Worst-case Error Research Articles

Two-machine flowshop scheduling in a physical examination center

In this paper, we formulate two important operational processes of the physical examination center as a group scheduling problem in two-machine flowshop. The objective is to minimize the makespan. We first analyze the complexity of the problem with a fixed number of release dates and present a pseudo-polynomial-time algorithm. Then we provide an approximation algorithm and analyze its worst-case error bounds for the problem with two release dates and an optimal algorithm for a very practical special case of the problem.

Mean–variance hedging with model risk

This paper studies a hedging problem of a derivative security in a one-period model when there is the model risk. The hedging error is measured by a quadratic criterion. The model risk means that the true model is uncertain and there are many candidates for the true model. The true model is assumed to be in a set of models. We study an optimal strategy which minimizes the worst-case hedging error over all models in the set. We show how to calculate an optimal strategy and the minimum hedging error effectively. Finally we give some numerical examples to demonstrate the usefulness of our method.

Multivariate approximation for analytic functions with Gaussian kernels

We approximate d-variate analytic functions defined on Rd which belong to a tensor product reproducing kernel Hilbert space. The kernel of this space is Gaussian with non-increasing positive shape parameters γj2 for j=1,2,…,d. The error of approximation is defined in the L2 sense with the standard Gaussian weight. We study the worst case error of algorithms that use at most n arbitrary linear functionals on d-variate functions. We prove that for arbitrary shape parameters there are algorithms enjoying exponential convergence, but the exponent of exponential convergence depends on d and goes to zero as d approaches infinity. We study the absolute and normalized error criteria, in which the information complexity n(ε,d) is defined as the minimal number of linear functionals which are needed to find an algorithm whose worst case error is at most ε or at most ε times the norm of the approximation operator. We study different notions of tractability which describe how the information complexity behaves as a function of d and lnε−1. We find necessary and sufficient conditions on various notions of tractability in terms of shape parameters γj2. Surprisingly enough, these conditions are the same for the absolute and normalized error criteria although the norm of the approximation operator may be exponentially small in d making the normalized error much harder than the absolute one. In particular, for any positive t and κ we find conditions on γj2 for which limd+ε−1→∞lnn(ε,d)dt+[lnε−1]κ=0.This holds •for all positive γj2 if max(t,κ)>1,•for t=1 and κ=1 iff limjγj2=0,•for t<1 and κ=1 iff limjln(j)∕ln(1∕γj2)=0,•for t≤1 and κ<1 iff limj→∞j(1−κ)∕κ∕ln(1∕γj2)=0.

Analytical and experimental approaches to study elastic deflection of thin strip in ultrasonic-assisted drilling process

During machining of thin material, workpiece endures deflection due to cutting forces that negatively influence dimensional tolerances and machined surface quality. An applicable method for the restriction of such problems is machining in the presence of ultrasonic vibration. However, understanding of how this improvement is caused by this type of vibration needs extensive studies. This paper presents a workpiece deflection model for conventional drilling and ultrasonic-assisted drilling of 1 mm thickness aluminum 7075. The workpiece considered was a thin beam and the deflection was determined by the use of beam vibration theory in continuous system. The deflection of workpiece during both the conventional and ultrasonic-assisted drilling processes were determined by means of developed model and compared with those derived from experiments. It was found that the average prediction error in worst case was less than 20%. Also, it is observed from both the experimental and theoretical approaches that applying ultrasonic vibration to drilling tool significantly reduces the workpiece deflection. Furthermore, from both approaches, it is identified that increase in vibration amplitude causes a decrease in beam deflection. Restriction of beam deflection by exerting high-amplitude ultrasonic vibration causes elimination of cap burr in the exit hole due to enhancement in the rigidity of workpiece.

Comparison of Measurement Models for 3D Magnetic Localization and Tracking.

In this paper, we consider magnetic positioning and tracking of objects and provide a comparison of the characteristics of two major measurement models: the magnetic dipole model and the mutual inductance model. The numerical results obtained by applying these models to a short-range position measurement application, with a maximum operating distance of approximately 50 cm, are compared. Based on the results of this comparison, a prototype 9-sensor array is developed, experimental tests are performed, and extensive measurement results are presented. Outcomes show the feasibility of tracking the position and orientation of a mobile coil in real time with a median positioning error below 1 cm and a worst-case error of about 2 cm and 11 degrees inside a spatial region of 30 × 30 × 30 cm3 operational volume.

Modelling of the effective dielectric constant of planar spoof surface plasmon polariton waveguides

The modelling of the effective dielectric constant of a planar spoof surface plasmon polariton waveguide for X-band frequencies is presented. The waveguide has a single conductor structure that is corrugated on both sides. A methodology to extract the effective dielectric constant of waveguides with complex patterns and a novel closed-form formula based on complex analysis of the structure and empirical methods are suggested. The suggested formula is a function of the corrugation depth, aperture, periodicity, thickness of the substrate material and permittivity. The worst case error between the formula and measurement result is 8.4%, which is much lower than the previously reported coplanar stripline formulation approach having an error of 14.5%.

Conservative classical and quantum resolution limits for incoherent imaging

I propose classical and quantum limits to the statistical resolution of two incoherent optical point sources from the perspective of minimax parameter estimation. Unlike earlier results based on the Cramér-Rao bound (CRB), the limits proposed here, based on the worst-case error criterion and a Bayesian version of the CRB, are valid for any biased or unbiased estimator and obey photon-number scalings that are consistent with the behaviours of actual estimators. These results prove that, from the minimax perspective, the spatial-mode demultiplexing measurement scheme recently proposed by Tsang, Nair, and Lu [Phys. Rev. X 2016, 6 031033.] remains superior to direct imaging for sufficiently high photon numbers.

Approximation of Quasi-Monte Carlo worst case error in weighted spaces of infinitely times smooth functions

In this paper, we consider Quasi-Monte Carlo (QMC) worst case error of weighted smooth function classes in $C^\infty[0,1]^s$ by a digital net over $\mathbb F_2$. We show that the ratio of the worst case error to the QMC integration error of an exponential function is bounded above and below by constants. This result provides us with a simple interpretation that a digital net with small QMC integration error for an exponential function also gives the small integration error for any function in this function space.

Lattice-based integration algorithms: Kronecker sequences and rank-1 lattices

We prove upper bounds on the order of convergence of lattice-based algorithms for numerical integration in function spaces of dominating mixed smoothness on the unit cube with homogeneous boundary condition. More precisely, we study worst-case integration errors for Besov spaces of dominating mixed smoothness mathring{mathbf {B}}^s_{p,theta }, which also comprise the concept of Sobolev spaces of dominating mixed smoothness mathring{mathbf {H}}^s_{p} as special cases. The considered algorithms are quasi-Monte Carlo rules with underlying nodes from T_Nleft( mathbb {Z}^dright) cap [0,1)^d, where T_N is a real invertible generator matrix of size d. For such rules, the worst-case error can be bounded in terms of the Zaremba index of the lattice mathbb {X}_N=T_Nleft( mathbb {Z}^dright) . We apply this result to Kronecker lattices and to rank-1 lattice point sets, which both lead to optimal error bounds up to log N-factors for arbitrary smoothness s. The advantage of Kronecker lattices and classical lattice point sets is that the run-time of algorithms generating these point sets is very short.

Observing the Permanent-Magnet Temperature of Synchronous Motors Based on Electrical Fundamental Wave Model Quantities

Estimating the magnet temperature of permanent-magnet synchronous motors on a model basis is the key for both the highest device utilization and preventing irreversible demagnetization. Using an exact flux linkage observer in the fundamental wave domain is an elegant way to indirectly retrieve the magnet temperature without requiring any additional sensors or signal injection. This contribution addresses several important issues for the real-world applicability of a permanent magnet temperature observer: thermal expansion affecting the radial air gap thickness, operation at low speeds or even standstill, and estimating the fundamental content of the motor's terminal voltage within one switching cycle. The proposed observer is comprehensively validated in the entire electrical and thermal operation range achieving a satisfying worst case error less than 10 K.

Error-Resilient Video Encoding Using Parallel Independent Signature Processing

Soft errors resulting from encoding video sequences on unreliable hardware can create significant artifacts in decoded video sequences, contributing to extreme video quality degradation. Modern systems are required to operate under increasingly challenging constraints, including smaller feature sizes and lower operating voltage, increasing the likelihood of soft errors in the video encoding hardware. These conditions are of particular concern for energy-limited battery-operated systems since they may be required to operate in nonideal environments and/or continue operating with a practically depleted energy source. The proposed parallel independent signature processing design performs error detection and mitigation in video encoding hardware, enabling a graceful degradation of quality when encoding by using unreliable hardware. The effects of soft errors are minimized by preventing the error propagation normally associated with errors in encoded video sequences. This allows for the recovery of quality when errors are present in the video encoding system. Conventional video encoding techniques are designed to handle worst case error rates by increasing gate sizes and/or increasing the operating voltage of the system. Such designs have error-rate limits, and when these limits are reached, the systems tend to fail catastrophically, resulting in an unrecoverable signal. The proposed design allows for single upset events to translate into single transient artifacts in a decoded video sequence.

Tractability of $\mathbb{L}_2$-approximation in hybrid function spaces

We consider multivariate $\mathbb{L}_2$-approximation in reproducing kernel Hilbert spaces which are tensor products of weighted Walsh spaces and weighted Korobov spaces. We study the minimal worst-case error $e^{\mathbb{L}_2-\mathrm{app},\Lambda}(N,d)$ of all algorithms that use $N$ information evaluations from the class $\Lambda$ in the $d$-dimensional case. The two classes $\Lambda$ considered in this paper are the class $\Lambda^{{\rm all}}$ consisting of all linear functionals and the class $\Lambda^{{\rm std}}$ consisting only of function evaluations. The focus lies on the dependence of $e^{\mathbb{L}_2-\mathrm{app},\Lambda}(N,d)$ on the dimension $d$. The main results are conditions for weak, polynomial, and strong polynomial tractability.

Assessing the Quality of Convex Approximations for Two-Stage Totally Unimodular Integer Recourse Models

We consider two types of convex approximations of two-stage totally unimodular integer recourse models. Although worst-case error bounds are available for these approximations, their actual performance has not yet been investigated, mainly because this requires solving the original recourse model. In this paper we assess the quality of the approximating solutions using Monte Carlo sampling, or more specifically, using the so-called multiple replications procedure. Based on numerical experiments for an integer newsvendor problem, a fleet allocation and routing problem, and a stochastic activity network investment problem, we conclude that the error bounds are reasonably sharp if the variability of the random parameters in the model is either small or large; otherwise, the actual error of using the convex approximations is much smaller than the error bounds suggest. Moreover, we conclude that the solutions obtained using the convex approximations are good only if the variability of the random parameters is medium to large. In case this variability is small, however, typically sampling methods perform best, even with modest sample sizes. In this sense, the convex approximations and sampling methods can be considered as complementary solution methods. Moreover, as required for our applications, we extend our approach to derive new error bounds dealing with deterministic second-stage side constraints and relatively complete recourse, and perfect dependencies in the right-hand side vector.

Construction of interlaced polynomial lattice rules for infinitely differentiable functions

We study multivariate integration over the $s$-dimensional unit cube in a weighted space of infinitely differentiable functions. It is known from a recent result by Suzuki that there exists a good quasi-Monte Carlo (QMC) rule which achieves a super-polynomial convergence of the worst-case error in this function space, and moreover, that this convergence behavior is independent of the dimension under a certain condition on the weights. In this paper we provide a constructive approach to finding a good QMC rule achieving such a dimension-independent super-polynomial convergence of the worst-case error. Specifically, we prove that interlaced polynomial lattice rules, with an interlacing factor chosen properly depending on the number of points $N$ and the weights, can be constructed using a fast component-by-component algorithm in at most $O(sN(\log N)^2)$ arithmetic operations to achieve a dimension-independent super-polynomial convergence. The key idea for the proof of the worst-case error bound is to use a variant of Jensen's inequality with a purposely-designed concave function.

Deletion Codes in the High-Noise and High-Rate Regimes

The noise model of deletions poses significant challenges in coding theory, with basic questions like the capacity of the binary deletion channel still being open. In this paper, we study the harder model of worst case deletions, with a focus on constructing efficiently decodable codes for the two extreme regimes of high-noise and high-rate. Specifically, we construct polynomial-time decodable codes with the following tradeoffs (for any $ \varepsilon > 0$ ): 1) codes that can correct a fraction $1- \varepsilon $ of deletions with rate $ \mathop {\mathrm {poly}}\nolimits ( \varepsilon )$ over an alphabet of size $ \mathop {\mathrm {poly}}\nolimits (1/ \varepsilon )$ ; 2) binary codes of rate $1-\tilde {O}(\sqrt { \varepsilon })$ that can correct a fraction $ \varepsilon $ of deletions; and 3) Binary codes that can be list-decoded from a fraction $(1/2- \varepsilon )$ of deletions with rate $ \mathop {\mathrm {poly}}\nolimits ( \varepsilon )$ . This paper gives the first efficient constructions which meet the qualitative goals of correcting a deletion fraction approaching 1 over bounded alphabets, and correcting a constant fraction of bit deletions with rate approaching 1 over a fixed alphabet. The above-mentioned results bring our understanding of deletion code constructions in these regimes to a similar level as worst case errors.

ALEA

Energy efficiency is becoming increasingly important, yet few developers understand how source code changes affect the energy and power consumption of their programs. To enable them to achieve energy savings, we must associate energy consumption with software structures, especially at the fine-grained level of functions and loops. Most research in the field relies on direct power/energy measurements taken from on-board sensors or performance counters. However, this coarse granularity does not directly provide the needed fine-grained measurements. This article presents ALEA, a novel fine-grained energy profiling tool based on probabilistic analysis for fine-grained energy accounting. ALEA overcomes the limitations of coarse-grained power-sensing instruments to associate energy information effectively with source code at a fine-grained level. We demonstrate and validate that ALEA can perform accurate energy profiling at various granularity levels on two different architectures: Intel Sandy Bridge and ARM big.LITTLE. ALEA achieves a worst-case error of only 2% for coarse-grained code structures and 6% for fine-grained ones, with less than 1% runtime overhead. Our use cases demonstrate that ALEA supports energy optimizations, with energy savings of up to 2.87 times for a latency-critical option pricing workload under a given power budget.

Development and Experimental Evaluation of Concurrent Control of a Robotic Arm and Continuum Manipulator for Osteolytic Lesion Treatment.

This paper presents the development and evaluation of concurrent control of a robotic system for less-invasive treatment of osteolytic lesions behind an acetabular implant. This system implements safety constraints including a remote center of motion (RCM), virtual walls, and joint limits while operating through the screw holes of the acetabular implant. The formulated linear constrained optimization problem ensures these constraints are satisfied while maintaining precise control of the tip of a Continuum Dexterous Manipulator (CDM) attached to a positioning robot. Experiments evaluated the performance of the tip control method within an acetabular cup. The controller reliably reached a series of goal points with a mean error of 0.42 mm and a worst-case error of straying 1.0 mm from our path.

Attack-Resilient State Estimation for Noisy Dynamical Systems

Several recent incidents have clearly illustrated the susceptibility of cyberphysical systems (CPS) to attacks, raising attention to security challenges in these systems. The tight interaction between information technology and the physical world has introduced new vulnerabilities that cannot be addressed with the use of standard cryptographic security techniques. Accordingly, the problem of state estimation in the presence of sensor and actuator attacks has attracted significant attention in the past. Unlike the existing work, in this paper, we consider the problem of attack-resilient state estimation in the presence of bounded-size noise. We focus on the most general model for sensor attacks where any signal can be injected via compromised sensors. Specifically, we present an $l_0$ -based state estimator that can be formulated as a mixed-integer linear program and its convex relaxation based on the $l_1$ norm. For both attack-resilient state estimators, we derive rigorous analytic bounds on the state-estimation errors caused by the presence of noise. Our analysis shows that the worst-case error is linear with the size of the noise and, thus, the attacker cannot exploit the noise to introduce unbounded state-estimation errors. Finally, we show how the $l_0$ and $l_1$ -based attack-resilient state estimators can be used for sound attack detection and identification; we provide conditions on the size of attack vectors that ensure correct identification of compromised sensors.

Infinite-Dimensional $$\ell ^1$$ ℓ 1 Minimization and Function Approximation from Pointwise Data

We consider the problem of approximating a smooth function from finitely many pointwise samples using \(\ell ^1\) minimization techniques. In the first part of this paper, we introduce an infinite-dimensional approach to this problem. Three advantages of this approach are as follows. First, it provides interpolatory approximations in the absence of noise. Second, it does not require a priori bounds on the expansion tail in order to be implemented. In particular, the truncation strategy we introduce as part of this framework is independent of the function being approximated, provided the function has sufficient regularity. Third, it allows one to explain the key role weights play in the minimization, namely, that of regularizing the problem and removing aliasing phenomena. In the second part of this paper, we present a worst-case error analysis for this approach. We provide a general recipe for analyzing this technique for arbitrary deterministic sets of points. Finally, we use this tool to show that weighted \(\ell ^1\) minimization with Jacobi polynomials leads to an optimal method for approximating smooth, one-dimensional functions from scattered data.

On the diversity analysis of symbol remapping in cooperative BICM–OFDM scheme for frequency selective Rayleigh fading channels

In this paper, diversity analysis of a symbol remapping scheme in cooperative systems is considered. We employ Bit Interleaved Coded Modulation and Orthogonal Frequency Division Multiplexing, which are widely used at wireless networks. In this paper, at the cooperative scheme, relays operate in DF mode and the effect of their erroneous detection will be considered. We present a closed form of the asymptotic worst-case pairwise error probability at the destination for our proposed scheme. It will be shown that this scheme exploits full spatial and frequency diversities. After that, we work on designing a new precoder matrix. Moreover, we propose two new subcarrier allocations for the proposed scheme. All new methods have been suggested to achieve the available diversities with a better performance than current state of the art. Finally, the simulation results reveal the analytical derivations and confirm the obtained diversity gain of the proposed systems.