Improving performance when solving high-order and mixed-order boundary value problems in ODEs

Solving high-order or mixed-order boundary value problems by general purpose software often requires the system to be first converted to a larger equivalent first-order system. The cost of solving such problems is generally O(m ³), where m is the dimension of the equivalent first-order system. In this paper, we show how to reduce this cost by exploiting the special structure the “equivalent” first-order system inherits from the original associated mixed-order system. This technique applies to a broad class of boundary value methods. We illustrate the potential benefits by considering in detail a general purpose Runge–Kutta method and a multiple shooting method. This revised version was published online in June 2006 with corrections to the Cover Date.     

A class of optimal state-delay control problems

We consider a general nonlinear time-delay system with state-delays as control variables. The problem of determining optimal values for the state-delays to minimize overall system cost is a non-standard optimal control problem–called an optimal state-delay control problem–that cannot be solved using existing optimal control techniques. We show that this optimal control problem can be formulated as a nonlinear programming problem in which the cost function is an implicit function of the decision variables. We then develop an efficient numerical method for determining the cost function’s gradient. This method, which involves integrating an auxiliary impulsive system backwards in time, can be combined with any standard gradient-based optimization method to solve the optimal state-delay control problem effectively. We conclude the paper by discussing applications of our approach to parameter identification and delayed feedback control.     

Maximization of Manufacturing Yield of Systems with Arbitrary Distributions of Component Values

This paper presents a general method for maximizing manufacturing yield when the realizations of system components are independent random variables with arbitrary distributions. Design specifications define a feasible region which, in the nonlinear case, is linearized using a first-order approximation. The method attempts to place the given tolerance hypercube of the uncertain parameters such that the area with higher yield lies in the feasible region. The yield is estimated by using the joint cumulative density function over the portion of the tolerance hypercube that is contained in the feasible region. A double-bounded density function is used to approximate various bounded distributions for which optimal designs are demonstrated on a tutorial example. Monte Carlo simulation is used to evaluate the actual yields of optimal designs.     

一类非对称的“跳-停”奇异型随机控制

研究了一类带停时的非对称的奇异型随机控制的折扣问题,不论是从受控状态过程还是从费用函数均推广为较一般的情形,得到"跳-停"策略是其最优控制策略,并给出了"跳-停"策略存在的条件、最优费用函数以及控制方法,所得的结论在实际中有较深的应用背景。     

The use of stochastic analytic center for yield maximization of systems with general distributions of component values

This paper presents a new method for maximizing manufacturing yield when the realizations of system components are dependent random variables with general distributions. The method uses a new concept of stochastic analytic center introduced herein to design the unknown parameters of component values. Design specifications define a feasible region which, in the nonlinear case, is linearized using a first-order approximation. The resulting problem becomes a convex optimization problem. Monte Carlo simulation is used to evaluate the actual yield of the optimal designs of a tutorial example.     

Error Analysis of SAV Finite Element Method to Phase Field Crystal Model

In this paper, we construct and analyze an energy stable scheme by combining the latest developed scalar auxiliary variable (SAV) approach and linear finite element method (FEM) for phase field crystal (PFC) model, and show rigorously that the scheme is first-order in time and second-order in space for the $L^2$ and $H^{-1}$ gradient flow equations. To reduce efficiently computational cost and capture accurately the phase interface, we give a simple adaptive strategy, equipped with a posteriori gradient estimator, i.e., $L^2$ norm of the recovered gradient. Extensive numerical experiments are presented to verify our theoretical results and to demonstrate the effectiveness and accuracy of our proposed method.     

Optimal Noise Rejection in Structural Analysis by Means of Generalized Sampled-Data Hold Functions

In this paper, a technique for optimal noise rejection, based on generalized sampled-data hold functions is applied to the control of civil engineering structures. The technique consists in suitably modulating the sampled outputs of the system under control by periodically varying functions in order to attenuate the effect of the disturbances on the system states to an acceptable level, by minimizing a quadratic cost function. This minimization is performed by feeding back the outputs of the system, which are assumed to be corrupted by measurement noise. Moreover, in the present paper, the robustness properties of the GSHF based optimal regulator is analyzed and guaranteed stability margins, expressed in terms of elementary cost and system matrices, are proposed for such a type of optimal regulators. The effectiveness of the method is demonstrated by various simulation results. The results of the paper can be used to assess the detrimental effect of noise on the closed-loop system and the tradeoff involved in assuring good sampled-data performance and sufficient robustness.     

Logics for stable and unstable mereological relations

In this paper we present logics about stable and unstable versions of several well-known relations from mereology: part-of, overlap and underlap. An intuitive semantics is given for the stable and unstable relations, describing them as dynamic counterparts of the base mereological relations. Stable relations are described as ones that always hold, while unstable relations hold sometimes. A set of first-order sentences is provided to serve as axioms for the stable and unstable relations, and representation theory is developed in similar fashion to Stone’s representation theory for distributive lattices.     

Nonzero-sum stochastic games with unbounded costs: Discounted and average cost cases

We treat non-cooperative stochastic games with countable state space and with finitely many players each having finitely many moves available in a given state. As a function of the current state and move vector, each player incurs a nonnegative cost. Assumptions are given for the expected discounted cost game to have a Nash equilibrium randomized stationary strategy. These conditions hold for bounded costs, thereby generalizing Parthasarathy (1973) and Federgruen (1978). Assumptions are given for the long-run average expected cost game to have a Nash equilibrium randomized stationary strategy, under which each player has constant average cost. A flow control example illustrates the results. This paper complements the treatment of the zero-sum case in Sennott (1993a).     

Notes on the dynamics of pension funding

The authors follow up some previous work on the dynamics of pension funding by three notes. The first of these concerns contribution rates consisting of the normal cost plus a generalized amortization method for unfunded supplemental present value (actuarial accrued liability). The second note examines aggregate cost funding for active members when there exist consistent difference between the assumed and the actual rates of interest and of growth. The third note explores the operation of a variable annuity system in the context of our general model for pension funding dynamics.     

An improved general extra-gradient method with refined step size for nonlinear monotone variational inequalities

Extra-gradient method and its modified versions are direct methods for variational inequalities VI(Ω, F) that only need to use the value of function F in the iterative processes. This property makes the type of extra-gradient methods very practical for some variational inequalities arising from the real-world, in which the function F usually does not have any explicit expression and only its value can be observed and/or evaluated for given variable. Generally, such observation and/or evaluation may be obtained via some costly experiments. Based on this view of point, reducing the times of observing the value of function F in those methods is meaningful in practice. In this paper, a new strategy for computing step size is proposed in general extra-gradient method. With the new step size strategy, the general extra-gradient method needs to cost a relatively minor amount of computation to obtain a new step size, and can achieve the purpose of saving the amount of computing the value of F in solving VI(Ω, F). Further, the convergence analysis of the new algorithm and the properties related to the step size strategy are also discussed in this paper. Numerical experiments are given and show that the amount of computing the value of function F in solving VI(Ω, F) can be saved about 12–25% by the new general extra-gradient method.     

On the economic tender quantity

De Boer, van Dijkhuizen and Telgen (BDT) proposed the economic tender quantity (ETQ) model for determining how many suppliers should be invited to tender for the provision of certain types of product or service. They presented two general conclusions concerning the relationship between the optimal number of suppliers and the uncertainty in the bid price. We show, through counter-examples, that the BDT conclusions do not hold in general for all possible distributions of bid price. However, their first, and principal, conclusion does hold in those cases in which any change in the distribution of the bid price is equivalent to a positive linear transformation of the bid price. In addition, if this condition holds and an optimal policy is followed, we show that the ETQ value is an increasing function, and the expected total cost of tendering is a decreasing function, of the variance of the bid price. The degree to which the BDT conclusions can be said to be counter-intuitive for management decision-making, and the implications of the research presented in this paper, are also discussed.     

Global convergence of SSM for minimizing a quadratic over a sphere

In an earlier paper [Minimizing a quadratic over a sphere, SIAM J. Optim., 12 (2001), 188-208], we presented the sequential subspace method (SSM) for minimizing a quadratic over a sphere. This method generates approximations to a minimizer by carrying out the minimization over a sequence of subspaces that are adjusted after each iterate is computed. We showed in this earlier paper that when the subspace contains a vector obtained by applying one step of Newton's method to the first-order optimality system, SSM is locally, quadratically convergent, even when the original problem is degenerate with multiple solutions and with a singular Jacobian in the optimality system. In this paper, we prove (nonlocal) convergence of SSM to a global minimizer whenever each SSM subspace contains the following three vectors: (i) the current iterate, (ii) the gradient of the cost function evaluated at the current iterate, and (iii) an eigenvector associated with the smallest eigenvalue of the cost function Hessian. For nondegenerate problems, the convergence rate is at least linear when vectors (i)-(iii) are included in the SSM subspace.

     

带停时的奇异型随机控制问题在金融投资模型中的应用

以随机分析的知识和最优控制理论为基础,讨论了一类带停时的奇异型随机控制的折扣费用问题在金融投资模型中的应用,将该带停时的奇异型随机控制模型的受控状态过程和费用函数结构都推广到了最一般的形式,使该模型的应用范围更加广泛.通过讨论一组相应的变分不等式的解,分别对退化和非退化两种情况给出了此随机控制问题的最优策略,相应得出了投资模型中的最佳决策,并且证明了变分不等式的解即为最优费用函数.与以往不同的是,所得的相关结论应用到了金融投资模型中,从而解决了一类金融投资问题.     

Stepsize selection for tolerance proportionality in explicit Runge–Kutta codes

The potential for adaptive explicit Runge–Kutta (ERK) codes to produce global errors that decrease linearly as a function of the error tolerance is studied. It is shown that this desirable property may not hold, in general, if the leading term of the locally computed error estimate passes through zero. However, it is also shown that certain methods are insensitive to a vanishing leading term. Moreover, a new stepchanging policy is introduced that, at negligible extra cost, ensures a robust global error behaviour. The results are supported by theoretical and numerical analysis on widely used formulas and test problems. Overall, the modified stepchanging strategy allows a strong guarantee to be attached to the complete numerical process. This revised version was published online in June 2006 with corrections to the Cover Date.     

A stochastic program based lower bound for assemble-to-order inventory systems

In this paper we introduce a multi-stage stochastic program that provides a lower bound on the long-run average inventory cost of a general class of assemble-to-order (ATO) inventory systems. The stochastic program also motivates a replenishment policy for these systems. Our lower bound generalizes a previous result of Do?ru et al. (2010) [3] for systems with identical component replenishment lead times to those with general deterministic lead times. We provide a set of sufficient conditions under which our replenishment policy, coupled with an allocation policy, attains the lower bound (and is hence optimal). We show that these sufficient conditions hold for two examples, a single product system and a special case of the generalized W model.     

带有一个冷贮备部件的两部件串联系统机会维修策略

针对带有一个冷贮备部件的两部件串联系统,本文首先提出一种预防维修与机会维修相结合的维修策略,运用更新报酬定理求得长期运行情况下的单位时间期望维修成本函数的表达式,然后研究最优的机会维修阀值,运用微分学理论求解最优解,最后用实例验证理论的正确性,从实际例子说明本文提出的维修策略可明显节约维修成本,为相应的带有冷贮备的多部件串联系统的维修策略分析提供参考,对企业设备的维修有实际指导意义。     

Evolution equations of higher order in banach spaces

The present paper deals with a minimal extension of the classical semigroup theory for equations of any order in Banach spaces with closed densely defined linear operators as coefficients. We do not ask anymore from our operators than in the case of first-order equations, i.e., Semigroups. We present here a generalization of Myadera-Phillips--Feller theorem, of Hille theorem and some other results. The method is quite general. We focus our attention on a particular operator solution (main propagator or abstract initial value Green function) and we assume some properties about it. From this we can obtain all needed information about complementary operator solutions, among others.     

Regenerative structure of Markov chains simulated via common random numbers

A standard strategy in simulation, for comparing two stochastic systems, is to use a common sequence of random numbers to drive both systems. Since regenerative output analysis of the steady-state of a system requires that the process be regenerative, it is of interest to derive conditions under which the method of common random numbers yields a regenerative process. It is shown here that if the stochastic systems are positive recurrent Markov chains with countable state space, then the coupled system is necessarily regenerative; in fact, we allow couplings more general than those induced by common random numbers. An example is given which shows that the regenerative property can fail to hold in general state space, even if the individual systems are regenerative.     

Replacement Strategies

Components of a system function for a time and then fail. A central problem is to decide whether a strategy of scheduled replacement is preferable to the running of all components until failure. This is considered by comparing some simple strategies for component replacement. Considerations from probability theory and renewal theory are used to obtain expressions for the average cost of replacements per unit time for each strategy. After obtaining some general conditions for one strategy to be preferable to another, a detailed comparison is made when the interfailure times are independent samples from a gamma distribution.