Coercive singular perturbations II: Reduction and convergence

An algebra of pseudodifferential singular perturbations is introduced. It provides a constructive machinery in order to reduce an elliptic singularly perturbed operator (in $\text{R}$ⁿ or on a smooth manifold without border) to a regular perturbation. The technique developed is applied to some singularly perturbed boundary value problems as well. Special attention is given to a singular perturbation appearing in the linear theory of thin elastic plates. A Wiener-Hopf-type operator containing the small parameter reduces this singular perturbation to a regular one. It also gives rise to a natural recurrence process for the construction of high-order asymptotic formulae for the solution of the perturbed problem. The method presented can be extended to the general coercive singular perturbations.     

A numerical model for the Raman amplification for laser-plasma interaction

In this paper, we continue the study of the Raman amplification initiated in [M. Colin, T. Colin, On a quasi-linear Zakharov system describing laser-plasma interactions, Differential Integral Equations 17(3–4) (2004) 297–330]. We use a dispersive, quasi-linear system. The quasi-linear part is not hyperbolic and this difficulty is overcome using the dispersion. We give an asymptotic result on a reduced system. We then introduce a simple, robust and efficient numerical scheme on the whole system that takes into account the non-hyperbolicity of the quasi-linear part as well as the nonlinear saturation of the Raman growth. The scheme is validated thanks to the asymptotic result. Finally, we present 1-D and 2-D simulations.     

Nevanlinna Functions,Perturbation Formulas,and Triplets of Hilbert Spaces

Let S be a closed symmetric operator with defect numbers (1,1) in a Hilbert space ?? and let A be a selfadjoint operator extension of S in ??. Then S is necessarily a graph restriction of A and the selfadjoint extensions of S can be considered as graph perturbations of A, cf. [8]. Only when S is not densely defined and, in particular, when S is bounded, 5 is given by a domain restriction of A and the graph perturbations reduce to rank one perturbations in the sense of [23]. This happens precisely when the Q - function of S and A belongs to the subclass No of Nevanlinna functions. In this paper we show that by going beyond the Hilbert space ?? the graph perturbations can be interpreted as compressions of rank one perturbations. We present two points of view: either the Hilbert space ?? is given a one-dimensional extension, or the use of Hilbert space triplets associated with A is invoked. If the Q - function of S and A belongs to the subclass N₁ of Nevanlinna functions, then it is convenient to describe the selfadjoint extensions of S including its generalized Friedrichs extension (see [6]) by interpolating the original triplet, cf. [5]. For the case when A is semibounded, see also [4]. We prove some invariance properties, which imply that such an interpolation is independent of the (nonexceptional) extension.     

A survey of numerical techniques for solving singularly perturbed ordinary differential equations

This survey paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of the ideas and methods of singular perturbation theory. Starting from Prandtl's work a large amount of work has been done in the area of singular perturbations. This paper limits its coverage to some standard singular perturbation models considered by various workers and the numerical methods developed by numerous researchers after 1984–2000. The work done in this area during the period 1905–1984 has already been surveyed by the first author of this paper, see [Appl. Math. Comput. 30 (1989) 223] for details. Due to the space constraints we have covered only singularly perturbed one-dimensional problems.     

一类拟线性奇摄动流体模型的渐近分析

对一类拟线性流体模型进行研究,借助Linard变换将所研究问题转化为可以用边界函数法处理的问题,进而用边界函数法对该方程组进行分析,构造其(n+1)阶形式渐近解,并证明解的存在唯一性和形式渐近解的一致有效性.     

Zur Stabilität diskreter Teilspektren singulärer Differentialoperatoren zweiter Ordnung gegenüber Störungen der Koeffizienten und der Definitionsgebiete

The stability ofL ²-eigenvalues and associated eigenspaces of singular second order differential operators of Schrödinger-type is shown for asymptotic perturbations of the coefficients and the domain of definition. The perturbations involved are more general than those studied in [3] and [5], because we do not postulate the convergence of the coefficients “from above” or of the domains “from inside” or “from outside”. Moreover, the domain of definition is allowed to be perturbed in its interior. The underlying abstract perturbation theory was established in a previous paper [9].     

Ergodicity for Nonlinear Stochastic Equations in Variational Formulation

This paper is concerned with nonlinear partial differential equations of the calculus of variation (see [13]) perturbed by noise. Well-posedness of the problem was proved by Pardoux in the seventies (see [14]), using monotonicity methods. The aim of the present work is to investigate the asymptotic behaviour of the corresponding transition semigroup P_t. We show existence and, under suitable assumptions, uniqueness of an ergodic invariant measure ν. Moreover, we solve the Kolmogorov equation and prove the so-called "identite du carre du champs". This will be used to study the Sobolev space W^1,2(H,ν) and to obtain information on the domain of the infinitesimal generator of P_t.     

基于双尺度渐近分析的有限元算法

1．引言正如文山所说，由于复合材料和周期结构的材料系数ail（x）在局部区域内间断且跳跃性很大，加上区域内含有周期性洞穴或裂缝，且周期长度很小．一般而言，直接采用有限元方法进行数值模拟，其计算量大得惊人，甚至难以实现．文山针对这种特征，提出了一种可计算的双尺度渐近分析模式，本文在此基础上给出了相应的有限元算法，它包括：1．周期解在一个基本构造上的有限元计算；2．边界层的有限元计算．同时，给出了相应的误差分析．2．周期解的有限元计算首先考虑下列形式的边值问题；其中把，代E尸（on叫，iii（0关于E—（EI，ZZ…     

Cohomology of the moduli of parabolic vector bundles

The purpose of this paper is to compute the Betti numbers of the moduli space ofparabolic vector bundles on a curve (see Seshadri [7], [8] and Mehta & Seshadri [4]), in the case where every semi-stable parabolic bundle is necessarily stable. We do this by generalizing the method of Atiyah and Bott [1] in the case of moduli of ordinary vector bundles. Recall that (see Seshadri [7]) the underlying topological space of the moduli of parabolic vector bundles is the space of equivalence classes of certain unitary representations of a discrete subgroup Γ which is a lattice in PSL (2,R). (The lattice Γ need not necessarily be co-compact). While the structure of the proof is essentially the same as that of Atiyah and Bott, there are some difficulties of a technical nature in the parabolic case. For instance the Harder-Narasimhan stratification has to be further refined in order to get the connected strata. These connected strata turn out to have different codimensions even when they are part of the same Harder-Narasimhan strata. If in addition to ‘stable = semistable’ the rank and degree are coprime, then the moduli space turns out to be torsion-free in its cohomology. The arrangement of the paper is as follows. In § 1 we prove the necessary basic results about algebraic families of parabolic bundles. These are generalizations of the corresponding results proved by Shatz [9]. Following this, in § 2 we generalize the analytical part of the argument of Atiyah and Bott (§ 14 of [1]). Finally in § 3 we show how to obtain an inductive formula for the Betti numbers of the moduli space. We illustrate our method by computing explicitly the Betti numbers in the special case of rank = 2, and one parabolic point.     

Dirichlet Forms and Markov Processes: A Generalized Framework Including Both Elliptic and Parabolic Cases

We extend the framework of classical Dirichlet forms to a class of bilinear forms, called generalized Dirichlet forms, which are the sum of a coercive part and a linear unbounded operator as a perturbation. The class of generalized Dirichlet forms, in particular, includes symmetric and coercive Dirichlet forms (cf. [6], [10]) as well as time dependent Dirichlet forms (cf. [14]) as special cases and also many new examples. Among these are, e.g. transformations of time dependent Dirichlet forms by -excessive functions h (h-transformations), Dirichlet forms with time dependent linear drift and fractional diffusion operators. One of the main results is that we identify an analytic property of these forms which ensures the existence of associated strong Markov processes with nice sample path properties, and give an explicit construction for such processes. This construction extends previous constructions of the processes in the elliptic and the parabolic cases, is, in particular, carried out on general topological state spaces (as in [10]), and is applied to the above examples.     

Foliations on ℂℙ2 of degree 2 with degenerate singularities

In this paper we construct non-singular, locally-closed, algebraic varieties which are sets of foliations on ??² of degree 2 with a certain degenerate singularity. We obtain the dimension and closure of these varieties. To do that we construct a stratification (based on GIT, see [7]) of the space of foliations with respect to the action by change of coordinates. We prove that the set of unstable foliations has two irreducible components. We have the following corollary: a foliation of degree 2 defined by a pencil of conics is unstable if and only if the pencil is unstable. Finallywe give another proof of the fact that there are only 4 foliations of degree 2 with a unique singular point (see [5]).     

W2,2 interior convergence for some class of elliptic anisotropic singular perturbations problems

In this paper, we deal with anisotropic singular perturbations of some class of elliptic problems. We study the asymptotic behavior of the solution in a certain second-order pseudo Sobolev space.     

Continuity of sublinear operators on F-spaces

In this note, some of the fundamental theorems concerning the automatic continuity of positive linear functionals and positive linear operators (see [9,14,21]) as well as certain uniform boundedness type theorems (see [5,13,15,19]) will be extended to derive various continuity properties of (even discontinuous) sublinear operators from some metrizable topological vector space to some ordered topological vector space.     

On asymptotic equivalence of perturbed linear systems of differential and difference equations

Standard results on asymptotic integration of systems of linear differential equations give sufficient conditions which imply that a system is strongly asymptotically equivalent to its principal diagonal part. These involve certain dichotomy conditions on the diagonal part as well as growth conditions on the off-diagonal perturbation terms. Here, we study perturbations with a triangularly-induced structure and see that growth conditions can be substantially weakened. In addition, we give results for not necessarily triangular perturbations which in some sense “interpolate” between the classical theorems of Levinson and Hartman-Wintner. Some analogous results for systems of linear difference equations are also given.     

Dealing with linear dependence during the iterations of the restarted block Lanczos methods

The Lanczos method can be generalized to block form to compute multiple eigenvalues without the need of any deflation techniques. The block Lanczos method reduces a general sparse symmetric matrix to a block tridiagonal matrix via a Gram–Schmidt process. During the iterations of the block Lanczos method an off-diagonal block of the block tridiagonal matrix may become singular, implying that the new set of Lanczos vectors are linearly dependent on the previously generated vectors. Unlike the single vector Lanczos method, this occurrence of linearly dependent vectors may not imply an invariant subspace has been computed. This difficulty of a singular off-diagonal block is easily overcome in non-restarted block Lanczos methods, see [12,30]. The same schemes applied in non-restarted block Lanczos methods can also be applied in restarted block Lanczos methods. This allows the largest possible subspace to be built before restarting. However, in some cases a modification of the restart vectors is required or a singular block will continue to reoccur. In this paper we examine the different schemes mentioned in [12,30] for overcoming a singular block for the restarted block Lanczos methods, namely the restarted method reported in [12] and the Implicitly Restarted Block Lanczos (IRBL) method developed by Baglama et al. [3]. Numerical examples are presented to illustrate the different strategies discussed.     

A characterization of parallelepipeds related to weak derivatives

Summary We characterize parallelepipeds in R^m within the family of all convex bodies by a property of special measures on its boundary. We show that these measures are related to weak derivatives (in the sense of [5] and [8]) of convex-valued functions. The results can be applied (see [9]) to derive a generalization of a theorem of Lehmann (see [4]) on the comparison of uniform location experiments.     

Methods de laplace et de la phase stationnaire sur l'espace de wiener

We obtain, using large deviations principles, full asymptotic expansions of functionals of Laplace type on Wiener space for general (i.e. degenerate) diffusions extending the results of Schilder [12], Azencott [2] and Doss [7, 20]. The variational hypothesis (non degeneracy of the minima) used here is shown to be optimal. The first term of the expansion is explicitly computed. Using the integration by parts of Malliavin calculus the stationary phase method is also developed. The results of this paper are the basic fact used (in [5]) to obtain the asymptotic expansion for small time of the density of a degenerate diffusion, they are also relevant for semi-classical expansions     

Perturbazioni singolari ellittiche

We consider general boundary value problem for partial differential operators with small parameter ε in their coefficients, so-called singular perturbation. Both the perturbed and reduced (with ε=0) problems are supposed to be elliptic and satisfy the Shapiro-Lopatinsky coerciveness condition (see [9], [13]). We point out necessary and sufficient conditions on the operator in the region and the boundary operators for the singulary perturbed boundary value problem to be coercive, i.e. for a characteristic two-sided a priori estimate to hold for its solutions uniformly with respect to ε.     

Dixmier traces are weak dense in the set of all fully symmetric traces

We extend Dixmier's construction of singular traces (see [2]) to arbitrary fully symmetric operator ideals. In fact, we show that the set of Dixmier traces is weak^? dense in the set of all fully symmetric traces (that is, those traces which respect Hardy–Littlewood submajorization). Our results complement and extend earlier work of Wodzicki [22].     

On the determination of the worst sampling error in a communication system for pulse amplitude modulated signals. Part II: The finite dimensional model

This paper is dedicated to the problem of optimizing the transmission properties of a Pulse Amplitude Modulation (PAM) system. The system is disturbed by a random timing jitter in the sampling device which periodically evaluates the continuous output signal at discrete times. Mathematically the timing jitter is a random variable with unknown probability distribution. So, our optimization problem turns out to be actually a minimax problem, for which mathematical game theory with its powerful concepts becomes the suitable frame for our analysis.In the first part [4] we have established a general existence theorem for the minimax problem, and we have worked out some properties of solutions in the case that the feasible impulse responses form a space of infinite dimension.This part summarizes results which we obtain, if we allow only for impulse responses lying in a certainn-dimensional subspace of the original space (see [2, 3]). By general results from semi-infinite optimization (see [1]) we know that, writing the minimax problem as a semi-infinite optimization problem, we can reduce the number of restrictions from infinity to a numbersn+1. On the basis of our special model we present a theory of uniformly singular quadratic forms, which has been developed (see [3]) in order to get additional statements abouts.In this way we supplement the work of Krabs [6], who was the first to present such a finite dimensional model, arguing that it is impossible for an engineer to construct a system in a way that an arbitrary impulse response is realized, unless this impulse response has a simple structure (for instance a low pass filter).The first two paragraphs have been taken almost literally from part I in order to render the lecture more comfortable. The interesting parts, however, are the following ones, where the results specific for the finite dimensional case are worked out.