Comparison of the Hansen-Sengupta and the Frommer-Lang-Schnurr existence tests

The Krawczyk and the Hansen-Sengupta interval operators are closely related to the interval Newton operator. These interval operators can be used as existence tests to prove existence of solutions for systems of equations. It is well known that the Krawczyk operator existence test is less powerful that the Hansen-Sengupta operator existence test, the latter being less powerful than the interval Newton operator existence test. In 2004, Frommer et al. proposed an existence test based on the Poincaré-Miranda theorem and proved that it is more powerful than the Krawczyk existence test. In this paper, we complete the classification of these four existence tests showing that, in practice, the Hansen-Sengupta existence test is actually more powerful than the existence test proposed by Frommer et al.     

On obtaining quadratic and cubic error convergence using weighted Kronecker-sequences

Kronecker-sequences that use the fractional parts of multiples of irrationals, are well known to be one of the special type of low-discrepancy sequences that can be used for quasi–Monte Carlo integration. One simply takes the average of the function values evaluated in the points of the sequence to obtain an estimate for the integral value. In the past, it was shown that applying certain weights to the average of the function values increases the asymptotic error convergence order dramatically for certain classes of functions. In this work, we start from the above theoretical basis and we derive algorithms for obtaining quadratic and cubic error convergence. The algorithms are “open” in the sense that extra steps in the algorithm can easily be taken in order to improve the result. The amount of work for our algorithms increases linearly with the number of steps.     

Comparison of direct to shooting enclosures for an inverse-monotone boundary value problem with locally steep solution

For an inverse-monotone boundary value problem with the nonlinear ODE –ɛu" + sinh(u)=1,ɛ>0, u(0)=u(1)=0, applications of the following enclosure methods are presented and discussed:

The construction of good extensible Korobov rules

In this paper, we introduce construction algorithms for Korobov rules for numerical integration which work well for a given set of dimensions simultaneously. The existence of such rules was recently shown by Niederreiter. Here we provide a feasible construction algorithm and an upper bound on the worst-case error in certain reproducing kernel Hilbert spaces for such quadrature rules. The proof is based on a sieve principle recently used by the authors to construct extensible lattice rules. We only treat classical lattice rules. The same ideas apply for polynomial lattice rules. The second author is supported by the Austrian Research Foundation (FWF), Project S9609 that is part of the Austrian National Research Network ``Analytic Combinatorics and Probabilistic Number Theory'. The support of the Australian Research Council under its Center of Excellence Program is greatfully acknowledged.     

A Sparse Matrix Arithmetic Based on Cal H-Matrices. Part I: Introduction to {Cal H}-Matrices

A class of matrices (-matrices) is introduced which have the following properties. (i) They are sparse in the sense that only few data are needed for their representation. (ii) The matrix-vector multiplication is of almost linear complexity. (iii) In general, sums and products of these matrices are no longer in the same set, but their truncations to the -matrix format are again of almost linear complexity. (iv) The same statement holds for the inverse of an -matrix. This paper is the first of a series and is devoted to the first introduction of the -matrix concept. Two concret formats are described. The first one is the simplest possible. Nevertheless, it allows the exact inversion of tridiagonal matrices. The second one is able to approximate discrete integral operators. Received: July 30, 1998; revised December 28, 1998     

Guaranteed approach with the farthest of the runaways

Games of the family {Λ _N } _N?2 are formulated and studied with the application of generalized Isaacs’s approach. The game Λ _N is a simplest model of the counteraction of one persecutor P and coalition N of E ^N runaways for the case when the payoff is the distance up to the coalition of E ^N equal to the Euclidean distance between P and the farthest from the runaways; P is in command of the termination moment. Moreover, an approach within the limits of which in games with a smooth terminal payoff are generated strategies prescribing players’ motions in the directions of local gradients of the payoff is described. The approach is used for constructing pursuit strategies in games in which smooth approximations of the maximum of Euclidean distances up to the runaways are in place of payoffs. Pursuit strategies prescribing the motion in the direction of the farthest of the runaways are studied. A numerical simulation of the development of the games Λ₂ and Λ₃ is conducted in using different strategies by the players.     

Nonlinear parity relations: A logic-dynamic approach

Consideration was given to construction of the parity relations for systems described by the nonlinear dynamic models. To solve this problem, a logic-dynamic approach was proposed, and the realizability conditions providing insensitivity to the perturbing actions were given for it.     

Robust filtering under nonrandom disturbances: The invariant ellipsoid approach

We present a simple and universal observer-based approach to solving the problem of robust filtering of unknown-but-bounded exogenous disturbances. The heart of this approach is the method of invariant ellipsoids. Application of this technique allows for a reformulation of the original problem in terms of linear matrix inequalities with reduction to semidefinite programming and one-dimensional optimization, which are easy to solve numerically. Continuous-time and discrete-time cases are studied in equal detail. The efficacy of the approach is demonstrated via the double pendulum example.     

A Dynamic Method for Weighted Linear Least Squares Problems

A new method for solving the weighted linear least squares problems with full rank is proposed. Based on the theory of Liapunov's stability, the method associates a dynamic system with a weighted linear least squares problem, whose solution we are interested in and integrates the former numerically by an A-stable numerical method. The numerical tests suggest that the new method is more than comparative with current conventional techniques based on the normal equations. Received August 4, 2000; revised August 29, 2001 Published online April 25, 2002     

Descent Iterations for Improving Approximate Eigenpairs of Polynomial Eigenvalue Problems with General Complex Matrices

In this paper, we consider descent iterations with line search for improving an approximate eigenvalue and a corresponding approximate eigenvector of polynomial eigenvalue problems with general complex matrices, where an approximate eigenpair was obtained by some method. The polynomial eigenvalue problem is written as a system of complex nonlinear equations with nondifferentiable normalized condition. Convergence theorems for iterations are established. Finally, some numerical examples are presented to demonstrate the effectiveness of the iterative methods. Received April 9, 2001; revised October 2, 2001 Published online February 18, 2002     

The geometry of informational extensions

The paper studies strategy spaces in all possible informational extensions of a game with the natural metric. We show that all these extensions can be considered subspaces of a single space corresponding to a special quasi-informational extension that has a clear meaningful interpretation. We obtain an upper bound on the entropy of informational extensions.     

Invariance for impulsive control systems

The conventional concepts of invariance are extended in this article to include impulsive control systems represented by measure driven differential inclusions. Invariance conditions and some of their main features are derived. The solution concept plays a critical role in the extension of the conditions for conventional problems to the impulsive control context.     

Modernization of one criterion for absolute stability

A new form of a partial frequency criterion of absolute stability for nonlinear automatic control systems is obtained basing on a quadratic transformation of the state vector. As is shown, the obtained criterion is stronger than V.M. Popov’s one. An alternative formulation of the criterion is given in terms excluding the notions of quadratic transformation.     

Gaining insight through case-based explanation

Traditional explanation strategies in machine learning have been dominated by rule and decision tree based approaches. Case-based explanations represent an alternative approach which has inherent advantages in terms of transparency and user acceptability. Case-based explanations are based on a strategy of presenting similar past examples in support of and as justification for recommendations made. The traditional approach to such explanations, of simply supplying the nearest neighbour as an explanation, has been found to have shortcomings. Cases should be selected based on their utility in forming useful explanations. However, the relevance of the explanation case may not be clear to the end user as it is retrieved using domain knowledge which they themselves may not have. In this paper the focus is on a knowledge-light approach to case-based explanations that works by selecting cases based on explanation utility and offering insights into the effects of feature-value differences. In this paper we examine to two such a knowledge-light frameworks for case-based explanation. We look at explanation oriented retrieval (EOR) a strategy which explicitly models explanation utility and also at the knowledge-light explanation framework (KLEF) that uses local logistic regression to support case-based explanation.

Algorithm of multithreshold decoding for Gaussian channels

Major principles of the method of the linear codes multithreshhold decoding as search for the global functional extremum for a great number of variables are considered. It was demonstrated that the multithreshold decoding efficiency is close to the results ensured by the optimum exhaustive search methods. Decoding complexity under software and hardware realizations is evaluated.     

The formula of the solution for some classes of initial boundary value problems for the hyperbolic equation with two independent variables

A new representation is proved of the solutions of initial boundary value problems for the equation of the form u _xx (x, t) + r(x)u _x (x, t) ? q(x)u(x, t) = u _tt (x, t) + μ(x)u _t (x, t) in the section (under boundary conditions of the 1st, 2nd, or 3rd type in any combination). This representation has the form of the Riemann integral dependent on the x and t over the given section.     

Design of robust stable controls for nonlinear objects

We consider a problem of discrete control for a class of nonlinear time-varying objects. Only set estimations for object parameters are available. The aim is to design controls that ensure robust stability of closed-loop systems in a given domain of state space. Since the considered class of objects is large enough not to have a stabilizing control, the proposed design method has to verify at the last step if the obtained conditions of robust stability are satisfied for a nonlinear system “in a given domain.”