On discrete multivariate distributions symmetric in frequencies

In this paper we describe a new class of discrete multivariate distributions which verify that their probability mass function is invariant when their univariate variables are permuted. These distributions may be generated by a multivariate extension of the Gauss function₂ F ₁ with matrix argument. A methodology that permits the fit of these distributions to real data is developed. A fit of a distribution for bivariate real data is shown and is compared with fits obtained by means of other usual bivariate distributions generated by extensions of the Gauss function.     

Reconstruction of Piecewise Smooth Functions from Non-uniform Grid Point Data

Spectral series expansions of piecewise smooth functions are known to yield poor results, with spurious oscillations forming near the jump discontinuities and reduced convergence throughout the interval of approximation. The spectral reprojection method, most notably the Gegenbauer reconstruction method, can restore exponential convergence to piecewise smooth function approximations from their (pseudo-)spectral coefficients. Difficulties may arise due to numerical robustness and ill-conditioning of the reprojection basis polynomials, however. This paper considers non-classical orthogonal polynomials as reprojection bases for a general order (finite or spectral) reconstruction of piecewise smooth functions. Furthermore, when the given data are discrete grid point values, the reprojection polynomials are constructed to be orthogonal in the discrete sense, rather than by the usual continuous inner product. No calculation of optimal quadrature points is therefore needed. This adaptation suggests a method to approximate piecewise smooth functions from discrete non-uniform data, and results in a one-dimensional approximation that is accurate and numerically robust.      

Cost-efficient parallel programs based on set-distributions for polynomial interpolation

The paper presents parallel algorithms for Lagrange and Hermite interpolation methods formally derived from specifications, and using set-distributions. Set-distributions are based on set-valued mappings, and they assign a data object to more than one process. The derivation from specifications assures the correctness, and the set-distributions assure the efficiency of the programs. The obtained parallel algorithms have very good time complexities and speeds-up, and they are also cost-efficient. We consider the number of processes p to be a parameter of the algorithms, so, bounded parallelism is considered. The derivation of the algorithms is not ruled by any particular interconnection network. The possible mappings on different networks could be evaluated. The performance analysis is done considering a full-connected network, and other two interconnection networks: hypercube and multi-mesh hypercube, which preserve the cost-efficiency of the algorithms.     

多项式插值与局部拟合的滤波器设计研究

Deslauriers-Dubuc插值滤波器系统冗余较大,且对偶正则性较差,将Deslauriers-Dubuc插值滤波器推广到一般意义下的插值滤波器,对改善滤波器性能具有重要的意义.在研究Deslauriers-Dubuc插值滤波器的基础上,给出了基于多项式局部拟合的插值滤波器设计方法.通过比较二者之间的区别和联系,发现Deslauriers-Dubuc插值滤波器可以看作是多项式局部拟合插值滤波器的特例.信号去噪仿真试验表明,基于多项式局部拟合的插值滤波器能够充分利用数据的信息,对噪声具有较好的稳健性.     

Eddy currents in a gradient coil,modeled as circular loops of strips

As a model for the z-coil of an MRI-scanner a set of circular loops of strips, or rings, placed on one cylinder is chosen. The current in this set of thin conducting rings is driven by an external source current. The source, and all excited fields, are time-harmonic. The frequency is low enough to allow for an electro-quasi-static approach. The rings have a thin rectangular cross-section with a thickness so small that the current can be assumed uniformly distributed in the thickness direction. Due to induction, eddy currents occur resulting in an edge-effect. Higher frequencies cause stronger edge-effects. As a consequence, the resistance of the system increases and its self-inductance decreases. The Maxwell equations imply an integral equation for the current distribution in the rings. The Galerkin method with Legendre polynomials as global basis functions is applied. This method shows fast convergence, so only a very restricted number of basis functions is needed. The general method is worked out for N (N ≥ 1) rings, and explicit results are presented for N = 1, N = 2 and N = 24. The derived integral equation and the numerical results of the simulations show that sets of circular rings and plane strips describe the same electromagnetic behavior, thus demonstrating that inductance effects are local.     

Stability criteria of matrix polynomials

In this paper, the stability of matrix polynomials is investigated. First, upper and lower bounds are derived for the eigenvalues of a matrix polynomial. The bounds are based on the spectral radius and the norms of the related matrices, respectively. Then, by means of the argument principle, stability criteria are presented which are necessary and sufficient conditions for the stability of matrix polynomials. Furthermore, a numerical algorithm is provided for checking the stability of matrix polynomials. Numerical examples are given to illustrate the main results.     

Convex inner approximations of nonconvex semialgebraic sets applied to fixed-order controller design

We describe an elementary algorithm to build convex inner approximations of nonconvex sets. Both input and output sets are basic semialgebraic sets given as lists of defining multivariate polynomials. Even though no optimality guarantees can be given (e.g. in terms of volume maximisation for bounded sets), the algorithm is designed to preserve convex boundaries as much as possible, while removing regions with concave boundaries. In particular, the algorithm leaves invariant a given convex set. The algorithm is based on Gloptipoly 3, a public-domain Matlab package solving nonconvex polynomial optimisation problems with the help of convex semidefinite programming (optimisation over linear matrix inequalities, or LMIs). We illustrate how the algorithm can be used to design fixed-order controllers for linear systems, following a polynomial approach.     

Hermite matrix in Lagrange basis for scaling static output feedback polynomial matrix inequalities

Using Hermite's formulation of polynomial stability conditions, static output feedback (SOF) controller design can be formulated as a polynomial matrix inequality (PMI), a (generally nonconvex) nonlinear semidefinite programming problem that can be solved (locally) with PENNON, an implementation of a penalty and augmented Lagrangian method. Typically, Hermite SOF PMI problems are badly scaled and experiments reveal that this has a negative impact on the overall performance of the solver. In this note we recall the algebraic interpretation of Hermite's quadratic form as a particular Bézoutian and we use results on polynomial interpolation to express the Hermite PMI in a Lagrange polynomial basis, as an alternative to the conventional power basis. Numerical experiments on benchmark problem instances show the improvement brought by the approach, in terms of problem scaling, number of iterations and convergence behaviour of PENNON.     

A simple method for the inclusion of external and internal supports in the spline finite strip method (SFSM) of buckling analysis

The SFSM is an attractive numerical technique for the buckling analysis of folded-plate structures where general loading regimes and boundary conditions need to be modelled. In implementing splines as interpolation functions in the longitudinal direction of the strip, amended splines have been used conventionally to model the variety of end conditions that may occur. These amended splines are fairly difficult to implement, particularly so if internal restraints are also to be specified. A simple technique for replacing the specification of dedicated amended splines is presented in this paper. The method is then employed to study the local buckling of flat plates under longitudinally and transversely varying compression and bending with different boundary conditions at the ends.     

Computation of approximate optimal policies in a partially observed inventory model with rain checks

This paper proposes a new methodology to solve partially observed inventory problems. Generally, these problems have infinite-dimensional states that are conditional distributions of the inventory level. Our methodology involves linearizing the state transitions via unnormalized probabilities. It then uses an appropriate functional basis to represent the state. Considering the speed and stability of computations, we choose truncated Chebyshev polynomials as the basis. We use Fast Fourier Transforms along with an appropriate discretization of inventory levels to speed up the computations. These main ideas are blended to obtain an iterative algorithm to solve a partially observed inventory model with rain checks. In this model, the inventory manager (IM) does not know the inventory level when it is positive. Otherwise, the IM fully observes it. This model provides a context to illustrate our methodology, which applies to other such models. Although this model has been studied mathematically in the literature, the use of our algorithm provides a numerical approximation of the optimal order quantities. These are compared to the orders released under a base mean-stock policy, where the IM replaces the unobserved inventory level with its mean and applies the well-known base stock policy. We show numerically that the optimal order quantity is very close to the base mean-stock order quantity, when the variance of the inventory distribution is small. When the mean of the inventory distribution is large, the optimal order quantity is more than the base mean-stock quantity, and it is the other way around when the mean is small or negative. These insights are explained via uncertainty and information effects and their interplay. We expect this interplay to show up in other partially observed inventory models.