learning unions of ω 1 dimensional rectangles we consider the problem of learning unions of rectangles over the domain b n in the uniform distribution membership query learning setting where both b and n are large we obtain poly n logb time algorithms for the following classes poly nlogb way majority of o log nlogb loglog nlogb dimensional rectangles union of poly log nlogb many o log 2 nlogb loglog nlogb logloglog nlogb 2 dimensional rectangles poly nlogb way majority of poly nlogb or of disjoint o log nlogb loglog nlogb dimensional rectangles our main algorithmic tool is an extension of jackson s boosting and fourier based harmonic sieve algorithm j c jackson an efficient membership query algorithm for learning dnf with respect to the uniform distribution journal of computer and system sciences 55 3 1997 414 440 to the domain b n building on work of akavia et al a akavia s goldwasser s safra proving hard core predicates using list decoding in proc of the 44th annual ieee symposium on foundations of computer science focs 03 2003 pp 146 156 other ingredients used to obtain the results stated above are techniques from exact learning a beimel e kushilevitz learning boxes in high dimension algorithmica 22 1 2 1998 76 90 and ideas from recent work on learning augmented ac 0 circuits j c jackson a r klivans r a servedio learnability beyond ac 0 in proc of the 34th annual acm symposium on theory of computing stoc 02 2002 pp 776 784 and on representing boolean functions as thresholds of parities a r klivans r a servedio learning dnf in time 2 o n 1 3 journal of computer and system sciences 68 2 2004 303 318

optimal prediction with memory abstract optimal prediction methods estimate the solution of nonlinear time dependent problems when that solution is too complex to be fully resolved or when data are missing the initial conditions for the unresolved components of the solution are drawn from a probability distribution and their effect on a small set of variables that are actually computed is evaluated via statistical projection the formalism resembles the projection methods of irreversible statistical mechanics supplemented by the systematic use of conditional expectations and new methods of solution for an auxiliary equation the orthogonal dynamics equation needed to evaluate a non markovian memory term the result of the computations is close to the best possible estimate that can be obtained given the partial data we present the constructions in detail together with several useful variants provide simple examples and point out the relation to the fluctuation dissipation formulas of statistical physics

a stable three level finite difference scheme for solving the parabolic two step model in a 3d micro sphere heated by ultrashort pulsed lasers ultrashort pulsed lasers with pulse durations of the order of sub picosecond to femtosecond domain possess exclusive capabilities in limiting the undesirable spread of the thermal process zone in the heated sample parabolic two step micro heat transport equations have been widely applied for thermal analysis of thin metal films exposed to picosecond thermal pulses in this study we develop a three level finite difference scheme for solving the heat transport equations in a three dimensional micro sphere heated by ultrashort pulsed lasers it is shown that the scheme is unconditionally stable the method is illustrated by numerical examples

on the stability of boundary conditions for molecular dynamics we study the stability of boundary conditions for molecular dynamics simulations a general stability criterion is established we first consider a one dimensional model with nearest neighbor interaction and multiple neighbor interactions we then generalize the results to more realistic models in 3d with nonlinear atomic interaction

absorbing boundary conditions for difference approximations to the multi dimensional wave equation construction des conditions aux limites absorbantes pour l equation des ondes multidimensionnelle a l aide d approximations aux differences formulation sous forme canonique des conditions aux limites absorbantes analytiques

worst case analysis of the perceptron and exponentiated update algorithms abstract the absolute loss is the absolute difference between the desired and predicted outcome this paper demonstrates worst case upper bounds on the absolute loss for the perception learning algorithm and the exponentiated update learning algorithm which is related to the weighted majority algorithm the bounds characterize the behavior of the algorithms over any sequence of trials where each trial consists of an example and a desired outcome interval any value in the interval is an acceptable outcome the worst case absolute loss of both algorithms is bounded by the absolute loss of the best linear function in a comparison class plus a constant dependent on the initial weight vector plus a per trial loss the per trial loss can be eliminated if the learning algorithm is allowed a tolerance from the desired outcome for concept learning the worst case bounds lead to mistake bounds that are comparable to past results

convective scheme solution of the boltzmann transport equation for nanoscale semiconductor devices a model for the simulation of the electron energy distribution in nanoscale metal oxide semiconductor field effect transistor mosfet devices using a kinetic simulation technique is implemented the convective scheme cs a method of characteristics is an accurate method of solving the boltzmann transport equation a nonlinear integrodifferential equation for the distribution of electrons in a mosfet device the method is used to find probabilities for use in an iterative scheme which iterates to find collision rates in cells the cs is also a novel approach to 2d semiconductor device simulation the cs has been extended to handle boundary conditions in 2d as well as to calculation of polygon overlap for polygons of more than three sides electron energy distributions in the channel of a mosfet are presented

weak ill posedness of spatial discretizations of absorbing boundary conditions for schrodinger type equations when we wish to solve numerically a differential problem defined on an infinite domain it is necessary to consider a finite subdomain and to use artificial boundary conditions in such a way that the solutions in the finite subdomain approximate the original solution these boundary conditions are called absorbing when small reflections to the interior domain are allowed in this paper we develop a general class of absorbing boundary conditions for schrodinger type equations by using rational approximations to the transparent boundary conditions with this approach previous absorbing boundary conditions in the literature are included in this class we use the method of lines for the discretization of the initial boundary value problems obtained this way we show that the ordinary differential systems that arise after the spatial discretization are weakly ill posed explaining a previous conjecture of fevens and jiang the time discretization is carried out with a stable runge kutta methods where the high order ones may be used to compensate for the possible troubles present in the problems semidiscretized in space

a multiscale coupling method for the modeling of dynamics of solids with application to brittle cracks we present a multiscale model for numerical simulations of dynamics of crystalline solids the method combines the continuum nonlinear elasto dynamics model which models the stress waves and physical loading conditions and molecular dynamics model which provides the nonlinear constitutive relation and resolves the atomic structures near local defects the coupling of the two models is achieved based on a general framework for multiscale modeling the heterogeneous multiscale method hmm we derive an explicit coupling condition at the atomistic continuum interface application to the dynamics of brittle cracks under various loading conditions is presented as test examples

exponentiated gradient versus gradient descent for linear predictors we consider two algorithm for on line prediction based on a linear model the algorithms are the well known gradient descent gd algorithm and a new algorithm which we call eg they both maintain a weight vector using simple updates for the gd algorithm the update is based on subtracting the gradient of the squared error made on a prediction the eg algorithm uses the components of the gradient in the exponents of factors that are used in updating the weight vector multiplicatively we present worst case loss bounds for eg and compare them to previously known bounds for the gd algorithm the bounds suggest that the losses of the algorithms are in general incomparable but eg has a much smaller loss if only a few components of the input are relevant for the predictions we have performed experiments which show that our worst case upper bounds are quite tight already on simple artificial data

a bridging domain method for coupling continua with molecular dynamics abstract a bridging domain method for coupling continuum models with molecular models is described in this method the continuum and molecular domains are overlapped in a bridging subdomain where the hamiltonian is taken to be a linear combination of the continuum and molecular hamiltonians we enforce the compatibility in the bridging domain by lagrange multipliers or by the augmented lagrangian method an explicit algorithm for dynamic solutions is developed results show that this multiscale method can avoid spurious wave reflections at the molecular continuum interface without any additional filtering procedures even for problems with significant nonlinearities the method is also shown to naturally handle the coupling of the continuum energy equation with the molecular subdomain a multiple time step algorithm is also developed within this framework

parabolic two step model and accurate numerical scheme for nanoscale heat conduction induced by ultrashort pulsed laser heating abstract simulation of nanoscale heat transfer phenomena has attracted great attention particularly the nano scale heat conduction induced by ultrashort pulsed laser heating in this article we propose a nanoscale parabolic two step model and an accurate numerical scheme for thermal analysis of the nanoscale heat conduction induced by ultrashort pulsed laser heating to this end we first introduce the knudsen number k n into the original parabolic two step heat conduction equations and couple them with the k n dependent and temperature jump boundary condition we then develop a fourth order accurate compact finite difference method for solving the nanoscale model stability and convergence of the obtained numerical scheme are analyzed theoretically we finally test the accuracy and applicability of the nanoscale model and the obtained numerical scheme in three examples by choosing various values of the k n and the parameter in the boundary condition the simulation could be a tool for analyzing the nanoscale heat conduction induced by ultrashort pulsed laser heating

regularization techniques for learning with matrices there is growing body of learning problems for which it is natural to organize the parameters into a matrix as a result it becomes easy to impose sophisticated prior knowledge by appropriately regularizing the parameters under some matrix norm this work describes and analyzes a systematic method for constructing such matrix based regularization techniques in particular we focus on how the underlying statistical properties of a given problem can help us decide which regularization function is appropriate our methodology is based on a known duality phenomenon a function is strongly convex with respect to some norm if and only if its conjugate function is strongly smooth with respect to the dual norm this result has already been found to be a key component in deriving and analyzing several learning algorithms we demonstrate the potential of this framework by deriving novel generalization and regret bounds for multi task learning multi class learning and multiple kernel learning

exact learning of discretized geometric concepts we first present an algorithm that uses membership and equivalence queries to exactly identify a discretized geometric concept defined by the union of m axis parallel boxes in d dimensional discretized euclidean space where each coordinate can have n discrete values this algorithm receives at most md counterexamples and uses time and membership queries polynomial in m and log n for any constant d furthermore all equivalence queries can be formulated as the union of o md log m axis parallel boxes next we show how to extend our algorithm to efficiently learn from only equivalence queries any discretized geometric concept generated from any number of halfspaces with any number of known to the learner slopes in a constant dimensional space in particular our algorithm exactly learns from equivalence queries only unions of discretized axis parallel boxes in constant dimensional space in polynomial time furthermore this equivalence query only algorithm can be modified to handle a polynomial number of lies in the counterexamples provided by the environment finally we introduce a new complexity measure that better captures the complexity of the union of m boxes than simply the number of boxes and the dimension our new measure sigma is the number of segments in the target where a segment is a maximum portion of one of the sides of the target that lies entirely inside or entirely outside each of the other halfspaces defining the target we present a modification of our first algorithm that uses time and queries polynomial in sigma and log n in fact the time and queries both membership and equivalence used by this single algorithm are polynomial for either m or d constant

kernelization of matrix updates when and how abstract we define what it means for a learning algorithm to be kernelizable in the case when the instances are vectors asymmetric matrices and symmetric matrices respectively we can characterize kernelizability in terms of an invariance of the algorithm to certain orthogonal transformations if we assume that the algorithm s action relies on a linear prediction then we can show that in each case the linear parameter vector must be a certain linear combination of the instances we give a number of examples of how to apply our methods in particular we show how to kernelize multiplicative updates for symmetric instance matrices

constrained nonnegative matrix factorization for image representation nonnegative matrix factorization nmf is a popular technique for finding parts based linear representations of nonnegative data it has been successfully applied in a wide range of applications such as pattern recognition information retrieval and computer vision however nmf is essentially an unsupervised method and cannot make use of label information in this paper we propose a novel semi supervised matrix decomposition method called constrained nonnegative matrix factorization cnmf which incorporates the label information as additional constraints specifically we show how explicitly combining label information improves the discriminating power of the resulting matrix decomposition we explore the proposed cnmf method with two cost function formulations and provide the corresponding update solutions for the optimization problems empirical experiments demonstrate the effectiveness of our novel algorithm in comparison to the state of the art approaches through a set of evaluations based on real world applications

a restarted krylov subspace method for the evaluation of matrix functions we show how the arnoldi algorithm for approximating a function of a matrix times a vector can be restarted in a manner analogous to restarted krylov subspace methods for solving linear systems of equations the resulting restarted algorithm reduces to other known algorithms for the reciprocal and the exponential functions we further show that the restarted algorithm inherits the superlinear convergence property of its unrestarted counterpart for entire functions and present the results of numerical experiments

kernels for generalized multiple instance learning the multiple instance learning mil model has been successful in numerous application areas recently a generalization of this model and an algorithm for it were introduced showing significant advantages over the conventional mil model on certain application areas unfortunately that algorithm is not scalable to high dimensions we adapt that algorithm to one using a support vector machine with our new kernel k wedge this reduces the time complexity from exponential in the dimension to polynomial computing our new kernel is equivalent to counting the number of boxes in a discrete bounded space that contain at least one point from each of two multisets we show that this problem is p complete but then give a fully polynomial randomized approximation scheme fpras for it we then extend k by enriching its representation into a new kernel kmin and also consider a normalized version of k wedge that we call k which may or may not not be a kernel but whose approximation yielded positive semidefinite gram matrices in practice we then empirically evaluate all three measures on data from content based image retrieval biological sequence analysis and the musk data sets we found that our kernels performed well on all data sets relative to algorithms in the conventional mil model

adaptive spacetime method using riemann jump conditions for coupled atomistic continuum dynamics abstract we combine the spacetime discontinuous galerkin sdg method for elastodynamics with the mathematically consistent atomistic discontinuous galerkin adg method in a new scheme that concurrently couples continuum and atomistic models of dynamic response in solids the formulation couples non overlapping continuum and atomistic models across sharp interfaces by weakly enforcing jump conditions for both momentum balance and kinematic compatibility using riemann values to preserve the characteristic structure of the underlying hyperbolic system momentum balances to within machine precision accuracy over every element on each atom and over the coupled system with small controllable energy dissipation in the continuum region that ensures numerical stability when implemented on suitable unstructured spacetime grids the continuum sdg model offers linear computational complexity in the number of elements and powerful adaptive analysis capabilities that readily bridge between atomic and continuum scales in both space and time a special trace operator for the atomic velocities and an associated atomistic traction field enter the jump conditions at the coupling interface the trace operator depends on parameters that specify at the scale of the atomic spacing the position of the coupling interface relative to the atoms in a key finding we demonstrate that optimizing these parameters suppresses spurious reflections at the coupling interface without the use of non physical damping or special boundary conditions we formulate the implicit sdg adg coupling scheme in up to three spatial dimensions and describe an efficient iterative solution scheme that outperforms common explicit schemes such as the velocity verlet integrator numerical examples in 1 d time and employing both linear and nonlinear potentials demonstrate the performance of the sdg adg method and show how adaptive spacetime meshing reconciles disparate time steps and resolves atomic scale signals in the continuum

online passive aggressive algorithms we present a family of margin based online learning algorithms for various prediction tasks in particular we derive and analyze algorithms for binary and multiclass categorization regression uniclass prediction and sequence prediction the update steps of our different algorithms are all based on analytical solutions to simple constrained optimization problems this unified view allows us to prove worst case loss bounds for the different algorithms and for the various decision problems based on a single lemma our bounds on the cumulative loss of the algorithms are relative to the smallest loss that can be attained by any fixed hypothesis and as such are applicable to both realizable and unrealizable settings we demonstrate some of the merits of the proposed algorithms in a series of experiments with synthetic and real data sets

the perceptron algorithm versus winnow linear versus logarithmic mistake bounds when few input variables are relevant abstract we give an adversary strategy that forces the perceptron algorithm to make kn mistakes in learning monotone disjunctions over n variables with at most k literals in contrast littlestone s algorithm winnow makes at most o k log n mistakes for the same problem both algorithms use thresholded linear functions as their hypotheses however winnow does multiplicative updates to its weight vector instead of the additive updates of the perceptron algorithm in general we call an algorithm additive if its weight vector is always a sum of a fixed initial weight vector and some linear combination of already seen instances thus the perceptron algorithm is an example of an additive algorithm we show that an adversary can force any additive algorithm to make n k 1 2 mistakes in learning a monotone disjunction of at most k literals simple experiments show that for k n winnow clearly outperforms the perceptron algorithm also on nonadversarial random data

optimal prediction of underresolved dynamics abstract a method is presented for computing the average solution of problems that are too complicated for adequate resolution but where information about the statistics of the solution is available the method involves computing average derivatives by interpolation based on linear regression and an updating of a measure constrained by the available crude information examples are given

on the proper learning of axis parallel concepts we study the proper learnability of axis parallel concept classes in the pac learning and exact learning models these classes include union of boxes dnf decision trees and multivariate polynomials for constant dimensional axis parallel concepts c we show that the following problems have time complexities that are within a polynomial factor of each other c is properly exactly learnable with hypotheses of size at most times the target size from membership and equivalence queries c is properly pac learnable without membership queries under any product distribution there is an approximation algorithm for the minequic problem given a g c find a minimal size f c that is logically equivalent to g in particular if one has polynomial time complexity they all do using this we give the first proper learning algorithm of constant dimensional decision trees and the first negative results in proper learning from membership and equivalence queries for many classes for axis parallel concepts over a nonconstant dimension we show that with the equivalence oracle 1 3 we use this to show that binary decision trees are not properly learnable in polynomial time assuming p np and dnf is not se properly learnable e 1 in polynomial time even with an np oracle assuming 2p pnp

efficient learning with virtual threshold gates abstract we reduce learning simple geometric concept classes to learning disjunctions over exponentially many variables we then apply an online algorithm called winnow whose number of prediction mistakes grows only logarithmically with the number of variables the hypotheses of winnow are linear threshold functions with one weight per variable we find ways to keep the exponentially many weights of winnow implicitly so that the time for the algorithm to compute a prediction and update its virtual weights is polynomial our method can be used to learnd dimensional axis parallel boxes whendis variable and unions ofd dimensional axis parallel boxes whendis constant the worst case number of mistakes of our algorithms for the above classes is optimal to within a constant factor and our algorithms inherit the noise robustness of winnow we think that other online algorithms with multiplicative weight updates whose loss bounds grow logarithmically with the dimension are amenable to our methods

denominator assignment invariants and canonical forms under dynamic feedback compensation in linear multivariable systems a result originally reported by hammer for linear time invariant lti single input single output systems and concerning an invariant and a canonical form of the transfer function matrix of the closed loop system under dynamic feedback compensation is generalized for lti multivariable systems based on this result we characterize the class of transfer function matrices that are obtainable from an open loop transfer function matrix via the use of proper dynamic feedback compensators and show that if the closed loop transfer function matrix pc s has a desired denominator polynomial matrix which satisfies a certain sufficient condition then there exists a proper compensator giving rise to an internally stable closed loop system whose transfer function matrix is pc s

solving nonstationary classification problems with coupled support vector machines many learning problems may vary slowly over time in particular some critical real world applications when facing this problem it is desirable that the learning method could find the correct input output function and also detect the change in the concept and adapt to it we introduce the time adaptive support vector machine ta svm which is a new method for generating adaptive classifiers capable of learning concepts that change with time the basic idea of ta svm is to use a sequence of classifiers each one appropriate for a small time window but in contrast to other proposals learning all the hyperplanes in a global way we show that the addition of a new term in the cost function of the set of svms that penalizes the diversity between consecutive classifiers produces a coupling of the sequence that allows ta svm to learn as a single adaptive classifier we evaluate different aspects of the method using appropriate drifting problems in particular we analyze the regularizing effect of changing the number of classifiers in the sequence or adapting the strength of the coupling a comparison with other methods in several problems including the well known stagger dataset and the real world electricity pricing domain shows the good performance of ta svm in all tested situations

a coarse grained molecular dynamics model for crystalline solids a general mathematical framework for coarse graining molecular dynamics md model for solid system is presented the formulation is based directly on the full md model the reduction of the atomic degrees of freedom is accomplished using the mori zwanzig projection method we also demonstrate how to simplify the model under this framework to make the numerical implementation much easier copyright 2010 john wiley sons ltd

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

step size adaptation in reproducing kernel hilbert space this paper presents an online support vector machine svm that uses the stochastic meta descent smd algorithm to adapt its step size automatically we formulate the online learning problem as a stochastic gradient descent in reproducing kernel hilbert space rkhs and translate smd to the nonparametric setting where its gradient trace parameter is no longer a coefficient vector but an element of the rkhs we derive efficient updates that allow us to perform the step size adaptation in linear time we apply the online svm framework to a variety of loss functions and in particular show how to handle structured output spaces and achieve efficient online multiclass classification experiments show that our algorithm outperforms more primitive methods for setting the gradient step size

unresolved computation and optimal predictions we present methods for predicting the solution of time dependent partial differential equations when that solution is so complex that it cannot be properly resolved numerically but when prior statistical information can be found the sparse numerical data are viewed as constraints on the solution and the gist of our proposal is a set of methods for advancing the constraints in time so that regression methods can be used to reconstruct the mean future for linear equations we offer general recipes for advancing the constraints the methods are generalized to certain classes of nonlinear problems and the conditions under which strongly nonlinear problems and partial statistical information can be handled are briefly discussed our methods are related to certain data acquisition schemes in oceanography and meteorology c 1999 john wiley sons inc

the weighted majority algorithm we study the construction of prediction algorithms in a situation in which a learner faces a sequence of trials with a prediction to be made in each and the goal of the learner is to make few mistakes we are interested in the case where the learner has reason to believe that one of some pool of known algorithms will perform well but the learner does not know which one a simple and effective method based on weighted voting is introduced for constructing a compound algorithm in such a circumstance we call this method the weighted majority algorithm we show that this algorithm is robust in the presence of errors in the data we discuss various versions of the weighted majority algorithm and prove mistake bounds for them that are closely related to the mistake bounds of the best algorithms of the pool for example given a sequence of trials if there is an algorithm in the pool a that makes at most m mistakes then the weighted majority algorithm will make at most c log a m mistakes on that sequence where c is fixed constant

continued fraction absorbing boundary conditions for convex polygonal domains continued fraction absorbing boundary conditions cfabcs are highly effective boundary conditions for modelling wave absorption into unbounded domains they are based on rational approximation of the exact dispersion relationship and were originally developed for straight computational boundaries in this paper cfabcs are extended to the more general case of polygonal computational domains the key to the current development is the surprising link found between the cfabcs and the complex co ordinate stretching of perfectly matched layers pmls this link facilitates the extension of cfabcs to oblique corners and thus to polygonal domains it is shown that the proposed cfabcs are easy to implement expected to perform better than pmls and are effective for general polygonal computational domains in addition to the derivation of cfabcs a novel explicit time stepping scheme is developed for efficient numerical implementation numerical examples presented in the paper illustrate that effective absorption is attained with a negligible increase in the computational cost for the interior domain although this paper focuses on wave propagation its theoretical development can be easily extended to the more general class of problems where the governing differential equation is second order in space with constant coefficients copyright 2005 john wiley sons ltd

the concave convex procedure the concave convex procedure cccp is a way to construct discrete time iterative dynamical systems that are guaranteed to decrease global optimization and energy functions monotonically this procedure can be applied to almost any optimization problem and many existing algorithms can be interpreted in terms of it in particular we prove that all expectation maximization algorithms and classes of legendre minimization and variational bounding algorithms can be reexpressed in terms of cccp we show that many existing neural network and mean field theory algorithms are also examples of cccp the generalized iterative scaling algorithm and sinkhorn s algorithm can also be expressed as cccp by changing variables cccp can be used both as a new way to understand and prove the convergence of existing optimization algorithms and as a procedure for generating new algorithms

proper denominator assigning feedback compensators equivalent to state variable feedback for linear time invariant stabilizable multivariable systems we examine the problem of the existence and computation of proper denominator assigning and internally stabilizing feedback compensators which give rise to a closed loop system whose transfer function matrix is equal to one that can be obtained by the action of state variable feedback we establish a sufficient condition for the solution of this problem for the class of systems with the same number of inputs and outputs and non singular transfer function matrix with all its zeros located in the open left half of the complex plane

structure and smith macmillan form of a rational matrix at infinity the use of special row and column operations in the reduction of a rational matrix to its smith macmillan form at infinity is investigated the connections between this reduction procedure and the valuation approach are established a graphical method for finding the smith macmillan form of a rational matrix at infinity from its bode magnitude array and some new results on realization theory for polynomial matrices are presented

a generalized space time mathematical homogenization theory for bridging atomistic and continuum scales a generalized space time mathematical homogenization theory which constructs an equivalent continuum description directly from molecular dynamics md equations is developed the noteworthy theoretical findings of this work are i the coarse scale continuum equations pdes obtained from the homogenization of md equations are identical to those obtained from the classical homogenization of the fine scale continuum ii the lower order effective continuum properties such as the fourth order elasticity tensor have similar characteristics to those resulting from the homogenization of fine scale continuum iii the higher order effective properties such as polarization and dispersion tensors substantially differ from those of continuum due to the discreteness effect both the heterogeneous continuum and discrete media possess the size effect but only atomistic medium has the discreteness effect some demonstrative numerical examples are presented to verify the formulation copyright 2005 john wiley sons ltd

analysis of some krylov subspace approximations to the matrix exponential operator in this note a theoretical analysis of some krylov subspace approximations to the matrix exponential operation exp a v is presented and a priori and a posteriors error estimates are established several such approximations are considered the main idea of these techniquesis to approximately project the exponential operator onto a small krylov subspace and to carry out the resulting small exponential matrix computation accurately this general approach which has been used with success in several applications provides a systematic way of defining high order explicit type schemes for solving systems of ordinary differential equations or time dependent partial differential equations

sequential prediction over hierarchical structures we study sequential compound decision problems in the context of sequential prediction of real valued sequences in particular we consider finite state fs predictors that are constructed based on a hierarchical structure such as the order preserving patterns of the sequence history we define hierarchical equivalence classes by tying certain models at a hierarchy level in a recursive manner in order to mitigate undertraining problems these equivalence classes defined on a hierarchical structure are then used to construct a super exponential number of sequential fs predictors based on their combinations and permutations we then introduce truly sequential algorithms with computational complexity only linear in the pattern length that 1 asymptotically achieve the performance of the best fs predictor or the best linear combination of all the fs predictors in an individual sequence manner without any stochastic assumptions over any data length n under a wide range of loss functions 2 achieve the mean square error of the best linear combination of all fs filters or predictors in the steady state for certain nonstationary models we illustrate the superior convergence and tracking capabilities of our algorithm with respect to several state of the art methods in the literature through simulations over synthetic and real benchmark data

generalized mathematical homogenization of atomistic media at finite temperatures in three dimensions we derive thermo mechanical continuum equations from molecular dynamics md equations using the generalized mathematical homogenization gmh theory developed by the authors for 0 k applications gmh constructs an array of atomistic unit cell problems coupled with a thermo mechanical continuum problem the unit cell problem derived is a molecular dynamics problem defined for the perturbation from the average atomistic displacements subjected to the deformation gradient and temperature extracted from the continuum problem the coarse scale problem derived is a constitutive law free continuum thermo mechanical equation attention is restricted to heat transfer by lattice vibration phonons the method is verified on several model problems against the reference molecular dynamics solution 2006 elsevier b v all rights reserved

absorbing boundary conditions for the discretization schemes of the one dimensional wave equation when computing a partial differential equation it is often necessary to introduce artificial boundaries here we explain a systematic method to obtain boundary conditions for the wave equation in one dimension fitting to the discretization scheme and stable moreover we give error estimates on the reflected part

online markov decision processes we consider a markov decision process mdp setting in which the reward function is allowed to change after each time step possibly in an adversarial manner yet the dynamics remain fixed similar to the experts setting we address the question of how well an agent can do when compared to the reward achieved under the best stationary policy over time we provide efficient algorithms which have regret bounds with no dependence on the size of state space instead these bounds depend only on a certain horizon time of the process and logarithmically on the number of actions

spurious reflection of elastic waves due to gradually changing finite element size summary when a homogeneous elastic continuum is modelled by a non uniform finite element grid the differences in the size of adjacent finite elements can cause a spurious wave reflection which does not exist in the continuum extending previous studies in which the spurious wave reflection was investigated for the case of two uniform grids with a sudden element size change from one grid to the other the present study deals with the case when the two uniform grids of different element sizes are separated by a transition zone through which the element size varies gradually either by a geometric progression or by an arithmetic progression the solution is carried out in complex variables and numerical plots of the results are given when the ratio of wavelength to the largest element size is as low as 4 1 the spurious wave reflection is very significant while for the ratios over 10 1 it is insignificant inserting a transition zone with gradually varying element size between the two uniform parts of the grid somewhat mitigates the phenomenon of spurious reflection but this is significant only when the ratio of element sizes in the uniform grids is small less than about 1 5 1 varying the element size throughout the transition zone in an arithmetic progression seems slightly better than in a geometric progression the spurious wave reflection is less pronounced for higher order elements in particular for linear strain elements as compared to constant strain elements the spurious reflection is also less severe for consistent mass than for lumped mass

three dimensional perfectly matched layer for the absorption of electromagnetic waves the perfectly matched layer is a technique of free space simulation developed for solving unbounded electromagnetic problems with the finite difference time domain method referred to as pml this technique has been described in a previous paper for two dimensional problems the present paper is devoted to the three dimensional case the theory of the perfectly matched layer is generalized to three dimensions and some numerical experiments are shown to illustrate the efficiency of this new method of free space simulation

virtual bubbles and galerkin least squares type methods ga l s abstract the equivalence between stabilized finite element methods or galerkin least squares tyoe methods ga l s and the standard galerkin method with bubble functions is established in an abstract framework the results are applicable to various finite element spaces including high order elements and applications to the advective diffusive model and to the stokes problem are presented illustrating the potential of the abstract theory introduced here

online ranking by projecting we discuss the problem of ranking instances in our framework each instance is associated with a rank or a rating which is an integer in 1 to k our goal is to find a rank prediction rule that assigns each instance a rank that is as close as possible to the instance s true rank we discuss a group of closely related online algorithms analyze their performance in the mistake bound model and prove their correctness we describe two sets of experiments with synthetic data and with the eachmovie data set for collaborative filtering in the experiments we performed our algorithms outperform online algorithms for regression and classification applied to ranking

boltzmann equation description of flows at long mean free paths the flow of a gas around a rectangular object is simulated by solving the boltzmann equation for the gas the boltzmann equation is solved by means of a method of characteristics which we refer to as a convective scheme cs this paper focuses first on two computational issues we describe how the cs which is presented here is implemented so as to handle reflecting boundary conditions very accurately next a collision operator for the self collisions of the neutral gas has been developed which conserves momentum and energy exactly and which also preserves higher moments of the distribution so as to correctly calculate quantities such as viscosity finally the method is illustrated briefly by calculating flow patterns and drag coefficients for a low mach number flow around a rectangular obstacle over a range of knudsen numbers which spans the transitional regime and very accurate values of the drag coefficient are obtained across the whole range

optimal rates for zero order convex optimization the power of two function evaluations we consider derivative free algorithms for stochastic and nonstochastic convex optimization problems that use only function values rather than gradients focusing on nonasymptotic bounds on convergence rates we show that if pairs of function values are available algorithms for d dimensional optimization that use gradient estimates based on random perturbations suffer a factor of at most sqrt d in convergence rate over traditional stochastic gradient methods we establish such results for both smooth and nonsmooth cases sharpening previous analyses that suggested a worse dimension dependence and extend our results to the case of multiple m ge 2 evaluations we complement our algorithmic development with information theoretic lower bounds on the minimax convergence rate of such problems establishing the sharpness of our achievable results up to constant sometimes logarithmic factors

the p norm generalization of the lms algorithm for adaptive filtering recently much work has been done analyzing online machine learning algorithms in a worst case setting where no probabilistic assumptions are made about the data this is analogous to the h sup spl infin setting used in adaptive linear filtering bregman divergences have become a standard tool for analyzing online machine learning algorithms using these divergences we motivate a generalization of the least mean squared lms algorithm the loss bounds for these so called p norm algorithms involve other norms than the standard 2 norm the bounds can be significantly better if a large proportion of the input variables are irrelevant i e if the weight vector we are trying to learn is sparse we also prove results for nonstationary targets we only know how to apply kernel methods to the standard lms algorithm i e p 2 however even in the general p norm case we can handle generalized linear models where the output of the system is a linear function combined with a nonlinear transfer function e g the logistic sigmoid

online learning of complex prediction problems using simultaneous projections we describe and analyze an algorithmic framework for online classification where each online trial consists of multiple prediction tasks that are tied together we tackle the problem of updating the online predictor by defining a projection problem in which each prediction task corresponds to a single linear constraint these constraints are tied together through a single slack parameter we then introduce a general method for approximately solving the problem by projecting simultaneously and independently on each constraint which corresponds to a prediction sub problem and then averaging the individual solutions we show that this approach constitutes a feasible albeit not necessarily optimal solution of the original projection problem we derive concrete simultaneous projection schemes and analyze them in the mistake bound model we demonstrate the power of the proposed algorithm in experiments with synthetic data and with multiclass text categorization tasks

variational boundary conditions for molecular dynamics simulations treatment of the loading condition this paper aims to extend the variational boundary conditions for molecular dynamics simulation x li w e variational boundary conditions for molecular dynamics simulations of solids at low temperature commun comp phys 1 2006 136 176 x li w e boundary conditions for molecular dynamics simulations at finite temperature treatment of the heat bath phys rev b 76 2007 104107 to take into account external loading conditions two derivations of the exact boundary conditions are presented one with mori zwanzig projection procedure and the other using lattice green s functions approximate boundary conditions which are more efficient in practice are then discussed finally several numerical experiments are presented to demonstrate the effectiveness of these methods

on the convergence of galerkin approximation schemes for second order hyperbolic equations in energy and negative norms given certain semidiscrete and single step fully discrete galerkin approximations to the solution of an initial boundary value problem for a second order hyperbolic equation h and l2 error estimates are obtained these estimates are valid simultaneously when the approximation to the initial data is taken to be the projection onto the approximating space with respect to the inner product which induces the energy norm that is naturally associated with the problem the l2 estimate is obtained as a by product of the analysis of convergence in certain negative norms estimates are also obtained for the convergence of higher order time derivatives in the presence of sufficiently smooth data

molecular dynamics boundary conditions for regular crystal lattices we present a method for deriving molecular dynamics boundary conditions for use in multiple scale simulations that can be applied at a planar boundary for any solid that has a periodically repeating crystal lattice the method is based on a linearization in the vicinity of the boundary and utilizes a fourier and laplace transforms in space and time to eliminate the degrees of freedom associated with atoms outside the boundary this method is straightforward to implement numerically and thus can be automated for a general crystal lattice we show that this method reproduces the known kernel for a 1d linear chain and apply the approach to obtain the damping kernel matrices for two real crystal lattices the graphene and diamond structures of carbon

output feedback invariants and complex breakpoints abstract a complete characterization of the output feedback equivalence class of a single variate plant is formulated in terms of its zeros and complex breakpoints the potential applicability of this result to the minimum degree stabilizing compensator problem is discussed and a partial generalization of the thepry to the multivariate case presented

a finite difference scheme for solving a three dimensional heat transport equation in a thin film with microscale thickness heat transport at the microscale is of vital importance in microtechnology applications the heat transport equation is different from the traditional heat diffusion equation since a second order derivative of temperature with respect to time and a third order mixed derivative of temperature with respect to space and time are introduced in this study we develop a finite difference scheme with two levels in time for the three dimensional heat transport equation it is shown by the discrete energy method that the scheme is unconditionally stable the three dimensional implicit scheme is then solved by using a preconditioned richardson iteration so that only a tridiagonal linear system is solved each iteration numerical results show that the solution is accurate copyright 2001 john wiley sons ltd

matching boundary conditions for lattice dynamics summary we design a class of accurate and efficient absorbing boundary conditions for molecular dynamics simulations of crystalline solids in one space dimension the proposed matching boundary conditions take the form of a linear constraint of displacement and velocity at atoms near the boundary where the coefficients are determined by matching the dispersion relation with a minimal number of atoms involved bearing the nice features of compactness locality and high efficiency the matching boundary conditions are then extended to treat the out of plane wave problems in the square lattice we construct multidirectional absorbing boundary conditions via operator multiplications reflection coefficient analysis and numerical studies verify their effectiveness for spurious reflection suppression in all directions compact and local in both space and time they are directly applicable to nonlinear lattices and multiscale simulations copyright 2012 john wiley sons ltd

unconditionally stable discretization schemes of non reflecting boundary conditions for the one dimensional schrodinger equation this paper addresses the problem of the construction of stable approximation schemes for the one dimensional linear schrodinger equation set in an unbounded domain after a study of the initial boundary value problem in a bounded domain with a transparent boundary condition some unconditionally stable discretization schemes are developed for this kind of problem the main difficulty is linked to the involvement of a fractional integral operator defining the transparent operator the proposed semi discretization of this operator yields with a very different point of view the one proposed by yevick friese and schmidt j comput phys 168 2001 433 two possible choices of transparent boundary conditions based on the dirichlet neumann dn and neumann dirichlet nd operators are presented to preserve the stability of the fully discrete scheme conform galerkin finite element methods are employed for the spatial discretization finally some numerical tests are performed to study the respective accuracy of the different schemes

classification of proper bases of rational vector spaces minimal macmillian degree bases abstract the algebraic structure of the get of all proper rational vectors contained in a given rational vector space x1d4af s is shown to be that of a noothorian pr s motiulo m ropf pr a is the ring of proper rational functions the proper submodules mt of m form an ascending chain of submodules partially ordered by an invariant of mt defined as the valuation at s ofmt tho various bases of mt are examined and classified according to their property of column reduceness at s the concept of a prime column reduced at basis of mt is introduced it is shown that tho prime bases of mt can be further classified by their macmillan degrees and the existence of minimal macmillan degree bases for mt is established a prime and minimal macmillan degree basis ofmt extends forney s concept of a minimal polynomial basis of s for the rpr mt module the macmillan degrees of the columns of such bases form a set of invariants mt for which are defined as the x1d4af s generalized invariant dynamicol indices o

learning functions represented as multiplicity automata we study the learnability of multiplicity automata in angluin s exact learning model and we investigate its applications our starting point is a known theorem from automata theory relating the number of states in a minimal multiplicity automaton for a function to the rank of its hankel matrix with this theorem in hand we present a new simple algorithm for learning multiplicity automata with improved time and query complexity and we prove the learnability of various concept classes these include among others the class of disjoint dnf and more generally satisfy o 1 dnf the class of polynomials over finite fields the class of bounded degree polynomials over infinite fields the class of xor of terms certain classes of boxes in high dimensions in addition we obtain the best query complexity for several classes known to be learnable by other methods such as decision trees and polynomials over gf 2 while multiplicity automata are shown to be useful to prove the learnability of some subclasses of dnf formulae and various other classes we study the limitations of this method we prove that this method cannot be used to resolve the learnability of some other open problems such as the learnability of general dnf formulas or even k term dnf for k log n or satisfy s dnf formulas for s 1 these results are proven by exhibiting functions in the above classes that require multiplicity automata with super polynomial number of states

a full invariant under dynamic output feedback a generalized inverse approach a novel full invariant of linear time invariant multivariable systems under causal dynamic output feedback is derived it makes use of a generalized inverse approach

finite element approximations of nonlinear elastic waves in this paper we study finite element methods for a class of problems of nonlinear elastodynamics we discretize the equations in space using galerkin methods for the temporal discretization the construction of our schemes is based on rational approximations of cos x and ex we analyze semidiscrete as well as secondand fourth order accurate in time fully discrete methods for the approximation of the solution of the problem and prove optimal order l2 error estimates for some schemes a taylor type technique is used so that only linear systems of equations need be solved at each time step in the proofs we need various estimates for a nonlinear elliptic projection the proofs of which are also established in the paper

fractional parabolic two step model and its accurate numerical scheme for nanoscale heat conduction abstract simulation of nanoscale heat transfer phenomena has attracted great attention particularly the nanoscale heat conduction induced by ultrashort pulsed laser heating in this article we present a nanoscale fractional parabolic two step model and obtain an accurate numerical scheme for nanoscale heat conduction in metals the model is obtained by introducing the knudsen number k n and the fractional order derivative in time to the original parabolic two step energy transport and then coupling them with the k n dependent and temperature jump boundary condition the numerical scheme is developed based on the fourth order compact finite difference method and the l 1 approximation for fractional derivatives the well posedness of the model and the stability and convergence of the scheme are analyzed theoretically we finally test the accuracy and applicability of the new model and the obtained numerical method by two examples by changing values of the knudsen number and fractional order derivative as well as the parameter in the boundary condition the simulation could be a tool for analyzing the nanoscale heat conduction in porous media such as porous thin metal film exposed to ultrashort pulsed lasers

local coordinate concept factorization for image representation learning sparse representation of high dimensional data is a state of the art method for modeling data matrix factorization based techniques such as nonnegative matrix factorization and concept factorization cf have shown great advantages in this area especially useful for image representation both of them are linear learning problems and lead to a sparse representation of the images however the sparsity obtained by these methods does not always satisfy locality conditions for example the learned new basis vectors may be relatively far away from the original data thus we may not be able to achieve the optimal performance when using the new representation for other learning tasks such as classification and clustering in this paper we introduce a locality constraint into the traditional cf by requiring the concepts basis vectors to be as close to the original data points as possible each datum can be represented by a linear combination of only a few basis concepts thus our method is able to achieve sparsity and locality simultaneously we analyze the complexity of our novel algorithm and demonstrate the effectiveness in comparison with the state of the art approaches through a set of evaluations based on real world applications

an efficient scheme for convection dominated transport partial differential equations describing transport processes involving a significant effect of the flow velocity may be solved efficiently and easily using a simple algorithm the algorithm is based on the propagator s or green s functions for the equations of transport theory the numerical method employed is always at least as fast as finite differencing and it reduces to a finite difference method in the short time step limit but is especially efficient in cases where flow dominates over diffusion and is consequently widely applicable in kinetic theory and fluid dynamics using this method the ion distribution function and the potential in a plasma sheath were calculated in the presence of charge exchange collisions for a wide range of neutral densities

coarse graining molecular dynamics models using an extended galerkin projection method summary we present a new framework for coarse graining molecular dynamics models for crystalline solids the reduction method is based on a galerkin projection to a subspace whose dimension is much smaller than that of the full atomistic model to effectively reduce artificial reflections of phonons at the interface we construct extended subspaces with increasing accuracy by adding more coarse grained variables near the interface between lattice defects and the surrounding region this approach is equivalent to the generalized langevin model but it eliminates the need to precompute the memory function a well known practical difficulty further the variational formulation preserves the stability of the molecular models copyright 2014 john wiley sons ltd