Parametric Solution Set Research Articles

Parameterized outer estimation of AE-solution sets to parametric interval linear systems

We consider linear algebraic equations, where the elements of the matrix and of the right-hand side vector are linear functions of interval parameters, and their parametric AE-solution sets, which are defined by applying universal and existential quantifiers to the interval parameters. Usually, interval methods find numerical interval vector that contains an AE-solution set.In this work we propose a method that generates an outer estimate of a parametric AE-solution set in form of a linear parametric interval function, called parameterized outer solution (p-solution). Parameterized outer solution is proposed for the parametric united solution set in Kolev (2014) and takes precedence over the classical interval solution enclosure when the latter is part of other problems involving the same parameters. The method we present generalizes the method from Kolev (2016) for parametric AE-solution sets. It is also a parameterized analogue of a method from Popova and Hladík (2013) and produces the same interval enclosure as the method from the last reference.

Eigenvalue Embedding of Undamped Vibroacoustic Systems with No-spillover

In this paper, we consider the eigenvalue embedding problem of the undamped vibroacoustic system with no-spillover (EEP-UVA), which is to update the original system to a new undamped vibroacoustic system, such that some eigen-structures are replaced with newly measured ones, while the remaining eigen-structures are kept unchanged. We provide a set of parametric solutions to the EEP-UVA. The freedoms in the parametric matrices can be further exploited to achieve some other desirable properties. The performance of the proposed algorithms are illustrated by numerical examples.

Multi-objective optimization of suspension parameters for rail vehicles based on a virtual prototype surrogate model

This research is intended to develop a suspension parameter optimization approach based on a virtual prototype surrogate model of rail vehicles considering the coupling effects of suspension parameters. In order to analyze the effects on the dynamic indexes, which were affected by the suspension parameters, a virtual prototype model of a rail vehicle was established. The indexes of lateral ride quality and motion stability were obtained under different combinations of suspension parameters by design of experiment and simulation of virtual prototype. For constructing objective function of multi-objective optimization model for suspension parameters, the suspension parameters that have significant effects on ride quality and motion stability simultaneously were taken as the design variables, and thereafter Kriging models of lateral ride quality index, derailment coefficient, and reduction ratio of wheel load were obtained. On this basis, the multi-objective optimization model of suspension parameters was established, in which the objective function was combined with the three Kriging models. Then, the Pareto optimal solution set and concrete value of suspension parameters were sought using the NSGA-II algorithm. The dynamic simulation results indicated that both ride quality and motion stability of the rail vehicle had been improved after the multi-objective optimization of suspension parameters.

Parametric AE-solution sets to the parametric linear systems with multiple right-hand sides and parametric matrix equation A(p)X = B(p)

In this paper, the parametric matrix equation A(p)X = B(p) whose elements are linear functions of uncertain parameters varying within intervals are considered. In this matrix equation A(p) and B(p) are known m-by-m and m-by-n matrices respectively, and X is the m-by-n unknown matrix. We discuss the so-called AE-solution sets for such systems and give some analytical characterizations for the AE-solution sets and a sufficient condition under which these solution sets are bounded. We then propose a modification of Krawczyk operator for parametric systems which causes reduction of the computational complexity of obtaining an outer estimation for the parametric united solution set, considerably. Then we give a generalization of the Bauer-Skeel and the Hansen-Bliek-Rohn bounds for enclosing the parametric united solution set which also enables us to reduce the computational complexity, significantly. Also some numerical approaches based on Gaussian elimination and Gauss-Seidel methods to find outer estimations for the parametric united solution set are given. Finally, some numerical experiments are given to illustrate the performance of the proposed methods.

Maximal inner boxes in parametric AE-solution sets with linear shape

We consider linear systems of equations A(p)x=b(p), where the parameters p are linearly dependent and come from prescribed boxes, and the sets of solutions (defined in various ways) which have linear boundary. One fundamental problem is to compute a box being inside a parametric solution set. We first consider parametric tolerable solution sets (being convex polyhedrons). For such solution sets we prove that finding a maximal inner box is an NP-hard problem. This justifies our exponential linear programming methods for computing maximal inner boxes. We also propose a polynomial heuristic that yields a large, but not necessarily the maximal, inner box. Next, we discuss how to apply the presented linear programming methods for finding large inner estimations of general parametric AE-solution sets with linear shape. Numerical examples illustrate the properties of the methods and their application.

Optimization of woven fabric parameters for ultraviolet radiation protection and comfort using artificial neural network and genetic algorithm

Optimization of woven fabric parameters for ultraviolet protection factor (UPF) and comfort properties has been attempted using hybrid artificial neural network (ANN)---genetic algorithm (GA) system. ANN was used for developing the prediction models, and GA was employed as an optimization tool. Four feasible combinations of UPF, air permeability and moisture vapor transmission rate (MVTR) were chosen from the Pareto charts of UPF---air permeability and UPF---MVTR. Penalty function method was adopted to form a single objective function by combining the objectives and constraints related to UPF, air permeability and MVTR. The developed ANN---GA hybrid system was executed to obtain the solution set of input parameters for achieving the targeted fabric properties. To validate the developed ANN---GA-based fabric parameter optimization system, four fabric samples were woven using the solution sets of input parameters and functional properties of these engineered fabrics were evaluated. The targeted and achieved values of fabric properties of four validation samples were in reasonably good agreement.

On the unbounded parametric tolerable solution set

We consider a linear algebraic system A(p)x=b(q), where the elements of the matrix and the right-hand side vector are linear functions of uncertain parameters p,q varying within given intervals p, q. The so-called parametric tolerable solution set Σ t o l (A(p),b(q),p,q):={x?? n ?(?p?p)(?q?q)(A(p)x=b(q))} is studied for unboundedness. Basing on a characterization of the parametric tolerable solution set as a convex polyhedron, we present necessary and sufficient conditions (in both general and computable forms) for a nonempty parametric tolerable solution set to be unbounded. Every parametric tolerable solution set is represented as a sum of a linear subspace and a bounded convex polyhedron. The latter implies better estimations (outer and inner) for the unbounded parametric tolerable solution sets. Numerical examples illustrate the discussed methodology and the solution sets.

Improved enclosure for some parametric solution sets with linear shape

Consider linear algebraic systems where the elements of the matrix and the right-hand side vector depend linearly on a number of interval parameters. We prove some sufficient conditions for the united parametric solution set of such a system to have linear boundary. These conditions imply an equivalent representation of the parametric system where each parameter appears once in a diagonal matrix. The latter representation allows us to expand the scope of applicability of the best known so far interval method, developed by A. Neumaier and A. Pownuk, for enclosing the parametric solution set and to generalize the method for systems where the parameter dependencies connect the matrix and the right-hand side vector. Some examples demonstrate that: parametric solution sets with linear boundary appear in various application domains, the generalized method improves the solution enclosure and the proven sufficient conditions can be helpful for solving various other interval problems.

The Unbounded Parametric Tolerable Solution Set

We consider a linear algebraic system A(p)x=b(q), where the elements of the matrix and the right-hand side vector are linear functions of uncertain parameters p, q varying within given intervals [p], [q]. The so-called parametric tolerable solution set is studied for unboundedness. Basing on a characterization of the parametric tolerable solution set as a convex polyhedron, we present necessary and sufficient conditions (in both general and computable forms) for a nonempty parametric tolerable solution set to be unbounded. Every parametric tolerable solution set is represented as a sum of a linear subspace and a bounded convex polyhedron. The latter implies better estimations (outer and inner) for the unbounded parametric tolerable solution set. Numerical examples illustrate the discussed methodology and the solution sets.

Inner estimation of the parametric tolerable solution set

We consider a linear algebraic system A(p)x=b(q), where the elements of the matrix and the right-hand side vector are linear functions of uncertain parameters varying within given intervals. The linear tolerance problem for the so-called parametric tolerable solution set Σtol(A(p),b(q),[p],[q])={x∈Rn∣(∀p∈[p])(∃q∈[q])(A(p)x=b(q))} requires an inner estimation of this solution set, that is an interval vector [y], such that [y]⊆Σtol(A(p),b(q),[p],[q]). In this paper we consider the first methods for finding inner estimation of the parametric tolerable solution set, namely, we propose parametric generalization of the so-called centered approach and of the vertex approach. The results obtained by the two approaches are compared on some numerical examples. The advantages of the parametric approach are demonstrated on problems with independent nonparametric entries and in controllability analysis of linear dynamical systems involving interval uncertainties.

Outer enclosures to the parametric AE solution set

We consider systems of linear equations, where the elements of the matrix and of the right-hand side vector are linear functions of interval parameters. We study parametric AE solution sets, which are defined by universally and existentially quantified parameters, and the former precede the latter. Based on a recently obtained explicit description of such solution sets, we present three approaches for obtaining outer estimations of parametric AE solution sets. The first approach intersects inclusions of parametric united solution sets for all combinations of the end-points of the universally quantified parameters. Polynomially computable outer bounds for parametric AE solution sets are obtained by parametric AE generalization of a single-step Bauer---Skeel method. In the special case of parametric tolerable solution sets, we derive an enclosure based on linear programming approach; this enclosure is optimal under some assumption. The application of these approaches to parametric tolerable and controllable solution sets is discussed. Numerical examples accompanied by graphic representations illustrate the solution sets and properties of the methods.

Explicit Description of AE Solution Sets for Parametric Linear Systems

Consider linear systems whose input data are linear functions of uncertain parameters varying within given intervals. We are interested in an explicit description of the so-called AE parametric solution sets (where all universally quantified parameters precede all existentially quantified ones) by a set of inequalities not involving the parameters. This work presents how to obtain explicit description of AE parametric solution sets by combining a modified Fourier--Motzkin type elimination of existentially quantified parameters with the elimination of the universally quantified parameters. Some necessary (and sufficient) conditions for existence of nonempty AE parametric solution sets are discussed, as well as some properties of the parametric AE solution sets, e.g., shape of the solution set and some inclusion relations. Explicit descriptions of particular classes of AE parametric solution sets (tolerable, controllable, any two-dimensional) are given. Numerical examples illustrate the solution sets and their properties.

Improved zonotopic method to set membership identification for systems with time-varying parameters

This study proposes an improved zonotopic method to set membership identification for linear discrete-time systems with a bounded description of noise and parameter expansion. As in the previous method, the time-varying nature of parameters is taken into account in a non-conservative way by a zonotope. A new family of zonotopes containing the intersection of a zonotope with a strip is constructed in the observation update procedure. The minimal volume zonotope in this family is selected as the outer bound of the feasible solution set of parameters. It can be proven that this minimal volume zonotope is not larger than the one obtained by means of the previous method. Simulation experiments are conducted showing the performance of the improved method as compared to the previous method and the parallelotopic method. Two types of distribution of the parameter expansion are considered. The results expose the weakness of the previous method and illustrate the effectiveness of the improvement.

Explicit description of 2D parametric solution sets

Consider a linear system A(p)⋅x=b(p), where the elements of the matrix and the right-hand side vector depend linearly on a m-tuple of parameters p=(p 1,…,p m ), the exact values of which are unknown but bounded within given intervals. Apart from quantifier elimination, the only known general way of describing the solution set {x∈ℝ n ∣∃p∈[p],A(p)x=b(p)} is a lengthy and non-unique Fourier-Motzkin-type parameter elimination process that leads to a description of the solution set by exponentially many inequalities. In this work we modify the parameter elimination process in a way that has a significant impact on the representation of the inequalities describing the solution set and their number. An explicit minimal description of the solution set to 2D parametric linear systems is derived. It generalizes the Oettli-Prager theorem for non-parametric linear systems. The number of the inequalities describing the solution set grows linearly with the number of the parameters involved simultaneously in both equations of the system. The boundary of any 2D parametric solution set is described by polynomial equations of at most second degree. It is proven that when the general parameter elimination process is applied to two equations of a system in higher dimension, some inequalities become redundant.

Optimization Over the Efficient Set of Multi-objective Convex Optimal Control Problems

We consider multi-objective convex optimal control problems. First we state a relationship between the (weakly or properly) efficient set of the multi-objective problem and the solution of the problem scalarized via a convex combination of objectives through a vector of parameters (or weights). Then we establish that (i) the solution of the scalarized (parametric) problem for any given parameter vector is unique and (weakly or properly) efficient and (ii) for each solution in the (weakly or properly) efficient set, there exists at least one corresponding parameter vector for the scalarized problem yielding the same solution. Therefore the set of all parametric solutions (obtained by solving the scalarized problem) is equal to the efficient set. Next we consider an additional objective over the efficient set. Based on the main result, the new objective can instead be considered over the (parametric) solution set of the scalarized problem. For the purpose of constructing numerical methods, we point to existing solution differentiability results for parametric optimal control problems. We propose numerical methods and give an example application to illustrate our approach.

Parametric Solutions for Slide Impact on Pipelines

Pipelines are frequently subjected to active loading from slide events both on land and in the offshore environment. Whether the pipeline is initially buried or lying close to the surface, and whether it crosses the unstable region or lies in the path of debris originating from further away, the main principles are unchanged. The pipeline will be subjected to active loading over some defined length, related to the width of the slide, and as it deforms will be restrained by transverse and longitudinal resistance in adjacent passive zones. Ultimately the pipeline may come to a stable deformed shape where the continued active loading from the slide is equilibrated by the membrane tension in the pipeline in addition to the passive resistance. This problem has been explored by various writers and these principles are well established. However, to date no attempt has been made to develop a standard set of parametric solutions, which is the purpose of the current paper. Both analytical and numerical solutions of the problem have been developed, initially for slides acting normal to the pipeline but later extended to general conditions with the slide impacting the pipeline at some angle. It is shown that analytical solutions based on certain idealizations maintain their accuracy over a wide parameter range, and the net effect of the slide in terms of stresses induced in the pipe wall and maximum displacement of the pipeline may be captured in appropriate dimensionless groups. Design charts are presented for slide widths of up to 10,000 times the pipeline diameter for a practical range of other parameters such as the ratios of passive normal and frictional resistance to the active loading. Although the solutions are limited by some of the idealizations, they should provide a useful starting point in design, providing a framework for a more detailed numerical analysis for the particular governing conditions.

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.

Visualizing parametric solution sets

We characterize the boundary ∂Σp of the solution set Σp of a parametric linear system A(p)x=b(p) where the elements of the n×n matrix and the right-hand side vector depend on a number of parameters p varying within interval bounds. The characterization of ∂Σp is by means of pieces of parametric hypersurfaces, the latter represented by their coordinate functions depending on corresponding subsets of n-1 parameters. The presented approach has a direct application for efficient visualization of parametric solution sets by utilizing some plotting functions supported by Mathematica and Maple.

유전자 알고리즘을 이용한 구조물 손상 탐색기법에 관한 연구

In this paper, a method for identifying the location and extent of a damage in a structure using residual forces was presented. Element stiffness matrix reduction parameters in a finite element model were used to describe the damaged structure mathematically. The element stiffness matrix reduction parameters were determined by minimizing a global error derived from dynamic residual vectors, which were obtained by introducing a simulated experimental data into the eigenvalue problem. Genetic algorithm was used to get the solution set of element stiffness reduction parameters. The proposed scheme was verified using Euler-Bernoulli beam. The results were presented in the form of tables and charts.

Inner and outer bounds for the solution set of parametric linear systems

Consider linear systems involving affine-linear dependencies on interval parameters. Presented is a free C-XSC software implementing a generalized parametric fixed-point iteration method for verified enclosure of the parametric solution set. Some specific features of the corresponding algorithm concerning sharp enclosure of the contracting matrix and inner approximation of the solution enclosure are discussed.