Finite-difference stable stencils based on least-square quadric fitting

In this paper a procedure to obtain new finite difference stencils is given. We investigate the particular branch of cases where the order of the finite difference stencil is lower than the amount of grid points. New finite difference stencils are obtained and compared with traditional ones from literature. General properties and advantages of the proposed stencils are investigated, in particular the improvements obtained in stability when solving ill-posed problems.     

Improving angular error via systematically designed near-circular Gaussian-based feature extraction operators

In image filtering, the ‘circularity’ of an operator is an important factor affecting its accuracy. For example, circular differential edge operators are effective in minimising the angular error in the estimation of image gradient direction. We present a general approach to the computation of scalable circular low-level image processing operators that is based on the finite element method. We show that the use of Gaussian basis functions within the finite element method provides a framework for a systematic and efficient design procedure for operators that are scalable to near-circular neighbourhoods through the use of an explicit scale parameter. The general design technique may be applied to a range of operators. Here we evaluate the approach for the design of the image gradient operator. We illustrate that this design procedure significantly reduces angular error in comparison to other well-known gradient approximation methods.     

Localized Generalization Error of Gaussian-based Classifiers and Visualization of Decision Boundaries

In pattern classification problem, one trains a classifier to recognize future unseen samples using a training dataset. Practically, one should not expect the trained classifier could correctly recognize samples dissimilar to the training dataset. Therefore, finding the generalization capability of a classifier for those unseen samples may not help in improving the classifiers accuracy. The localized generalization error model was proposed to bound above the generalization mean square error for those unseen samples similar to the training dataset only. This error model is derived based on the stochastic sensitivity measure(ST-SM)of the classifiers. We present the ST-SMS for various Gaussian based classifiers: radial basis function neural networks and support vector machine in this paper. At the end of this work, we compare the decision boundaries visualization using the training samples yielding the largest sensitivity measures and the one using support vectors in the input space.     

Vectorizing matrix operations arising from PDE discretization on 9-point stencils

When solving a system of PDEs, discretized on 9-point stencils over a nonrectangular domain, the linear systems that arise will have matrices with an irregular block structure. In this paper we discuss the vectorization of the matrix-vector multiply and of the Incomplete LU factorization and backsolve for these types of matrices. The performance of the matrix-vector multiply is already optimal for a small number of grid points (one result per clock cycle). For the ILU factorization and backsolve the vector performance is not as satisfying, partly because the resulting vector length is generally small and partly because of the heavy use of indirect addressing. A comparison with the general-purpose routines from the SLAP library shows a significant gain in computational time.This work was supported by Cray Research, Inc., under grant CRG 92.05, via the Stichting Nationale Computerfeciliteiten (National Computing Facilities Foundation, NCF).     

Stable LQG controllers

The optimal LQG controller is known to stabilize the closed loop if certain mild conditions are satisfied. The controller itself, however, may be unstable. The paper presents a method of selecting the weighting and covariance matrices such that the optimal controller is internally asymptotically stable. The method is very easy to apply and for stable open-loop system involves the solution of a single Lyapunov equation     

Stable polygons of cyclic pursuit

In a companion paper [5] we resolved the question of whether cyclic pursuits can exhibit ‘non-mutual’ captures. Although, as we showed, non-mutual captures can occur, the set of initial conditions which lead to them has Lebesgue measure zero. Thus, generically, cyclic pursuits collapse into a mutual capture. In this paper we consider whether the pursuit configuration can asymptotically approach a regular one for a non-trivial set of initial conditions. More precisely, we study the stability of regular geometries of cyclic pursuit. We show that in all dimensions the only stable regular n-bug shapes are the regular two dimensional n-gons, n≥7, in which each vertex chases its neighboring vertex in some fixed orientation. We also analize the three bug cyclic pursuit in detail, proving that, except for the equilateral initial position, the triangle formed is asymptotically degenerate with the minimum interior angle tending to zero while the vertex at which the minimum is located rotates among the vertices infinitely often. This revised version was published online in August 2006 with corrections to the Cover Date.     

Stable redesign of predictive control

The paper deals with I/O versions of receding horizon controllers based on the minimization of multistep quadratic costs with the constraint that the terminal state goes to zero. The resulting control law yields stable closed-loop systems under sharp conditions. Simulation results are presented to both verify the theoretical analysis and relate the new control law with GPC     

Stable proper nth-order inverses

A stable proper right (left)nth-order inverse of a given linear time-invariant system of orderncan always be constructed, via a simple algorithm, if a proper right (left) inverse exists and the zeros of the given system are stable. Furthermore, it is shown that all of the poles of this inverse can be arbitrarily assigned except those which equal the zeros of the given system.     

Stable and controllable noise

We introduce a stable noise function with controllable properties. The well-known Perlin noise function is generated by interpolation of a pre-defined random number table. This table must be modified if user-defined constraints are to be satisfied, but modification can destroy the stability of the table. We integrate statistical tools for measuring the stability of a random number table with user constraints within an optimization procedure, so as to create a controlled random number table which nevertheless has a uniform random distribution, no periodicity, and a band-limited property.     

Stable marriages by coroutines

The stable marriage problem is an appealing version of many pairing problems. A solution by coroutines is given, based on the recursive algorithm of McVitie and Wilson (1971). There are few published algorithms where coroutines are really useful but they solve this problem very naturally.     

Stable hybrid adaptive control

This paper examines the asymptotic properties of a hybrid adaptive control scheme. While there are many different meanings of the word ‘hybrid’ control in the literature, we are referring to a scheme where the on-line parameter estimation and underlying control design are carried out in a discrete time using digital electronics and do not require any analog multiplications or divisions. This is of critical importance in practice since analog multipliers/dividers are subject to serious drift and offset errors. For such hybrid schemes we establish stability and an output tracking property without recourse to persistence of exitation. The proof of stability requires a sufficiently fast sampling rate for the discrete time parameter estimation and the underlying control design.     

Stable robust adaptive controller

This paper presents a globally stable adaptive controller for linear discrete systems in the presence of unmodelled dynamics and bounded disturbances. The proposed formulation uses an augmented plant representation that incorporates P, Q and R weighting polynomials for the system output, input and setpoint respectively into the predictive control law. The algorithm also includes a normalized estimation scheme, based on a least-squares estimator, and a parameter adaptation stopping criterion to guarantee stability.     

Stable Length Estimates of Tube-Like Shapes

Motivated by applications in biology, we present an algorithm for estimating the length of tube-like shapes in 3-dimensional Euclidean space. In a first step, we combine the tube formula of Weyl with integral geometric methods to obtain an integral representation of the length, which we approximate using a variant of the Koksma–Hlawka Theorem. In a second step, we use tools from computational topology to decrease the dependence on small perturbations of the shape. We present computational experiments that shed light on the stability and the convergence rate of our algorithm.     

Parameter Tuning of Stable Fuzzy Controllers

In this paper, it is shown that a fuzzy controller with singleton deffuzzification can be considered as the association of a regionwise constant term and of a regionwise non linear term, the latter being bounded by a linear controller. A systematic design method is presented that allows to design stable controlled systems. Stability conditions are given under the form of a set of inequalities which provides an admissible domain for the controller parameters.