Error bounds for Loop subdivision surfaces

In this paper, we obtain the error bounds on the distance between a Loop subdivision surface and its control mesh. Both local and global bounds are derived by means of computing and analysing the control meshes with two rounds of refinement directly. The bounds can be expressed with the maximum edge length of all triangles in the initial control mesh. Our results can be used as posterior estimates and also can be used to predict the subdivision depth for any given tolerance.     

Error bounds for polynomial tensor product interpolation

We provide estimates for the maximum error of polynomial tensor product interpolation on regular grids in ${\mathbb{R}^d}$ . The set of partial derivatives required to form these bounds depends on the clustering of interpolation nodes. Also bounds on the partial derivatives of the error are derived.     

Error bounds for the method of alternating projections

Given a collection of closed subspaces of a Hilbert space, the method of alternating projections produces a sequence which converges to the orthogonal projection onto the intersection of the subspaces. A large class of problems in medical and geophysical image reconstruction can be solved using this method. A sharp error bound will enable the userto estimate accurately the number of iterations necessary to achieve a desired relative error. We obtain the sharpest possible upper bound for the case of two subspaces, and the sharpest known upper bound for more than two subspaces. This work was supported by the Office of Naval Research under Contract N00014-85-K-0255.     

Error bounds for direct and composite cubature formulae-II

In this part II, narrow cubature error bounds given by Lether are used and made suitable for comparing direct and composite integration techniques for two-dimensional Gauss-Legendre formulae. The results we obtain support the conclusion that in many cases composite formulae turn out to be preferable in comparison with direct ones.

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Sommario In questa parte II vengono utilizzati limiti di errore di cubatura particolarmente stringenti dati da Lether, rendendoli adatti al confronto fra tecniche dirette e composte per formule di integrazione bidimensionale di Gauss-Legendre. I risultati confermano che, contrariamente ad una opinione abbastanza diffusa, le formule composte possono, in molti casi, risultare preferibili alle dirette.

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This work has been done within the GNAFA-CNR research activity.     

Error bounds for initial value problems by optimization

The computation of the error bounds for approximate solutions of initial value problems for ordinary differential equations has a long and successful history. This paper presents a new scheme to compute such bounds with uncertain initial conditions using preconditioned defect estimates and optimization techniques. These bounds are based on the newly developed concept of conditional differential inequalities. The scheme is implemented in MATLAB and AMPL. The resulting enclosures are compared with the packages VALENCIA-IVP, VNODE-LP and VSPODE for bounding solutions of ODEs. The current prototype uses heuristics to solve the global optimization subproblems. Hence the bounds obtained in the numerical experiments are not fully rigorous. The latter can be achieved by using rigorous global optimization and rounding error control, but the effect on the bounds is likely to be marginal only.     

Error bounds of optimization algorithms for semi-Markov decision processes

Cao's work shows that, by defining an α-dependent equivalent infinitesimal generator A _α, a semi-Markov decision process (SMDP) with both average- and discounted-cost criteria can be treated as an α-equivalent Markov decision process (MDP), and the performance potential theory can also be developed for SMDPs. In this work, we focus on establishing error bounds for potential and A _α-based iterative optimization methods. First, we introduce an α-uniformized Markov chain (UMC) for a SMDP via A _α and a uniformized parameter, and show their relations. Especially, we obtain that their performance potentials, as solutions of corresponding Poisson equations, are proportional, so that the studies of a SMDP and the α-UMC based on potentials are unified. Using these relations, we derive the error bounds for a potential-based policy-iteration algorithm and a value-iteration algorithm, respectively, when there exist various calculation errors. The obtained results can be applied directly to the special models, i.e., continuous-time MDPs and Markov chains, and can be extended to some simulation-based optimization methods such as reinforcement learning and neuro-dynamic programming, where estimation errors or approximation errors are common cases. Finally, we give an application example on the look-ahead control of a conveyor-serviced production station (CSPS), and show the corresponding error bounds.     

Error bounds for complementarity problems with tridiagonal nonlinear functions

In this paper we consider the complementarity problem NCP(f) with f(x) = Mx + φ(x), where M ∈ R ^n×n is a real matrix and φ is a so-called tridiagonal (nonlinear) mapping. This problem occurs, for example, if certain classes of free boundary problems are discretized. We compute error bounds for approximations \({\hat x}\) to a solution x* of the discretized problems. The error bounds are improved by an iterative method and can be made arbitrarily small. The ideas are illustrated by numerical experiments.     

多目标联合检测与估计的误差界

在随机有限集框架下提出了当杂波和漏检存在时基于最优子模式分配距离的多目标联合检测与估计(JDE) 误差界. 此处的JDE 是指同时估计目标个数和存活目标状态. 算例1 展示了该误差界随传感器检测概率和杂波密度的变化趋势; 算例2 利用多假设跟踪, 概率假设密度(PHD) 和势PHD 滤波器对该误差界的有效性进行了验证.

     

Error bounds for direct and composite cubature formulae—I

In this paper direct and composite two dimensional integration formulae are compared, and conditions are given under which certain error bounds turn out to be better for either one or the other approach.

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Sommario In questo articolo vengono confrontate formule dirette e formule composte di integrazione numerica in due dimensioni, determinando condizioni sotto cui risultano migliori certi limiti di errore per l'uno o per l'altro approccio.

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This work has been done within the GNAFA-CNR research activity.     

Error bounds of multi-graph regularized semi-supervised classification

In this paper, we investigate the generalization performance of the multi-graph regularized semi-supervised classification algorithm associated with the hinge loss. We provide estimates for the excess misclassification error of multi-graph regularized classifiers and show the relations between the generalization performance and the structural invariants of data graphs. Experiments performed on real database demonstrate the effectiveness of our theoretical analysis.     

Error bounds for joint detection and estimation of multiple unresolved target-groups

According to random finite set (RFS) and information inequality, this paper derives an error bound for joint detection and estimation (JDE) of multiple unresolved target-groups in the presence of clutters and missed detections. The JDE here refers to determining the number of unresolved target-groups and estimating their states. In order to obtain the results of this paper, the states of the unresolved target-groups are modeled as a multi-Bernoulli RFS first. The point cluster model proposed by Mahler is used to describe the observation likelihood of each group. Then, the error metric between the true and estimated state sets of the groups is defined by the optimal sub-pattern assignment distance rather than the usual Euclidean distance. The maximum a posteriori detection and unbiased estimation criteria are used in deriving the bound. Finally, we discuss some special cases of the bound when the number of unresolved target-groups is known a priori or is at most one. Example 1 shows the variation of the bound with respect to the probability of detection and clutter density. Example 2 verifies the effectiveness of the bound by indicating the performance limitations of the cardinalized probability hypothesis density and cardinality balanced multi-target multi-Bernoulli filters for unresolved target-groups. Example 3 compares the bound of this paper with the (single-sensor) bound of [4] for the case of JDE of a single unresolved target-group. At present, this paper only addresses the static JDE problem of multiple unresolved target-groups. Our future work will study the recursive extension of the bound proposed in this paper to the filtering problems by considering the group state evolutions.     

Error bounds for Newton refinement of solutions to algebraic riccati equations

Recent error bounds derived from the Schur method of solving algebraic Riccati equations (ARE) complement residual error bounds associated with Newton refinement of approximate solutions. These approaches to the problem of error estimation not only work well together but also represent the first computable error bounds for the solution of Riccati equations. In this paper the closed-loop Lyapunov operator is seen to be central to the question of whether Newton refinement will improve an approximate solution (region of convergence), as well as providing a means of bounding the actual error in terms of the residual error. In turn, both of these issues are related to the condition of the ARE and the damping of the associated closed-loop dynamical system. Numerical results are given for seven problems taken from the literature. This research was supported by the National Science Foundation (and AFOSR) under Grant No. ECS87-18897 and the National Science Foundation under Grant No. DMS88-00817.     

Error bounds in the averaging of hybrid systems

The authors analyze the error introduced by the averaging of hybrid systems. These systems involve linear systems which can take a number of different realizations based on the state of an underlying finite state process. The averaging technique (based on a formula from Lie algebras known as the Backer-Campbell-Hausdorff (BCH) formula) provides a single system matrix as an approximation to the hybrid system. The two errors discussed are: (1) the error induced by the truncation of the BCH series expansion and (2) the error between the actual hybrid system and its average. A simple sufficient stability test is proposed to check the asymptotic behavior of this error. In addition, conditions are derived that allow the use of state feedback instead of averaging to arrive at a time-invariant system matrix     

Error bounds for finite step approximations for solving infinite horizon controlled Markov set-chains

This note considers finite-step approximations for solving an infinite-horizon controlled Markov set-chain problem with finite state and action spaces. We develop a value-iteration type algorithm based on the optimality equation developed by Kurano et al. and analyze an error bound relative to the optimal value that satisfies the optimality equation from the successive approximation. We further analyze an error bound of the approximate control policy defined from a finite-step approximate value by applying the value-iteration type algorithm.     

Error probability bounds for nuclear detection: Improving accuracy through controlled mobility

A collection of static and mobile radiation sensors is tasked with deciding, within a fixed time interval, whether a moving target carries radioactive material. Formally, this is a problem of detecting weak time-inhomogeneous Poisson signals (target radiation) concealed in another Poisson signal (naturally occurring background radiation). Each sensor locally processes its observations to form a likelihood ratio, which is transmitted once—at the end of the decision interval—to a fusion center. The latter combines the transmitted information to optimally (in the Neyman–Pearson sense) decide whether the measurements contain a radiation signal, or just noise. We provide a set of analytically derived upper bounds for the probabilities of false alarm and missed detection, which are used to design threshold tests without the need for computationally intensive Monte Carlo simulations. These analytical bounds couple the physical quantities of interest to facilitate planning the motion of the mobile sensors for minimizing the probability of missed detection. The network reconfigures itself in response to the target motion, to allow more accurate collective decisions within the given time interval. The approach is illustrated in numerical simulations, and its effectiveness demonstrated in experiments that emulate the statistics of nuclear emissions using a pulsed laser.     

Ensembles of jittered association rule classifiers

The ensembling of classifiers tends to improve predictive accuracy. To obtain an ensemble with N classifiers, one typically needs to run N learning processes. In this paper we introduce and explore Model Jittering Ensembling, where one single model is perturbed in order to obtain variants that can be used as an ensemble. We use as base classifiers sets of classification association rules. The two methods of jittering ensembling we propose are Iterative Reordering Ensembling (IRE) and Post Bagging (PB). Both methods start by learning one rule set over a single run, and then produce multiple rule sets without relearning. Empirical results on 36 data sets are positive and show that both strategies tend to reduce error with respect to the single model association rule classifier. A bias–variance analysis reveals that while both IRE and PB are able to reduce the variance component of the error, IRE is particularly effective in reducing the bias component. We show that Model Jittering Ensembling can represent a very good speed-up w.r.t. multiple model learning ensembling. We also compare Model Jittering with various state of the art classifiers in terms of predictive accuracy and computational efficiency.