multilevel frames for sparse tensor product spaces for au f with an elliptic differential operator a mathcal h rightarrow mathcal h and stochastic data f the m point correlation function mathcal m m u of the random solution u satisfies a deterministic equation with the m fold tensor product operator a m of a sparse tensor products of hierarchic fe spaces in mathcal h are known to allow for approximations to mathcal m m u which converge at essentially the rate as in the case m 1 i e for the deterministic problem they can be realized by wavelet type fe bases von petersdorff and schwab in appl math 51 2 145 180 2006 schwab and todor in computing 71 43 63 2003 if wavelet bases are not available we show here how to achieve the fast computation of sparse approximations of mathcal m m u for galerkin discretizations of a by multilevel frames such as bpx or other multilevel preconditioners of any standard fem approximation for a numerical examples illustrate feasibility and scope of the method

a practical finite element approximation of a semi definite neumann problem on a curved domain this paper considers the finite element approximation of the semi definite neumann problem u f in a curved domain n n 2 or 3 sigma frac partial u partial v g on and int limits omega u dx q a given constant for dataf andg satisfying the compatibility condition int limits omega f dx int limits partial omega g ds 0 due to perturbation of domain errors h the standard galerkin approximation to the above problem may not have a solution a remedy is to perturb the right hand side so that a discrete form of the compatibility condition holds using this approach we show that for a finite element space defined overd h a union of elements with approximation powerh k in thel 2 norm and with dist h ch k one obtains optimal rates of convergence in theh 1 andl 2 norms whether h is fitted h d h or unfitted h d h provided the numerical integration scheme has sufficient accuracy

semilinear copulas a family of copulas called semilinear is constructed starting with some assumptions about the linearity of the copulas along some segments of the unit square this family contains some other known families of copulas e g cuadras auge frechet and has a nice statistical interpretation several construction methods are provided especially concerning aggregation of semilinear copulas and a special form of ordinal sum construction is introduced some results about related families of quasi copulas and semicopulas are hence given

preconditioned conjugate gradients for solving singular systems abstract in this paper the preconditioned conjugate gradient method is used to solve the system of linear equations ax b where a is a singular symmetric positive semi definite matrix the method diverges if b is not exactly in the range r a of a if the null space n a of a is explicitly known then this divergence can be avoided by subtracting from b its orthogonal projection onto n a as well as analysing this subtraction conditions necessary for the existence of a nonsingular incomplete cholesky decomposition are given finally the theory is applied to the discretized semi definite neumann problem

least action principle and the incompressible euler equations with variable density in this article we study a variational formulation proposed by v i arnold and by y brenier for the incompressible euler equations with variable density we consider the problem of minimizing an action functional the integral in time of the kinetic energy among fluid motions considered as trajectories in the group of volume preserving diffeomorphisms beginning at the identity and ending at some fixed diffeomorphism at a given time we show that a relaxed version of this variational problem always has a solution and we derive an euler lagrange system for the relaxed minimization problem which we call the relaxed euler equations finally we prove consistency between the relaxed euler equations and the classical euler system showing that weak solutions of the relaxed euler equations with the appropriate geometric structure give rise to classical euler solutions and that classical solutions of the euler system induce weak solutions of the relaxed euler equations the first consistency result is new even in the constant density case the remainder of our analysis is an extension of the work of y brenier 1999 to the variable density case

semistability of steepest descent with momentum for quadratic functions two steepest descent algorithms with momentum for quadratic functions are considered for a given learning rate the sufficient and necessary conditions for the semistability of the steepest descent algorithms with momentum are established moreover the optimal momentum factors that generally make for the fastest convergence are obtained

analysis of the momentum lms algorithm several modifications of the well known lms algorithm have been proposed for improved operation this work analyzes one such algorithm that corresponds to the standard lms algorithm with an additional update term parameterized by the scalar factor alpha where mod alpha mod

characterization of all copulas associated with non continuous random variables we introduce a constructive method by means of a doubly stochastic measure to describe all the copulas that in view of sklar s theorem are able to connect a bivariate distribution to its marginals we use this to give the lower and upper optimal bounds for all the copulas that extend a given subcopula

copulas with given track and opposite track sections solution to a problem on diagonals abstract in this note we provide conditions for the existence of a copula with given track and opposite track sections generalizing conditions for the existence of copulas with given diagonal and opposite diagonal sections in de amo et al 2016 3 we also use this construction to characterize a class of copulas with a fixed diagonal section providing as a consequence a solution to the problem 2 4 posed by c sempi in mesiar and sarkoci 2010 16

dynamic learning rate optimization of the backpropagation algorithm it has been observed by many authors that the backpropagation bp error surfaces usually consist of a large amount of flat regions as well as extremely steep regions as such the bp algorithm with a fixed learning rate will have low efficiency this paper considers dynamic learning rate optimization of the bp algorithm using derivative information an efficient method of deriving the first and second derivatives of the objective function with respect to the learning rate is explored which does not involve explicit calculation of second order derivatives in weight space but rather uses the information gathered from the forward and backward propagation several learning rate optimization approaches are subsequently established based on linear expansion of the actual outputs and line searches with acceptable descent value and newton like methods respectively simultaneous determination of the optimal learning rate and momentum is also introduced by showing the equivalence between the momentum version bp and the conjugate gradient method since these approaches are constructed by simple manipulations of the obtained derivatives the computational and storage burden scale with the network size exactly like the standard bp algorithm and the convergence of the bp algorithm is accelerated with in a remarkable reduction typically by factor 10 to 50 depending upon network architectures and applications in the running time for the overall learning process numerous computer simulation results are provided to support the present approaches

copulas based time series combined forecasters time series combined forecasters have been superior to the respective single models in statistical terms in this way the linear combination functions e g the simple average sa and the minimal variance mv approaches have been the main alternatives for aggregation in the literature in this work it is presented a copulas based method for combining biased single models copulas are multivariate functions that operate on marginal probability distributions and have the specific advantage of generalizing mv by flexibly modelling the forecasters residuals and then the dependence among them a typical divide and conquer framework that can result in superior combined forecasters the usefulness of the copulas based combination method is highlighted by means of a comparison with sa and mv models based on a number of simulated cases and a real world time series

new constructions of diagonal patchwork copulas we present a method for constructing symmetric copulas which generalizes the diagonal patchwork construction of copulas procedure we also show how it is related to a new construction of a generalized farlie gumbel morgenstern distribution and to the copula transforms

conic aggregation functions inspired by the notion of conic t norms we introduce conic aggregation functions such aggregation functions are completely characterized by their zero set special classes of binary conic aggregation functions such as conic quasi copulas and conic copulas are considered we provide the necessary and sufficient conditions on the boundary curve of the zero set of a conic aggregation function to obtain a conic quasi copula and conclude that the class of conic copulas is a proper subclass of the class of conic quasi copulas moreover we characterize the class of singular conic copulas we investigate some aggregations of conic quasi copulas some examples are also provided

zero linear copulas abstract we introduce a new method for constructing copulas c a special type of aggregation functions based on the idea that c u v 0 if u g v for any continuous and decreasing function g 0 1 0 1 such that g 1 0 and c is linear if u g v 1 we also study the relationship between these copulas and other known families of copulas we provide some salient properties and measure the dependence and asymmetry in this class of copulas

on parameter acceleration methods for saddle point problems for solving saddle point problems parameter acceleration methods which include uzawa type methods are investigated by many researchers in the literature in this paper we introduce the inexact uzawa method with another parameter acceleration that is the so called momentum acceleration method for solving saddle point problems we discuss the convergence conditions of the inexact uzawa iteration with momentum acceleration and give the optimal momentum factors which minimize the spectral radii of the associated iteration matrices numerical results demonstrate the effectiveness of the inexact uzawa method with momentum acceleration and the mixed parameter acceleration methods for solving saddle point problems

the distribution of the probability mass of biconic copulas we study the probability mass distribution of biconic copulas provide a counterexample to a result given in jwaid et al 2012 29 concerning a characterization of singular biconic copulas and provide a new one

a study on the optimal double parameters for steepest descent with momentum this letter presents the stability analysis for two steepest descent algorithms with momentum for quadratic functions the corresponding local optimal parameters in torii and hagan 2002 and zhang 2013 are extended to the global optimal parameters that is both the optimal learning rates and the optimal momentum factors are obtained simultaneously which make for the fastest convergence

parameter convergence and learning curves for neural networks we revisit the oft studied asymptotic in sample size behavior of the parameter or weight estimate returned by any member of a large family of neural network training algorithms by properly accounting for the characteristic property of neural networks that their empirical and generalization errors possess multiple minima we rigorously establish conditions under which the parameter estimate converges strongly into the set of minima of the generalization error convergence of the parameter estimate to a particular value cannot be guaranteed under our assumptions we then evaluate the asymptotic distribution of the distance between the parameter estimate and its nearest neighbor among the set of minima of the generalization error results on this question have appeared numerous times and generally assert asymptotic normality the conclusion expected from familiar statistical arguments concerned with maximum likelihood estimators these conclusions are usually reached on the basis of somewhat informal calculations although we shall see that the situation is somewhat delicate the preceding results then provide a derivation of learning curves for generalization and empirical errors that leads to bounds on rates of convergence

on the convergence of the lms algorithm with adaptive learning rate for linear feedforward networks we consider the problem of training a linear feedforward neural network by using a gradient descent like lms learning algorithm the objective is to find a weight matrix for the network by repeatedly presenting to it a finite set of examples so that the sum of the squares of the errors is minimized kohonen showed that with a small but fixed learning rate or stepsize some subsequences of the weight matrices generated by the algorithm will converge to certain matrices close to the optimal weight matrix in this paper we show that by dynamically decreasing the learning rate during each training cycle the sequence of matrices generated by the algorithm will converge to the optimal weight matrix we also show that for any given 0 the lms algorithm with decreasing learning rates will generate an optimal weight matrix i e a matrix of distance at most away from the optimal matrix after o 1 training cycles this is in contrast to 1 log 1 training cycles needed to generate an optima

a comparison of deflation and the balancing preconditioner in this paper we compare various preconditioners for the numerical solution of partial differential equations we compare the well known balancing preconditioner used in domain decomposition methods with a so called deflation preconditioner we prove that the effective condition number of the deflated preconditioned system is always i e for all deflation vectors and all restrictions and prolongations below the condition number of the system preconditioned by the balancing preconditioner even more we establish that both preconditioners lead to almost the same spectra the zero eigenvalues of the deflation preconditioned system are replaced by eigenvalues which are one if the balancing preconditioner is used moreover we prove that the a norm of the errors of the iterates built by the deflation preconditioner is always below the a norm of the errors of the iterates built by the balancing preconditioner depending on the implementation of the balancing preconditioner the amount of work of one iteration of the deflation preconditioned system is less than or equal to the amount of work of one iteration of the balancing preconditioned system if the amount of work is equal both preconditioners are sensitive with respect to inexact computations finally we establish that the deflation preconditioner and the balancing preconditioner produce the same iterates if one uses certain starting vectors numerical results for porous media flows emphasize the theoretical results

lower eigenvalue bounds for singular pencils of matrices abstract beauwens procedure for obtaining lower eigenvalue bounds for regular pencils of matrices a b is simplified and extended to the singular case the theory is then compared through a particular perturbed modified incomplete factorization with notay s generalization of another approach initiated by gustafsson and developed by axelsson and barker and by wilmet

on the regularity of the pressure field of relaxed solutions to euler equations with variable density abstract in this paper we improve the regularity in time of the gradient of the pressure field in the solution of relaxed version of variational formulation proposed by v i arnold and by y brenier for the incompressible euler equations with variable density we obtain that the pressure field is not only a measure but a function in l l o c 2 0 t b v l o c d as an extension of the work of ambrosio and figalli 2008 in 1 to the variable density case

analysis of the back propagation algorithm with momentum in this letter the back propagation algorithm with the momentum term is analyzed it is shown that all local minima of the sum of least squares error are stable other equilibrium points are unstable

aggregation functions construction methods conjunctive disjunctive and mixed classes in this second part of our state of the art overview on aggregation theory based again on our recent monograph on aggregation functions we focus on several construction methods for aggregation functions and on special classes of aggregation functions covering the well known conjunctive disjunctive and mixed aggregation functions some fields of applications are included

an online gradient method with momentum for two layer feedforward neural networks an online gradient method with momentum for two layer feedforward neural networks is considered the momentum coefficient is chosen in an adaptive manner to accelerate and stabilize the learning procedure of the network weights corresponding convergence results are proved that is the weak convergence result is proved under the uniformly boundedness assumption of the activation function and its derivatives moreover if the number of elements of the stationary point set for the error function is finite then strong convergence result holds

minimal geodesics on groups of volume preserving maps and generalized solutions of the euler equations the three dimensional motion of an incompressible inviscid fluid is classically described by the euler equations but can also be seen following arnold 1 as a geodesic on a group of volume preserving maps local existence and uniqueness of minimal geodesics have been established by ebin and marsden 16 in the large for a large class of data the existence of minimal geodesics may fail as shown by shnirelman 26 for such data we show that the limits of approximate solutions are solutions of a suitable extension of the euler equations or equivalently are sharp measure valued solutions to the euler equations in the sense of diperna and majda 14 1999 john wiley sons inc

stability of steepest descent with momentum for quadratic functions this paper analyzes the effect of momentum on steepest descent training for quadratic performance functions we demonstrate that there always exists a momentum coefficient that will stabilize the steepest descent algorithm regardless of the value of the learning rate we also demonstrate how the value of the momentum coefficient changes the convergence properties of the algorithm

the least action principle and the related concept of generalized flows for incompressible perfect fluids the link between the euler equations of perfect incompressible flows and the least action principle has been known for a long time i solu tions can be considered as geodesic curves along the manifold of volume pre serving mappings here the shortest path problem is investigated given two different volume preserving mappings at two different times find for the inter mediate times an incompressible flow map that minimizes the kinetic energy or more generally the action in its classical formulation this problem has been solved 7 only when the two different mappings are sufficiently close in some very strong sense in this paper a new framework is introduced where gen eralized flows are defined in the spirit of l c young as probability measures on the set of all possible trajectories in the physical space then the minimiza tion problem is generalized as the continuous linear programming problem that is much easier to handle the existence problem is completely solved in this content downloaded from 157 55 39 159 on sun 18 sep 2016 05 02 20 utc all use subject to http about jstor org terms the least action principle 255 the case of the d dimensional torus it is also shown that under natural re strictions a classical solution to the euler equations is the unique optimal flow in the generalized framework finally a link is established with the concept of measure valued solutions to the euler equations 6 and an example is provided where the unique generalized solution can be explicitly computed and turns out to be genuinely probabilistic inria bp 105 78153 le chesnay cedex france this content downloaded from 157 55 39 159 on sun 18 sep 2016 05 02 20 utc all use subject to http about jstor org terms

biconic aggregation functions we introduce a new method to construct aggregation functions these aggregation functions are called biconic aggregation functions with a given diagonal resp opposite diagonal section and their construction is based on linear interpolation on segments connecting the diagonal resp opposite diagonal of the unit square to the points 0 1 and 1 0 resp 0 0 and 1 1 subclasses of biconic aggregation functions such as biconic semi copulas biconic quasi copulas and biconic copulas are studied in detail

a universal integral as common frame for choquet and sugeno integral the choquet and the sugeno integral provide a useful tool in many problems in engineering and social choice where the aggregation of data is required however their applicability is restricted because of the special operations used in the construction of these integrals therefore we provide a concept of integrals generalizing both the choquet and the sugeno case for functions with values in the nonnegative real numbers universal integrals are introduced and investigated which can be defined on arbitrary measurable spaces and for arbitrary monotone measures for a fixed pseudo multiplication on the nonnegative real numbers the smallest and the greatest universal integrals are given finally another construction method for obtaining universal integrals is introduced and the restriction to the unit interval i e to fuzzy integrals is considered

short communication a note on the notion of singular copula we clarify the link between the notion of singular copula and the concept of support of the measure induced by a copula

on copulas and their diagonals we formulate the necessary and sufficient conditions for a function d 0 1 0 1 to be the diagonal section of a multivariate absolutely continuous copula moreover we provide some simple analytic formulas for copulas having given diagonal section or given distribution functions of order statistics

on representations of 2 increasing binary aggregation functions in this paper we provide two different representations of 2 increasing binary aggregation functions by means of their lower and upper margins and a suitable copula

piecewise linear aggregation functions based on triangulation we introduce a method to construct piecewise linear binary aggregation functions on the unit interval based on a triangulation of the unit square with one additional vertex we derive conditions under which such piecewise linear aggregation functions possess additional interesting properties such as idempotence symmetry lipschitz continuity and 2 monotonicity this construction method can also be used to approximate binary aggregation functions in this way copulas and quasi copulas are approximated by singular copulas

aggregation functions means this two part state of the art overview on aggregation theory summarizes the essential information concerning aggregation issues an overview of aggregation properties is given including the basic classification on aggregation functions in this first part the stress is put on means i e averaging aggregation functions both with fixed arity n ary means and with multiple arities extended means

guidelines for the usage of incomplete decompositions in solving sets of linear equations as they occur in practical problems abstract this paper presents incomplete decompositions for various types of matrices as they occur in the implicit discretisation of practical problems a review is given of methods for the usual five point discretisation of a self adjoint elliptic second order partial differential equation in two dimensions on a square the matrices which occur in this type of problem are symmetric m matrices of very regular structure the convergence behaviour of the different decompositions for this case is demonstrated by numerical experiments the paper also gives decompositions for the following type of matrices i symmetric m matrices of a different structure ii symmetric positive definite matrices iii non symmetric matrices

necessary and sufficient conditions for the convergence of orthomin k on singular and inconsistent linear systems we consider the convergence of orthomin k on singular and inconsistent linear systems criteria for the breakdown of orthomin k are discussed and analyzed moreover necessary and sufficient conditions for the convergence of orthomin k for any right hand side are given and a rate of convergence is provided as well finally numerical experiments are shown to confirm the convergence theorem

robust multigrid for high order discontinuous galerkin methods we present a polynomial multigrid method for nodal interior penalty and local discontinuous galerkin formulations of the poisson equation on cartesian grids for smoothing we propose two classes of overlapping schwarz methods the first class comprises element centered and the second face centered methods within both classes we identify methods that achieve superior convergence rates prove robust with respect to the mesh spacing and the polynomial order at least up to p 32 consequent structure exploitation yields a computational complexity of o pn where n is the number of unknowns further we demonstrate the suitability of the face centered method for element aspect ratios up to 32 p multigrid for discontinuous formulations of poisson problems on cartesian grids robust with respect to the mesh size and polynomial order at least up to p 32 convergence rates between 0 02 and 0 003 improving with increasing p computational complexity of pn for n unknowns linear complexity in runtime excellent performance up to aspect ratios of 1 16

deterministic convergence of an online gradient method for neural networks the online gradient method has been widely used as a learning algorithm for neural networks we establish a deterministic convergence of online gradient methods for the training of a class of nonlinear feedforward neural networks when the training examples are linearly independent we choose the learning rate to be a constant during the training procedure the monotonicity of the error function in the iteration is proved a criterion for choosing the learning rate is also provided to guarantee the convergence under certain conditions similar to those for the classical gradient methods an optimal convergence rate for our online gradient methods is proved

convergence of an online gradient method for feedforward neural networks with stochastic inputs in this paper we study the convergence of an online gradient method for feed forward neural networks the input training examples are permuted stochastically in each cycle of iteration a monotonicity and a weak convergence of deterministic nature are proved

projected gradient methods for nonnegative matrix factorization nonnegative matrix factorization nmf can be formulated as a minimization problem with bound constraints although bound constrained optimization has been studied extensively in both theory and practice so far no study has formally applied its techniques to nmf in this letter we propose two projected gradient methods for nmf both of which exhibit strong optimization properties we discuss efficient implementations and demonstrate that one of the proposed methods converges faster than the popular multiplicative update approach a simple matlab code is also provided