an almost fourth order uniformly convergent difference scheme for a semilinear singularly perturbed reaction diffusion problem this paper is concerned with a high order convergent discretization for the semilinear reaction diffusion problem varepsilon 2 u b x u 0 for x in 0 1 subject to u 0 u 1 0 where varepsilon in 0 1 we assume that b u x u b 0 2 0 on 0 1 times bbb r 1 which guarantees uniqueness of a solution to the problem asymptotic properties of this solution are discussed we consider a polynomial based three point difference scheme on a simple piecewise equidistant mesh of shishkin type existence and local uniqueness of a solution to the scheme are analysed we prove that the scheme is almost fourth order accurate in the discrete maximum norm uniformly in the perturbation parameter varepsilon we present numerical results in support of this result

on the finite element method for singularly perturbed reaction diffusion problems in two and one dimensions second order elliptic boundary value problems which are allowed to degenerate into zero order equations are considered the behavior of the ordinary galerkin finite element method without special arrangements to treat singularities is studied as the problem ranges from true second order to singularly perturbed

an adaptive uniformly convergent numerical method for a semilinear singular perturbation problem a singularly perturbed semilinear two point boundary value problem is considered without any restriction on its turning points a difference scheme is presented for solving this problem on an arbitrary locally quasi uniform mesh it is shown that the solution of the scheme is first order accurate uniformly in the perturbation parameter in a discrete l 1 norm numerical results are presented for an adaptive method based on the scheme

random magnetic fields and instantons in replica space the authors study a ferromagnet in a random magnetic field using the replica method they find that the mean field equations have localized solutions i e instantons which are not invariant under rotations in replica space the relevance and the properties of these solutions are studied

a uniformly convergent method for a singularly perturbed semilinear reaction diffusion problem with multiple solutions this paper considers a simple central difference scheme for a singularly perturbed semilinear reaction diffusion problem which may have multiple solutions asymptotic properties of solutions to this problem are discussed and analyzed to compute accurate approximations to these solutions we consider a piecewise equidistant mesh of shishkin type which contains o n points on such a mesh we prove existence of a solution to the discretization and show that it is accurate of order n 2 in 2 n in the discrete maximum norm where the constant factor in this error estimate is independent of the perturbation parameter e and n numerical results are presented that verify this rate of convergence

non polynomial spline solution of singularly perturbed boundary value problems a generalized scheme based on quartic non polynomial spline functions is proposed this scheme is designed for numerical solution of singularly perturbed two point boundary value problems arising in the study of science and engineering the scheme leads to a five diagonal linear system of equations convergence analysis of the method is briefly discussed two numerical examples each of constant and variable coefficients are given to show practical usefulness of the method

the fully frustrated ising model in infinite dimensions the authors solve subject to the validity of some reasonable assumptions the fully frustrated ising model in the limit of infinite dimensions using an extension of the tap theory for spin glasses in contrast to the tap theory of the infinite range spin glass an infinite summation of diagrams is required to recover the gibbs free energy for this model the model undergoes a first order transition the method used to solve the model should have many applications to other physical problems

information geometry of mean field approximation i present a general theory of mean field approximation based on information geometry and applicable not only to boltzmann machines but also to wider classes of statistical models using perturbation expansion of the kullback divergence or plefka expansion in statistical physics a formulation of mean field approximation of general orders is derived it includes in a natural way the naive mean field approximation and is consistent with the thouless anderson palmer tap approach and the linear response theorem in statistical physics

a uniformly accurate finite element method for a singularly perturbed one dimensional reaction diffusion application d une methode a elements finis avec des elements de base exponentiels a un probleme autoadjoint a perturbation singuliere et a deux points limites generation d un schema aux differences tridiagonal uniformement precis du second ordre

error estimates for a class of partial functional differential equation with small dissipation this article deals with a class of partial functional differential equation with a small dissipating parameter on a rectangular domain classical numerical methods for solving this type of problems reveal disappointing behavior or are tremendously expensive in computer memory and processor time this arises because the precision of an approximate solution depends inversely on perturbation parameter values and thus it deteriorates as a parameter decreases therefore it is of particular interest to develop numerical methods whose error estimates would be independent of the perturbation parameter contaminating the solution in order to overcome the said difficulty we derive robust parameter uniform error estimates for a class of partial functional differential equations the analysis presented in this paper uses a suitable decomposition of the error into smooth and a singular component combined with the appropriate barrier functions and comparison principle

a uniformly convergent scheme for a system of reaction diffusion equations in this work a system of two parabolic singularly perturbed equations of reaction diffusion type is considered the asymptotic behaviour of the solution and its partial derivatives is given a decomposition of the solution in its regular and singular parts has been used for the asymptotic analysis of the spatial derivatives to approximate the solution we consider the implicit euler method for time stepping and the central difference scheme for spatial discretization on a special piecewise uniform shishkin mesh we prove that this scheme is uniformly convergent with respect to the diffusion parameters having first order convergence in time and almost second order convergence in space in the discrete maximum norm numerical experiments illustrate the order of convergence proved theoretically

high order fitted operator numerical method for self adjoint singular perturbation problems we consider self adjoint singularly perturbed two point boundary value problems in conservation form highest possible order of uniform convergence for such problems achieved hitherto via fitted operator methods was one see e g doolan et al uniform numerical methods for problems with initial and boundary layers boole press dublin 1980 p 121 reducing the original problem into the normal form and then using the theory of inverse monotone matrices a fitted operator finite difference method is derived via the standard numerov s method the scheme thus derived is fourth order accurate for moderate values of the perturbation parameter e whereas for very small values of this parameter the method is e uniformly convergent with order two numerical examples are given in support of the theory

improvement of numerical solution by boundary value technique for singularly perturbed one dimensional reaction diffusion problem a class of singularly perturbed two point boundary value problems bvps for second order ordinary differential equations in self adjoint form arising in the study of chemical catalysis and michaelis menten process in biology is considered in order to solve them a numerical method as in appl math comput 55 1993 31 is proposed the essential idea in this method is to divide the domain of the differential equation into three non overlapping subdomains and solve the given equations over these regions separately as three two point bvps numerically the inner region problems are solved using a fitted operator method uniform numerical methods for problems with initial and boundary layers boole press dublin 1980 whereas the outer region problem is solved using a classical central difference method the boundary conditions at the transition points are obtained using the zero order asymptotic expansion approximation to the solution of the problem this method is well suited for parallel computing and an algorithm for the same is given error estimates are derived for the numerical solution some schemes for self adjoint equations in conservation form are given using newton s method of quasilinearization a class of semilinear problems are also solved numerical experiments are conducted it is found that the present method performs better than the fitted mesh method fitted numerical methods for singular perturbation problems world scientific singapore 1996 and higher order method numeriche mathematik 56 1990 675

a finite element method with a large mesh width for a stiff two point boundary value problem abstract we describe a finite element method for computation of numerical approximations of the solution of the second order singularly perturbed two point boundary value problem on 1 1 u pu f u 1 u 1 0 0 d dx on a quasi uniform mesh we construct exponentially fitted trial spaces which consist of piece wise polynomials and of exponentials which fit locally to the singular solution of the equation or its adjoint we discretise the galerkin form for the boundary problem using such exponentially fitted trial spaces we derive rigorous bounds for the error of discretisation with respect to the energy norm and we obtain superconvergence at the mesh points the error depending on the mesh width and the degree of the piece wise polynomials

replica trick to calculate means of absolute values applications to stochastic equations a version of the replica trick can be used to evaluate means of the absolute values of functions that are not positive definite and with some modifications the mean logarithm of the absolute value this trick can be applied in the context of stochastic equations where the absolute value of the jacobian of the stochastic function in each solution must be computed the calculation of the average number of solutions of the naive mean field equations for the sk spin glass is discussed an extension of the considerations to other mean field equations such as tap is straightforward the brst supersymmetry is found to provide a useful tool in uncovering some puzzling aspects in these calculations the trick presented provides a possible solution to these problems

memory free dynamics for the thouless anderson palmer equations of ising models with arbitrary rotation invariant ensembles of random coupling matrices we propose an iterative algorithm for solving the thouless anderson palmer equations of ising models with arbitrary rotation invariant random coupling matrices in the thermodynamic limit we prove by means of the dynamical functional method that the proposed algorithm converges when the so called de almeida thouless criterion is fulfilled moreover we give exact analytical expressions for the rate of the convergence

a second order parameter uniform overlapping schwarz method for reaction diffusion problems with boundary layers the problem of constructing a parameter uniform numerical method for a singularly perturbed self adjoint ordinary differential equation is considered it is shown that a suitably designed discrete schwarz method based on a standard finite difference operator with a uniform mesh on each subdomain gives numerical approximations which converge in the maximum norm to the exact solution uniformly with respect to the singular perturbation parameter this parameter uniform convergence is shown to be essentially second order that this new discrete schwarz method is efficient in practice is demonstrated by numerical experiments

adaptive and self averaging thouless anderson palmer mean field theory for probabilistic modeling we develop a generalization of the thouless anderson palmer tap mean field approach of disorder physics which makes the method applicable to the computation of approximate averages in probabilistic models for real data in contrast to the conventional tap approach where the knowledge of the distribution of couplings between the random variables is required our method adapts to the concrete set of couplings we show the significance of the approach in two ways our approach reproduces replica symmetric results for a wide class of toy models assuming a nonglassy phase with given disorder distributions in the thermodynamic limit on the other hand simulations on a real data model demonstrate that the method achieves more accurate predictions as compared to conventional tap approaches

improvement of numerical solution of selfadjoint singular perturbation problems by incorporation of asymptotic approximations a numerical method for singularly perturbed two point boundary value problems for second order ordinary differential equations without a first derivative term arising in the study of chemical catalysis and michaelis menten process in biology is proposed in this method booster method an asymptotic approximation is incorporated into a suitable finite difference scheme to improve the numerical solution uniform error estimates are derived for this method when implemented in known difference schemes an improvement by a factor of n 1 can be obtained where is the small parameter and n is the order of the asymptotic approximation for a small amount of extra work numerical results are presented in support of this claim

uniformly convergent non standard finite difference methods for self adjoint singular perturbation problems we design non standard finite difference schemes for self adjoint singularly perturbed two point boundary value problems essential physical properties e g dissipativity of the solutions of such problems are captured in the schemes by an appropriate renormalization of the denominator of the discrete derivative the schemes are analyzed for e uniform convergence several numerical examples are given to support the predicted theory

how to expand around mean field theory using high temperature expansions high temperature expansions performed at a fixed order parameter provide a simple and systematic way to derive and correct mean field theories for statistical mechanical models for models like spin glasses which have general couplings between spins the authors show that these expansions generate the thouless anderson palmer equations at low order they explicitly calculate the corrections to tap theory for these models for ferromagnetic models they show that their expansions can easily be converted into 1 d expansions around mean field theory where d is the number of spatial dimensions only a small finite number of graphs need to be calculated to generate each order in 1 d for thermodynamic quantities like free energy or magnetization unlike previous 1 d expansions the expansions are valid in the low temperature phases of the models considered they consider alternative ways to expand around mean field theory besides 1 d expansions in contrast to the 1 d expansion for the critical temperature which is presumably asymptotic these schemes can be used to devise convergent expansions for the critical temperature they also appear to give convergent series for thermodynamic quantities and critical exponents they test the schemes using the spherical model where their properties can be studied using exact expressions

a parameter robust numerical method for a two dimensional reaction diffusion problem in this paper a singularly perturbed reaction diffusion partial differential equation in two space dimensions is examined by means of an appropriate decomposition we describe the asymptotic behaviour of the solution of problems of this kind a central finite difference scheme is constructed for this problem which involves an appropriate shishkin mesh we prove that the numerical approximations are almost second order uniformly convergent in the maximum norm with respect to the singular perturbation parameter some numerical experiments are given that illustrate in practice the theoretical order of convergence established for the numerical method

mean field behaviour of spin systems with orthogonal interaction matrix for the long range deterministic spin models with glassy behaviour of marinari parisi and ritort we prove weighted factorization properties of the correlation functions which represent the natural generalization of the factorization rules valid for the curie weiss case

high order parameter robust numerical method for singularly perturbed reaction diffusion problems we present a high order parameter robust finite difference method for singularly perturbed reaction diffusion problems the problem is discretized using a suitable combination of fourth order compact difference scheme and central difference scheme on generalized shishkin mesh the convergence analysis is given and the method is proved to be almost fourth order uniformly convergent in maximum norm with respect to singular perturbation parameter e numerical experiments are conducted to demonstrate the theoretical results

optimal b spline collocation method for self adjoint singularly perturbed boundary value problems we present a b spline collocation method of higher order for a class of self adjoint singularly perturbed boundary value problems the essential idea in this method is to divide the domain of the differential equation into three non overlapping subdomains and solve the regular problems obtained by transforming the differential equation with respective boundary conditions on these subdomains using the present higher order b spline collocation method the boundary conditions at the transition points are obtained by the asymptotic approximation of order zero to the solution of the problem the convergence analysis is given and the method is shown to have optimal order convergence by collocating the perturbed differential equation which is satisfied by a special cubic spline interpolate of the true solution numerical experiments are conducted to demonstrate the efficiency of the method

a numerical method for a system of singularly perturbed reaction diffusion equations a dirichlet problem for a system of two coupled singularly perturbed reaction diffusion ordinary differential equations is examined a numerical method whose solutions converge pointwise at all points of the domain independently of the singular perturbation parameters is constructed and analysed numerical results are presented which illustrate the theoretical results

b spline collocation method for the singular perturbation problem using artificial viscosity in this paper we develop a b spline collocation method using artificial viscosity for solving singularly perturbed equations given by eu x a x u x b x u x f x a x a 0 b x b 0 0 1 u 0 a u 1 b we use the artificial viscosity to capture the exponential features of the exact solution on a uniform mesh and use b spline collocation method which leads to a tridiagonal linear system the convergence analysis is given and the method is shown to have uniform convergence of second order the design of artificial viscosity parameter is confirmed to be a crucial ingredient for simulating the solution of the problem known test problems have been studied to demonstrate the accuracy of the method numerical results show the behaviour of the method with emphasis on treatment of boundary conditions results shown by the method are found to be in good agreement with the exact solution

a coprime factorisation and its application in ℋ2 control given a plant transfer matrix as a fraction of two polynomial matrices we present a doubly coprime factorisation identity over the proper stable rational matrices of the plant transfer matrix and

replica field theory for deterministic models i binary sequences with low autocorrelation we study systems without quenched disorder with a complex landscape and we use replica symmetry theory to describe them we discuss the golay bernasconi derrida approximation of the low autocorrelation model and we reconstruct it by using replica calculations we then consider the full model its low t properties with the help of number theory and a hartree fock resummation of the high temperature series we show that replica theory allows us to solve the model in the high t phase our solution is based on one link integral techniques and is based on substituting a fourier transform with a generic unitary transformation we discuss this approach as a powerful tool to describe systems with a complex landscape in the absence of quenched disorder

a survey of numerical techniques for solving singularly perturbed ordinary differential equations this survey paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of the ideas and methods of singular perturbation theory starting from prandtl s work a large amount of work has been done in the area of singular perturbations this paper limits its coverage to some standard singular perturbation models considered by various workers and the numerical methods developed by numerous researchers after 1984 2000 the work done in this area during the period 1905 1984 has already been surveyed by the first author of this paper see appl math comput 30 1989 223 for details due to the space constraints we have covered only singularly perturbed one dimensional problems

convergence condition of the tap equation for the infinite ranged ising spin glass model it is shown that the power expansion of the gibbs potential of the sk model up to second order in the exchange couplings leads to the tap equation this result remains valid for the general including a ferromagnetic exchange sk model theorems of power expansions and resolvent techniques are employed to solve the convergence problem the convergence condition is presented for the whole temperature range and for general distributions of the local magnetisations

on a uniformly accurate finite difference approximation of a singulary peturbed reaction diffusion problem using grid equidistribution we examine the convergence properties of a finite difference approximation of a singularly perturbed reaction diffusion boundary value problem using a nonuniform grid the grid is based on the equidistribution of a positive monitor function that is a linear combination of a constant floor and a power of the second derivative of the solution analysis shows how the monitor function can be chosen to ensure that the accuracy of the numerical approximation is insensitive to the size of the singular perturbation parameter the use of equidistribution principles appears in many practical grid adaption schemes and our analysis provides insight into the convergence behaviour on such grids numerical results are given that confirm the uniform convergence rates

mean field equations for spin models with orthogonal interaction matrices we study the metastable states in ising spin models with orthogonal interaction matrices we focus on three realizations of this model the random case and two non random cases i e the fully frustrated model on an infinite dimensional hypercube and the so called sine model we use the mean field or tap equations which we derive by resumming the high temperature expansion of the gibbs free energy in some special non random cases we can find the absolute minimum of the free energy for the random case we compute the average number of solutions to the tap equations we find that the configurational entropy or complexity is extensive in the range trsb t tm we also present an apparently unrelated replica calculation which reproduces the analytical expression for the total number of tap solutions

global uniformly convergent schemes for a singularly perturbed boundary value problem using patched base spline functions abstract for a self adjoint second order model singular perturbation problem global uniformly convergent schemes are constructed the basic idea consists in replacing the coefficients by piecewise polynomials and solving the resulting problems exactly

a theory of solving tap equations for ising models with general invariant random matrices we consider the problem of solving tap mean field equations by iteration for ising models with coupling matrices that are drawn at random from general invariant ensembles we develop an analysis of iterative algorithms using a dynamical functional approach that in the thermodynamic limit yields an effective dynamics of a single variable trajectory our main novel contribution is the expression for the implicit memory term of the dynamics for general invariant ensembles by subtracting these terms that depend on magnetizations at previous time steps the implicit memory terms cancel making the iteration dependent on a gaussian distributed field only the tap magnetizations are stable fixed points if a de almeida thouless stability criterion is fulfilled we illustrate our method explicitly for coupling matrices drawn from the random orthogonal ensemble

rational decisions random matrices and spin glasses we consider the problem of rational decision making in the presence of nonlinear constraints by using tools borrowed from spin glass and random matrix theory we focus on the portfolio optimisation problem we show that the number of optimal solutions is generally exponentially large and each of them is fragile rationality is in this case of limited use in addition this problem is related to spin glasses with levy like long ranged couplings for which we show that the ground state is not exponentially degenerate

high order parameter uniform numerical method for singular perturbation problems in this paper we systematically describe how to derive a method of higher order for the numerical solution of singularly perturbed ordinary differential equations first we apply this idea to derive a fourth order method for a self adjoint singularly perturbed two point boundary value problem this method is uniformly convergent on a piecewise uniform mesh of shishkin type after we have developed and analyzed a fourth order method we explain with appropriate details how can one obtain the methods of order higher than four which looks straightforward but has not been seen in the literature so far besides these the fourth order e uniformity in the theoretical estimate has been justified by some numerical experiments

accurate discretization for singular perturbations the one dimensional case this paper develops a discretization method for one dimensional singular perturbation problems based on a petrov galerkin finite element or an equivalent finite volume scheme the method is unique in that its discretization error has a bound that is second order in the mesh size and uniform in the perturbation parameter it satisfies a local discrete conservation law and it exhibits a discrete maximum principle numerical results are included for comparison with the theory