On the Convergence of the Variable Metric Algorithm

The variable metric algorithm is a frequently used method forcalculating the least value of a function of several variables.However it has been proved only that the method is successfulif the objective function is quadratic, although in practiceit treats many types of objective functions successfully. Thispaper extends the theory, for it proves that successful convergenceis obtained provided that the objective function has a strictlypositive definite second derivative matrix for all values ofits variables. Moreover it is shown that the rate of convergenceis super-linear.     

Weighted Uniform Sampling -- a Monte Carlo Technique for Reducing Variance

A new Monte Carlo technique for reducing variance is presentedand analysed. It obtains gains in accuracy comparable to thoserealized by importance sampling (Hammersley & Handscomb,1964), without requiring the use of non-uniform probabilitydistributions. Some numerical examples illustrate the theory.     

A NOVEL METHOD OF FITTING SPATIO‐TEMPORAL MODELS TO DATA,WITH APPLICATIONS TO THE DYNAMICS OF MOUNTAIN PINE BEETLES

Abstract We develop a modular landscape model for the mountain pine beetle (Dendroctonus ponderosae Hopkins) infestation of a stage‐structured forest of lodgepole pine (Pinus contorta Douglas). Beetle attack dynamics are modeled using response functions and beetle movement using dispersal kernels. This modeling technique yields four model candidates. These models allow discrimination between four broad possibilities at the landscape scale: whether or not beetles are subject to an Allee effect at the landscape scale and whether or not host selection is random or directed. We fit the models with aerial damage survey data to the Sawtooth National Recreation Area using estimating functions, which allows for more rapid and complete parameter determination. We then introduce a novel model selection procedure based on facial recognition technology to compliment traditional nonspatial selection metrics. Together with these we are able to select a best model and draw inferences regarding the behavior of the beetle in outbreak conditions.     

Assessment of the Maintenance of the Effectiveness of Safety Schemes by Inspection and Age-replacement of Protective Devices

Expressions are derived for quantities measuring deficienciesoccurring in the protective capabilities of safety schemes comprisedof devices which are subject to random failure. The devicesare assumed to be prone to two modes of failure, one which isreadily detectable in service, and one which is undetectablein service. The expressions take into account the effects ofpreventive maintenance procedures in which devices are inspectedat regularly spaced times to discover and renew devices whichhave failed "detectably", and also devices are replaced aftera specified age in service to mitigate the effects of "undetectable"failures. These expressions are evaluated for devices havingconstant failure rate (i.e. exponential failure distributions)for both modes of failure; and also for a constant "detectable"-failurerate and an increasing "undetectable"-failure rate (i.e. representedby a second order gamma function failure distribution).     

Univariate Interpolation on a Regular Finite Grid by a Multiquadric Plus a Linear Polynomial

Univariate multiquadric interpolation to a twice continuouslydifferentiable function on a regular infinite grid enjoys secondorder convergence and some excellent localization properties,but numerical calculations suggest that, if the grid is finite,then usually the convergence rate deteriorates to first ordernear the grid boundaries, ibis conjecture is proved. It is alsoshown that one can recover superlinear convergence by addinga linear polynomial term to the multiquadric approximation.Making such additions is a standard technique, but we find thatthe usual way of choosing the polynomial fails to provide superlinearconvergence m general. Therefore some new procedures are giventhat pick a suitable polynomial automatically. Thus it is notunusual to reduce the maximum error of the interpolation bya factor of 10³. Further, it is straightforward to include oneof the new procedures in multiquadric interpolation to functionsof several variables when the data points are in general position.     

在BaYF5中Eu2+和Ho3+的荧光和能量传递

在脉冲N₂分子激光激发下,研究了在BaYF,中Eu²⁺和Ho³⁺的发射光谱,激发光谱和荧光寿命,以及Ho³⁺的超灵敏跃迁(⁵I₈→⁵G₆)。在Ho³⁺发射光谱中,⁵S₂→⁵I₈跃迁占主导地位。从Eu²⁺ 关键词：     

The Minimum Sum of Squares Change to Univariate Data that gives Convexity

Let n measurements of a real valued function of one variablebe given. If the function is convex but the data have lost convexitydue to the errors of the measuring process, then the least sumof squares change to the data that provides nonnegative seconddivided differences may be required. An algorithm is proposedfor this highly structured quadratic programming calculation.First a procedure that requires only O(n) computer operationsgenerates a starting point for the main calculation, and thena version of the iterative method of Goldfarb & Idnani (1983)is applied. It is proved that the algorithm converges, the analysisbeing a special case of the theory of Goldfarb & Idnani.The algorithm is efficient because the matrices that occur arebanded due to representing the required fit as a linear combinationof B-splines. Some numerical results illustrate the method.They suggest that the algorithm can be used when n is very large,because the O(n) starting procedure identifies most of the convexityconstraints that are active at the solution.     

On the Estimation of Sparse Jacobian Matrices

We show how to use known constant elements in a Jacobian matrixto reduce the work required to estimate the remaining elementsby finite differences.     

Least Squares Smoothing of Univariate Data to achieve Piecewise Monotonicity

If measurements of a univariate function include uncorrelatederrors, then it is usual for the first-order divided differencesof the measurements to show far more sign changes than the correspondingdifferences of the underlying function. Therefore we addressthe problem of making the least sum of squares change to thedata so that the piecewise linear interpolant to the smootheddata is composed of at most k monotonic sections, k being aprescribed positive integer. The positions of the joins of thesections are integer variables whose optimal values are determinedautomatically, which is a combinatorial problem that can haveO(n^k) local minima, where n is the number of data. Fortunatelywe find that a dynamic programming procedure can calculate theglobal minimum of the sum of squares in at most O(n² + kn logn) computer operations. Further, the complexity reduces to onlyO(n) when k = 1 or k = 2, this result being well known in themonotonic case (k = 1). Algorithms that achieve these efficienciesare described. They perform well in practice, but a discussionof complexity suggests that there is still room for improvementwhen k 3.     

QR Factorization for Linear Least-Squares Problems on a Hypercube Multiprocessor

Least squares problems occur in many branches of science. Typicallythere may be a large number of data points or observations andonly a small to moderate number of variables. On sequentialmachines these problems can be time-consuming and thereforethe use of parallel machines to solve large least-squares problemsmay well yield substantial savings. The solution of least-squaresproblems by a QR factorization using Givens rotations seemsto be particularly suitable for a parallel machine, becausethere is much choice in the order of the Givens rotations andmany Givens rotations can be carried out in parallel. In this paper, an implementation of a QR factorization on theIntel hypercube is described. Each row of the least-squaresmatrix is assigned to a processor and most of the rotationsinvolve rows within one processor in the usual case when eachprocessor receives several rows. However, it is also necessaryto carry out rotations involving rows in different processorsand we call these rotations merges. Two ways of implementingthe merges are described and they are compared on the groundsof load balance and the number of communications required. Onefeature of the implementations is that processors can continueto do Givens rotations on rows within the processor while waitingfor messages that are required for merges. There is also someflexibility in the order of the merges and this can be incorporatedinto the algorithm. For each column, the merges are carriedout according to a tree structure and the choices of trees andtheir roots are discussed. Numerical results are given to showthe usefulness and efficiency of the proposed algorithms.