Robust viability of hybrid systems

Robust viability of hybrid systems is examined based on the controllability operator subject to transition dynamics uncertainty. Two modifications to the controllability operator are introduced, these being the uncertain controllability operator and the uncertainty operator. It is shown how existing fixed-point approximation algorithms can be generalized and applied to compute robustly viable sets for hybrid systems. The three-tank control problem is considered. Robust viable sets are computed for this system subject to transition dynamics uncertainties.     

Maximal B-Regular Integro-Differential Equation

By using Fourier multiplier theorems, the maximal B-regularity of ordinary integro-differential operator equations is investigated. It is shown that the corresponding differential operator is positive and satisfies coercive estimate. Moreover, these results are used to establish maximal regularity for infinite systems of integro-differential equations.     

Asymptotic solutions of Fredholm integro-differential equations with rapidly varying kernels and irreversible limit operator

We consider a system of singularly perturbed integro-differential Fredholm equations with rapidly varying kernel in the case of irreversible operator of differential part. We develop an algorithm for constructing regularized asymptotic solutions. It is shown that in the presence of rapidly decreasing multiplier in the kernel the original problem is not on the spectrum (i.e, it is solvable for any right-hand side). We study the limit transition (with small parameter tending to zero), and solve the problem of initialization, i.e., the problem of extracting of the source data for which an exact solution to the system tends to the limit at all duration (including a zone of boundary layer).     

Algorithms for vector field generation in mass consistent models

Diagnostic models in meteorology are based on the fulfillment of some time independent physical constraints as, for instance, mass conservation. A successful method to generate an adjusted wind field, based on mass conservation equation, was proposed by Sasaki and leads to the solution of an elliptic problem for the multiplier. Here we study the problem of generating an adjusted wind field from given horizontal initial velocity data, by two ways. The first one is based on orthogonal projection in Hilbert spaces and leads to the same elliptic problem but with natural boundary conditions for the multiplier. We derive from this approach the so called E–algorithm. An innovative alternative proposal is obtained from a second approach where we consider the saddle–point formulation of the problem—avoiding boundary conditions for the multiplier— and solving this problem by iterative conjugate gradient methods. This leads to an algorithm that we call the CG–algorithm, which is inspired from Glowinsk's approach to solve Stokes–like problems in computational fluid dynamics. Finally, the introduction of new boundary conditions for the multiplier in the elliptic problem generates better adjusted fields than those obtained with the original boundary conditions. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Representation of Operators in the Time-Frequency Domain and Generalized Gabor Multipliers

Starting from a general operator representation in the time-frequency domain, this paper addresses the problem of approximating linear operators by operators that are diagonal or band-diagonal with respect to Gabor frames. A characterization of operators that can be realized as Gabor multipliers is given and necessary conditions for the existence of (Hilbert-Schmidt) optimal Gabor multiplier approximations are discussed and an efficient method for the calculation of an operator’s best approximation by a Gabor multiplier is derived. The spreading function of Gabor multipliers yields new error estimates for these approximations. Generalizations (multiple Gabor multipliers) are introduced for better approximation of overspread operators. The Riesz property of the projection operators involved in generalized Gabor multipliers is characterized, and a method for obtaining an operator’s best approximation by a multiple Gabor multiplier is suggested. Finally, it is shown that in certain situations, generalized Gabor multipliers reduce to a finite sum of regular Gabor multipliers with adapted windows.     

EXTENSION OF MULTIPLIERS AND INJECTIVE HILBERT MODULES

The author studies the problem whether a multiplier of a hereditary C^*-subalgebra B of a C^*-algebra A can be extended to a multiplier of A. One related problem is the Hahn-Banach extension theorem for Hilbert modules over C^*-algebras. It is shown that every self-dual Hilbert module over W^*-algebra or an injective C^*-algebra is injective.     

Estimation of vertical wind velocity using doppler radar data with vorticity constraints

The vorticity equation and the mass continuity equation are used as constraints to aid in calculation of the vertical component of a wind field from horizontal wind components. Typically the horizontal wind components result from estimations obtained from radar data. Using a Hilbert space minimization formulation, we characterize the estimated vertical velocity as a solution of an elliptic boundary value problem whose coefficients are functions of the horizontal wind components. Differentiable dependence of the vertical component of the wind field with respect to the horizontal fields is established. Finite dimensional approximating problems are obtained. A numerical study is presented using a sample problem based on the Beltrami flow to compare accuracy of the estimated vertical wind component with the exact. Also, sensitivity to perturbations in horizontal wind components are observed.     

Approximate source conditions in Tikhonov–Phillips regularization and consequences for inverse problems with multiplication operators

The object of this paper is threefold. First, we investigate in a Hilbert space setting the utility of approximate source conditions in the method of Tikhonov–Phillips regularization for linear ill‐posed operator equations. We introduce distance functions measuring the violation of canonical source conditions and derive convergence rates for regularized solutions based on those functions. Moreover, such distance functions are verified for simple multiplication operators in L²(0, 1). The second aim of this paper is to emphasize that multiplication operators play some interesting role in inverse problem theory. In this context, we give examples of non‐linear inverse problems in natural sciences and stochastic finance that can be written as non‐linear operator equations in L²(0, 1), for which the forward operator is a composition of a linear integration operator and a non‐linear superposition operator. The Fréchet derivative of such a forward operator is a composition of a compact integration and a non‐compact multiplication operator. If the multiplier function defining the multiplication operator has zeros, then for the linearization an additional ill‐posedness factor arises. By considering the structure of canonical source conditions for the linearized problem it could be expected that different decay rates of multiplier functions near a zero, for example the decay as a power or as an exponential function, would lead to completely different ill‐posedness situations. As third we apply the results on approximate source conditions to such composite linear problems in L²(0, 1) and indicate that only integrals of multiplier functions and not the specific character of the decay of multiplier functions in a neighbourhood of a zero determine the convergence behaviour of regularized solutions. Copyright © 2005 John Wiley & Sons, Ltd.     

Domaines fictifs, éléments finis H(div) et condition de Neumann: le problème de la condition inf-sup

Approximation of the Neumann problem for a second order elliptic operator by a fictitious domain method with a Lagrange multiplier on the boundary is considered. The problem is written in its vectorial dual formulation and H(div) mixed finite elements for the vector unknown and H^1/2 conforming elements for the multiplier are used. The uniform inf-sup condition is demonstrated under a compatibility condition between surface and volume meshes.     

On robust maximum flow with polyhedral uncertainty sets

We address the problem of determining a robust maximum flow value in a network with uncertain link capacities taken in a polyhedral uncertainty set. Besides a few polynomial cases, we focus on the case where the uncertainty set is taken to be the solution set of an associated (continuous) knapsack problem. This class of problems is shown to be polynomially solvable for planar graphs, but NP-hard for graphs without special structure. The latter result provides evidence of the fact that the problem investigated here has a structure fundamentally different from the robust network flow models proposed in various other published works.     

Reduction of a Minimization Problem of a Separable Convex Function Under Linear Constraints to a Fixed Point Problem

The paper is devoted to studying a constrained nonlinear optimization problem of a special kind. The objective functional of the problem is a separable convex function whose minimum is sought for on a set of linear constraints in the form of equalities. It is proved that, for this type of optimization problems, the explicit form can be obtained of a projection operator based on a generalized projection matrix. The projection operator allows us to represent the initial problem as a fixed point problem. The explicit form of the fixed point problem makes it possible to run a process of simple iteration. We prove the linear convergence of the obtained iterative method and, under rather natural additional conditions, its quadratic convergence. It is shown that an important application of the developed method is the flow assignment in a network of an arbitrary topology with one pair of source and sink.     

Sales and operations planning in systems with order configuration uncertainty

This paper addresses the problem of aligning demand and supply in configure-to-order systems. Using stochastic programming methods, this study demonstrates the value of accounting for the uncertainty associated with how orders are configured. We also demonstrate the value of component supply flexibility in the presence of order configuration uncertainty. We present two stochastic models: an explosion problem model and an implosion problem model. These models are positioned sequentially within a popular business process called sales and operations planning. Both models are formulated as two-stage stochastic programs with recourse and are solved using the sample average approximation method. Computational analyses were performed using data obtained from IBM System and Technology Group. The problem sets used in our analysis are created from actual industry data and our results show that significant improvements in revenue and serviceability can be achieved by appropriately accounting for the uncertainty associated with order configurations.     

On Correctness Conditions for Algebra of Recognition Algorithms with μ-Operators over Pattern Problems with Binary Data

The concept of an Ω-weakly regular problem is introduced. On the basis of the Zhuravlev operator approach combined with the neural network paradigm, it is shown that, for each such problem, a correct algorithm and a six-level spatial neural network reproducing the computations executed by this algorithm can be constructed. Moreover, the set of Ω-weakly regular problems includes the set of Ω-regular problems. It turns out that a three-level spatial network (μ-block) is a forward propagation network whose inner loop under estimation of the class membership for each test object consists of a single iteration. As a result, the amount of computations required for the six-level network is reduced noticeably.     

Inversion in finite fields of characteristic 2 using logarithmic depth

It is shown that inversion in a polynomial basis of a finite field of characteristic 2 can be realized by a network of logarithmic depth with the constant multiplier 6.44 for the principal term and also by a network of almost quadratic complexity and of logarithmic depth.     

Solving the Segregated Storage Problem with Benders' Partitioning

We investigate the application of Benders' partitioning to a mixed integer programming formulation of the segregated storage problem. The dual subproblem reduces to an efficiently-solvable network flow problem. This approach is compared empirically to Neebe's multiplier adjustment procedure. Benders' procedure is shown to be computationally effective for an important class of practical applications having high demand-to-capacity ratios and fewer products than compartments.     

Impact of compressor failures on gas transmission network capability

National Grid, the gas operator in the United Kingdom, has experienced challenges in evaluating the capability of its gas transmission network to maintain function in the event of risks particularly to withstand the impact of compressor failures. We propose a mathematical programming model to support the operator in dealing with the problem. Several solution techniques are developed to solve the various versions of the problem efficiently. In the case of little data on compressor failure, an uncertainty theory is applied to solve this problem if the compressor failures are independent; while a robust optimisation technique is developed to solve it when they are not. Otherwise, when there are data on compressor failure, Monte Carlo simulation is applied to find the expected capability of the gas transmission network. Computational experiments, carried out on a case study at National Grid, demonstrate the efficiency of the proposed model and solution techniques. A further analysis is performed to determine the impact of compressor failures and suggest efficient maintenance policies for National Grid.     

Order sequencing on a unidirectional cyclical picking line

A real life order-picking configuration that requires multiple pickers to cyclically move around fixed locations in a single direction is considered. This configuration is not the same, but shows similarities to, unidirectional carousel systems described in literature. The problem of minimising the pickers’ travel distance to pick all orders on this system is a variant of the clustered travelling salesman problem. An integer programming (IP) formulation of this problem cannot be solved in a realistic time frame for real life instances of the problem. A relaxation of this IP formulation is proposed that can be used to determine a lower bound on an optimal solution. It is shown that the solution obtained from this relaxation can always be transformed to a feasible solution for the IP formulation that is, at most, within one pick cycle of the lower bound. The computational results and performance of the proposed methods as well as adapted order sequencing approaches for bidirectional carousel systems from literature are compared to one another by means of real life historical data instances obtained from a retail distribution centre.     

Numerical solution of a class of integral equations arising in two-dimensional aerodynamics

We consider the numerical solution of a class of integral equations arising in the determination of the compressible flow about a thin airfoil in a ventilated wind tunnel. The integral equations are of the first kind with kernels having a Cauchy singularity. Using appropriately chosen Hilbert spaces, it is shown that the kernel gives rise to a mapping which is the sum of a unitary operator and a compact operator. This enables us to study the problem in terms of an equivalent integral equation of the second kind. Using Galerkin's method, we are able to derive a convergent numerical algorithm for its solution. It is shown that this algorithm is numerically equivalent to Bland's collocation method, which is then used as our method of computation. Extensive numerical calculations are presented establishing the validity of the theory.This paper was prepared with support of the National Aeronautics and Space Administration, Grant No. NSG-2140.The authors would like to acknowledge the help of Messrs. Tuli Haromy, Charles Doughty, Karl Kuopus, and Steven Sedlacek in the preparation of this paper.     

Some Unitary Operators on Hoop-algebras

In this paper, we introduce the notions of multiplier, $\odot$-closure operator and modal operator on hoop-algebras. After that, we investigate some algebraic properties of multiplier, $\odot$-closure operator and modal operator on hoop-algebras. Then we study relationships between multiplier and closure operator on hoop-algebras showing that, any monotone modal operator on hoop-algebra serves as a closure operator. Finally we represent image of hoop-algebra under the closure operator can be a hoop-algebra.     

Theory of a class of Wiener-Hopf equations of the first kind: Application to the Sommerfeld problem

A new theory of a class of Wiener-Hopf equations of the first kind in a space of distributions is presented. It is shown that the corresponding Wiener-Hopf operator is a Fredholm operator. This result is obtained by an appropriate modification of the standard Wiener-Hopf technique used for equations of the second kind. The nullity and defect numbers of the operator are determined from a factorization of the symbol. An application to the Sommerfeld problem is briefly considered.