the method of fundamental solutions for the numerical solution of the biharmonic equation abstract the method of fundamental solutions mfs is a relatively new technique for the numerical solution of certain elliptic boundary value problems it falls in the class of methods generally called boundary methods and like the well known boundary integral equation method is applicable when a fundamental solution of the differential equation is known in the mfs the approximate solution is a linear combination of fundamental solutions with singularities placed outside the domain of the problem the locations of the singularities are either preassigned or determined along with the coefficients of the fundamental solutions so that the approximate solution satisfies the boundary conditions as well as possible in many applications these quantities are determined by a least squares fit of the boundary conditions a nonlinear problem which is solved using standard software in this paper the mfs is formulated for biharmonic problems and is applied to a variety of standard test problems as well as to problems arising in elasticity and fluid flow

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 homogenization and scaling limit of some gradient perturbations of a massless free field we study the continuum scaling limit of some statistical mechanical models defined by convex hamiltonians which are gradient perturbations of a massless free field by proving a central limit theorem for these models we show that their long distance behavior is identical to a new homogenized continuum massless free field we shall also obtain some new bounds on the 2 point correlation functions of these models

finite range decomposition for families of gradient gaussian measures abstract let a family of gradient gaussian vector fields on z d be given we show the existence of a uniform finite range decomposition of the corresponding covariance operators that is the covariance operator can be written as a sum of covariance operators whose kernels are supported within cubes of diameters l k in addition we prove natural regularity for the subcovariance operators and we obtain regularity bounds as we vary within the given family of gradient gaussian measures

a meshless method for solving the cauchy problem in three dimensional elastostatics the application of the method of fundamental solutions to the cauchy problem in three dimensional isotropic linear elasticity is investigated the resulting system of linear algebraic equations is ill conditioned and therefore its solution is regularized by employing the first order tikhonov functional while the choice of the regularization parameter is based on the l curve method numerical results are presented for both under and equally determined cauchy problems in a piece wise smooth geometry the convergence accuracy and stability of the method with respect to increasing the number of source points and the distance between the source points and the boundary of the solution domain and decreasing the amount of noise added into the input data respectively are analysed

primal dual weak galerkin finite element methods for elliptic cauchy problems abstract the authors propose and analyze a well posed numerical scheme for a type of ill posed elliptic cauchy problem by using a constrained minimization approach combined with the weak galerkin finite element method the resulting euler lagrange formulation yields a system of equations involving the original equation for the primal variable and its adjoint for the dual variable and is thus an example of the primal dual weak galerkin finite element method this new primal dual weak galerkin algorithm is consistent in the sense that the system is symmetric well posed and is satisfied by the exact solution a certain stability and error estimates were derived in discrete sobolev norms including one in a weak l 2 topology some numerical results are reported to illustrate and validate the theory developed in the paper

finite range decompositions of positive definite functions abstract we give sufficient conditions for a positive definite function to admit decomposition into a sum of positive definite functions which are compactly supported within disks of increasing diameters l n more generally we consider positive definite bilinear forms f v f f defined on c 0 we say v has a finite range decomposition if v can be written as a sum v g n of positive definite bilinear forms g n such that g n f g 0 when the supports of the test functions f g are separated by a distance greater or equal to l n we prove that such decompositions exist when v is dual to a bilinear form b 2 where b is a vector valued partial differential operator satisfying some regularity conditions

the boundary element solution of the cauchy steady heat conduction problem in an anisotropic medium in this paper the iterative algorithm proposed by kozlov et al for the cauchy problem for the laplace equation is extended in order to solve the cauchy steady state heat conduction problem in an anisotropic medium the iterative algorithm is numerically implemented using the boundary element method bem the convergence and the stability of the numerical method as well as various types of accuracy convergence and stopping criteria are investigated the numerical results obtained confirm that provided an appropriate stopping regularization criterion is imposed then the iterative bem produces a convergent and stable numerical solution with respect to increasing the number of boundary elements and decreasing the amount of noise added into the input data an efficient stopping regularization criterion to cease the iterative process is proposed and the rate of convergence of the algorithm is improved by using various relaxation procedures between iterations a new concept of a variable relaxation factor is proposed analytical formulae for the coefficients of the matrices resulting from the direct application of the bem in an anisotropic medium are also presented copyright 2000 john wiley sons ltd

a renormalisation group method v a single renormalisation group step this paper is the fifth in a series devoted to the development of a rigorous renormalisation group method applicable to lattice field theories containing boson and or fermion fields and comprises the core of the method in the renormalisation group method increasingly large scales are studied in a progressive manner with an interaction parametrised by a field polynomial which evolves with the scale under the renormalisation group map in our context the progressive analysis is performed via a finite range covariance decomposition perturbative calculations are used to track the flow of the coupling constants of the evolving polynomial but on their own perturbative calculations are insufficient to control error terms and to obtain mathematically rigorous results in this paper we define an additional non perturbative coordinate which together with the flow of coupling constants defines the complete evolution of the renormalisation group map we specify conditions under which the non perturbative coordinate is contractive under a single renormalisation group step our framework is essentially combinatorial but its implementation relies on analytic results developed earlier in the series of papers the results of this paper are applied elsewhere to analyse the critical behaviour of the 4 dimensional continuous time weakly self avoiding walk and of the 4 dimensional n component varphi 4 model in particular the existence of a logarithmic correction to mean field scaling for the susceptibility can be proved for both models together with other facts about critical exponents and critical behaviour

grad o perturbations of massless gaussian fields we investigate weak perturbations of the continuum massless gaussian measure by a class of approximately local analytic functionals and use our general results to give a new proof that the pressure of the dilute dipole gas is analytic in the activity

a simple method for finite range decomposition of quadratic forms and gaussian fields we present a simple method to decompose the green forms corresponding to a large class of interesting symmetric dirichlet forms into integrals over symmetric positive semi definite and finite range properly supported forms that are smoother than the original green form this result gives rise to multiscale decompositions of the associated gaussian free fields into sums of independent smoother gaussian fields with spatially localized correlations our method makes use of the finite propagation speed of the wave equation and chebyshev polynomials it improves several existing results and also gives simpler proofs

finite range decomposition for gaussian measures with improved regularity abstract we consider a family of gradient gaussian vector fields on the torus z l n z d adams koteck muller and independently bauerschmidt established the existence of a uniform finite range decomposition of the corresponding covariance operators i e the covariance can be written as a sum of covariance operators supported on increasing cubes with diameter l k we improve this result and show that the decay behaviour of the kernels in fourier space can be controlled then we show the regularity of the integration map that convolves functionals with the partial measures of the finite range decomposition in particular the new finite range decomposition avoids the loss of regularity which arises in the renormalisation group approach to anisotropic problems in statistical mechanics

the method of fundamental solutions for the cauchy problem in two dimensional linear elasticity in this paper the application of the method of fundamental solutions to the cauchy problem in two dimensional isotropic linear elasticity is investigated the resulting system of linear algebraic equations is ill conditioned and therefore its solution is regularised by employing the first order tikhonov functional while the choice of the regularisation parameter is based on the l curve method numerical results are presented for both smooth and piecewise smooth geometries as well as for constant and linear stress states the convergence and the stability of the method with respect to increasing the number of source points and the distance between the source points and the boundary of the solution domain and decreasing the amount of noise added into the input data respectively are analysed

a meshless method for the numerical solution of the cauchy problem associated with three dimensional helmholtz type equations in this paper the application of the method of fundamental solutions to the cauchy problem associated with three dimensional helmholtz type equations is investigated the resulting system of linear algebraic equations is ill conditioned and therefore its solution is regularized by employing the zeroth order tikhonov functional while the choice of the regularization parameter is based on the l curve method numerical results are presented for under equally and over determined cauchy problems in a piecewise smooth geometry the convergence accuracy and stability of the method with respect to increasing the number of source points and the distance between the source points and the boundary of the solution domain and decreasing the amount of noise added into the input data respectively are analysed

finite range decomposition of gaussian processes let be the finite difference laplacian associated to the lattice z d for dimension d 3 a 0 and l a sufficiently large positive dyadic integer we prove that the integral kernel of the resolvent g a a 1 can be decomposed as an infinite sum of positive semi definite functions v n of finite range v n x y 0 for x y o l n equivalently the gaussian process on the lattice with covariance g a admits a decomposition into independent gaussian processes with finite range covariances for a 0 v n has a limiting scaling form l n left d 2 right gamma c left tfrac x y l n right as n as a corollary such decompositions also exist for fractional powers 2 0 2 the results of this paper give an alternative to the block spin renormalization group on the lattice

ising game nonequilibrium steady states of resource allocation systems abstract resource allocation systems are ubiquitous in the human society but how external fields affect the state of such systems remains poorly explored due to the lack of a suitable model because the behavior of spins pursuing energy minimization required by physical laws is similar to that of humans chasing payoff maximization studied in game theory here we combine the ising model with the market directed resource allocation game yielding an ising game based on the ising game we show theoretical simulative and experimental evidences for a formula which offers a clear expression of nonequilibrium steady states nesss interestingly the formula also reveals a convertible relationship between the external field exogenous factor and resource ratio endogenous factor and a class of saturation as the external field exceeds certain limits this work suggests that the ising game could be a suitable model for studying external field effects on resource allocation systems and it could provide guidance both for seeking more relations between nesss and equilibrium states and for regulating human systems by choosing nesss appropriately

meshless methods for the inverse problem related to the determination of elastoplastic properties from the torsional experiment abstract the problem of determining the elastoplastic properties of a prismatic bar from the given experimental relation between the torsional moment m and the angle of twist per unit length of the rod s length is investigated as an inverse problem the proposed method to solve the inverse problem is based on the solution of some sequences of the direct problem by applying the levenberg marquardt iteration method in the direct problem these properties are known and the torsional moment is calculated as a function of the angle of twist from the solution of a non linear boundary value problem this non linear problem results from the saint venant displacement assumption the ramberg osgood constitutive equation and the deformation theory of plasticity for the stress strain relation to solve the direct problem in each iteration step the kansa method is used for the circular cross section of the rod or the method of fundamental solutions mfs and the method of particular solutions mps are used for the prismatic cross section of the rod the non linear torsion problem in the plastic region is solved using the picard iteration

an alternating iterative algorithm for the cauchy problem associated to the helmholtz equation in this paper the iterative algorithm proposed by kozlov et al comput maths math phys 31 1991 45 for obtaining approximate solutions to the ill posed cauchy problem for the helmholtz equation is analysed the technique is then numerically implemented using the boundary element method bem the numerical results confirm that the iterative bem produces a convergent and stable numerical solution with respect to increasing the number of boundary elements and decreasing the amount of noise added into the input data an efficient stopping regularising criterion is also proposed

numerical solution of the cauchy problem for steady state heat transfer in two dimensional functionally graded materials the application of the method of fundamental solutions to the cauchy problem for steady state heat conduction in two dimensional functionally graded materials fgms is investigated the resulting system of linear algebraic equations is ill conditioned and therefore regularization is required in order to solve this system of equations in a stable manner this is achieved by employing the zeroth order tikhonov functional while the choice of the regularization parameter is based on the l curve method numerical results are presented for both smooth and piecewise smooth geometries the convergence and the stability of the method with respect to increasing the number of source points and the distance between the source points and the boundary of the solution domain and decreasing the amount of noise added into the input data respectively are analysed

logarithmic correction for the susceptibility of the 4 dimensional weakly self avoiding walk a renormalisation group analysis we prove that the susceptibility of the continuous time weakly self avoiding walk on mathbb z d in the critical dimension d 4 has a logarithmic correction to mean field scaling behaviour as the critical point is approached with exponent frac 1 4 for the logarithm the susceptibility has been well understood previously for dimensions d 5 using the lace expansion but the lace expansion does not apply when d 4 the proof begins by rewriting the walk two point function as the two point function of a supersymmetric field theory the field theory is then analysed via a rigorous renormalisation group method developed in a companion series of papers by providing a setting where the methods of the companion papers are applied together the proof also serves as an example of how to assemble the various ingredients of the general renormalisation group method in a coordinated manner