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Multiple Solutions (multiple + solution)

Distribution by Scientific Domains

Mathematics and Statistics	46%
Engineering	23%
Chemistry	23%

Selected Abstracts

An asymptotic-induced one-dimensional model to describe fires in tunnels II: the stationary problem

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 15 2003
Ingenuin Gasser
Abstract We study stationary solutions of a one-dimensional low-Mach-number model derived in Gasser and Struckmeier (Math. Meth. Appl. Sci. 2002; 25(14): 1231) to describe fire events in long tunnels. The existence of solutions of the corresponding stationary model is shown to be equivalent to the existence of solutions of an algebraic problem. Multiple solutions are shown to be possible. The relation between different formulations of the problem is analysed. Weak and special distributional solutions are considered. Finally, numerical examples of realistic tunnel data with single and multiple solutions of the stationary problem are given. Copyright © 2003 John Wiley & Sons, Ltd. [source]

A connectionist inference model for pattern-directed knowledge representation

EXPERT SYSTEMS, Issue 2 2000
I Mitchell
In this paper we propose a connectionist model for variable binding. The model is topology dependent on the graph it builds based on the predicates available. The irregular connections between perceptron-like assemblies facilitate forward and backward chaining. The model treats the symbolic data as a sequence and represents the training set as a partially connected network using basic set and graph theory to form the internal representation. Inference is achieved by opportunistic reasoning via the bidirectional connections. Consequently, such activity stabilizes to a multigraph. This multigraph is composed of isomorphic subgraphs which all represent solutions to the query made. Such a model has a number of advantages over other methods in that irrelevant connections are avoided by superimposing positionally dependent sub-structures that are identical, variable binding can be encoded and multiple solutions can be extracted simultaneously. The model also has the ability to adapt its existing architecture when presented with new clauses and therefore add new relationships/rules to the model explicitly; this is done by some partial retraining of the network due to the superimposition properties. [source]

Numerical direct kinematic analysis of fully parallel linearly actuated platform type manipulators

JOURNAL OF FIELD ROBOTICS (FORMERLY JOURNAL OF ROBOTIC SYSTEMS), Issue 8 2002
Li-Chun T. Wang
This article presents a new numerical approach for solving the direct kinematics problem of fully parallel, linearly actuated platform manipulators. The solution procedure consists of two stages. The first stage transforms the direct kinematics problem into an equivalent nonlinear programming program, and a robust search algorithm is developed to bring the moving platform from arbitrary initial approximation to a feasible configuration that is near to the true solution. The second stage uses the Newton-Raphson iterative method to converge the solution to the desired accuracy. This approach is numerically stable and computationally efficient. In addition, by randomly perturbing the initial approximations, it can be implemented successively to find multiple solutions to the direct kinematics problem. Two numerical examples are presented to demonstrate the stability and efficiency of this approach. © 2002 Wiley Periodicals, Inc. [source]

Efficient optimization strategies with constraint programming,

AICHE JOURNAL, Issue 2 2010
Prakash R. Kotecha
Abstract In this article, we propose novel strategies for the efficient determination of multiple solutions for a single objective, as well as globally optimal pareto fronts for multiobjective, optimization problems using Constraint Programming (CP). In particular, we propose strategies to determine, (i) all the multiple (globally) optimal solutions of a single objective optimization problem, (ii) K -best feasible solutions of a single objective optimization problem, and (iii) globally optimal pareto fronts (including nonconvex pareto fronts) along with their multiple realizations for multiobjective optimization problems. It is shown here that the proposed strategy for determining K -best feasible solutions can be tuned as per the requirement of the user to determine either K -best distinct or nondistinct solutions. Similarly, the strategy for determining globally optimal pareto fronts can also be modified as per the requirement of the user to determine either only the distinct set of pareto points or determine the pareto points along with all their multiple realizations. All the proposed techniques involve appropriately modifying the search techniques and are shown to be computationally efficient in terms of not requiring successive re-solving of the problem to obtain the required solutions. This work therefore convincingly addresses the issue of efficiently determining globally optimal pareto fronts; in addition, it also guarantees the determination of all the possible realizations associated with each pareto point. The uncovering of such solutions can greatly aid the designer in making informed decisions. The proposed approaches are demonstrated via two case studies, which are nonlinear, combinatorial optimization problems, taken from the area of sensor network design. © 2009 American Institute of Chemical Engineers AIChE J, 2010 [source]

Computation of an extractive distillation column with affine arithmetic

AICHE JOURNAL, Issue 7 2009
Ali Baharev
Abstract The need of reliably solving systems of nonlinear equations often arises in the everyday practice of chemical engineering. In general, standard methods cannot provide theoretical guarantee for convergence to a solution, cannot reliably find multiple solutions, and cannot prove nonexistence of solutions. Interval methods provide tools to overcome these problems, thus achieving reliability. To the authors' best knowledge, computation of distillation columns with interval methods have not yet been considered in the literature. This article presents significant enhancements compared with a previously published interval method of the authors. The proposed branch-and-prune algorithm is guaranteed to converge, and is fairly general at the same time. If no solution exists then this information is provided by the method as a result. Power of the suggested method is demonstrated by solving, with guaranteed convergence, even the MESH equations of a 22 stage extractive distillation column with a ternary mixture. © 2009 American Institute of Chemical Engineers AIChE J, 2009 [source]

Detecting local convexity on the pareto surface

JOURNAL OF MULTI CRITERIA DECISION ANALYSIS, Issue 1 2002
Georges Fadel
Abstract Recent regain in interest in multi-criteria optimization approaches to provide a designer with multiple solutions to select from and support decisions has led to various methodologies to generate such solutions and possibly approximate the Pareto set. This paper introduces the notions of H - and w -convexities, and develops a simple method to identify local convexity of Pareto hyper-surfaces since their shape can dictate the choice of the method to use to obtain Pareto solutions and possibly to build an approximation of that set. The method is based on comparing the results from the weighting method to those of the Tchebycheff method at any point on the Pareto hyper-surface. If, under some conditions, the points obtained from the two methods are identical or not, a local convexity or its lack can be assumed at that location and in its immediate neighbourhood. A numerical example is included. Copyright © 2002 John Wiley & Sons, Ltd. [source]

An asymptotic-induced one-dimensional model to describe fires in tunnels II: the stationary problem

Existence of multiple positive solutions for ,,u,,(u/,x,2)=u2*,1+,f(x)

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 15 2002
Ting Cheng
Abstract In this paper, we consider the semilinear elliptic problem where ,,,N (N,3) is a bounded smooth domain such that 0,,, ,>0 is a real parameter, and f(x) is some given function in L,(,) such that f(x),0, f(x),,0 in ,. Some existence results of multiple solutions have been obtained by implicit function theorem, monotone iteration method and Mountain Pass Lemma. Copyright © 2002 John Wiley & Sons, Ltd. [source]

On a non-uniqueness in fragmentation models

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 7 2002
J. Banasiak
Abstract Equations of fragmentation describing e.g. the polymer degradation, despite their apparent simplicity, have numerous non-typical features. One of them is the existence of multiple solutions. In this paper, applying classical results of Hille and Phillips, we explain their occurrence and prove that nevertheless the uniqueness holds for a large class of physically reasonable solutions. Copyright © 2002 John Wiley & Sons, Ltd. [source]

Positive solutions and multiple solutions for periodic problems driven by scalar p -Laplacian

MATHEMATISCHE NACHRICHTEN, Issue 12 2006
Shouchuan Hu
Abstract In this paper we study a nonlinear second order periodic problem driven by a scalar p -Laplacian and with a nonsmooth, locally Lipschitz potential function. Using a variational approach based on the nonsmooth critical point theory for locally Lipschitz functions, we first prove the existence of nontrivial positive solutions and then establish the existence of a second distinct solution (multiplicity theorem) by strengthening further the hypotheses. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

Dynamic optimization and Skiba sets in economic examples

OPTIMAL CONTROL APPLICATIONS AND METHODS, Issue 5-6 2001
Wolf-Jürgen Beyn
Abstract We discuss two optimization problems from economics. The first is a model of optimal investment and the second is a model of resource management. In both cases the time horizon is infinite and the optimal control variables are continuous. Typically, in these optimal control problems multiple steady states and periodic orbits occur. This leads to multiple solutions of the state,costate system each of which relates to a locally optimal strategy but has its own limiting behaviour (stationary or periodic). Initial states that allow different optimal solutions with the same value of the objective function are called Skiba points. The set of Skiba points is of interest, because it provides thresholds for a global change of optimal strategies. We provide a systematic numerical method for calculating locally optimal solutions and Skiba points via boundary value problems. In parametric or higher dimensional systems Skiba curves (or manifolds) appear and we show how to follow them by a continuation process. We apply our method to the models above where Skiba sets consist of points or curves and where optimal solutions have different stationary or periodic asymptotic behaviour. Copyright © 2001 John Wiley & Sons, Ltd. [source]

Non-Linear Model Predictive Control: A Personal Retrospective,

THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, Issue 4 2007
B. Wayne Bequette
Abstract An overview of non-linear model predictive control (NMPC) is presented, with an extreme bias towards the author's experiences and published results. Challenges include multiple solutions (from non-convex optimization problems), and divergence of the model and plant outputs when the constant additive output disturbance (the approach of dynamic matrix control, DMC) is used. Experiences with the use of fundamental models, multiple linear models (MMPC), and neural networks are reviewed. Ongoing work in unmeasured disturbance estimation, prediction and rejection is also discussed. On présente un aperçu général du contrôle prédictif par modèles non linéaires (NMPC), en mettant l'accent en particulier sur les expériences des auteurs et les résultats publiés. Les défis incluent des solutions multiples (à partir des problèmes d'optimisation non convexes), ainsi que la divergence entre les sorties de modèle et d'installation lorsque la perturbation de sortie additive constante (la méthode du contrôle de matrice dynamique, DMC) est utilisée. Les expériences avec les modèles fondamentaux, les modèles linéaires multiples (MMPC) et les réseaux neuronaux sont examinées. Le travail actuellement mené sur l'estimation, la prédiction et le rejet des perturbations non mesurées est également examiné. [source]

Efficiency measure, modelling and estimation in combined array designs

APPLIED STOCHASTIC MODELS IN BUSINESS AND INDUSTRY, Issue 4 2003
Tak Mak
Abstract In off-line quality control, the settings that minimize the variance of a quality characteristic are unknown and must be determined based on an estimated dual response model of mean and variance. The present paper proposes a direct measure of the efficiency of any given design-estimation procedure for variance minimization. This not only facilitates the comparison of different design-estimation procedures, but may also provide a guideline for choosing a better solution when the estimated dual response model suggests multiple solutions. Motivated by the analysis of an industrial experiment on spray painting, the present paper also applies a class of link functions to model process variances in off-line quality control. For model fitting, a parametric distribution is employed in updating the variance estimates used in an iteratively weighted least squares procedure for mean estimation. In analysing combined array experiments, Engel and Huele (Technometrics, 1996; 39:365) used log-link to model process variances and considered an iteratively weighted least squares leading to the pseudo-likelihood estimates of variances as discussed in Carroll and Ruppert (Transformation and Weighting in Regression, Chapman & Hall: New York). Their method is a special case of the approach considered in this paper. It is seen for the spray paint data that the log-link may not be satisfactory and the class of link functions considered here improves substantially the fit to process variances. This conclusion is reached with a suggested method of comparing ,empirical variances' with the ,theoretical variances' based on the assumed model. Copyright © 2003 John Wiley & Sons, Ltd. [source]