a posteriori error estimation and adaptive mesh refinement techniques abstract we analyse three different a posteriori error estimators for elliptic partial differential equations they are based on the evaluation of local residuals with respect to the strong form of the differential equation on the solution of local problems with neumann boundary conditions and on the solution of local problems with dirichlet boundary conditions we prove that all three are equivalent and yield global upper and local lower bounds for the true error thus adaptive mesh refinement techniques based on these estimators are capable to detect local singularities of the solution and to appropriately refine the grid near these singularities some numerical examples prove the efficiency of the error estimators and the mesh refinement techniques

quantitative error estimate in modeling the laminar stationary flow of a bingham fluid we present a quantitative a posteriori error estimate when the linear laminar stationary flow of a bingham fluid in a cylindrical pipe is modeled as a newtonian fluid the estimate is proved by using a complementary variational principle for the bingham fluid flow examples show that the estimate is effective

applications of fixed point methods to discrete variational and quasi variational inequalities in this paper discrete analogues of variational inequalities v i and quasi variational inequalities q v i encountered in stochastic control and mathematical physics are discussed it is shown that those discrete v i s and q v i s can be written in the fixed point formx tx such that eithert or some power oft is a contraction this leads to globally convergent iterative methods for the solution of discrete v i s and q v i s which are very suitable for implementation on parallel computers with single instruction multiple data architecture particularly on massively parallel processors m p p s

flow of a bingham fluid in contact with a newtonian fluid abstract we study the problem modeling the flow of a bingham fluid in contact with a newtonian fluid playing the role of lubricant this is a free boundary problem coupled by means of diffraction conditions with a boundary value problem of parabolic type we examine the steady state solutions the evolutive case with particular regard for the asymptotic behaviour of the solution and a regularized model related to the appearance of a new rigid zone

applications of fixed point technique in solving certain dynamic programming and variational inequalities abstract in this paper we introduce the concepts of compatible maps of type t type i and weakly compatible maps of type t type i for hybrid functions to produce common fixed point theorems for set valued functions our results extend some known results for point valued and set valued functions as applications of fixed point technique the existence and uniqueness of common solutions for certain class of the functional equations in dynamic programming and variational inequalities arises in two point obstacle problem are discussed

error estimates for the numerical approximation of time dependent flow of bingham fluid in cylindrical pipes by the regularization method the flow of a bingham fluid in a cylindrical pipe can give rise to free boundary problems the fluid behaves like a viscous fluid if the shear stress expressed as a linear function of the shear rate exceeds a yield value and like a rigid body otherwise the surfaces dividing fluid and rigid zones are the free boundaries therefore the solution for such highly nonlinear problems can in general only be obtained by numerical methods considerable progress has been made in the development of numerical algorithms for bingham fluids 2 20 27 29 30 32 however very little research can be found in the literature regarding the rate of convergence of the numerical solution to the true continuous solution that is the error estimate of these numerical methods error estimates are a critically important issue because they tells us how to control the error by appropriately choosing the grid sizes and other related parameters this paper concerns the error estimates of a unsteady bingham fluid modeled as a variational inequality due to duvaut lions 16 and glowinski 30 the difficulty both in the analytical and numerical treatment of the mathematical model is due to the fact that it contains a nondifferentiable term a common technique called the regularization method is to replace the non differentiable term by a perturbed differentiable term which depends on a small regularization parameter the regularization method effectively reduces the variational inequality to an equation a regularized problem which is much easier to cope with this paper has achieved the following 1 error estimates are derived for a continuous time galerkin method in suitable norms 2 we give an estimate of the difference between the true solution and the regularized solution in terms of 3 some regularity properties for both regularized solution and the true solution are proved 4 the error estimates for full discretization of the regularized problem using piecewise linear finite elements in space and backward differencing in time are established for the first time by coupling the regularization parameter and the discretization parameters h and t 5 we are able to improve our estimates in the one dimensional case or under stronger regularity assumptions on the true solution the estimates for the one dimensional case are optimal and confirmed by numerical experiment the estimates from 4 and 5 provide very important information on the measure of the error and give us a powerful mechanism to properly choose the parameters h t and in order to control the error see corollary 4 4 the above estimates extend the error bounds derived in glowinski lions and tremolieres 32 chapter 5 pp 348 404 for the stationary bingham fluid to the time dependent one which is the main contribution of this paper

error estimates for the approximation of semicoercive variational inequalities an abstract error estimate for the approximation of semicoercive variational inequalities is obtained provided a certain condition holds for the exact solution this condition turns out to be necessary as is demonstrated analytically and numerically the results are applied to the finite element approximation of poisson s equation with signorini boundary conditions and to the obstacle problem for the beam with no fixed boundary conditions for second order variational inequalities the condition is always satisfied whereas for the beam problem the condition holds if the center of forces belongs to the interior of the convex hull of the contact set applying the error estimate yields optimal order of convergence in terms of the mesh size h the numerical convergence rates observed are in good agreement with the predicted ones

anl2 error estimate for an approximation of the solution of a parabolic variational inequality we estimate the order of the difference between the numerical approximation and the solution of a parabolic variational inequality the numerical approximation is obtained using a finite element discretization in space and a finite difference discretization in time which is more general than is used in the literature we obtain better error estimates than those given in the literature the error estimates are compared with numerical experiments

a posteriori error estimator for obstacle problems in this paper we consider a posteriori error estimators for obstacle problems the variational inequality is reformulated as a mixed problem in terms of a discrete nodewise defined but variationally consistent lagrange multiplier locally defined equilibrated fluxes and an h mathrm div conforming lifting define our estimator to obtain a better local upper bound for the estimator we introduce a different elementwise defined lagrange multiplier although the upper and lower bounds are established for affine obstacles we present generalizations to nonsmooth obstacles and to nonmatching meshes different numerical examples show the efficiency and reliability of our estimator due to its flexible construction principle and abstract framework it can be also applied as an error indicator to more complex obstacle problems such as e g american option pricing in financial mathematics

a bi projection method for bingham type flows we propose and study a new numerical scheme to compute the isothermal and unsteady flow of an incompressible viscoplastic bingham medium the main difficulty for both theoretical and numerical approaches is due to the non differentiability of the plastic part of the stress tensor in regions where the rate of strain tensor vanishes this is handled by reformulating the definition of the plastic stress tensor in terms of a projection a new time scheme based on the classical incremental projection method for the newtonian navier stokes equations is proposed the plastic tensor is treated implicitly in the first sub step of the projection scheme and is computed by using a fixed point procedure a pseudo time relaxation is added into the bingham projection whose effect is to ensure a geometric convergence of the fixed point algorithm this is a key feature of the bi projection scheme which provides a fast and accurate computation of the plastic tensor stability and error analyses of the numerical scheme are provided the error induced by the pseudo time relaxation term is controlled by a prescribed numerical parameter so that a first order estimate of the time error is derived for the velocity field a second order cell centred finite volume scheme on staggered grids is applied for the spatial discretisation the scheme is assessed against previously published benchmark results for both newtonian and bingham flows in a two dimensional lid driven cavity for reynolds number equals 1000 moreover the proposed numerical scheme is able to reproduce the fundamental property of cessation in finite time of a viscoplastic medium in the absence of any energy source term in the equations for a fixed value 100 of the bingham number various numerical simulations for a range of reynolds numbers up to 200000 were performed with the bi projection scheme on a grid with 1024 2 mesh points the effect of this physical parameter on the flow behaviour is discussed

dynamic programming and maximum principle for discrete goursat systems abstract we obtain the dynamic programming equations for discrete goursat systems we prove that they are necessary and sufficient for optimality and we prove formulas for the exact or approximate solution of these equations using dynamic programming arguments we obtain semi dynamic programming equations and then we prove a maximum principle of pontryagin s type

neural networks with a redundant representation detecting the undetectable we consider a three layer sejnowski machine and show that features learnt via contrastive divergence have a dual representation as patterns in a dense associative memory of order p 4 the latter is known to be able to hebbian store an amount of patterns scaling as n p ensuremath 1 where n denotes the number of constituting binary neurons interacting p wisely we also prove that by keeping the dense associative network far from the saturation regime namely allowing for a number of patterns scaling only linearly with n while pg2 such a system is able to perform pattern recognition far below the standard signal to noise threshold in particular a network with p 4 is able to retrieve information whose intensity is mathcal o 1 even in the presence of a noise mathcal o sqrt n in the large n limit this striking skill stems from a redundancy representation of patterns which is afforded given the relatively low load information storage and it contributes to explain the impressive abilities in pattern recognition exhibited by new generation neural networks the whole theory is developed rigorously at the replica symmetric level of approximation and corroborated by signal to noise analysis and monte carlo simulations

exponential rosenbrock integrators for option pricing in this paper we are concerned with the time integration of differential equations modeling option pricing in particular we consider the black scholes equation for american options as an alternative to existing methods we present exponential rosenbrock integrators these integrators require the evaluation of the exponential and related functions of the jacobian matrix the resulting methods have good stability properties they are fully explicit and do not require the numerical solution of linear systems in contrast to standard integrators we have implemented some numerical experiments in matlab showing the reliability of the new method

resummed mean field inference for strongly coupled data we present a resummed mean field approximation for inferring the parameters of an ising or a potts model from empirical noisy one and two point correlation functions based on a resummation of a class of diagrams of the small correlation expansion of the log likelihood the method outperforms standard mean field inference methods even when they are regularized the inference is stable with respect to sampling noise contrarily to previous works based either on the small correlation expansion on the bethe free energy or on the mean field and gaussian models because it is mostly analytic its complexity is still very low requiring an iterative algorithm to solve for n auxiliary variables that resorts only to matrix inversions and multiplications we test our algorithm on the sherrington kirkpatrick model submitted to a random external field and large random couplings and demonstrate that even without regularization the inference is stable across the whole phase diagram in addition the calculation leads to a consistent estimation of the entropy of the data and allows us to sample form the inferred distribution to obtain artificial data that are consistent with the empirical distribution

robust identification and galois sequences abstract worst case l1 identification is studied for bibo stable linear shift invariant systems it is shown that the chebyshev identification method when used with galois input designs satisfies a certain robust convergence property and provides l1 model error bounds in worst case identification of bibo stable systems with a uniformly bounded noise set up the robust identification methodology developed is compatible with the modelling requirements of modern robust control design

implicit explicit runge kutta methods for financial derivatives pricing models abstract implicit explicit runge kutta methods are investigated for application to financial derivatives pricing models in the partial differential equations approach the methods are showed to be an alternative to other existing procedures for the numerical valuation of american type contracts we follow the method of lines in order to have a numerical method that can be used with a variety of state variable discretizations including finite elements finite differences and finite volume methods some numerical experiments are presented

nonlinear black box models in system identification mathematical foundations we discuss several aspects of the mathematical foundations of the nonlinear black box identification problem we shall see that the quality of the identification procedure is always a result of a certain trade off between the expressive power of the model we try to identify the larger the number of parameters used to describe the model the more flexible is the approximation and the stochastic error which is proportional to the number of parameters a consequence of this trade off is the simple fact that a good approximation technique can be the basis of a good identification algorithm from this point of view we consider different approximation methods and pay special attention to spatially adaptive approximants we introduce wavelet and neuron approximations and show that they are spatially adaptive then we apply the acquired approximation experience to estimation problems finally we consider some implications of these theoretical developments for the practically implemented versions of the spatially adaptive algorithms

broken replica symmetry bounds in the mean field spin glass model by using a simple interpolation argument in previous work we have proven the existence of the thermodynamic limit for mean field disordered models including the sherrington kirkpatrick model and the derrida p spin model here we extend this argument in order to compare the limiting free energy with the expression given by the parisi ansatz and including full spontaneous replica symmetry breaking our main result is that the quenched average of the free energy is bounded from below by the value given in the parisi ansatz uniformly in the size of the system moreover the difference between the two expressions is given in the form of a sum rule extending our previous work on the comparison between the true free energy and its replica symmetric sherrington kirkpatrick approximation we give also a variational bound for the infinite volume limit of the ground state energy per site

convergence past singularities for a fully discrete approximation of curvature drive interfaces consider a closed surface in rn of codimension 1 which propagates in the normal direction with velocity proportional to its mean curvature plus a forcing term this geometric problem is first approximated by a singularly perturbed parabolic double obstacle problem with small parameter eps 0 conforming piecewise linear finite elements over a quasi uniform and strongly acute mesh of size h are further used for space discretization and combined with backward differences for time discretization with uniform time step tau it is shown that the zero level set of the fully discrete solution converges past singularities to the true interface provided tau h 2 approx o eps 3 and no fattening occurs if the more stringent relations tau h 2 approx o eps 4 are enforced then a linear rate of convergence o eps for interfaces is derived in the vicinity of regular points namely those for which the underlying viscosity solution is nondegenerate singularities and their smearing effect are also studied the analysis is based on constructing discrete barriers via a parabolic projection lipschitz dependence of viscosity solutions with respect to perturbations of data and discrete nondegeneracy these issues are proven along with quasi optimality in two dimensions of the parabolic projection in l infty with respect to both order and regularity requirements for functions in w 2 1 p

sharpl error estimates for semilinear elliptic problems with free boundaries a numerical scheme to approximate a semilinear pde involving a singular maximal monotone graph is analyzed inl a preliminary regularization is combined with piecewise linear finite elements defined on a triangulation which is not assumed to be acute the discrete maximum principle is thus avoided sharp pointwise error estimates are derived for both the smoothing and the discretization procedures an optimal choice of the regularization parameter as a function of the mesh size leads to a sharp global rate of convergence these error estimates for solutions in conjunction with nondegeneracy properties of continuous problems provide sharp interface error estimates two model examples are discussed the obstacle problem and a combustion equation

multiplier methods for nonlinear optimal control error estimates are derived for an augmented lagrangian approximation to an optimal control problem convex control constraints are treated explicitly while a lagrange multiplier is introduced for the nonlinear system dynamics the nonlinear optimal control problem does not fit the classical theory for estimating the error in multiplier approximations since the natural coercivity assumption is formulated in a hilbert space where the cost is not differentiable this discrepancy between the function space setting needed for coercivity and that needed for differentiability is compensated for by regularity results associated with the necessary conditions the paper concludes with an analysis of the optimal penalty parameter corresponding to a given finite element discretization

neural networks retrieving boolean patterns in a sea of gaussian ones restricted boltzmann machines are key tools in machine learning and are described by the energy function of bipartite spin glasses from a statistical mechanical perspective they share the same gibbs measure of hopfield networks for associative memory in this equivalence weights in the former play as patterns in the latter as boltzmann machines usually require real weights to be trained with gradient descent like methods while hopfield networks typically store binary patterns to be able to retrieve the investigation of a mixed hebbian network equipped with both real e g gaussian and discrete e g boolean patterns naturally arises we prove that in the challenging regime of a high storage of real patterns where retrieval is forbidden an additional load of boolean patterns can still be retrieved as long as the ratio between the overall load and the network size does not exceed a critical threshold that turns out to be the same of the standard amit gutfreund sompolinsky theory assuming replica symmetry we study the case of a low load of boolean patterns combining the stochastic stability and hamilton jacobi interpolating techniques the result can be extended to the high load by a non rigorous but standard replica computation argument

pointwise a posteriori error control for elliptic obstacle problems we consider a finite element method for the elliptic obstacle problem over polyhedral domains in d which enforces the unilateral constraint solely at the nodes we derive novel optimal upper and lower a posteriori error bounds in the maximum norm irrespective of mesh fineness and the regularity of the obstacle which is just assumed to be holder continuous they exhibit optimal order and localization to the non contact set we illustrate these results with simulations in 2d and 3d showing the impact of localization in mesh grading within the contact set along with quasi optimal meshes

a decomposition dualization approach for solving constrained convex minimization problems with applications to discretized obstacle problems in this paper we shall be concerned with the solution of constrained convex minimization problems the constrained convex minimization problems are proposed to be transformable into a convex additively decomposed and almost separable form e g by decomposition of the objective functional and the restrictions unconstrained dual problems are generated by using fenchel rockafellar duality this decomposition dualization concept has the advantage that the conjugate functionals occuring in the derived dual problem are easily computable moreover the minimum point of the primal constrained convex minimization problem can be obtained from any maximum point of the corresponding dual unconstrained concave problem via explicit return formulas in quadratic programming the decomposition dualization approach considered here becomes applicable if the quadratic part of the objective functional is generated byh matrices numerical tests for solving obstacle problems in 1 discretized by using piecewise quadratic finite elements and in 2 by using the five point difference approximation are presented

optimal convergence for discrete variational inequalities modelling signorini contact in 2d and 3d without additional assumptions on the unknown contact set the basic h 1 finite element error estimate of order h with only h 2 regularity on the solution has not been yet established for the simplest two dimensional signorini problem approximated by a discrete variational inequality or the equivalent mixed method and linear finite elements to obtain an optimal error bound in this basic case and also when considering more general cases e g the three dimensional problem quadratic finite elements additional assumptions on the exact solution in particular on the unknown contact set see z belhachmi and f ben belgacem math comp 72 2003 pp 83 104 s hueber and b wohlmuth siam j numer anal 43 2005 pp 156 173 b wohlmuth a popp m gee and w wall comput mech 49 2012 pp 735 747 had to be used in this paper we consider finite element approximations of the two dimensional and three dimensional signorini problems with linear and quadratic finite elements in the analysis we remove all the additional assumptions and pr

an interactive term approach to non parametric fir nonlinear system identification in this technical note a framework for designing specially structured input sequences for non parametric nonlinear system identification is presented so that interaction terms which describe interactions among variables can be identified separately in a sense the approach decomposes a general difficult nonlinear identification problem into a number of problems that are of lower orders corresponding identification algorithms are proposed

representation and identification of non parametric nonlinear systems of short term memory and low degree of interaction in this paper we propose a new representation which is particularly useful for a class of non parametric nonlinear systems that have short term memory and low degree of interaction advantages and disadvantages of this representation are discussed and compared to existing methods both theoretically and numerically furthermore results regarding structural estimation based on the analysis of variance and on full scale identification are also provided

l error estimate for an approximation of a parabolic variational inequality almost optimall convergence of an approximation of a variational inequality of parabolic type is proved under regularity assumptions which are met by the solution of a one phase stefan problem the discretization employs piecewise linear finite elements in space and the backward euler scheme in time by means of a maximum principle the problem is reduced to an error estimate for an auxiliary parabolic equation the latter bound is obtained by using the smoothing property of the galerkin method

error probabilities for binary direct sequence spread spectrum communications with random signature sequences binary direct sequence spread spectrum multiple access communications an additive white gaussian noise channel and a coherent correlation receiver are considered an expression for the output of the receiver is obtained for the case of random signature sequences and the corresponding characteristic function is determined the expression is used to study the density function of the multiple access interference and to determine arbitrarily tight upper and lower bounds on the average probability of error the bounds which are obtained without making a gaussian approximation are compared to results obtained using a gaussian approximation the effects of transmitter power the length of the signature sequences and the number of interfering transmitters are illustrated each transmitter is assumed to have the same power although the general approach can accommodate the case of transmitters with unequal powers

adaptive finite element methods for hydrodynamic lubrication with cavitation we present an adaptive finite element method for a cavitation model based on reynolds equation a posteriori error estimates and adaptive algorithms are discussed and numerical examples illustrating the theory are supplied

convergence of a penalty finite element approximation for an obstacle problem this study establishes an error estimate for a penalty finite element approximation of the variational inequality obtained by a class of obstacle problems by special identification of the penalty term we first show that the penalty solution converges to the solution of a mixed formulation of the variational inequality the rate of convergence of the penalization is where is the penalty parameter to obtain the error of finite element approximation we apply the results obtained by brezzi hager and raviart for the mixed finite element method to the variational inequality

error analysis for implicit approximations to solutions to cauchy problems exact error analyses for two implicit approximation schemes to the cauchy problem du t dt mathcal a u t mathrel backepsilon 0 u 0 u 0 in d mathcal a are presented the operator mathcal a is required to be maximal monotone on a hilbert space mathcal h the approximations considered are u h kh i h mathcal a k u 0 and solutions u lambda to the approximating problems du lambda t dt mathcal a lambda u lambda 0 u lambda 0 u 0 where a lambda is the yosida approximation of mathcal a the special case of a being a subgradient is considered and tight estimates of the convergence of both u h and mathcal a h u h or u h and a h u h are given the new results in this case are the linear rates of convergence of u h in l infty 0 infty mathcal h and in the energy norm and the sublinear rate of convergence of a h u h in l 2 0 infty mathcal h applications to both linear and

multilevel projection algorithm for solving obstacle problems abstract obstacle problems are nonlinear free boundary problems and the computation of approximate solutions can be difficult and expensive little work has been done on effective numerical methods of such problems this paper addresses some aspects of this issue discretizing the problem in a continuous piecewise linear finite element space gives a quadratic programming problem with inequality constraints a new method called the multilevel projection mp method is established in this paper the mp algorithm extends the multigrid method for linear equations to nonlinear obstacle problems the convergence theorems of this method are also proved a numerical example presented shows our error estimate is sharp and the mp algorithm is robust

galerkin least squares finite element method for the obstacle problem we construct a consistent multiplier free method for the finite element solution of the obstacle problem the method is based on an augmented lagrangian formulation in which we eliminate the multip

conforming and discontinuous galerkin fem in space for solving parabolic obstacle problem abstract in this article we propose and analyze conforming and discontinuous galerkin dg finite element methods for numerical approximation of the solution of the parabolic variational inequality associated with a general obstacle in r d d 2 3 for fully discrete conforming method we use globally continuous and piecewise linear finite element space whereas for the fully discrete dg scheme we employ piecewise linear finite element space for spatial discretization the time discretization has been done by using the implicit backward euler method we present the error analysis for the conforming and the dg fully discrete schemes and derive an error estimate of optimal order o h t in a certain energy norm defined precisely in the article the analysis is performed without any assumptions on the speed of propagation of the free boundary but only assumes the pragmatic regularity that u t l 2 0 t l 2 the obstacle constraints are incorporated at the lagrange nodes of the triangular mesh and the analysis exploits the lagrange interpolation we present some numerical experiment to illustrate the performance of the proposed methods

application of a first order operator splitting method to bingham fluid flow simulation abstract we discuss here a first order operator splitting method for the solution of the time dependent variational inequality modeling the flow of a bingham visco plastic medium at each time step we shall have to solve a nonlinear elliptic problem then a nondifferentiable minimization problem and finally a stokes type problem the first problem is solved by a fixed point method the second one happens to be a saddle point problem for a lagrangian functional and we shall use an uzawa algorithm to solve it the last problem is solved by a conjugate gradient method we have applied this splitting method to two test problems the first one has a forcing term with homogeneous dirichlet boundary conditions while the second one concerns the simulation of the flow of a bingham fluid in a wall driven cavity our results compare very well with the ones obtained by solving directly the variational inequality mentioned above

on deterministic control problems an approximation procedure for the optimal cost i the stationary problem we study deterministic optimal control problems having stopping time continuous and impulse controls in each strategy we obtain the optimal cost considered as the maximum element of a suitable set of subsolutions of the associated hamilton jacobi equation using an approximation method a particular derivative discretization scheme is employed convergence of approximate solutions is shown taking advantage of a discrete maximum principle which is also proved for the numerical solutions of approximate problems we use a method of relaxation type the algorithm is very simple it can be run on computers with small central memory in part i we study the stationary case in part ii siam j control optim 23 1985 pp 267 285 we study the nonstationary case

residual type a posteriori error estimates for elliptic obstacle problems a posteriori error estimators of residual type are derived for piecewise linear finite element approximations to elliptic obstacle problems an instrumental ingredient is a new interpolation operator which requires minimal regularity exhibits optimal approximation properties and preserves positivity both upper and lower bounds are proved and their optimality is explored with several examples sharp a priori bounds for the a posteriori estimators are given and extensions of the results to double obstacle problems are briefly discussed

small correlation expansions for the inverse ising problem we present a systematic small correlation expansion to solve the inverse ising problem and find a set of couplings and fields corresponding to a given set of correlations and magnetizations couplings are calculated up to the third order in the correlations for generic magnetizations and to the seventh order in the case of zero magnetizations in addition we show how to sum some useful classes of diagrams exactly the resulting expansion outperforms existing algorithms on the sherrington kirkpatrick spin glass model

best simultaneous approximations asymptotically nonexpansive mappings and variational inequalities in banach spaces abstract the existence of common fixed points is established for three mappings where t is either generalized f g nonexpansive or asymptotically f g nonexpansive on a nonempty subset of a banach space as applications the invariant best simultaneous approximation results are proved and the existence of solution of variational inequalities is obtained our results unify and substantially improve several recent results existing in the current literature

the space of interactions in neural network models the typical fraction of the space of interactions between each pair of n ising spins which solve the problem of storing a given set of p random patterns as n bit spin configurations is considered the volume is calculated explicitly as a function of the storage ratio alpha p n of the value kappa 0 of the product of the spin and the magnetic field at each site and of the magnetisation m here m may vary between 0 no correlation and 1 completely correlated the capacity increases with the correlation between patterns from alpha 2 for correlated patterns with kappa 0 and tends to infinity as m tends to 1 the calculations use a saddle point method and the order parameters at the saddle point are assumed to be replica symmetric this solution is shown to be locally stable a local iterative learning algorithm for updating the interactions is given which will converge to a solution of given kappa provided such solutions exist

phase diagram of restricted boltzmann machines and generalized hopfield networks with arbitrary priors restricted boltzmann machines are described by the gibbs measure of a bipartite spin glass which in turn can be seen as a generalized hopfield network this equivalence allows us to characterize the state of these systems in terms of their retrieval capabilities both at low and high load of pure states we study the paramagnetic spin glass and the spin glass retrieval phase transitions as the pattern i e weight distribution and spin i e unit priors vary smoothly from gaussian real variables to boolean discrete variables our analysis shows that the presence of a retrieval phase is robust and not peculiar to the standard hopfield model with boolean patterns the retrieval region becomes larger when the pattern entries and retrieval units get more peaked and conversely when the hidden units acquire a broader prior and therefore have a stronger response to high fields moreover at low load retrieval always exists below some critical temperature for every pattern distribution ranging from the boolean to the gaussian case

numerical solution of the obstacle problem by the penalty method part ii time dependent problems in a previous paper we considered the possibility of using the penalty method to get approximations of the solution of elliptic variational inequalities in this note we show that the same idea can be used for the treatment of parabolic problems by coupling the penalty parameter e and the discretization parametersh and t quasi optimal error estimates are derived in suitable norms

on a system of first order quasi variational inequalities connected with the optimal switching problem abstract in this paper we prove the existence of the maximum bounded lipschitz continuous solution to a system of first order quasi variational inequalities the method is to discretize by an euler scheme the characteristic lines of the differential operator appearing in the system and to solve by iteration the corresponding approximate problem the solution is interpreted as the value function of a deterministic optimal switching problem

mixed and stabilized finite element methods for the obstacle problem we discretize the lagrange multiplier formulation of the obstacle problem by mixed and stabilized finite element methods a priori and a posteriori error estimates are derived and numerically verified

approximations to the multiplier method we analyze approximations to the multiplier method for solving an equality constrained optimization problem the multiplier method replaces the constrained problem by the unconstrained optimization of a penalized lagrangian over a hilbert space we consider the effect of replacing this hilbert space by a subspace the analysis is closely connected to an analysis of the galerkin approximation to the original optimization problem our estimates are essentially corollaries of a theorem which bounds the error in the galerkin approximation to a nonlinear equation some applications to problems in optimal control are discussed

common fixed point theorems for pd operator pairs under relaxed conditions with applications in this paper we define pd operator pair of single valued mappings and obtain some common fixed point theorems for this class of maps under relaxed conditions our theorem generalizes results of bhatt et al a bhatt h chandra d r sahu common fixed point theorems for occasionally weakly compatible mappings under relaxed conditions nonlinear anal 73 2010 176 182 jungck and rhoades g jungck b e rhoades fixed point theorems for occasionally weakly compatible mappings fixed point theory 7 2 2006 287 296 and hussain et al n hussain m a khamsi a latif common fixed points for jh operators and occasionally weakly biased pairs under relaxed conditions nonlinear anal 74 2011 2133 2140 and others as applications we also study the solution of functional equations arising from dynamic programming and variational inequalities arises in two point obstacle problem we also discuss some illustrative examples to highlight the realized improvements

common fixed points for banach operator pairs with applications the existence of common fixed point results for a banach operator pair under certain generalized contractions is established the invariant best approximation results are proved as applications and the existence of solutions of variational inequalities is obtained we also study the solution of functional equations arising from dynamic programming 2007 elsevier ltd all rights reserved

a nitsche based method for unilateral contact problems numerical analysis we introduce a nitsche based formulation for the finite element discretization of the unilateral contact problem in linear elasticity it features a weak treatment of the non linear contact conditions through a consistent penalty term without any additional assumption on the contact set we can prove theoretically its fully optimal convergence rate in the h1 norm for linear finite elements in two dimensions which is o h 1 2 when the solution lies in h 3 2 0 1 2 an interest of the formulation is that conversely to lagrange multiplier based methods no other unknown is introduced and no discrete inf sup condition needs to be satisfied

semi discrete stabilized finite element methods for navier stokes equations with nonlinear slip boundary conditions based on regularization procedure based on the pressure projection stabilized methods the semi discrete finite element approximation to the time dependent navier stokes equations with nonlinear slip boundary conditions is considered in this paper because this class of boundary condition includes the subdifferential property then the variational formulation is the navier stokes type variational inequality problem using the regularization procedure we obtain a regularized problem and give the error estimate between the solutions of the variational inequality problem and the regularized problem with respect to the regularized parameter varepsilon which means that the solution of the regularized problem converges to the solution of the navier stokes type variational inequality problem as the parameter varepsilon longrightarrow 0 moreover some regularized estimates about the solution of the regularized problem are also derived under some assumptions about the physical data the pressure projection stabilized finite element methods are used to the regularized problem and some optimal error estimates of the finite element approximation solutions are derived

a residual based error estimator for mortar finite element discretizations a residual based error estimator for the approximation of linear elliptic boundary value problems by nonconforming finite element methods is introduced and analyzed in particular we consider mortar finite element techniques restricting ourselves to geometrically conforming domain decomposition methods using p1 approximations in each subdomain additionally a residual based error estimator for crouzeix raviart elements of lowest order is presented and compared with the error estimator obtained in the more general mortar situation it is shown that the computational effort of the error estimator can be considerably reduced if the special structure of the lagrange multiplier is taken into account

a system of nonlinear partial differential equations arising in the optimal control of stochastic systems with switching costs the problem of optimal switching control of a diffusion process with costly switchings leads to a system of fully nonlinear elliptic partial differential equations with implicit obstacles we obtain results on the existence uniqueness and regularity of the solution of this system

convergence analysis of a conforming adaptive finite element method for an obstacle problem the adaptive algorithm for the obstacle problem presented in this paper relies on the jump residual contributions of a standard explicit residual based a posteriori error estimator each cycle of the adaptive loop consists of the steps solve estimate mark and refine the techniques from the unrestricted variational problem are modified for the convergence analysis to overcome the lack of galerkin orthogonality we establish r linear convergence of the part of the energy above its minimal value if there is appropriate control of the data oscillations surprisingly the adaptive mesh refinement algorithm is the same as in the unconstrained case of a linear pde in fact there is no modification near the discrete free boundary necessary for r linear convergence the arguments are presented for a model obstacle problem with an affine obstacle and homogeneous dirichlet boundary conditions the proof of the discrete local efficiency is more involved than in the unconstrained case numerical results are given to illustrate the performance of the error estimator

efficient and reliable a posteriori error estimators for elliptic obstacle problems a posteriori error estimators are derived for linear finite element approximations to elliptic obstacle problems these estimators yield global upper and local lower bounds for the discretization error here discretization error means the sum of two contributions the distance between continuous and discrete solution in the energy norm and some quantity that is related to the distance of continuous and discrete contact set moreover the local error indicators in the interior of the discrete contact set reduce to quantities that measure only data resolution

regularized model of post touchdown configurations in electrostatic mems equilibrium analysis abstract in canonical models of micro electro mechanical systems mems an event called touchdown whereby the electrical components of the device come into contact is characterized by a blow up in the governing equations and a non physical divergence of the electric field in the present work we propose novel regularized governing equations whose solutions remain finite at touchdown and exhibit additional dynamics beyond this initial event before eventually relaxing to new stable equilibria we employ techniques from variational calculus dynamical systems and singular perturbation theory to obtain a detailed understanding of the properties and equilibrium solutions of the regularized family of equations

adaptive finite elements for elastic bodies in contact to avoid interpenetration of matter under the small strain assumption the linear contact condition is frequently applied where the distance of bodies is controlled only along a certain direction the standard direction is the normal on the surface where interpenetration might occur in this paper we allow other directions as well we address questions such as the correct mathematical model existence of solutions the penalty method for regularization of the variational inequality finite element discretization and a priori and a posteriori error estimates but exclude the error of penalization the computable upper error bound leads to a criterion for automatic mesh refinements within a finite element method numerical simulations of the hertzian contact problem and a supported cantilever beam are included

convergence analysis of finite element method for a parabolic obstacle problem abstract a conforming finite element method is proposed and analyzed for numerical approximation of the solution of a parabolic variational inequality of the obstacle problem the model problem which is originally proposed using a general obstacle is reframed as a model problem with zero obstacle but with non homogeneous dirichlet boundary conditions subsequently the discrete problem is reframed and the error analysis proving the convergence of the method is performed the analysis requires a positive preserving interpolation with non homogeneous dirichlet boundary condition and a post processed solution that satisfies the boundary conditions sharply the results in the article extend the results of johnson sinum 1976 for a zero obstacle function to a more general obstacle function

equilibrium statistical mechanics of bipartite spin systems the aim of this paper is to give an extensive treatment of bipartite mean field spin systems pure and disordered at first bipartite ferromagnets are investigated and an explicit expression for the free energy is achieved through a new minimax variational principle then via the hamilton jacobi technique the same structure of the free energy is obtained together with the existence of its thermodynamic limit and the minimax principle is connected to a standard max one the same is investigated for bipartite spin glasses by the borel cantelli lemma we obtain the control of the high temperature regime while via the double stochastic stability technique we also obtain the explicit expression of the free energy in the replica symmetric approximation uniquely defined by a minimax variational principle again we also obtain a general result that states that the free energies of these systems are convex linear combinations of their independent one party model counterparts for the sake of completeness we show further that at zero temperature the replica symmetric entropy becomes negative and consequently such a symmetry must be broken the treatment of the fully broken replica symmetry case is deferred to a forthcoming paper as a first step in this direction we start deriving the linear and quadratic constraints to overlap fluctuations

nonlinear black box modeling in system identification a unified overview a nonlinear black box structure for a dynamical system is a model structure that is prepared to describe virtually any nonlinear dynamics there has been considerable recent interest in this area with structures based on neural networks radial basis networks wavelet networks and hinging hyperplanes as well as wavelet transform based methods and models based on fuzzy sets and fuzzy rules this paper describes all these approaches in a common framework from a user s perspective it focuses on what are the common features in the different approaches the choices that have to be made and what considerations are relevant for a successful system identification application of these techniques it is pointed out that the nonlinear structures can be seen as a concatenation of a mapping form observed data to a regression vector and a nonlinear mapping from the regressor space to the output space these mappings are discussed separately the latter mapping is usually formed as a basis function expansion the basis functions are typically formed from one simple scalar function which is modified in terms of scale and location the expansion from the scalar argument to the regressor space is achieved by a radial or a ridge type approach basic techniques for estimating the parameters in the structures are criterion minimization as well as two step procedures where first the relevant basis functions are determined using data and then a linear least squares step to determine the coordinates of the function approximation a particular problem is to deal with the large number of potentially necessary parameters this is handled by making the number of used parameters considerably less than the number of offered parameters by regularization shrinking pruning or regressor selection

quadratic convergence for valuing american options using a penalty method the convergence of a penalty method for solving the discrete regularized american option valuation problem is studied sufficient conditions are derived which both guarantee convergence of the nonlinear penalty iteration and ensure that the iterates converge monotonically to the solution these conditions also ensure that the solution of the penalty problem is an approximate solution to the discrete linear complementarity problem the efficiency and quality of solutions obtained using the implicit penalty method are compared with those produced with the commonly used technique of handling the american constraint explicitly convergence rates are studied as the timestep and mesh size tend to zero it is observed that an implicit treatment of the american constraint does not converge quadratically as the timestep is reduced if constant timesteps are used a timestep selector is suggested which restores quadratic convergence