adaptive bem with optimal convergence rates for the helmholtz equation abstract we analyze an adaptive boundary element method for the weakly singular and hypersingular integral equations for the 2d and 3d helmholtz problem the proposed adaptive algorithm is steered by a residual error estimator and does not rely on any a priori information that the underlying meshes are sufficiently fine we prove convergence of the error estimator with optimal algebraic rates independently of the coarse initial mesh as a technical contribution we prove certain local inverse type estimates for the boundary integral operators associated with the helmholtz equation

the conditioning of boundary element equations on locally refined meshes and preconditioning by diagonal scaling consider a boundary integral operator on a bounded d dimensional surface in mbox smallbbb r d 1 suppose that the operator is a pseudodifferential operator of order 2m m in mbox smallbbb r and that the associated bilinear form is symmetric and positive definite the surface may be open or closed and m may be positive or negative let b denote the stiffness matrix arising from a galerkin boundary element method with standard nodal basis functions if local mesh refinement is used then the partition may contain elements of very widely differing sizes and consequently b may be very badly conditioned in fact if the elements are nondegenerate and 2 m d then the ell 2 condition number of b satisfies rm cond mbox bf em b leq c n 2 m d left frac h rm max h rm min right d 2m where hmax and h min are the sizes of the largest and smallest elements in the partition and n is the number of degrees of freedom however if b is preconditioned using a simple diagonal scaling then the condition number is reduced to cal o n 2 m d that is diagonal scaling restores the condition number of the linear system to the same order of growth as that for a uniform partition the growth in the critical case 2 m d is worse by only a logarithmic factor

equilibrium distributions of stochastic networks without detailed balance abstract we present results for the determination of the equilibrium microstate probability distribution of a class of strongly interacting systems obeying stochastic dynamics but without the necessity of detailed balance the specific case of ising systems is highlighted with applications to non symmetric synaptic neural networks and more general recurrent boolean networks

a primal dual active set algorithm for three dimensional contact problems with coulomb friction in this paper efficient algorithms for contact problems with tresca and coulomb friction in three dimensions are presented and analyzed the numerical approximation is based on mortar methods for nonconforming meshes with dual lagrange multipliers using a nonsmooth complementarity function for the three dimensional friction conditions a primal dual active set algorithm is derived the method determines active contact and friction nodes and at the same time resolves the additional nonlinearity originating from sliding nodes no regularization and no penalization are applied and superlinear convergence can be observed locally in combination with a multigrid method it defines a robust and fast strategy for contact problems with tresca or coulomb friction the efficiency and flexibility of the method is illustrated by several numerical examples

feed forward chains of recurrent attractor neural networks near saturation we perform a stationary state replica analysis for a layered network of ising spin neurons with recurrent hebbian interactions within each layer in combination with strictly feed forward hebbian interactions between successive layers this model interpolates between the fully recurrent and symmetric attractor network studied by amit et al and the strictly feed forward attractor network studied by domany et al due to the absence of detailed balance it is as yet solvable only in the zero temperature limit the built in competition between two qualitatively different modes of operation feed forward ergodic within layers versus recurrent non ergodic within layers is found to induce interesting phase transitions

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

feed forward chains of recurrent attractor neural networks with finite dilution near saturation abstract a stationary state replica analysis for a dual neural network model that interpolates between a fully recurrent symmetric attractor network and a strictly feed forward layered network studied by coolen and viana is extended in this work to account for finite dilution of the recurrent hebbian interactions between binary ising units within each layer gradual dilution is found to suppress part of the phase transitions that arise from the competition between recurrent and feed forward operation modes of the network despite that a long chain of layers still exhibits a relatively good performance under finite dilution for a balanced ratio between inter layer and intra layer interactions

approximation and numerical realization of 2d contact problems with coulomb friction and a solution dependent coefficient of friction the paper analyzes discrete contact problems with the coulomb law of friction which involves a solution dependent coefficient of friction f solutions to these problems are defined as fixed points of an auxiliary mapping it is shown that there exists at least one solution provided that f is bounded and continuous in r 1 further conditions guaranteeing uniqueness of the solution are studied the paper is completed by numerical results of several model examples

external fields in attractor neural networks with different learning rules the effects of external fields on the retrieval properties of highly dilute attractor neural networks with general classes of learning rules are examined it can be shown that external fields increase basins of attraction making even perfect retrieval possible for relatively high loading the application of different classes of noise distributions on the stimulus field indicates that certain first order transitions occurring are peculiarities of the type of noise optimally adapted networks in the presence of external fields extend the critical loading above which perfect retrieval is impossible in the presence of external fields in the high temperature regime hebb networks also retrieve better than rules with more optimal performances at low temperatures

inverse type estimates on hp finite element spaces and applications this work is concerned with the development of inverse type inequalities for piecewise polynomial functions and in particular functions belonging to hp finite element spaces the cases of positive and negative sobolev norms are considered for both continuous and discontinuous finite element functions the inequalities are explicit both in the local polynomial degree and the local mesh size the assumptions on the hp finite element spaces are very weak allowing anisotropic shape irregular elements and varying polynomial degree across elements finally the new inverse type inequalities are used to derive bounds for the condition number of symmetric stiffness matrices of hp boundary element method discretisations of integral equations with element wise discontinuous basis functions constructed via scaled tensor products of legendre polynomials

rapid solution of integral equations of classical potential theory an algorithm is described for rapid solution of classical boundary value problems dirichlet an neumann for the laplace equation based on iteratively solving integral equations of potential theory cpu time requirements for previously published algorithms of this type are proportional to n2 where n is the number of nodes in the discretization of the boundary of the region the cpu time requirements for the algorithm of the present paper are proportional to n making it considerably more practical for large scale problems

approximation and numerical realization of 3d contact problems with coulomb friction and a solution dependent coefficient of friction this paper analyzes 3d discrete contact problems with coulomb friction and a coefficient of friction depending on the solution the formulation of this problem is based on a fixed point approach existence of at least one discrete solution is guaranteed for any continuous non negative and bounded function the uniqueness result is established for small enough and lipschitz continuous with a sufficiently small modulus of lipschitz continuity results of several numerical experiments are shown copyright 2009 john wiley sons ltd

optimal additive schwarz preconditioning for adaptive 2d iga boundary element methods abstract we define and analyze local multilevel diagonal preconditioners for isogeometric boundary elements on locally refined meshes in two dimensions hypersingular and weakly singular integral equations are considered we prove that the condition number of the preconditioned systems of linear equations is independent of the mesh size and the refinement level therefore the computational complexity when using appropriate iterative solvers is optimal our analysis is carried out for closed and open boundaries and numerical examples confirm our theoretical results

anisotropic mesh refinement the conditioning of galerkin boundary element matrices and simple preconditioners in this paper we obtain upper and lower bounds on the spectrum of the stiffness matrix arising from a finite element galerkin approximation using nodal basis functions of a bounded symmetric bilinear form which is elliptic on a sobolev space of real index m in 1 1 the key point is that the finite element mesh is required to be neither quasi uniform nor shape regular so that our theory allows anisotropic meshes often used in practice however we assume that the polynomial degree of the elements is fixed our bounds indicate the ill conditioning which can arise from anisotropic mesh refinement in addition we obtain spectral bounds for the diagonally scaled stiffness matrix which indicate the improvement provided by this simple preconditioning for the special case of boundary integral operators on a two dimensional screen in r3 numerical experiments show that our bounds are sharp we find that diagonal scaling essentially removes the ill conditioning due to mesh degeneracy leading to the same asymptotic growth in the condition number as arises for a quasi uniform mesh refinement our results thus generalize earlier work by bank and scott siam j numer anal 26 1989 pp 1383 1394 and ainsworth mclean and tran siam j numer anal 36 1999 pp 1901 1932 for the shape regular case

layered neural networks with non monotonic transfer functions abstract we investigate storage capacity and generalization ability for two types of fully connected layered neural networks with non monotonic transfer functions random patterns are embedded into the networks by a hebbian learning rule one of them is a layered network in which a non monotonic transfer function of even layers is different from that of odd layers the other is a layered network with intra layer connections in which the non monotonic transfer function of inter layer is different from that of intra layer and inter layered neurons and intra layered neurons are updated alternately we derive recursion relations for order parameters for those layered networks by the signal to noise ratio method we clarify that the storage capacity and the generalization ability for those layered networks are enhanced in comparison with those with a conventional monotonic transfer function when non monotonicity of the transfer functions is selected optimally we also point out that some chaotic behavior appears in the order parameters for the layered networks when non monotonicity of the transfer functions increases

qualitative analysis of 3d elastostatic contact problems with orthotropic coulomb friction and solution dependent coefficients of friction this paper analyzes a discrete form of 3d contact problems with local orthotropic coulomb friction and coefficients of friction which may depend on the solution itself the analysis is based on the fixed point reformulation of the original problem conditions guaranteeing the existence and uniqueness of discrete solutions are established finally numerical results of a model example are presented

optimal panel clustering in the presence of anisotropic mesh refinement in this paper we consider the numerical solution of discrete boundary integral equations on polyhedral surfaces in three dimensions when the solution contains typical edge singularities highly stretched meshes are preferred to uniform meshes since they reduce the number of degrees of freedom needed to obtain a fixed accuracy the classical panel clustering method can still be applied in the presence of such highly stretched meshes however we will show that the savings in computation time and storage become suboptimal because the nearfield matrix arising in the panel clustering algorithm is no longer as sparse as it is in the case of uniform meshes hence a natural question arises as to whether a new enhanced panel clustering algorithm can be designed which performs efficiently even in the presence of highly stretched meshes the main result of this paper is to formulate such an enhanced version of the panel clustering algorithm the key features of the algorithm are i the employment of partial analytic integration in the direction of stretching yielding a new kernel function on a one dimensional manifold where the influence of high aspect ratios in the stretched elements is removed and ii the introduction of a generalized admissibility condition with respect to the partially integrated kernel which ensures that certain stretched clusters which are inadmissible in the classical sense now become admissible in the context of a model problem we prove that our algorithm yields an accurate up to the discretization error matrix vector multiplication which requires cal o n log k n operations where n is the number of degrees of freedom and k is small and independent of the aspect ratio the generalized admissibility condition can be viewed as an addition to the classical method which may be useful in general when stretched meshes are present we also have performed a numerical experiment which shows that the sparsity of the nearfield matrix for the enhanced panel clustering method is not negatively affected by stretched elements and the method will perform optimally

order parameter evolution in a feedforward neural network we consider layered neural networks in which the weights are trained with the pseudoinverse rule to store a set of random patterns using many body diagrammatic techniques the evolution in the network can be described by the overlap order parameter m and the noise parameter delta looping effects are shown to be significant in contrast to a previous conjecture order parameter pairs corresponding to various input conditions are found to collapse on a universal curve

convergence rate of an optimization algorithm for minimizing quadratic functions with separable convex constraints a new active set algorithm for minimizing quadratic functions with separable convex constraints is proposed by combining the conjugate gradient method with the projected gradient it generalizes recently developed algorithms of quadratic programming constrained by simple bounds a linear convergence rate in terms of the hessian spectral condition number is proven numerical experiments including the frictional three dimensional 3d contact problems of linear elasticity illustrate the computational performance

enlarging the attractor basins of neural networks with noisy external fields a neural network model with optimal connections trained with ensembles of external discrete noisy fields is studied allowing for non zero errors in the storage novel behaviour is observed which is reflected in the model s retrieval map improvement in the model s content addressability is determined by comparing the maximum storage level at which there is a near 100 basin of attraction the cases presented have the external field applied during training during retrieval and during both with statistically equal parameters in all three the content addressability is improved over the zero external field network with the equal training and retrieval fields case having the largest improvement however the apparent domination of the retrieval over the training field suggests this simple equality is perhaps not the optimal relationship

1 dimensional attractor neural networks we solve a class of attractor neural network models with a mixture of 1d nearest neighbour interactions and infinite range interactions which are both of a hebbian type form our solution is based on a combination of mean field methods transfer matrices and 1d random field techniques and is obtained both for boltzmann type equilibrium following sequential glauber dynamics and peretto type equilibrium following parallel dynamics competition between the alignment forces mediated via short range interactions and those mediated via infinite range ones is found to generate novel phenomena such as multiple locally stable pure states first order transitions between recall states 2 cycles and non recall states and domain formation leading to extremely long relaxation times we test our results against numerical simulations and simple benchmark cases and find excellent agreement

an optimal algorithm for minimization of quadratic functions with bounded spectrum subject to separable convex inequality and linear equality constraints an in a sense optimal algorithm for minimization of quadratic functions subject to separable convex inequality and linear equality constraints is presented its unique feature is an error bound in terms of bounds on the spectrum of the hessian of the cost function if applied to a class of problems with the spectrum of the hessians in a given positive interval the algorithm can find approximate solutions in a uniformly bounded number of simple iterations such as matrix vector multiplications moreover if the class of problems admits a sparse representation of the hessian it simply follows that the cost of the solution is proportional to the number of unknowns theoretical results are illustrated by numerical experiments