we construct numerical schemes using radial basis functions rbfs to solve sturm liouville eigenvalue problems in which the so called fractional derivatives are involved we take into account the fact that the solutions to such problems might be nonsmooth and consequently we augment the space generated by rbf base functions with some fractional polynomials to overcome this problem we show through some examples the effectiveness of our method

fundamental solutions method for elliptic boundary value problems we consider a procedure for solving boundary value problems for elliptic homogeneous equations known as the fundamental solutions method we prove its applicability for some second order operators as well as for fourth order ones the boundary conditions of an elliptic problem are approximated by using fundamental solutions of the corresponding operator with singularities located outside the domain of interest in a specific case of the laplacian an estimate shows that the method discussed possesses convergence properties as good as those of any method using harmonic polynomials as trial functions numerical examples are given among which are simple signorini s problems for harmonic and lame operators

on the inverse spectral problem for euclidean triangles we consider the inverse spectral problem for the laplace operator on triangles with dirichlet boundary conditions providing numerical evidence to the effect that the eigenvalue triplet 1 2 3 is sufficient to determine a triangle uniquely on the other hand we show that other combinations such as 1 2 4 will not be enough and that there will exist at least two triangles with the same values on these triplets

optimization of spectral functions of dirichlet laplacian eigenvalues we consider the shape optimization of spectral functions of dirichlet laplacian eigenvalues over the set of star shaped symmetric bounded planar regions with smooth boundary the regions are represented using fourier cosine coefficients and the optimization problem is solved numerically using a quasi newton method the method is applied to maximizing two particular nonsmooth spectral functions the ratio of the nth to first eigenvalues and the ratio of the nth eigenvalue gap to first eigenvalue both of which are generalizations of the payne polya weinberger ratio the optimal values and attaining regions for n 13 are presented and interpreted as a study of the range of the dirichlet laplacian eigenvalues for both spectral functions and each n the optimal attaining region has multiplicity two nth eigenvalue

isospectral shapes with neumann and alternating boundary conditions the isospectrality of a well known pair of shapes constructed from two arrangements of seven congruent right isosceles triangles with the neumann boundary condition is verified numerically to high precision equally strong numerical evidence for isospectrality is presented for the eigenvalues of this standard pair in new boundary configurations with alternating dirichlet and neumann boundary conditions along successive edges good agreement with theory is obtained for the corresponding spectral staircase functions strong numerical evidence is also presented for isospectrality in an example of a different pair of shapes whose basic building block triangle is not isosceles some possible confirmatory experiments involving fluids are suggested

localized mfs for the inverse cauchy problems of two dimensional laplace and biharmonic equations abstract this paper makes a first attempt to use a new localized method of fundamental solutions lmfs to accurately and stably solve the inverse cauchy problems of two dimensional laplace and biharmonic equations in complex geometries the lmfs firstly divides the whole physical domain into several small overlapping subdomains and then employs the traditional method of fundamental solutions mfs formulation in every local subdomain for calculating the unknown coefficients on the local fictitious boundary after that a sparse linear system is formed by using the governing equation for interior nodes and the nodes on under specified boundary and by using the given boundary conditions for the nodes on over specified boundary finally the numerical solutions of the inverse problems can be obtained by solving the resultant sparse system compared with the traditional mfs with the global boundary discretization the lmfs requires less computational cost which may make the lmfs suitable for solving large scale problems numerical experiments demonstrate the validity and accuracy of the proposed lmfs for the inverse cauchy problems of two dimensional laplace and biharmonic equations with noisy boundary data

the method of fundamental solutions applied to boundary value problems on the surface of a sphere abstract in this work we propose using the method of fundamental solutions mfs to solve boundary value problems for the helmholtz beltrami equation on a sphere we prove density and convergence results that justify the proposed mfs approximation several numerical examples are considered to illustrate the good performance of the method

reviving the method of particular solutions fox henrici and moler made famous a method of particular solutions for computing eigenvalues and eigenmodes of the laplacian in planar regions such as polygons we explain why their formulation of this method breaks down when applied to regions that are insufficiently simple and propose a modification that avoids these difficulties the crucial changes are to introduce points in the interior of the region as well as on the boundary and to minimize a subspace angle rather than just a singular value or a determinant similar methods may be used to improve other mesh free algorithms for a variety of computational problems

asymptotic rate of quantum ergodicity in chaotic euclidean billiards the quantum unique ergodicity que conjecture of rudnick and sarnak is that every eigenfunction n of the laplacian on a manifold with uniformly hyperbolic geodesic flow becomes equidistributed in the semiclassical limit eigenvalue en that is strong scars are absent we study numerically the rate of equidistribution for a uniformly hyperbolic sinai type planar euclidean billiard with dirichlet boundary condition the drum problem at unprecedented high e and statistical accuracy via the matrix elements n m of a piecewise constant test function a by collecting 30 000 diagonal elements up to level n 7 105 we find that their variance decays with eigenvalue as a power 0 48 0 01 close to the semiclassical estimate of feingold and peres this contrasts with the results of existing studies which have been limited to en a factor 102 smaller we find strong evidence for que in this system we also compare off diagonal variance as a function of distance from the diagonal against feingold peres or spectral measure at the highest accuracy 0 7 thus far in any chaotic system we outline the efficient scaling method and boundary integral formulae used to calculate eigenfunctions 2006 wiley periodicals inc

depinning from the defect located at a finite distance from the surface in the planar abraham model a recently developed renormalisation group approach to interface pinning problems is generalised to deal with the depinning from the defect at a finite distance z removed from the surface in the planar abraham model the phase diagrams and the incremental defect free energies over the whole range of temperatures and distances z are obtained the agreement with exact results when available is quite satisfactory

on the application of the method of fundamental solutions to boundary value problems with jump discontinuities abstract two meshfree methods are proposed for the numerical solution of boundary value problems bvps for the laplace equation coupled with boundary conditions with jump discontinuities in the first case the bvp is solved in two steps using a subtraction of singularity approach here the singular subproblem is solved analytically while the classical method of fundamental solutions mfs is applied for the solution of the regular subproblem in the second case the total bvp is solved using a variant of the mfs where its approximation basis is enriched with a set of harmonic functions with singular traces on the boundary of the domain the same singularity capturing functions motivated by the boundary element method bem are used for the singular part of the solution in the first method and for augmenting the mfs basis in the second method comparative numerical results are presented for 2d problems with discontinuous dirichlet boundary conditions in particular the inappropriate oscillatory behavior of the classical mfs solution due to the gibbs phenomenon is shown to vanish

stress intensity factor computation using the method of fundamental solutions mixed mode problems the method of fundamental solutions is applied to the computation of stress intensity factors in linear elastic fracture mechanics the displacements are approximated by linear combinations of the fundamental solutions of the cauchy navier equations of elasticity and the leading terms for the displacement near the crack tip two algorithms are developed one using a single domain and one using domain decomposition numerical results are given copyright 2006 john wiley sons ltd

coupling finite elements and auxiliary sources abstract we consider second order scalar elliptic boundary value problems on unbounded domains which model for instance electrostatic fields we propose a discretization that relies on a trefftz approximation by multipole auxiliary sources in some parts of the domain and on standard mesh based primal lagrangian finite elements in other parts several approaches are developed and based on variational saddle point theory rigorously analyzed to couple both discretizations across the common interface 1 least squares based coupling using techniques from pde constrained optimization 2 coupling through dirichlet to neumann operators 3 three field variational formulation in the spirit of mortar finite element methods we compare these approaches in a series of numerical experiments

the method of fundamental solutions applied to boundary eigenvalue problems we develop methods based on fundamental solutions to compute the steklov wentzell and laplace beltrami eigenvalues in the context of shape optimization in the class of smooth simply connected two dimensional domains the numerical method is accurate and fast a theoretical error bound is given along with comparisons with mesh based methods we illustrate the use of this method in the study of a wide class of shape optimization problems in two dimensions we extend the method to the computation of the laplace beltrami eigenvalues on surfaces and we investigate some spectral optimal partitioning problems

a modal spectral model for flanking transmissions abstract a model for the prediction of direct and indirect flanking sound transmissions is presented it can be applied to geometries with extrusion symmetry the structures are modelled with spectral finite elements the acoustic domains are described by means of a modal expansion of the pressure field and must be cuboid shaped these reasonable simplifications in the geometry allow the use of more efficient numerical methods consequently the coupled vibroacoustic problem in structures such as junctions is efficiently solved the vibration reduction index of t junctions with acoustic excitation and with point force excitation is compared the differences due to the excitation type obey quite general trends that could be taken into account by prediction formulas however they are smaller than other uncertainties not considered in practice the model is also used to check if the sound transmissions of a fully vibroacoustic problem involving several flanking paths can be reproduced by superposition of independent paths there exist some differences caused by the interaction between paths which are more important at low frequencies

the method of fundamental solutions for helmholtz type equations in composite materials in this paper we introduce and develop the method of fundamental solutions mfs for solving helmholtz type elliptic partial differential equations in composite materials this study builds upon the previous developments and applications of the mfs to linear and nonlinear heat conduction elasticity and functionally graded composite layered materials numerical results are presented and discussed for four examples involving both the modified helmholtz and the helmholtz equations in two dimensional or three dimensional bounded or unbounded smooth or non smooth composite domains it was found that the method produces numerical results which are in good agreement with the analytical solutions where available

zero sum average semi markov games fixed point solutions of the shapley equation this paper deals with zero sum average semi markov games with borel state and action spaces unbounded payoffs and mean holding times a solution to the shapley equation is obtained via the banach fixed point theorem assuming that the model satisfies a lyapunov like condition a growth hypothesis on the payoff function and the mean holding times besides standard continuity and compactness requirements

the method of fundamental solutions in solving coupled boundary value problems for m eeg the estimation of neuronal activity in the human brain from electroencephalography eeg and magnetoencephalography meg signals is a typical inverse problem whose solution process requires an accurate and fast forward solver in this paper the method of fundamental solutions is for the first time proposed as a meshfree boundary type and easy to implement alternative to the boundary element method bem for solving the m eeg forward problem the solution of the forward problem is obtained by numerically solving a set of coupled boundary value problems for the three dimensional laplace equation numerical accuracy convergence and computational load are investigated the proposed method is shown to be a competitive alternative to the state of the art bem for m eeg forward solving

depinning transition and non universal behaviour of defects in the two dimensional ising model a unified treatment the author develops a unified renormalisation group rg framework for studying depinning and or non universal behaviour associated with linear defects in two dimensional ising models non universality is characterised by a fixed line in the rg recursion relations a generalised version of the depinning transition is proposed and a nearly exact phase diagram is calculated

an adaptive method of fundamental solutions for solving the laplace equation abstract in this paper we propose a residual type adaptive method of fundamental solutions amfs for solving the two dimensional laplace equation an error estimator is defined only on the boundary of the domain initial distributions of source points and collocation points are determined by using approaches proposed in chen et al 2006 the adding removing and stopping strategies are designed so that the required accuracy can be satisfied within finite steps numerical experiments reveal that amfs improves the accuracy of the mfs approximation obtained from uniformly distributed sources and collocation points which makes the mfs more practical for non harmonic and non smooth boundary conditions moreover it is shown that the error estimator becomes equidistributed after an adaptive iteration a detailed comparison between amfs and mfs using uniformly distributed points is also presented for each numerical example