Tracking poles and representing Hankel operators directly from data

Summary We propose and analyse a method of estimating the poles near the unit circleT of a functionG whose values are given at a grid of points onT: we give an algorithm for performing this estimation and prove a convergence theorem. The method is to identify the phase for an estimate by considering the peaks of the absolute value ofG onT, and then to estimate the modulus by seeking a bestL ² fit toG over a small arc by a first order rational function. These pole estimates lead to the construction of a basis ofL ² which is well suited to the numerical representation of the Hankel operator with symbolG and thereby to the numerical solution of the Nehari problem (computing the bestH , analytic, approximation toG relative to theL norm), as analysed in [HY]. We present the results of numerical tests of these algorithms.Partially supported by grants from the AFOSR and NSF     

低场脉冲核磁共振测定二醋酸纤维素丝束中油剂

研究了采用低场脉冲核磁共振(NMR)测定二醋酸纤维素丝束油剂的新方法，给出NMR法测定二醋酸纤维素丝束油荆含量的标准曲线。同时对NMR法测量二醋酸纤维素丝束油荆不确定度进行了评估。结果表明，NMR法具有分析速度快、无污染、无损伤和准确度高的优点，适合于二醋酸纤维素丝束油荆的分析。     

Asymptotic expansions andL ∞-error estimates for mixed finite element methods for second order elliptic problems

Summary Asymptotic expansions for mixed finite element approximations of the second order elliptic problem are derived and Richardson extrapolation can be applied to increase the accuracy of the approximations. A new procedure, which is called the error corrected method, is presented as a further application of the asymptotic error expansion for the first order BDM approximation of the scalar field. The key point in deriving the asymptotic expansions for the error is an establishment ofL ¹-error estimates for mixed finite element approximations for the regularized Green's functions. As another application of theL ¹-error estimates for the regularized Green's functions, we shall present maximum norm error estimates for mixed finite element methods for second order elliptic problems.     

A fast ‘Monte-Carlo cross-validation’ procedure for large least squares problems with noisy data

Summary We propose a fast Monte-Carlo algorithm for calculating reliable estimates of the trace of the influence matrixA involved in regularization of linear equations or data smoothing problems, where is the regularization or smoothing parameter. This general algorithm is simply as follows: i) generaten pseudo-random valuesw ₁, ...,w _n , from the standard normal distribution (wheren is the number of data points) and letw=(w ₁, ...,w _n ) ^T , ii) compute the residual vectorw–A w, iii) take the normalized inner-product (w ^T (w–A w))/(w ^T w) as an approximation to (1/n)tr(I–A ). We show, both by theoretical bounds and by numerical simulations on some typical problems, that the expected relative precision of these estimates is very good whenn is large enough, and that they can be used in practice for the minimization with respect to of the well known Generalized Cross-Validation (GCV) function. This permits the use of the GCV method for choosing in any particular large-scale application, with only a similar amount of work as the standard residual method. Numerical applications of this procedure to optimal spline smoothing in one or two dimensions show its efficiency.     

Spectral methods with sparse matrices

Summary Spectral methods employ global polynomials for approximation. Hence they give very accurate approximations for smooth solutions. Unfortunately, for Dirichlet problems the matrices involved are dense and have condition numbers growing asO(N ⁴) for polynomials of degree N in each variable. We propose a new spectral method for the Helmholtz equation with a symmetric and sparse matrix whose condition number grows only asO(N ²). Certain algebraic spectral multigrid methods can be efficiently used for solving the resulting system. Numerical results are presented which show that we have probably found the most effective solver for spectral systems.     

On the asymptotic stability ofθ-methods for delay differential equations

Stability regions of -methods for the linear delay differential test equations

Truncation error bounds for modified continued fractions with applications to special functions

Much recent work has been done to investigate convergence of modified continued fractions (MCF's), following the proof by Thron and Waadeland [35] in 1980 that a limit-periodic MCFK(a _n , 1;x ₁), with andnth approximant

Uniformly convergent difference schemes for singularly perturbed parabolic diffusion-convection problems without turning points

We consider first the initial-boundary value problem for the parabolic equation

Comparison theorems forL-monosplines of minimal norm

Summary LetLM _N be the set of allL-monosplines withN free knots, prescribed by a pair (x;E) of pointsx = {x _i } ₁ ⁿ ,a <x ₁ < ... <x _n <b and an incidence matrixE = (e _ij ) _i=1 ⁿ , _r-1 ^j=0 with Denote byLM _N ^O the subset ofLM _N consisting of theL-monosplines withN simple knots (n=N). We prove that theL-monosplines of minimalL _p-norms inLM _N belong toLM _N ^O .The results are reformulated as comparison theorems for quadrature formulae.     

A method for computing the tension parameters in convexity-preserving spline-in-tension interpolation

Summary This paper is concerned with the problem of convexity-preservng (orc-preserving) interpolation by using Exponential Splines in Tension (or EST's). For this purpose the notion of ac-preserving interpolant, which is usually employed in spline-in-tension interpolation, is refined and the existence ofc-preserving EST's is established for the so-calledc-admissible data sets. The problem of constructing ac-preserving and visually pleasing EST is then treated by combining a generalized Newton-Raphson method, due to Ben-Israel, with a step-length technique which serves the need for visual pleasantness. The numerical performance of the so formed iterative scheme is discussed for several examples.