fast methods for the eikonal and related hamilton jacobi equations on unstructured meshes the fast marching method is a numerical algorithm for solving the eikonal equation on a rectangular orthogonal mesh in o m log m steps where m is the total number of grid points the scheme relies on an upwind finite difference approximation to the gradient and a resulting causality relationship that lends itself to a dijkstra like programming approach in this paper we discuss several extensions to this technique including higher order versions on unstructured meshes in rn and on manifolds and connections to more general static hamilton jacobi equations

aspects of an adaptive finite element method for the fractional laplacian a priori and a posteriori error estimates efficient implementation and multigrid solver abstract we develop all of the components needed to construct an adaptive finite element code that can be used to approximate fractional partial differential equations on non trivial domains in d 1 dimensions our main approach consists of taking tools that have been shown to be effective for adaptive boundary element methods and where necessary modifying them so that they can be applied to the fractional pde case improved a priori error estimates are derived for the case of quasi uniform meshes which are seen to deliver sub optimal rates of convergence owing to the presence of singularities attention is then turned to the development of an a posteriori error estimate and error indicators which are suitable for driving an adaptive refinement procedure we assume that the resulting refined meshes are locally quasi uniform and develop efficient methods for the assembly of the resulting linear algebraic systems and their solution using iterative methods including the multigrid method the storage of the dense matrices along with efficient techniques for computing the dense matrix vector products needed for the iterative solution is also considered the performance and efficiency of the resulting algorithm is illustrated for a variety of examples

adaptive bem with inexact pcg solver yields almost optimal computational costs we consider the preconditioned conjugate gradient method pcg with optimal preconditioner in the frame of the boundary element method for elliptic first kind integral equations our adaptive algorithm steers the termination of pcg as well as the local mesh refinement besides convergence with optimal algebraic rates we also prove almost optimal computational complexity in particular we provide an additive schwarz preconditioner which can be computed in linear complexity and which is optimal in the sense that the condition numbers of the preconditioned systems are uniformly bounded as model problem serves the 2d or 3d laplace operator and the associated weakly singular integral equation with energy space widetilde h 1 2 gamma the main results also hold for the hyper singular integral equation with energy space h 1 2 gamma

effects of domain characteristics on instance based learning algorithms this paper presents average case analyses of instance based learning algorithms the algorithms analyzed employ a variant of k nearest neighbor classifier k nn our analysis deals with a monotone m of n target concept with irrelevant attributes and handles three types of noise relevant attribute noise irrelevant attribute noise and class noise we formally represent the expected classification accuracy of k nn as a function of domain characteristics including the number of training instances the number of relevant and irrelevant attributes the threshold number in the target concept the probability of each attribute the noise rate for each type of noise and k we also explore the behavioral implications of the analyses by presenting the effects of domain characteristics on the expected accuracy of k nn and on the optimal value of k for artificial domains

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

generalization learning in a perceptron with binary synapses we consider the generalization problem for a perceptron with binary synapses implementing the stochastic belief propagation inspired sbpi learning algorithm which we proposed earlier and perform a mean field calculation to obtain a differential equation which describes the behaviour of the device in the limit of a large number of synapses n we show that the solving time of sbpi is of order n sqrt log n while the similar well known clipped perceptron cp algorithm does not converge to a solution at all in the time frame we considered the analysis gives some insight into the ongoing process and shows that in this context the sbpi algorithm is equivalent to a new simpler algorithm which only differs from the cp algorithm by the addition of a stochastic unsupervised meta plastic reinforcement process whose rate of application must be less than sqrt 2 pi n for the learning to be achieved effectively the analytical results are confirmed by simulations

preconditioners for pseudodifferential equations on the sphere with radial basis functions in a previous paper a preconditioning strategy based on overlapping domain decomposition was applied to the galerkin approximation of elliptic partial differential equations on the sphere in this paper the methods are extended to more general pseudodifferential equations on the sphere using as before spherical radial basis functions for the approximation space and again preconditioning the ill conditioned linear systems of the galerkin approximation by the additive schwarz method numerical results are presented for the case of hypersingular and weakly singular integral operators on the sphere mathbb s 2

adaptive boundary element methods with convergence rates this paper presents adaptive boundary element methods for positive negative as well as zero order operator equations together with proofs that they converge at certain rates the convergence rates are quasi optimal in a certain sense under mild assumptions that are analogous to what is typically assumed in the theory of adaptive finite element methods in particular no saturation type assumption is used the main ingredients of the proof that constitute new findings are some results on a posteriori error estimates for boundary element methods and an inverse type inequality involving boundary integral operators on locally refined finite element spaces

some improvements of the fast marching method the fast marching method published by sethian proc natl acad sci usa 93 1996 pp 1591 1595 is an optimally efficient algorithm for solving problems of front evolution where the front speed is monotonic it has been used in a wide variety of applications such as robotic path planning r kimmel and j sethian fast marching methods for computing distance maps and shortest paths tech report 669 cpam university of california berkeley 1996 crack propagation m stolarska et al modelling crack growth by level sets in the extended finite element method comput methods appl mech engrg to appear internat j numer methods engrg 51 2001 pp 943 960 n sukumar d l chopp and b moran extended finite element method and fast marching method for three dimensional fatigue crack propagation j comput phys submitted seismology j sethian and a popovici geophysics 64 1999 pp 516 523 photolithography j sethian fast marching level set methods for three dimensional photolithography development in proceedings of the spie 1996 international symposium on microlithography santa clara ca 1996 and medical imaging r malladi and j sethian proc natl acad sci usa 93 1996 pp 9389 9392 it has also been a valuable tool for the implementation of modern level set methods where it is used to efficiently compute the distance to the front and or an extended velocity function in this paper we improve upon the second order fast marching method of sethian siam rev 41 1999 pp 199 235 by constructing a second order approximation of the interface generated from local data on the mesh the data is interpolated on a single box of the mesh using a bicubic approximation the distance to the front is then calculated by using a variant of newton s method to solve both the level curve equation and the orthogonality condition for the nearest point to a given node the result is a second order approximation of the distance to the interface which can then be used to produce second order accurate initial conditions for the fast marching method and a third order fast marching method

estimating spot volatility with high frequency financial data we construct a spot volatility estimator for high frequency financial data which contain market microstructure noise we prove consistency and derive the asymptotic distribution of the estimator a data driven method is proposed to select the scale parameter and the bandwidth parameter in the estimator in monte carlo simulations we compare the finite sample performance of our estimator with some existing estimators empirical examples are given to illustrate the potential applications of the estimator

numerical solution of bernoulli type free boundary value problems by variable domain method the variable domain method for a free surface problem is considered as a weighted residual method this considerably simplifies discretization of the problem and at the same time allows implementation of any standard discretization technique discretization of the problem results in a non linear system of equations for discrete values of a field variable and co ordinates of the nodes on the free boundary variables corresponding to a field variable are eliminated and the resulting non linear system is of considerable smaller size some new update strategies of the jacobian together with discussion of the cost effectiveness and covergence rate are introduced and compared to broyden s method

estimating the degree of activity of jumps in high frequency data we define a generalized index of jump activity propose estimators of that index for a discretely sampled process and derive the estimators properties these estimators are applicable despite the presence of brownian volatility in the process which makes it more challenging to infer the characteristics of the small infinite activity jumps when the method is applied to high frequency stock returns we find evidence of infinitely active jumps in the data and estimate their index of activity

variational approach to shape derivatives for a class of bernoulli problems the shape derivative of a functional related to a bernoulli problem is derived without using the shape derivative of the state the gradient information is combined with level set ideas in a steepest descent algorithm numerical examples show the feasibility of the approach

fully discrete spectral methods for boundary integral equations on slender spheroids we develop fully discrete spectral boundary integral algorithms to solve interior and exterior potential boundary value problems on three dimensional slender spheroids our approach is based on identifying and using analytical solutions of associated generalised eigenvalue problems of boundary integral operators we generalise approximation theory results on the sphere using spheroidally appropriate eigenfunctions and solve an open problem of stability of spectral methods for first kind boundary integral equations on spheroids

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

learnability and the vapnik chervonenkis dimension valiant s learnability model is extended to learning classes of concepts defined by regions in euclidean space e n the methods in this paper lead to a unified treatment of some of valiant s results along with previous results on distribution free convergence of certain pattern recognition algorithms it is shown that the essential condition for distribution free learnability is finiteness of the vapnik chervonenkis dimension a simple combinatorial parameter of the class of concepts to be learned using this parameter the complexity and closure properties of learnable classes are analyzed and the necessary and sufficient conditions are provided for feasible learnability

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

the one phase free boundary problem for the p laplacian with non constant bernoulli boundary condition our objective here is to generalize our earlier results on the existence of classical convex solution to a free boundary problem with a bernoulli type boundary gradient condition and with the p laplacian as the governing operator the main theorems of this paper assert that the exterior and the interior free boundary problem with a bernoulli law i e with a prescribed pressure a x on the free streamline of the flow have convex solutions provided the initial domains are convex the continuous function a x is subject to certain convexity properties in our earlier results we have considered the case of constant a x in the lines of the proof of the main results we also prove the semi continuity up to the boundary of the gradient of the p capacitary potentials in convex rings with c 1 boundaries

computational study of fast methods for the eikonal equation a computational study of the fast marching and the fast sweeping methods for the eikonal equation is given it is stressed that both algorithms should be considered as direct as opposed to iterative methods on realistic grids fast sweeping is faster than fast marching for problems with simple geometry for strongly nonuniform problems and or complex geometry the situation may be reversed finally fully second order generalizations of methods of this type for problems with obstacles are proposed and implemented

on learning perceptrons with binary weights we present an algorithm that pac learns any perceptron with binary weights and arbitrary threshold under the family of product distributions the sample complexity of this algorithm is of o n e 4 ln n and its running time increases only linearly with the number of training examples the algorithm does not try to find an hypothesis that agrees with all of the training examples rather it constructs a binary perceptron based on various probabilistic estimates obtained from the training examples we show that under the restricted case of the uniform distribution and zero threshold the algorithm reduces to the well known clipped hebb rule we calculate exactly the average generalization rate i e the learning curve of the algorithm under the uniform distribution in the limit of an infinite number of dimensions we find that the error rate decreases exponentially as a function of the number of training examples hence the average case analysis gives a sample complexity of o n ln 1 e a large improvement over the pac learning analysis the analytical expression of the learning curve is in excellent agreement with the extensive numerical simulations in addition the algorithm is very robust with respect to classification noise

existence of classical solutions to a free boundary problem for the p laplace operator i the exterior convex case in this paper we prove under convexity assumptions for the data the existence of classical solutions for a bernoulli type free boundary problem with the p laplacian as the governing operator the method employed here originates from a pioneering work of a beurling where he proves the existence for the harmonic case in the plane though with no geometrical restrictions

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

a hypersingular boundary integral method for two dimensional screen and crack problems we analyze hypersingular integral equations on a curved open smooth arc in r 2 that model either curved cracks in an elastic medium or the scattering of acoustic and elastic waves at a hard screen by using the mellin transformation we obtain sharp regularity results for the solution of these equations in sobolev spaces in the form of singular expansions in particular we show that the expansions do not contain logarithmic singularities

the distribution of realized exchange rate volatility using high frequency data on deutschemark and yen returns against the dollar we construct model free estimates of daily exchange rate volatility and correlation that cover an entire decade our estimates termed realized volatilities and correlations are not only model free but also approximately free of measurement error under general conditions which we discuss in detail hence for practical purposes we may treat the exchange rate volatilities and correlations as observed rather than latent we do so and we characterize their joint distribution both unconditionally and conditionally noteworthy results include a simple normality inducing volatility transformation high contemporaneous correlation across volatilities high correlation between correlation and volatilities pronounced and persistent dynamics in volatilities and correlations evidence of long memory dynamics in volatilities and correlations and remarkably precise scaling laws under temporal aggregation

designing realised kernels to measure the ex post variation of equity prices in the presence of noise this paper shows how to use realised kernels to carry out efficient feasible inference on the expost variation of underlying equity prices in the presence of simple models of market frictions the issue is subtle with only estimators which have symmetric weights delivering consistent estimators with mixed gaussian limit theorems the weights can be chosen to achieve the best possible rate of convergence and to have an asymptotic variance which is close to that of the maximum likelihood estimator in the parametric version of this problem realised kernels can also be selected to i be analysed using endogenously spaced data such as that in databases on transactions ii allow for market frictions which are endogenous iii allow for temporally dependent noise the finite sample performance of our estimators is studied using simulation while empirical work illustrates their use in practice

ordered upwind methods for static hamilton jacobi equations theory and algorithms we develop a family of fast methods for approximating the solutions to a wide class of static hamilton jacobi pdes these fast methods include both semi lagrangian and fully eulerian versions numerical solutions to these problems are typically obtained by solving large systems of coupled nonlinear discretized equations our techniques which we refer to as ordered upwind methods oums use partial information about the characteristic directions to decouple these nonlinear systems greatly reducing the computational labor our techniques are considered in the context of control theoretic and front propagation problems we begin by discussing existing oums focusing on those designed for isotropic problems we then introduce a new class of oums which decouple systems for general anisotropic problems we prove convergence of one such scheme to the viscosity solution of the corresponding hamilton jacobi pde next we introduce a set of finite differences methods based on an analysis of the role played by anisotropy in the context of front propagation and optimal trajectory problems the performance of the methods is analyzed and computational experiments are performed using test problems from computational geometry and seismology

predictive inference for integrated volatility numerous volatility based derivative products have been engineered in recent years this has led to interest in constructing conditional predictive densities and confidence intervals for integrated volatility in this article we propose nonparametric estimators of the aforementioned quantities based on model free volatility estimators we establish consistency and asymptotic normality for the feasible estimators and study their finite sample properties through a monte carlo experiment finally using data from the new york stock exchange we provide an empirical application to volatility directional predictability

numerical solution of the free boundary bernoulli problem using a level set formulation we present a numerical method based on a level set formulation to solve the bernoulli problem the formulation uses time as a parameter of boundary evolution the level set formulation enables to consider nonconnected domains numerical experiments show the efficiency of the method if boundary conditions are handled accurately in particular the case of multiple solutions is treated

predictive density estimators for daily volatility based on the use of realized measures abstract the main objective of this paper is to propose a feasible model free estimator of the predictive density of integrated volatility in this sense we extend recent papers by andersen et al andersen t g bollerslev t diebold f x labys p 2003 modelling and forecasting realized volatility econometrica 71 579 626 and by andersen et al andersen t g bollerslev t meddahi n 2004 analytic evaluation of volatility forecasts international economic review 45 1079 1110 andersen t g bollerslev t meddahi n 2005 correcting the errors volatility forecast evaluation using high frequency data and realized volatilities econometrica 73 279 296 who address the issue of pointwise prediction of volatility via arma models based on the use of realized volatility our approach is to use a realized volatility measure to construct a non parametric kernel estimator of the predictive density of daily volatility we show that by choosing an appropriate realized measure one can achieve consistent estimation even in the presence of jumps and microstructure noise in prices more precisely we establish that four well known realized measures i e realized volatility bipower variation and two measures robust to microstructure noise satisfy the conditions required for the uniform consistency of our estimator furthermore we outline an alternative simulation based approach to predictive density construction finally we carry out a simulation experiment in order to assess the accuracy of our estimators and provide an empirical illustration that underscores the importance of using microstructure robust measures when using high frequency data

on the ability of the optimal perceptron to generalise a linearly separable boolean function is derived from a set of examples by a perceptron with optimal stability the probability to reconstruct a pattern which is not learnt is calculated analytically using the replica method

moments of markov switching models this paper derives the moments for a range of markov switching models we characterize in detail the patterns of volatility skewness and kurtosis that these models can produce as a function of the transition probabilities and parameters of the underlying state densities entering the switching process the autocovariance of the level and squares of time series generated by markov switching processes is also derived and we use these results to shed light on the relationship between volatility clustering regime switches and structural breaks in time series models

some dichotomy theorems for neural learning problems the computational complexity of learning from binary examples is investigated for linear threshold neurons we introduce combinatorial measures that create classes of infinitely many learning problems with sample restrictions we analyze how the complexity of these problems depends on the values for the measures the results are established as dichotomy theorems showing that each problem is either np complete or solvable in polynomial time in particular we consider consistency and maximum consistency problems for neurons with binary weights and maximum consistency problems for neurons with arbitrary weights we determine for each problem class the dividing line between the np complete and polynomial time solvable problems moreover all efficiently solvable problems are shown to have constructive algorithms that require no more than linear time on a random access machine model similar dichotomies are exhibited for neurons with bounded threshold the results demonstrate on the one hand that the consideration of sample constraints can lead to the discovery of new efficient algorithms for non trivial learning problems on the other hand hard learning problems may remain intractable even for severely restricted samples

limit theorems for moving averages of discretized processes plus noise this paper presents some limit theorems for certain functionals of moving averages of semimartingales plus noise which are observed at high frequency our method generalizes the pre averaging approach see bernoulli15 2009 634 658 stochastic process appl 119 2009 2249 2276 and provides consistent estimates for various characteristics of general semimartingales furthermore we prove the associated multidimensional stable central limit theorems as expected we find central limit theorems with a convergence rate n 1 4 if n is the number of observations

empirical evidence on the importance of aggregation asymmetry and jumps for volatility prediction abstract many recent modeling advances in finance topics ranging from the pricing of volatility based derivative products to asset management are predicated on the importance of jumps or discontinuous movements in asset returns in light of this a number of recent papers have addressed volatility predictability some from the perspective of the usefulness of jumps in forecasting volatility key papers in this area include andersen et al 2003 corsi 2004 andersen et al 2007 corsi et al 2008 barndorff et al 2010 patton and shephard 2011 and the references cited therein in this paper we review the extant literature and then present new empirical evidence on the predictive content of realized measures of jump power variations including upside and downside risk jump asymmetry and truncated jump variables constructed using instantaneous returns i e r t q 0 q 6 in the spirit of ding et al 1993 and ding and granger 1996 we also present new empirical evidence on the predictive content of realized measures of truncated large jump variations constructed using truncated squared instantaneous return i e r t 2 i r t where is the threshold jump size our prediction experiments use high frequency price returns constructed using s p500 futures data as well as stocks in the dow 30 and our empirical implementation involves estimating linear and nonlinear heterogeneous autoregressive realized volatility har rv type models we find that past large jump power variations help less in the prediction of future realized volatility than past small jump power variations additionally we find evidence that past realized signed jump power variations which have not previously been examined in this literature are strongly correlated with future volatility and that past downside jump variations matter in prediction finally incorporation of downside and upside jump power variations does improve predictability albeit to a limited extent

business cycle asymmetries in stock returns evidence from higher order moments and conditional densities markov switching models with time varying means variances and mixing weights are applied to characterize business cycle variation in the probability distribution and higher order moments of stock returns this allows us to provide a comprehensive characterization of risk that goes well beyond the mean and variance of returns several mixture models with different specifications of the state transition are compared and we propose a new mixture of gaussian and student t distributions that captures outliers in returns the models produce very similar expected returns and volatilities but imply very different time series for conditional skewness kurtosis and predictive density consistent with economic theory the gains in predictive accuracy from considering two state mixture models rather than a single state specification are higher for small firms than for large firms

testing for jumps in a discretely observed process we propose a new test to determine whether jumps are present in asset returns or other discretely sampled processes as the sampling interval tends to 0 our test statistic converges to 1 if there are jumps and to another deterministic and known value such as 2 if there are no jumps the test is valid for all ito semimartingales depends neither on the law of the process nor on the coefficients of the equation which it solves does not require a preliminary estimation of these coefficients and when there are jumps the test is applicable whether jumps have finite or infinite activity and for an arbitrary blumenthal getoor index we finally implement the test on simulations and asset returns data

generalization performance of bayes optimal classification algorithm for learning a perceptron the generalization error of the bayes optimal classification algorithm when learning a perceptron from noise free random training examples is calculated exactly using methods of statistical mechanics it is shown that if an assumption of replica symmetry is made then in the thermodynamic limit the error of the bayes optimal algorithm is less than the error of a canonical stochastic learning algorithm by a factor approaching ensuremath surd 2 as the ratio of the number of training examples to perceptron weights grows in addition it is shown that approximations to the generalization error of the bayes optimal algorithm can be achieved by learning algorithms that use a two layer neutral net to learn a perceptron

testing for jumps when asset prices are observed with noise a swap variance approach abstract this paper proposes a new test for jumps in asset prices that is motivated by the literature on variance swaps formally the test follows by a direct application of ito s lemma to the semi martingale process of asset prices and derives its power from the impact of jumps on the third and higher order return moments intuitively the test statistic reflects the cumulative gain of a variance swap replication strategy which is known to be minimal in the absence of jumps but substantial in the presence of jumps simulations show that the jump test has nice properties and is generally more powerful than the widely used bi power variation test an important feature of our test is that it can be applied in analytically modified form to noisy high frequency data and still retain power as a by product of our analysis we obtain novel analytical results regarding the impact of noise on bi power variation an empirical illustration using ibm trade data is also included

strongly agree or strongly disagree in linear classifiers such as the support vector machine svm a score is associated with each feature and objects are assigned to classes based on the linear combination of the scores and the values of the features inspired by discrete psychometric scales which measure the extent to which a factor is in agreement with a statement we propose the discrete level support vector machine dilsvm where the feature scores can only take on a discrete number of values defined by the so called feature rating levels the dilsvm classifier benefits from interpretability and it has visual appeal since it can be represented as a collection of likert scales one for each feature where we rate the level of agreement with the positive class to construct the dilsvm classifier we propose a mixed integer linear programming approach as well as a collection of strategies to reduce computational cost our numerical experiments show that the three point and the five point dilsvm classifiers have comparable accuracy to the svm with a substantial gain in interpretability and visual appeal but also in sparsity thanks to the appropriate choice of the feature rating levels

on the conditioning of finite element equations with highly refined meshes it is proven that the condition number of the linear system representing a finite element discretization of an elliptic boundary value problem does not degrade significantly as the mesh is refined locally provided the mesh remains nondegenerate and a natural scaling of the basis functions is used bounds for the euclidean condition number as a function of the number of degrees f freedom are derived in n geq 2 dimensions when n geq 3 the bound is the same as for the regular mesh case but when n 2 a factor appears in the bound for the condition number that is logarithmic in the ratio of the maximum and minimum mesh sizes applications of the results to the conjugate gradient iterative method for solving such linear systems are given

boundary element preconditioners for problems defined on slender domains within the framework of two dimensional potential theory we consider problems defined on slender domains the boundary element matrices for such problems are poorly conditioned due to the presence of two disparate length scales a new class of preconditioners is proposed to remedy this problem so that the preconditioned boundary element matrices become insensitive to the domain slenderness the preconditioners are constructed using the spectral properties of the boundary integral operators defined on the elliptical domains

computational limitations on learning from examples the computational complexity of learning boolean concepts from examples is investigated it is shown for various classes of concept representations that these cannot be learned feasibly in a distribution free sense unless r np these classes include a disjunctions of two monomials b boolean threshold functions and c boolean formulas in which each variable occurs at most once relationships between learning of heuristics and finding approximate solutions to np hard optimization problems are given

learning from examples in large neural networks a statistical mechanical theory of learning from examples in layered networks at finite temperature is studied when the training error is a smooth function of continuously varying weights the generalization error falls off asymptotically as the inverse number of examples by analytical and numerical studies of single layer perceptrons we show that when the weights are discrete the generalization error can exhibit a discontinuous transition to perfect generalization for intermediate sizes of the example set the state of perfect generalization coexists with a metastable spin glass state

fast numerical methods for bernoulli free boundary problems the numerical solution of the free boundary bernoulli problem is addressed an iterative method based on a level set formulation and boundary element method is proposed issues related to the implementation the accuracy and the generality of the method are discussed the efficiency of the approach is illustrated by numerical results