Worst-case Error Research Articles

Digital inversive vectors can achieve polynomial tractability for the weighted star discrepancy and for multivariate integration

We study high-dimensional numerical integration in the worst-case setting. The subject of tractability is concerned with the dependence of the worst-case integration error on the dimension. Roughly speaking, an integration problem is tractable if the worst-case error does not grow exponentially fast with the dimension. Many classical problems are known to be intractable. However, sometimes tractability can be shown. Often such proofs are based on randomly selected integration nodes. Of course, in applications, true random numbers are not available and hence one mimics them with pseudorandom number generators. This motivates us to propose the use of pseudorandom vectors as underlying integration nodes in order to achieve tractability. In particular, we consider digital inverse vectors and present two examples of problems, the weighted star discrepancy and integration of Hölder continuous, absolute convergent Fourier and cosine series, where the proposed method is successful.

Optimal Order Quadrature Error Bounds for Infinite-Dimensional Higher-Order Digital Sequences

Quasi-Monte Carlo (QMC) quadrature rules using higher order digital nets and sequences have been shown to achieve the almost optimal rate of convergence of the worst-case error in Sobolev spaces of arbitrary fixed smoothness $\alpha\in \mathbb{N}$, $\alpha\geq 2$. In a recent paper by the authors, it was proved that randomly-digitally-shifted order $2\alpha$ digital nets in prime base $b$ achieve the best possible rate of convergence of the root mean square worst-case error of order $N^{-\alpha}(\log N)^{(s-1)/2}$ for $N=b^m$, where $N$ and $s$ denote the number of points and the dimension, respectively, which implies the existence of an optimal order QMC rule. More recently, the authors provided an explicit construction of such an optimal order QMC rule by using Chen-Skriganov's digital nets in conjunction with Dick's digit interlacing composition. These results were for fixed number of points. In this paper we give a more general result on an explicit construction of optimal order QMC rules for arbitrary fixed smoothness $\alpha\in \mathbb{N}$ including the endpoint case $\alpha=1$. That is, we prove that the projection of any infinite-dimensional order $2\alpha +1$ digital sequence in prime base $b$ onto the first $s$ coordinates achieves the best possible rate of convergence of the worst-case error of order $N^{-\alpha}(\log N)^{(s-1)/2}$ for $N=b^m$. The explicit construction presented in this paper is not only easy to implement but also extensible in both $N$ and $s$.

Formula omitted]-Approximation in Korobov spaces with exponential weights

We study multivariate L∞-approximation for a weighted Korobov space of periodic functions for which the Fourier coefficients decay exponentially fast. The weights are defined, in particular, in terms of two sequences a={aj} and b={bj} of positive real numbers bounded away from zero. We study the minimal worst-case error eL∞−app,Λ(n,s) of all algorithms that use n information evaluations from a class Λ in the s-variate case. We consider two classes Λ in this paper: the class Λall of all linear functionals and the class Λstd of only function evaluations.We study exponential convergence of the minimal worst-case error, which means that eL∞−app,Λ(n,s) converges to zero exponentially fast with increasing n. Furthermore, we consider how the error depends on the dimension s. To this end, we define the notions of κ-EC-weak, EC-polynomial and EC-strong polynomial tractability, where EC stands for “exponential convergence”. In particular, EC-polynomial tractability means that we need a polynomial number of information evaluations in s and 1+logε−1 to compute an ε-approximation. We derive necessary and sufficient conditions on the sequences a and b for obtaining exponential error convergence, and also for obtaining the various notions of tractability. The results are the same for both classes Λ.L2-approximation for functions from the same function space has been considered in Dick et al. (2014). It is surprising that most results for L∞-approximation coincide with their counterparts for L2-approximation. This allows us to deduce also results for Lp-approximation for p∈[2,∞].

Construction of Quasi-Monte Carlo Rules for Multivariate Integration in Spaces of Permutation-Invariant Functions

We study multivariate integration of functions that are invariant under the permutation (of a subset) of their arguments. Recently, in Nuyens et al. (Adv Comput Math 42(1):55–84, 2016), the authors derived an upper estimate for the nth minimal worst case error for such problems and showed that under certain conditions this upper bound only weakly depends on the dimension. We extend these results by proposing two (semi-) explicit construction schemes. We develop a component-by-component algorithm to find the generating vector for a shifted rank-1 lattice rule that obtains a rate of convergence arbitrarily close to \(\mathcal {O}(n^{-\alpha })\), where \(\alpha >1/2\) denotes the smoothness of our function space and n is the number of cubature nodes. Further, we develop a semi-constructive algorithm that builds on point sets that can be used to approximate the integrands of interest with a small error; the cubature error is then bounded by the error of approximation. Here the same rate of convergence is achieved while the dependence of the error bounds on the dimension d is significantly improved.

Tight error bounds for rank-1 lattice sampling in spaces of hybrid mixed smoothness

We consider the approximate recovery of multivariate periodic functions from a discrete set of function values taken on a rank-1 lattice. Moreover, the main result is the fact that any (non-)linear reconstruction algorithm taking function values on any integration lattice of size M has a dimension-independent lower bound of $$2^{-(\alpha +1)/2} M^{-\alpha /2}$$ when considering the optimal worst-case error with respect to function spaces of (hybrid) mixed smoothness $$\alpha >0$$ on the d-torus. We complement this lower bound with upper bounds that coincide up to logarithmic terms. These upper bounds are obtained by a detailed analysis of a rank-1 lattice sampling strategy, where the rank-1 lattices are constructed by a component–by–component method. The lattice (group) structure allows for an efficient approximation of the underlying function from its sampled values using a single one-dimensional fast Fourier transform. This is one reason why these algorithms keep attracting significant interest. We compare our results to recent (almost) optimal methods based upon samples on sparse grids.

Concentration inequalities of the cross-validation estimator for empirical risk minimizer

ABSTRACTWe derive concentration inequalities for the cross-validation estimate of the generalization error for empirical risk minimizers. In the general setting, we show that the worst-case error of this estimate is not much worse that of training error estimate see Kearns M, Ron D. [Algorithmic stability and sanity-check bounds for leave-one-out cross-validation. Neural Comput. 1999;11:1427–1453]. General loss functions and class of predictors with finite VC-dimension are considered. Our focus is on proving the consistency of the various cross-validation procedures. We point out the interest of each cross-validation procedure in terms of rates of convergence. An interesting consequence is that the size of the test sample is not required to grow to infinity for the consistency of the cross-validation procedure.

MEASUREMENT METHODOLOGIES FOR REDUCING ERRORS IN THE ASSESSMENT OF EMF BY EXPOSIMETER

Objective: As well known, using a single body worn sensor exposimeter introduces systematic errors on the measurement of the incident free space electric field strength. This is because the body creates around it high, intermediate and low level field zones, which depend on the direction of arrival of the incident field. The goal of this work is to propose an efficient method for the reduction of these errors. Methods: After classifying the perturbations induced by the body on the measured electric and magnetic fields thanks to realistic numerical simulations, we then propose a two-sensor setup in conjunction with simple semi-empirical correction formulas, in order to compensate these perturbations. Results: At 942 MHz, when the two sensors are placed in any opposite sides of the body at chest height, the worst case, maximum and average errors respectively decrease to 12% and 3% compared to 83% and 22% for measurement techniques using a single sensor, or 32% and 11% when using the average value of the measurements. Conclusion: The error related to the measurement in the presence of the body was significantly reduced by the proposed method making use of two opposite sensors, E-field and H-field at the chest. Significance: The conformity of exposure to EMF in terms of reference values according to the ICNIRP is given in the abscence of the human body. The interest of this work lies in the reduction of the errors made when measuring the field in the presence of the body.

Single-Event Characterization of Common 1<formula> <tex>$^{\hbox{st}}$</tex> </formula> and 2<formula> <tex>$^{\hbox{nd}}$</tex> </formula>-order All-digital Phase-locked Loops (ADPLLs)

The single-event upset (SEU) vulnerability of common first- and second-order all-digital-phase-locked loops (ADPLLs) is investigated through field-programmable gate array-based fault injection experiments. SEUs in the highest order pole of the loop filter and fraction-based phase detectors (PDs) may result in the worst case error response, i.e., limit cycle errors, often requiring system restart. SEUs in integer-based linear PDs may result in loss-of-lock errors, while SEUs in bang-bang PDs only result in temporary-frequency errors. ADPLLs with the same frequency tuning range but fewer bits in the control word exhibit better overall SEU performance.

On the lifting of deterministic convergence rates for inverse problems with stochastic noise

Both for the theoretical and practical treatment of Inverse Problems, the modeling of the noise is crucial. One either models the measurement via a deterministic worst-case error assumption or assumes a certain stochastic behavior of the noise. Although some connections between both models are known, the communities develop rather independently. In this paper we seek to bridge the gap between the deterministic and the stochastic approach and show convergence and convergence rates for Inverse Problems with stochastic noise by lifting the theory established in the deterministic setting into the stochastic one. This opens the wide field of deterministic regularization methods for stochastic problems without having to do an individual stochastic analysis for each problem.

A New Adaptive H-Infinity Filtering Algorithm for the GPS/INS Integrated Navigation.

The Kalman filter is an optimal estimator with numerous applications in technology, especially in systems with Gaussian distributed noise. Moreover, the adaptive Kalman filtering algorithms, based on the Kalman filter, can control the influence of dynamic model errors. In contrast to the adaptive Kalman filtering algorithms, the H-infinity filter is able to address the interference of the stochastic model by minimization of the worst-case estimation error. In this paper, a novel adaptive H-infinity filtering algorithm, which integrates the adaptive Kalman filter and the H-infinity filter in order to perform a comprehensive filtering algorithm, is presented. In the proposed algorithm, a robust estimation method is employed to control the influence of outliers. In order to verify the proposed algorithm, experiments with real data of the Global Positioning System (GPS) and Inertial Navigation System (INS) integrated navigation, were conducted. The experimental results have shown that the proposed algorithm has multiple advantages compared to the other filtering algorithms.

A Note on 1.5 Sigma Shift in Performance Evaluation

Understanding variation is critical to quality (CTQ) of product or service delivery, which is the key to success. Six Sigma is the business process improvement strategy that extensively focuses on variation reduction thereby reducing number of defects. One of the major constituents of Six Sigma definitions is 1.5 sigma shift, which is attributable to random error. It is not possible to understand Six Sigma thoroughly by overlooking the concept of 1.5 sigma shift. A conventional 3.4 defect per million opportunity (DPMO) capability of Six Sigma process is based on 1.5 sigma shift. This paper aimed at explaining ancestry of 1.5 sigma shift in connection with quality engineering methods. Origin of 1.5 sigma shift factor with reference to producibility analysis, worst-case sampling error and other quality engineering methods has been discussed in this paper.

FEC Code Anchored Robust Design of Massive MIMO Receivers

Massive multiple-input-multiple-output (MIMO) systems have been proposed to support high rate multiple access. As channel estimation in massive MIMO suffers from the well-known impairment of pilot contamination, we propose a novel approach to multi-user detection by exploiting forward error correction (FEC) code diversity. Unlike traditional approaches solely based on worst-case or probabilistic channel estimation errors, we develop a joint quadratic-programming (QP) receiver anchored with a set of FEC code constraints. Exploiting the user signatures presented by FEC channel codes of distinct permutations, our receiver can effectively recover signals from pilot-interfering users. The code-anchored robust design (CARD) method can also be applied to a chance-constrained receiver, which shows further performance gain compared with the direct integration of FEC code constraints in joint QP receiver. The effectiveness of CARD receivers is demonstrated by numerical results that establish substantial performance gain of the proposed receivers over existing robust designs. In addition, we present a distributed multi-cell processing scheme for enhanced performance via alternating direction method of multipliers.

Prediction of proton-induced SEE error rates for the VATA160 ASIC

We predict proton single event effect (SEE) error rates for the VATA160 ASIC chip on the Dark Matter Particle Explorer (DAMPE) to evaluate its radiation tolerance. Lacking proton test facilities, we built a Monte Carlo simulation tool named PRESTAGE to calculate the proton SEE cross-sections. PRESTAGE is based on the particle transport toolkit Geant4. It adopts a location-dependent strategy to derive the SEE sensitivity of the device from heavy-ion test data, which have been measured at the HI-13 tandem accelerator of the China Institute of Atomic Energy and the heavy-ion research facility in Lanzhou. The AP-8, SOLPRO, and August 1972 worst-case models are used to predict the average and peak proton fluxes on the DAMPE orbit. Calculation results show that the averaged proton SEE error rate for the VATA160 chip is approximately 2.17 × 10−5/device/day. Worst-case error rates for the Van Allen belts and solar energetic particle events are 1–3 orders of magnitude higher than the averaged error rate.

Approximate Euclidean Steiner Trees

An approximate Steiner tree is a Steiner tree on a given set of terminals in Euclidean space such that the angles at the Steiner points are within a specified error from 120^{circ }. This notion arises in numerical approximations of minimum Steiner trees. We investigate the worst-case relative error of the length of an approximate Steiner tree compared to the shortest tree with the same topology. It has been conjectured that this relative error is at most linear in the maximum error at the angles, independent of the number of terminals. We verify this conjecture for the two-dimensional case as long as the maximum angle error is sufficiently small in terms of the number of terminals. In the two-dimensional case we derive a lower bound for the relative error in length. This bound is linear in terms of the maximum angle error when the angle error is sufficiently small in terms of the number of terminals. We find improved estimates of the relative error in length for larger values of the maximum angle error and calculate exact values in the plane for three and four terminals.

Noise tailoring for scalable quantum computation via randomized compiling

Quantum computers are poised to radically outperform their classical counterparts by manipulating coherent quantum systems. A realistic quantum computer will experience errors due to the environment and imperfect control. When these errors are even partially coherent, they present a major obstacle to achieving robust computation. Here, we propose a method for introducing independent random single-qubit gates into the logical circuit in such a way that the effective logical circuit remains unchanged. We prove that this randomization tailors the noise into stochastic Pauli errors, leading to dramatic reductions in worst-case and cumulative error rates, while introducing little or no experimental overhead. Moreover we prove that our technique is robust to variation in the errors over the gate sets and numerically illustrate the dramatic reductions in worst-case error that are achievable. Given such tailored noise, gates with significantly lower fidelity are sufficient to achieve fault-tolerant quantum computation, and, importantly, the worst case error rate of the tailored noise can be directly and efficiently measured through randomized benchmarking experiments. Remarkably, our method enables the realization of fault-tolerant quantum computation under the error rates observed in recent experiments.

Complexity of oscillatory integrals on the real line

We analyze univariate oscillatory integrals defined on the real line for functions from the standard Sobolev space $H^s({\mathbb{R}})$ and from the space $C^s({\mathbb{R}})$ with an arbitrary integer $s\ge1$. We find tight upper and lower bounds for the worst case error of optimal algorithms that use $n$ function values. More specifically, we study integrals of the form \[ I_k^\rho (f) = \int_{ {\mathbb{R}}} f(x) \,e^{-i\,kx} \rho(x) \, {\rm d} x\ \ \ \mbox{for}\ \ f\in H^s({\mathbb{R}})\ \ \mbox{or}\ \ f\in C^s({\mathbb{R}}) \] with $k\in {\mathbb{R}}$ and a smooth density function $\rho$ such as $ \rho(x) = \frac{1}{\sqrt{2 \pi}} \exp( -x^2/2) $. The optimal error bounds are $\Theta((n+\max(1,|k|))^{-s})$ with the factors in the $\Theta$ notation dependent only on $s$ and $\rho$.

([formula omitted])-weak tractability of linear problems

We introduce a new notion of tractability for multivariate problems, namely (s,lnκ)-weak tractability for positive s and κ. This allows us to study the information complexity of a d-variate problem with respect to different powers of d and the bits of accuracy lnε−1. We consider the worst case error for the absolute and normalized error criteria. We provide necessary and sufficient conditions for (s,lnκ)-weak tractability for general linear problems and linear tensor product problems defined over Hilbert spaces. In particular, we show that non-trivial linear tensor product problems cannot be (s,lnκ)-weakly tractable when s∈(0,1] and κ∈(0,1]. On the other hand, they are (s,lnκ)-weakly tractable for κ>1 and s>1 if the univariate eigenvalues of the linear tensor product problem enjoy a polynomial decay. Finally, we study (s,lnκ)-weak tractability for the remaining combinations of the values of s and κ.

Worst-case multi-objective error estimation and adaptivity

This paper introduces a new computational methodology for determining a-posteriori multi-objective error estimates for finite-element approximations, and for constructing corresponding (quasi-)optimal adaptive refinements of finite-element spaces. As opposed to the classical goal-oriented approaches, which consider only a single objective functional, the presented methodology applies to general closed convex subsets of the dual space and constructs a worst-case error estimate of the finite-element approximation error. This worst-case multi-objective error estimate conforms to a dual-weighted residual, in which the dual solution is associated with an approximate supporting functional of the objective set at the approximation error. We regard both standard approximation errors and data-incompatibility errors associated with incompatibility of boundary data with the trace of the finite-element space. Numerical experiments are presented to demonstrate the efficacy of applying the proposed worst-case multi-objective error estimate in adaptive refinement procedures.

Super-polynomial convergence and tractability of multivariate integration for infinitely times differentiable functions

We investigate multivariate integration for a space of infinitely times differentiable functions Fs,u:={f∈C∞[0,1]s∣‖f‖Fs,u<∞}, where ‖f‖Fs,u:=supα=(α1,…,αs)∈N0s‖f(α)‖L1/∏j=1sujαj, f(α):=∂∣α∣∂x1α1⋯∂xsαsf and u={uj}j≥1 is a sequence of positive decreasing weights. Let e(n,s) be the minimal worst-case error of all algorithms that use n function values in the s-variate case. We prove that for any u and s considered e(n,s)≤C(s)exp(−c(s)(logn)2) holds for all n, where C(s) and c(s) are constants which may depend on s. Further we show that if the weights u decay sufficiently fast then there exist some 1<p<2 and absolute constants C and c such that e(n,s)≤Cexp(−c(logn)p) holds for all s and n. These bounds are attained by quasi-Monte Carlo integration using digital nets. These convergence and tractability results come from those for the Walsh space into which Fs,u is embedded.

Robust Localization Using Time Difference of Arrivals

We investigate a localization problem using time-difference-of-arrival measurements with unknown and bounded measurement errors. Different from most existing algorithms, we consider the minimization of the worst-case position estimation error to improve the robustness of the algorithm. The localization problem is formulated as a nonconvex optimization problem. We adopt semidefinite relaxation to relax the original problem into a convex optimization problem, which can be solved using existing semidefinite program solvers. Simulation results show that our proposed algorithm has lower worst-case position estimation error than other existing algorithms.