Recursive state estimation for hybrid systems

The paper deals with recursive state estimation for hybrid systems. An unobservable state of such systems is changed both in a continuous and a discrete way. Fast and efficient online estimation of hybrid system state is desired in many application areas. The presented paper proposes to look at this problem via Bayesian filtering in the factorized (decomposed) form. General recursive solution is proposed as the probability density function, updated entry-wise. The paper summarizes general factorized filter specialized for (i) normal state-space models; (ii) multinomial state-space models with discrete observations; and (iii) hybrid systems. Illustrative experiments and comparison with one of the counterparts are provided.     

Setting targets for surrogate-based optimization

In the context of surrogate-based optimization (SBO), most designers have still very little guidance on when to stop and how to use infill measures with target requirements (e.g., one-stage approach for goal seeking and optimization); the reason: optimum estimates independent of the surrogate and optimization strategy are seldom available. Hence, optimization cycles are typically stopped when resources run out (e.g., number of objective function evaluations/time) or convergence is perceived, and targets are empirically set which may affect the effectiveness and efficiency of the SBO approach. This work presents an approach for estimating the minimum (target) of the objective function using concepts from extreme order statistics which relies only on the training data (sample) outputs. It is assumed that the sample inputs are randomly distributed so the outputs can be considered a random variable, whose density function is bounded (a, b), with the minimum (a) as its lower bound. Specifically, an estimate of the minimum (a) is obtained by: (i) computing the bounds (using training data and the moment matching method) of a selected set of analytical density functions (catalog), and (ii) identifying the density function in the catalog with the best match to the sample outputs distribution and corresponding minimum estimate (a). The proposed approach makes no assumption about the nature of the objective functions, and can be used with any surrogate, and optimization strategy even with high dimensional problems. The effectiveness of the proposed approach was evaluated using a compact catalog of Generalized Beta density functions and well-known analytical optimization test functions, i.e., F2, Hartmann 6D, and Griewangk 10D and in the optimization of a field scale alkali-surfactant-polymer enhanced oil recovery process. The results revealed that: (a) the density function (from a catalog) with the best match to a function outputs distribution, was the same for both large and reduced samples, (b) the true optimum value was always within a 95% confidence interval of the estimated minimum distribution, and (c) the estimated minimum represents a significant improvement over the present best solution and an excellent approximation of the true optimum value.     

Uniform CLT for empirical process

Empirical processes indexed by classes of functions based on dependent observations are considered. Sufficient conditions in order to satisfy stochastic equicontinuity are given. The derived conditions are in terms of bracketing numbers with respect to a norm arising from a Rosenthal type moment inequality satisfied by the process. The application involves mixing sequences and improves on the result of Andrews and Pollard (Int. Statist. Rev. 62 (1) (1994) 119) for strong mixing, Shao and Yu (Ann. Probab. 24 (4) (1996) 2098) for $\rho$-mixing sequences, and Csörgő and Mielniczuk (Probab. Theory Relat. Fields 104 (1) (1996) 15) for functions of Gaussian sequences.     

Polytope Lyapunov Functions for Stable and for Stabilizable LSS

We present a new approach for constructing polytope Lyapunov functions for continuous-time linear switching systems (LSS). This allows us to decide the stability of LSS and to compute the Lyapunov exponent with a good precision in relatively high dimensions. The same technique is also extended for stabilizability of positive systems by evaluating a polytope concave Lyapunov function (“antinorm”) in the cone. The method is based on a suitable discretization of the underlying continuous system and provides both a lower and an upper bound for the Lyapunov exponent. The absolute error in the Lyapunov exponent computation is estimated from above and proved to be linear in the dwell time. The practical efficiency of the new method is demonstrated in several examples and in the list of numerical experiments with randomly generated matrices of dimensions up to 10 (for general linear systems) and up to 100 (for positive systems). The development of the method is based on several theoretical results proved in the paper: the existence of monotone invariant norms and antinorms for positively irreducible systems, the equivalence of all contractive norms for stable systems and the linear convergence theorem.     

An IMEX scheme for reaction-diffusion equations: application for a PEM fuel cell model

An implicit-explicit (IMEX) method is developed for the numerical solution of reaction-diffusion equations with pure Neumann boundary conditions. The corresponding method of lines scheme with finite differences is analyzed: explicit conditions are given for its convergence in the ‖·‖_∞ norm. The results are applied to a model for determining the overpotential in a proton exchange membrane (PEM) fuel cell.     

A group majorization ordering for Euclidean distance matrices

This paper investigates some properties of Euclidean distance matrices (EDMs) with focus on their ordering structure. The ordering treated here is the group majorization ordering induced by the group of permutation matrices. By using this notion, we establish two monotonicity results for EDMs: (i) The radius of a spherical Euclidean distance matrix (spherical EDM) is increasing with respect to the group majorization ordering. (ii) The larger an EDM is in terms of the group majorization ordering, the more spread out its eigenvalues are. Minimal elements with respect to this ordering are also described.     

Approximate Controllability for Semilinear Systems

The relations between the (strong) reachable sets of the semilinear evolution equation systems x′(t) + A(t)x(t) = f(t, x(t), u(t)) + Hu(t), x′(t) + A(t)x(t) = f(t, x(t), Hu(t)) + Hu(t) on a Banach space, and their corresponding linear systems are studied. Compared with previous results, the systems considered here are more general (f is not independent of the control u), no compactness assumptions on A or f are imposed in some of our main results, and we suppose f is a set-contraction rather than Lipschitz and have less restriction on the contraction coefficient. Other kinds of conditions are involved to guarantee the approximate controllability.     

Gallacetophenone as an analytical reagent for tantalum

Gallacetophenone is proposed as a new specific reagent for the determination of tantalum in the presence of niobium. It gave an yellow colour with tantalum (pH 1.2–3.0) and orange-yellow to orange-red colour with niobium (pH 5.0–8.0). The limits of identification (and dilution) were 10 γ (1:10⁴) and 0.5 γ (1:10⁵) for tantalum and niobium pentoxides respectively. The reagent is specific for tantalum at pH 1.5. The absorption maxima for gallacetophenone were 215 and 280 nm. The maximum absorption for the tantalum and niobium complexes were 395 and 385 nm respectively. The effects of pH, time, ammonium oxalate and reagent on the absorbancy of tantalum complex were studied. The tolerance limits for other ions were determined. Beer’s law was obeyed in the range of 0.5–8.0 mg of tantalum pentoxide per 25 ml. Tantalum pentoxide could be determined colorimetrically in presence of ten times its weight of niobium pentoxide. Gallacetophenone formed 1:1 complex with tantalum (pH 1.5) and 1:2 complex with niobium (pH 6.4). The probable structures have been suggested.     

隐式极限状态方程可靠性研究

针对非线性隐式极限状态方程失效概率的计算,提出了精度更高的改进的均值二次法,并提出了与响应面法相结合的改进均值法,给出了所提方法的实现策略．具体算例表明,改进的均值二次法的精度较改进的均值一次法有明显提高,而改进均值法与响应面法结合后的精度改善更为明显,并且这种结合方法对响应面法的插值点位置不敏感,插值点在较大范围内变化均能得到稳健的高精度结果,从而说明所提方法的有效性．     

Flux-splitting schemes for parabolic problems

Splitting with respect to space variables can be used in solving boundary value problems for second-order parabolic equations. Classical alternating direction methods and locally one-dimensional schemes could be examples of this approach. For problems with rapidly varying coefficients, a convenient tool is the use of fluxes (directional derivatives) as independent variables. The original equation is written as a system in which not only the desired solution but also directional derivatives (fluxes) are unknowns. In this paper, locally one-dimensional additional schemes (splitting schemes) for second-order parabolic equations are examined. By writing the original equation in flux variables, certain two-level locally one-dimensional schemes are derived. The unconditional stability of locally one-dimensional flux schemes of the first and second approximation order with respect to time is proved.     

A posteriori error estimation for hp-version time-stepping methods for parabolic partial differential equations

The aim of this paper is to develop an hp-version a posteriori error analysis for the time discretization of parabolic problems by the continuous Galerkin (cG) and the discontinuous Galerkin (dG) time-stepping methods, respectively. The resulting error estimators are fully explicit with respect to the local time-steps and approximation orders. Their performance within an hp-adaptive refinement procedure is illustrated with a series of numerical experiments.     

An IB-LBM implementation for fluid-solid interactions with an MLS approximation for implicit coupling

This paper presents a combination of the implicit immersed boundary (IB) method and the lattice Boltzmann method (LBM) for a fluid-solid interaction involving an immersed body with moving boundaries and complex geometries. In the implicit IB-LBM method, the flow is computed using LBM. The body force caused by the immersed body is not pre-calculated, but implicitly determined using a corrected velocity field that accurately satisfies the no-slip and no-penetration conditions on the interface between the fluid and solid. The information transfer is not using the traditional Dirac delta function and moving-least-square (MLS). But an improved MLS (IMLS) approximation has been developed to combined the implicit IB-LBM method, where the orthogonal function system with a weight function is used as the basis function and overcome these disadvantages of traditional MLS, in which it may cause numerical instabilities because the matrix inversion must be solved and the final equations system is easily ill-conditioned or singular. Several different flow problems (a two-dimensional flow past a static, rotating and oscillating cylinder, and the vortex-induced vibration of a cylinder as well as the free movement of flapping foil) are simulated to validate the accuracy and efficiency of the present implicit IB-LBM method. The simulation results show good agreement with previous numerical and experimental results. It is found that the present IB-LBM is efficient and reliable in dealing with fluid–solid interaction problems.     

A bound for the optimum relaxation factor for the successive overrelaxation method

Summary An upper bound on the optimum relaxation factor for use with the successive overrelaxation method is derived for a class of linear systems arising from the numerical solution by finite difference methods of a boundary value problem involving the self-adjoint differential equationWork on this paper was supported in part by the U. S. Army Research Office (Durham) through Grant DA-ARO(D)-31-124-G 1050 and by the National Science Foundation through Grant GP-8442 with The University of Texas at Austin.     

Algorithms for the CMRH method for dense linear systems

The CMRH (Changing Minimal Residual method based on the Hessenberg process) method is a Krylov subspace method for solving large linear systems with non-symmetric coefficient matrices. CMRH generates a (non orthogonal) basis of the Krylov subspace through the Hessenberg process, and minimizes a quasi-residual norm. On dense matrices, the CMRH method is less expensive and requires less storage than other Krylov methods. In this work, we describe Matlab codes for the best of these implementations. Fortran codes for sequential and parallel implementations are also presented.     

An improvement for the large sieve for square moduli

We establish a result on the large sieve with square moduli. These bounds improve recent results by S. Baier [S. Baier, On the large sieve with sparse sets of moduli, J. Ramanujan Math. Soc. 21 (2006) 279-295] and L. Zhao [L. Zhao, Large sieve inequality for characters to square moduli, Acta Arith. 112 (3) (2004) 297-308].     

A computer program for fuzzy reasoning

An interactive computer program is described which implements the procedure proposed in “A Formal System for Fuzzy Reasoning” [1]. The problem in question is that of deciding what conclusions may be drawn in the presence of (posibly conflicting) evidence provided, generally with associated partial degrees of belief, by several sources of differing reliability. In using the program, each piece of evidence is entered as a sentence (using the terms NOT, AND, OR, IMPLIES as necessary), with an associated ‘degree of belief’ and ‘weight’; followed by a tentative conclusion. The system returns the degree(s) of belief and weight(s) which may rationally be attached to the conclusion. Copies of the program, written in FORTRAN IV (870 lines) have been lodged with the program libraries CUBE, DECUS, and SHARE, or may be obtained by writing to the author.     

Error estimates for a Galerkin method for a class of model equations for long waves

The Galerkin method, together with a second order time discretization, is applied to the periodic initial value problem for $$\frac{\partial }{{\partial t}}(u - (a(x)u_x )_x ) + (f(x,u))_x = 0$$ . Heref(x, ·) may be highly nonlinear, but a certain cancellation effect is assumed for∫f(x, u) _x u. Optimal order error estimates inL ₂,H ₁, andL _∞ are derived for a general class of piecewise polynomial spaces.     

A multi-objective approach for solving a replacement policy problem for equipment subject to imperfect repairs

This paper proposes a multi-objective approach to model a replacement policy problem applicable to equipment with a predetermined period of use (a planning horizon), which may undergo critical and non-critical failures. Corrective replacements and imperfect repairs are taken to restore the system to operation respectively when critical and non-critical failures occur. Generalized Renewal Process (GRP) is used to model imperfect repairs. The proposed model supports decisions on preventive replacement intervals and the number of spare parts purchased at the beginning of the planning horizon. A Multi-Objective Genetic Algorithm (MOGA) coupled with discrete event simulation (DES) is proposed to provide a set of solutions (Pareto-optimum set) committed to the different objectives of a maintenance manager in the face of a replacement policy problem, that is, maintenance cost, rate of occurrence of failures, unavailability, and investment on spare parts. The proposed MOGA is validated by an application example against the results obtained via the exhaustive approach. Moreover, examples are presented to evaluate the behavior of objective functions on Pareto set (trade-off analysis) and the impact of the repair effectiveness on the decision making.     

Hermite polynomial ansatz for a multidimensional well

The Schrödinger operator in ?^d with analytic potential that has a nondegenerate minimum (a well) at the origin is considered. Under the additional Diophantine condition on the frequencies, the full asymptotic expansions (as Planck's constant h tends to zero) of a set of eigenfunctions (Ansatz with Hermite polynomials) and of eigenvalues with given quantum numbers (n ∈ ?^d, |n|=0,1,2...), located at the bottom of the potential well are constructed in a neighborhood of the origin, which is independent of h. The asymptotics obtained can be prolonged onto a larger domain by using the ray method. A method of approximately describing the zero-sets of eigenfunctions (and of their intersections) is discussed. Some simple examples in the two-dimensional case are considered. Bibliography: 22 titles.