Andrzej Pownuk

From Interval Computations to Constraint-Related Set Computations: Towards Faster Estimation of Statistics and ODEs Under Interval, p-Box, and Fuzzy Uncertainty

In interval computations, at each intermediate stage of the computation, we have intervals of possible values of the corresponding quantities. In our previous papers, we proposed an extension of this technique to set computations, where on each stage, in addition to intervals of possible values of the quantities, we also keep sets of possible values of pairs (triples, etc.). In this paper, we show that in several practical problems, such as estimating statistics (variance, correlation, etc.) and solutions to ordinary differential equations (ODEs) with given accuracy, this new formalism enables us to find estimates in feasible (polynomial) time.

A global optimisation method for computing interval hull solution for parametric linear systems

An algorithm for computing the interval solution for a parametric interval linear system is presented. The basic idea behind the research is to combine an interval global optimisation method with the Direct Method for Checking the Monotonicity (MCM) of the parametric solution. The MCM is used to perform the monotonicity test to speed up the convergence of the global optimisation. Other acceleration techniques such as subdivision direction selection rules, multisection and the midpoint test are involved as well and checked for usefulness. By using the proposed algorithm, several examples of parametric linear systems are solved. The research proves that the proposed monotonicity test is crucial for the convergence of the interval global optimisation used for computing the interval hull for parametric solution sets, whereas other accelerating techniques are not relevant. The presented algorithm can be useful for solving real-life problems concerning structure mechanics.

Solution of the interval equations of dynamics by using adaptive approximation

In this paper a new method for solution of the interval equations of dynamics will be presented. In this approach, in order to estimate upper (y) and lower bound (y) of the solution y = y(t, p) it is necessary to create approximation y^approx = y^approx(t, p<inf>0</inf>, …, p<inf>n</inf>, p) (p<inf>0</inf>, …, p<inf>n</inf> are some point values, and y ≈ y^approx). Then this approximation can be applied for calculation of y^approx and y^approx. It is also possible to get reliable inner estimation y^inner and y^inner of the solution. Using the differences y^approx — y^inner, y^approx — y^inner it is possible to control accuracy of the calculations. This method gives the possibility to calculate combinations of parameters (p^min<inf>approx</inf>(t), p^max<inf>approx</inf> (t), p<inf>min</inf><inf>inner</inf>(t), p^max<inf>inner</inf> (t)), which generate both interval solutions (i.e. y^approx = y(t, p^min<inf>approx</inf> (t)), y^approx = y(t, p^max<inf>approx</inf> (t)), y^inner = y(t, p^min<inf>inner</inf>(t)), y^inner = y(t, p^max<inf>inner</inf> (t))). This is very useful in applications of the interval methods.

Stress analysis of a singly reinforced concrete beam with uncertain structural parameters

This paper presents the effect of interval uncertainty of input parameters in the stress analysis of reinforced concrete flexural members. A singly reinforced concrete beam with interval values of steel reinforcement and corresponding Youngs modulus and subjected to an interval bending moment is taken up for analysis. The internal moment of resistance of the beam is expressed as a function of interval values of stresses in concrete and steel. The internal moment of resistance is equated to the external bending moment due to interval loads acting on the beam. The stresses in concrete and steel are obtained as interval values for various combinations of interval values of structural parameters. The interval stresses and strains in concrete and steel obtained using combinatorial solution, search-based algorithm, sensitivity analysis and interval global optimisation are found to be in excellent agreement.

Design of truss and frame structures with interval and fuzzy parameters

In this paper a method of designing a structure with interval parameters and fuzzy set parameters is presented. This paper also outlines a procedure for designing a structure with random set parameters. All procedures use solutions of the interval equations which are based on the earlier works of both authors. Safety of the structures is determined by using interval limit state equations.

Fuzzy Data Processing Beyond Min t-Norm

Usual algorithms for fuzzy data processing—based on the usual form of Zadeh’s extension principle—implicitly assume that we use the \(\min \) “and”-operation (t-norm). It is known, however, that in many practical situations, other t-norms more adequately describe human reasoning. It is therefore desirable to extend the usual algorithms to situations when we use t-norms different from \(\min \). Such an extension is provided in this chapter.

Hypothetical) Negative Probabilities Can Speed Up Uncertainty Propagation Algorithms

One of the main features of quantum physics is that, as basic objects describing uncertainty, instead of (non-negative) probabilities and probability density functions, we have complex-valued probability amplitudes and wave functions. In particular, in quantum computing, negative amplitudes are actively used. In the current quantum theories, the actual probabilities are always non-negative. However, there have been some speculations about the possibility of actually negative probabilities. In this paper, we show that such hypothetical negative probabilities can lead to a drastic speed up of uncertainty propagation algorithms.

In system identification, interval (and fuzzy) estimates can lead to much better accuracy than the traditional statistical ones: General algorithm and case study

In many real-life situations, we know the upper bound of the measurement errors, and we also know that the measurement error is the joint result of several independent small effects. In such cases, due to the Central Limit Theorem, the corresponding probability distribution is close to Gaussian, so it seems reasonable to apply the standard Gaussian-based statistical techniques to process this data — in particular, when we need to identify a system. Yes, in doing this, we ignore the information about the bounds, but since the probability of exceeding them is small, we do not expect this to make a big difference on the result. Surprisingly, it turns out that in some practical situations, we get a much more accurate estimates if we, vice versa, take into account the bounds — and ignore all the information about the probabilities. In this paper, we explain the corresponding algorithms. and we show, on a practical example, that using this algorithm can indeed lead to a drastic improvement in estimation accuracy.

How to Gauge the Accuracy of Fuzzy Control Recommendations: A Simple Idea

Fuzzy control is based on approximate expert information, so its recommendations are also approximate. However, the traditional fuzzy control algorithms do not tell us how accurate are these recommendations. In contrast, for the probabilistic uncertainty, there is a natural measure of accuracy: namely, the standard deviation. In this paper, we show how to extend this idea from the probabilistic to fuzzy uncertainty and thus, to come up with a reasonable way to gauge the accuracy of fuzzy control recommendations.

Practical Need for Algebraic (Equality-Type) Solutions of Interval Equations and for Extended-Zero Solutions

One of the main problems in interval computations is solving systems of equations under interval uncertainty. Usually, interval computation packages consider united, tolerance, and control solutions. In this paper, we explain the practical need for algebraic (equality-type) solutions, when we look for solutions for which both sides are equal. In situations when such a solution is not possible, we provide a justification for extended-zero solutions, in which we ignore intervals of the type \([-a,a]\).