computing orthogonal polynomials in sobolev spaces numerical methods are considered for generating polynomials orthogonal with respect to an inner product of sobolev type i e one that involves derivatives up to some given order each having its own positive measure associated with it the principal objective is to compute the coefficients in the increasing order recurrence relation that these polynomials satisfy by virtue of them forming a sequence of monic polynomials with degrees increasing by 1 from one member to the next as a by product of this computation one gains access to the zeros of these polynomials via eigenvalues of an upper hessenberg matrix formed by the coefficients generated two methods are developed one is based on the modified moments of the constitutive measures and generalizes what for ordinary orthogonal polynomials is known as modified chebyshev algorithm the other a generalization of stieltjes s procedure expresses the desired coefficients in terms of a sobolev inner product involving the orthogonal polynomials in question whereby the inner product is evaluated by numerical quadrature and the polynomials involved are computed by means of the recurrence relation already generated up to that point the numerical characteristics of these methods are illustrated in the case of sobolev orthogonal polynomials of old as well as new types based on extensive numerical experimentation a number of conjectures are formulated with regard to the location and interlacing properties of the respective zeros

ergodicity of the 2d navier stokes equations with degenerate stochastic forcing the stochastic 2d navier stokes equations on the torus driven by degenerate noise are studied we characterize the smallest closed invariant subspace for this model and show that the dynamics restricted to that subspace is ergodic in particular our results yield a purely geometric characterization of a class of noises for which the equation is ergodic in l 2 0 t 2 unlike previous works this class is independent of the viscosity and the strength of the noise the two main tools of our analysis are the asymptotic strong feller property introduced in this work and an approximate integration by parts formula the first when combined with a weak type of irreducibility is shown to ensure that the dynamics is ergodic the second is used to show that the first holds un

ergodicity of the finite dimensional approximation of the 3d navier stokes equations forced by a degenerate noise we prove ergodicity of the finite dimensional approximations of the three dimen sional navier stokes equations driven by a random force the forcing noise acts only on a few modes and some algebraic conditions on the forced modes are found that imply the ergodicity the convergence rate to the unique invariant measure is shown to be exponential the uniqueness of statistical steady states for the navier stokes equations is a less known but nevertheless important problem in the mathematical theory of turbulence the question is completely open in dimension three mainly because due to the lack of uniqueness of the equations there is no way yet to give meaning to the mathematical objects involved in the subject in the present paper the property of ergodicity is proved for the finite dimensional approxi mations of the three dimensional navier stokes equations driven by a random force the same problem has been solved in two dimensions by e and mattingly 2 such result can have a qualitative interest for the statistical behaviour of an incompressible fluid indeed if the kolmogorov theory of turbulence is taken into account one can believe that the cascade of energy responsible of the transport of the energy through the scales is effective in the inertial range so that at smaller scales only the dissipation ends up to be relevant hence the long time statistical properties of the fluid can be sufficiently depicted by the low modes of the velocity field in some sense if the ultraviolet cut off is sufficiently large in order to capture all the important modes the corresponding invariant measure gives the real behaviour of the fluid in view of these considerations the conclusions of the paper can give also both a hint and a possible starting point for the analysis of the infinite dimensional case we consider a finite dimensional truncation of the three dimensional navier stokes equations driven by a random force with periodic boundary conditions the proof of ergodicity is classical and it is developed in two steps firstly we prove that the transition probability densities are regular by checking that the diffusion operator is hypoelliptic the hormander condition then we show that the markov process is irreducible in the sense that each open set is visited with positive probability at each time for this aim we study the associated control problem see section 6 irreducibility for the infinite dimensional equations was firstly proved by flandoli 4 under the assumption that the noise acts on all modes

continuous methods for extreme and interior eigenvalue problems in this paper continuous methods are introduced to compute both the extreme and interior eigenvalues and their corresponding eigenvectors for real symmetric matrices the main idea is to convert the extreme and interior eigenvalue problems into some optimization problems then a continuous method which includes both a merit function and an ordinary differential equation ode is introduced for each resulting optimization problem the convergence of each ode solution is proved for any starting point the limit of each ode solution for any starting point is fully studied both the extreme and the interior eigenvalues and their corresponding eigenvectors can be easily obtained under a very mild condition promising numerical results are also presented

zeros of sobolev orthogonal polynomials following from coherent pairs let sn denote the monic orthogonal polynomial sequence with respect to the sobolev inner product f g s fgd 0 f g d 1 where d 0 d 1 is a so called coherent pair and 0 then sn has n different real zeros the position of these zeros with respect to the zeros of other orthogonal polynomials in particular laguerre and jacobi polynomials is investigated coherent pairs are found where the zeros of sn 1 separte the zeros of sn

existence of densities for the 3d navier stokes equations driven by gaussian noise we prove three results on the existence of densities for the laws of finite dimensional functionals of the solutions of the stochastic navier stokes equations in dimension 3 in particular under very mild assumptions on the noise we prove that finite dimensional projections of the solutions have densities with respect to the lebesgue measure which have some smoothness when measured in a besov space this is proved thanks to a new argument inspired by an idea introduced in fournier and printems 2010

on rational transformations of linear functionals direct problem abstract let u be a quasi definite linear functional we find necessary and sufficient conditions in order to the linear functional v satisfying x a u x a v be a quasi definite one also we analyze some linear relations linking the polynomials orthogonal with respect to u and v

malliavin calculus for the stochastic 2d navier stokes equation we consider the incompressible two dimensional navier stokes equation with periodic boundary conditions under the effect of an additive white in time stochastic forcing under mild restrictions on the geometry of the scales forced we show that any finite dimensional projection of the solution possesses a smooth strictly positive density with respect to lebesgue measure in particular our conditions are viscosity independent we are mainly interested in forcing that excites a very small number of modes all of the results rely on proving the nondegeneracy of the infinite dimensional malliavin matrix c 2006 wiley periodicals inc

analysis of equilibrium states of markov solutions to the 3d navier stokes equations driven by additive noise we prove that every markov solution to the three dimensional navier stokes equations with periodic boundary conditions driven by additive gaussian noise is uniquely ergodic the convergence to the unique invariant measure is exponentially fast

compressible fluids driven by stochastic forcing the relative energy inequality and applications we show the relative energy inequality for the compressible navier stokes system driven by a stochastic forcing as a corollary we prove the weak strong uniqueness property pathwise and in law and convergence of weak solutions in the inviscid incompressible limit in particular we establish a yamada watanabe type result in the context of the compressible navier stokes system that is pathwise weak strong uniqueness implies weak strong uniqueness in law

irreducibility of the 3 d stochastic navier stokes equation abstract a 3 dimensional navier stokes equation with random force is investigated a form of irreducibility of interest in ergodic theory is proved under a full noise assumption the basic tool is the fact that even if the equation is a priori non well posed the solutions depend continuously on the noise around regular solutions

existence of densities for stable like driven sdeʼs with holder continuous coefficients consider a multidimensional stochastic differential equation driven by a stable like levy process we prove that the law of the solution immediately has a density in some besov space under some non degeneracy condition on the driving levy process and some very light holder continuity assumptions on the drift and diffusion coefficients

geometric properties of runge kutta discretizations for index 2 differential algebraic equations we analyze runge kutta discretizations applied to index 2 differential algebraic equations daes the asymptotic features of the numerical and the exact solutions are compared it is shown that runge kutta methods satisfying the first order constraint condition of the dae correctly reproduce the geometric properties of the continuous system the proof combines embedding techniques of index 2 daes and ordinary differential equations odes with some invariant manifolds results of nipp and stoffer attractive invariant manifolds for maps sam research report 92 11 eth zurich switzerland 1992 the results support the favorable behavior of these runge kutta methods applied to index 2 daes for t ge 0

one and multistep discretizations of index 2 differential algebraic systems and their use in optimization an approach to solve constrained minimization problems is to integrate a corresponding index 2 differential algebraic equation dae here corresponding means that the limit sets of the dae dynamics are local solutions of the minimization problem in order to obtain an efficient optimization code we analyze the behavior of certain runge kutta and linear multistep discretizations applied to these daes it is shown that the discrete dynamics reproduces the geometric properties and the long time behavior of the continuous system correctly finally we compare the dae approach with a classical sqp method

partial regularity for the stochastic navier stokes equations the effects of random forces on the emergence of singularities in the navier stokes equations are investigated in spite of the presence of white noise the paths of a martingale suitable weak solution have a set of singular points of one dimensional hausdorff measure zero furthermore statistically stationary solutions with finite mean dissipation rate are analysed for these stationary solutions it is proved that at any time t the set of singular points is empty the same result holds true for every martingale solution starting from 0 a e initial condition u 0 where 0 is the law at time zero of a stationary solution finally the previous result is non trivial when the noise is sufficiently non degenerate since for any stationary solution the measure 0 is supported on the whole space h of initial conditions

sobolev orthogonal polynomials and m n coherent pairs of measures we introduce the notion of m n coherent pair of measures as a generalization of the concept of coherent pair of measures introduced by iserles et al a iserles p e koch s p norsett j m sanz serna on polynomials orthogonal with respect to certain sobolev inner products j approx theory 65 2 1991 151 175 and subsequently generalized by several authors a pair of measures email protected 0 email protected 1 is called m n coherent if the corresponding orthogonal polynomial sequences p n n and q n n resp satisfy a non zero structure relation such as i 0nr i np n i x email protected i 0ms i nq n i 1 x for all n 0 1 2 where m and n are fixed non negative integer numbers and r i n and s i n are given real parameters satisfying some natural conditions we prove that the regular moment linear functionals associated to an m n coherent pair are semiclassical and they are related by a rational modification in the usual sense of distribution theory we also discuss the converse statement under the assumption that email protected 0 email protected 1 form an m n coherent pair we study the sequence s n l n of the monic orthogonal polynomials with respect to the sobolev inner product l email protected email protected email protected f g email protected 1 defined in the space of all polynomials with real coefficients where l 0 an efficient algorithm is stated to compute the coefficients in the fourier sobolev type series f x email protected n 0 c n ls n l x with respect to l for suitable smooth functions f such that email protected l m 0 2 r and f l m 1 2 r finally some illustrative computational examples are presented

what is beyond coherent pairs of orthogonal polynomials abstract usually coherent pairs of orthogonal polynomials have been considered in the wider context of sobolev orthogonality in this paper we focus our attention on the problem of coherence between two orthogonal polynomial sequences in terms of the corresponding linear functionals we deduce some conditions about the linear functionals in order that the corresponding orthogonal polynomial sequences constitute a coherent pair

ssc eke semi supervised classification with extensive knowledge exploitation we introduce a new semi supervised classification method that extensively exploits knowledge the method has three steps first the manifold regularization mechanism adapted from the laplacian support vector machine lapsvm is adopted to mine the manifold structure embedded in all training data especially in numerous label unknown data meanwhile by converting the labels into pairwise constraints the pairwise constraint regularization formula pcrf is designed to compensate for the few but valuable labelled data second by further combining the pcrf with the manifold regularization the precise manifold and pairwise constraint jointly regularized formula mpcjrf is achieved third by incorporating the mpcjrf into the framework of the conventional svm our approach referred to as semi supervised classification with extensive knowledge exploitation ssc eke is developed the significance of our research is fourfold 1 the mpcjrf is an underlying adjustment with respect to the pairwise constraints to the graph laplacian enlisted for approximating the potential data manifold this type of adjustment plays the correction role as an unbiased estimation of the data manifold is difficult to obtain whereas the pairwise constraints converted from the given labels have an overall high confidence level 2 by transforming the values of the two terms in the mpcjrf such that they have the same range with a trade off factor varying within the invariant interval 0 1 the appropriate impact of the pairwise constraints to the graph laplacian can be self adaptively determined 3 the implication regarding extensive knowledge exploitation is embodied in ssc eke that is the labelled examples are used not only to control the empirical risk but also to constitute the mpcjrf moreover all data both labelled and unlabelled are recruited for the model smoothness and manifold regularization 4 the complete framework of ssc eke organically incorporates multiple theories such as joint manifold and pairwise constraint based regularization smoothness in the reproducing kernel hilbert space empirical risk minimization and spectral methods which facilitates the preferable classification accuracy as well as the generalizability of ssc eke

orthogonal polynomials associated with an inverse quadratic spectral transform let p n n 0 be a sequence of monic orthogonal polynomials with respect to a quasi definite linear functional u and q n n 0 a sequence of polynomials defined by q n x p n x s np n 1 x t np n 2 x n 1 with t n 0 for n 2 we obtain a new characterization of the orthogonality of the sequence q n n 0 with respect to a linear functional v in terms of the coefficients of a quadratic polynomial h such that h x v u we also study some cases in which the parameters s n and t n can be computed more easily and give several examples finally the interpretation of such a perturbation in terms of the jacobi matrices associated with p n n 0 and q n n 0 is presented

the small scales of the stochastic navier stokes equations under rough forcing we prove that the small scale structures of the stochastically forced navier stokes equations approach those of the naturally associated ornstein uhlenbeck process as the scales get smaller precisely we prove that the rescaled kth spatial fourier mode converges weakly on path space to an associated ornstein uhlenbeck process as k in addition we prove that the navier stokes equations and the naturally associated ornstein uhlenbeck process induce equivalent transition densities if the viscosity is replaced with hyper viscosity this gives a simple proof of unique ergodicity for the hyperviscous navier stokes system we show how different strengthened hyperviscosity produce varying levels of equivalence

on linearly related sequences of derivatives of orthogonal polynomials abstract we discuss an inverse problem in the theory of standard orthogonal polynomials involving two orthogonal polynomial families p n n and q n n whose derivatives of higher orders m and k resp are connected by a linear algebraic structure relation such as i 0 n r i n p n i m m x i 0 m s i n q n i k k x for all n 0 1 2 where m and n are fixed nonnegative integer numbers and r i n and s i n are given complex parameters satisfying some natural conditions let u and v be the moment regular functionals associated with p n n and q n n resp assuming 0 m k we prove the existence of four polynomials m m i and n k i of degrees m m i and n k i resp such that d k m m m i u n k i v i 0 1 the k m th derivative as well as the left product of a functional by a polynomial being defined in the usual sense of the theory of distributions if k m then u and v are connected by a rational modification if k m 1 then both u and v are semiclassical linear functionals which are also connected by a rational modification when k m the stieltjes transform associated with u satisfies a non homogeneous linear ordinary differential equation of order k m with polynomial coefficients

qualitative properties of stationary measures for three dimensional navier stokes equations abstract the paper is devoted to studying the distribution of stationary solutions for 3d navier stokes equations perturbed by a random force under a non degeneracy assumption we show that the support of such a distribution coincides with the entire phase space and its finite dimensional projections are minorised by a measure possessing an almost surely positive smooth density with respect to the lebesgue measure similar assertions are true for weak solutions of the cauchy problem with a regular initial function the results of this paper were announced in the short note a shirikyan controllability of three dimensional navier stokes equations and applications in semin equ deriv partielles 2005 2006 ecole polytech palaiseau 2006

trust region algorithms and timestep selection unconstrained optimization problems are closely related to systems of ordinary differential equations odes with gradient structure in this work we prove results that apply to both areas we analyze the convergence properties of a trust region or levenberg marquardt algorithm for optimization the algorithm may also be regarded as a linearized implicit euler method with adaptive timestep for gradient odes from the optimization viewpoint the algorithm is driven directly by the levenberg marquardt parameter rather than the trust region radius this approach is discussed for example in r fletcher practical methods of optimization 2nd ed john wiley new york 1987 but no convergence theory is developed we give a rigorous error analysis for the algorithm establishing global convergence and an unusual extremely rapid type of superlinear convergence the precise form of superlinear convergence is exhibited the ratio of successive displacements from the limit point is bounded above and below by geometrically decreasing sequences we also show how an inexpensive change to the algorithm leads to quadratic convergence from the ode viewpoint this work contributes to the theory of gradient stability by presenting an algorithm that reproduces the correct global dynamics and gives very rapid local convergence to a stable steady state

on an extension of symmetric coherent pairs of orthogonal polynomials given two symmetric and positive definite linear functionals w and v we study the coefficients in the recurrence relation for the system of monic polynomials orthogonal with respect to the second linear functional assuming that the first one is classical and that there exists an algebraic differential relation between these two families of polynomials moreover we determine this companion linear functional as a rational modification of the classical one

absolute continuity of the invariant measures for some stochastic pdes we consider a free system and an interacting systems having invariant measures and respectively under suitable assumptions we prove an explicit formula relating with and implying the absolute continuity of with respect to we apply our result to a reaction diffusion equation and to the burgers equation

on the linear functionals associated to linearly related sequences of orthogonal polynomials abstract an inverse problem is solved by stating that the regular linear functionals u and v associated to linearly related sequences of monic orthogonal polynomials p n n and q n n respectively in the sense p n x i 1 n r i n p n i x q n x i 1 m s i n q n i x for all n 0 1 2 where r i n and s i n are complex numbers satisfying some natural conditions are connected by a rational modification i e there exist polynomials and with degrees m and n respectively such that u v we also make some remarks concerning the corresponding direct problem stating a characterization theorem in the case n 1 and m 2 as an example we give a linear relation of the above type involving jacobi polynomials with distinct parameters

exponential mixing for the 3d stochastic navier stokes equations we study the navier stokes equations in dimension 3 ns3d driven by a noise which is white in time we establish that if the noise is at the same time sufficiently smooth and non degenerate in space then the weak solutions converge exponentially fast to equilibrium

on linearly related orthogonal polynomials and their functionals let pn be a sequence of polynomials orthogonal with respect a linear functional u and qn a sequence of polynomials defined by pn x snpn 1 x qn x tnqn 1 x we find necessary and sufficient conditions in order to qn be a sequence of polynomials orthogonal with respect to a linear functional v furthermore we prove that the relation between these linear functionals is x a u x a v even more if u and v are linked in this way we get that pn and qn satisfy a formula as above

martingale and stationary solutions for stochastic navier stokes equations we prove the existence of martingale solutions and of stationary solutions of stochastic navier stokes equations under very general hypotheses on the diffusion term the stationary martingale solutions yield the existence of invariant measures when the transition semigroup is well defined the results are obtained by a new method of compactness

a method based on rayleigh quotient gradient flow for extreme and interior eigenvalue problems abstract recently a continuous method has been proposed by golub and liao as an alternative way to solve the minimum and interior eigenvalue problems according to their numerical results their method seems promising this article is an extension along this line in this article firstly we convert an eigenvalue problem to an equivalent constrained optimization problem secondly using the karush kuhn tucker conditions of this equivalent optimization problem we obtain a variant of the rayleigh quotient gradient flow which is formulated by a system of differential algebraic equations thirdly based on the rayleigh quotient gradient flow we give a practical numerical method for the minimum and interior eigenvalue problems finally we also give some numerical experiments of our method the golub and liao method and eigs a matlab implementation for computing eigenvalues using restarted arnoldi s method for some typical eigenvalue problems our numerical experiments indicate that our method seems promising for most test problems

markov selections for the 3d stochastic navier stokes equations we investigate the markov property and the continuity with respect to the initial conditions strong feller property for the solutions to the navier stokes equations forced by an additive noise first we prove by means of an abstract selection principle that there are markov solutions to the navier stokes equations due to the lack of continuity of solutions in the space of finite energy the markov property holds almost everywhere in time then depending on the regularity of the noise we prove that any markov solution has the strong feller property for regular initial conditions we give also a few consequences of these facts together with a new sufficient condition for well posedness