We investigate a problem of finding the minimum, in which each user has a real value, and we want to estimate the minimum of these values under the local differential privacy constraint. We reveal that this problem is fundamentally difficult, and we cannot construct a consistent mechanism in the worst case. Instead of considering the worst case, we aim to construct a private mechanism whose error rate is adaptive to the easiness of estimation of the minimum. As a measure of easiness, we introduce a parameter α that characterizes the fatness of the minimum-side tail of the user data distribution. As a result, we reveal that the mechanism can achieve O((ln6N/ε2N)1/2α) error without knowledge of α and the error rate is near-optimal in the sense that any mechanism incurs Ω((1/ε2N)1/2α) error. Furthermore, we demonstrate that our mechanism outperforms a naive mechanism by empirical evaluations on synthetic datasets. Also, we conducted experiments on the MovieLens dataset and a purchase history dataset and demonstrate that our algorithm achieves Õ((1/N)1/2α) error adaptively to α.

In this paper, we proposed that an area- and speed-effective fixed-point pipelined divider be used for reducing the bit-width of a division unit to fit a mobile rendering processor. To decide the bit-width of a division unit, error analysis has been carried out in various ways. As a result, when the original bit-width was 31-bit, the proposed method reduced the bit-width to 24-bit and reduced the area by 42% with a maximum error of 0.00001%.

Error analysis of the multilevel fast multipole algorithm is studied for electromagnetic scattering problems. We propose novel error prediction and control methods and verify that the computational error for scattering problems with over one million unknowns can be precisely controlled under desired digits of accuracy. Optimum selection of truncation numbers to minimize computational error also will be discussed.

We introduce a novel method to optimize field decomposition for a mode matching technique. Using our method, expanded mode numbers can be minimized to achieve the desired digits of computational accuracy.

The waveguide-penetration method is a method to measure the electrical properties of materials. In this method, a cylindrical object pierces a rectangular waveguide through a pair of holes at the centre of its broad walls. Then, the complex permittivity and permeability of the object are estimated from measured S-parameters after TRL calibration. This paper proposes a new calibration algorithm for the waveguide-penetration method. Reference materials with known electrical properties are fabricated in cylindrical shapes to fit into the holes in the waveguide and are used as calibration standards. The algorithm is formulated using the property of equal traces in similar matrices, and we show that at least two reference materials are needed to calibrate the system. The proposed algorithm yields a simpler means of calibration compared to TRL and is verified using measurements in the S-band. Also, the error sensitivity coefficients are derived. These coefficients give valuable information for the selection of reference materials.

In the paper, a technique of the numerical inversion of multidimensional Laplace transforms (nD NILT), based on a complex Fourier series approximation is elaborated in light of a possible ralative error achievable. The detailed error analysis shows a relationship between the numerical integration of a multifold Bromwich integral and a complex Fourier series approximation, and leads to a novel formula relating the limiting relative error to the nD NILT technique parameters.

The Waveguide-Penetration method is a permittivity measurement technique where a columnar object pierces a rectangular waveguide through a pair of holes at the center of its broad walls. The permittivity of the object is estimated from measured S-parameters . This paper demonstrates a scheme for analyzing permittivity measurement errors in the Waveguide-Penetration method. The sources of errors are categorized into systematic and random error sources. Systematic errors in the values of the sample and waveguide holes diameters, the effect of sample's length, and the influence of ambient temperature are investigated and corrected for. Potential random error sources such as imperfect TRL calibration elements, VNA thermal noise, sample loading, and test-port cable flexure are analyzed and their contribution to random errors are estimated.

The fast H∞ filter is developed by one of the authors, and its practical use in industries is expected. This paper derives a linear propagation model of numerical errors in the recursive variables of the fast H∞ filter, and then theoretically analyzes the stability of the filter. Based on the analyzed results, a numerical stabilization method of the fast H∞ filter is proposed with the error feedback control in the backward prediction. Also, the effectiveness of the stabilization method is verified using numerical examples.

This paper proposes a hybrid multiband (MB) ultra wideband (UWB) system with direct sequence (DS) spreading. The theoretical error analysis for the DS-MB-UWB multiple access system with Rake receiver in the presence of multipath and narrowband interference is developed. The developed theoretical framework models the multiple access interference (MAI), multipath interference (MI) and narrowband interference for the designed UWB system. It is shown that the system error performance corresponding to the combining effects of these interference can be accurately modeled and calculated. Monte Carlo simulation results are provided to validate the accuracy of the model. Additionally, it is found that narrowband interference can be mitigated effectively in the multiband UWB system by suppressing the particular UWB sub-band co-existing with the interfering narrowband signal. A typical improvement of 5 dB can be achieved with 75% sub-band power suppression. On the other hand, suppression of UWB sub-band is also found to decrease frequency diversity, thus facilitating the increase of MAI. In this paper, the developed model is utilized to determine the parameters that optimize the UWB system performance by minimizing the effective interference.

The Green's function of free space for the fast inhomogeneous plane wave algorithm is represented by an integration in the complex plane. The error in the computational process is determined by the number of sampling points, the truncation of the integration path, and the extrapolation. Therefore, the error control method is different from that for the fast multipole method. We will discuss the worst-case interactions of the fast inhomogeneous plane wave algorithm for the box implementation and define the upper and lower bounds of the computational error.

A new paradigm in the design of optical coatings connected with an outstanding computational efficiency of modern design techniques is discussed. Several other topics including pre-production error analysis, monitoring of coating production, and computational manufacturing of optical coatings are considered.

The computational error of the multilevel fast multipole algorithm is studied. The error convergence rate, achievable minimum error, and error bound are investigated for various element distributions. We will discuss the boundary between the large and small buffer cases in terms of machine precision. The needed buffer size to reach double precision accuracy will be clarified.

For fitting an ellipse to a point sequence, ML (maximum likelihood) has been regarded as having the highest accuracy. In this paper, we demonstrate the existence of a "hyperaccurate" method which outperforms ML. This is made possible by error analysis of ML followed by subtraction of high-order bias terms. Since ML nearly achieves the theoretical accuracy bound (the KCR lower bound), the resulting improvement is very small. Nevertheless, our analysis has theoretical significance, illuminating the relationship between ML and the KCR lower bound.

In this paper, ultra-wideband (UWB) multiple access systems are introduced by using direct-sequence (DS) and hybrid direct-sequence time-hopping (DS/TH) code division multiple access (CDMA) that use arbitrary chip-duty of the spreading sequences. The bit error probabilities are presented. First of all, the variances of the multiple access interference are developed by investigating the collision properties of the signals. Afterward, various approximations are applied. The standard Gaussian approximation (SGA) for the DS system is shown to become extremely optimistic as the chip-duty becomes low. Though the hybrid system performs better, the SGA still remains optimistic. To obtain accurate results, Holtzman's simplified improved Gaussian approximation (SIGA) and Morrow and Lehnert's improved Gaussian approximation (IGA) are used. A shortcoming of the SIGA is rediscovered that renders it unusable for low-duty DS systems, especially, at high signal-to-noise ratio. However, for the hybrid system, the SIGA works as an excellent tool. The IGA is used to get accurate results for the low-duty DS systems. It is shown that lowering of chip-duty by keeping chip rate and chip length unchanged improves performance for asynchronous DS and both asynchronous and synchronous hybrid systems. However, under the same processing gain, a high-duty system performs better than a low-duty system. Performance of synchronous DS system remains independent of chip-duty.

We propose using SQP (Sequential Quadratic Programming) to directly recover 3D quadratic surface parameters from multiple views. A surface equation is used as a constraint. In addition to the sum of squared reprojection errors defined in the traditional bundle adjustment, a Lagrangian term is added to force recovered points to satisfy the constraint. The minimization is realized by SQP. Our algorithm has three advantages. First, given corresponding features in multiple views, the SQP implementation can directly recover the quadratic surface parameters optimally instead of a collection of isolated 3D points coordinates. Second, the specified constraints are strictly satisfied and the camera parameters and 3D coordinates of points can be determined more accurately than that by unconstrained methods. Third, the recovered quadratic surface model can be represented by a much smaller number of parameters instead of point clouds and triangular patches. Experiments with both synthetic and real images show the power of this approach.

Computers are increasingly used for implementing control algorithms in safety-critical embedded applications, such as engine control, braking control and flight surface control. Consequently, computer errors can have severe impact on the safety of such systems. Addressing the coupling of control performance with computer related errors, this paper develops a methodology for analyzing the impacts data errors have on control system dependability. The impact of a data error is measured as the resulting control error. We use maximum bounds on this measure as the criterion for control system failure (i.e., if the control error exceeds a certain threshold, the system has failed). In this paper we a) develop suitable models of computer faults for analysis of control level effects and related analysis methods, and b) apply traditional control theory analysis methods for understanding the impacts of data errors on system dependability. An automobile slip-control brake-system is used as an example showing the viability of our approach.

The application of fault injection in the context of dependability benchmarking is far from being straightforward. One decisive issue to be addressed is to what extent injected faults are representative of the considered faults. This paper proposes an approach to analyze the effects of real and injected faults.

In an optical time domain reflectometer type strain measurement system, we theoretically derive the shape of the Brillouin gain spectrum produced in an optical fiber under a parabolic strain distribution which is formed in a uniformly loaded beam. Based on the derived result, we investigate the effects of the parabolic strain distribution parameters and the measurement conditions such as the launched pulse width and the measurement position on the beam on the deformation of the Brillouin backscattered-light power spectrum shape. In addition, we investigate the strain measurement error resulting from the deformation of the power spectrum shape by analyzing the peak-power frequency at which the power spectrum is maximized.

With a view to virtual studio applications, we design an optimal grid pattern such that the observed image of a small portion of it can be matched to its corresponding position in the pattern easily. The grid shape is so determined that the cross ratio of adjacent intervals is different everywhere. The cross ratios are generated by an optimal Markov process that maximizes the accuracy of matching. We test our camera calibration system using the resulting grid pattern in a realistic setting and show that the performance is greatly improved by applying techniques derived from the designed properties of the pattern.

Three multiplicative algorithms for the floating-point divide operation are compared: the Newton-Raphson method, Goldschmidt's algorithm, and a naive method that simply calculates a form of the Taylor series expansion of a reciprocal. The series also provides a theoretical basis for Goldschmidt's algorithm. It is well known that, of the Newton-Raphson method and Goldschmidt's algorithm, the former is the more accurate while the latter is the faster on a pipelined unit. However, little is reported about the naive method. In this report, we analyze the speed and accuracy of each method and present the results of numerical tests, which we conducted to confirm the validity of the accuracy analysis. Basically, the comparison are made in the context of software implementation (e. g. , a macro library) and compliance with the IEEE Standard 754 rounding is not considered. It is shown that the naive method is useful in a realistic setting where the number of iterations is small and the method is implemented on a pipelined floating-point unit with a multiply-accumulate configuration. In such a situation, the naive method gives a more accurate result with a slightly lower latency, as compared with Goldschmidt's algorithm, and is much faster than but slightly inferior in accuracy to the Newton-Raphson method.