Uniform Error Research Articles

On the Kolmogorov Distance for the Least Squares Estimator in the Fractional Ornstein-Uhlenbeck Process

The paper shows that the distribution of the normalized least squares estimator of the drift parameter in the fractional Ornstein-Uhlenbeck process observed over [0, T] converges to the standard normal distribution with an uniform optimal error bound of the order O(T −1/2) for 0.5 ≤ H ≤ 0.63 and of the order O(T4H-3) for 0.63 < H < 0.75 where H is the Hurst exponent of the fractional Brownian motion driving the Ornstein-Uhlenbeck process. For the normalized quasi-least squares estimator, the error bound is of the order O(T−1/4) for 0.5 ≤ H ≤ 0.69 and of the order O(T4H−3) for 0.69 < H < 0.75.

Parameter uniform fitted mesh finite difference scheme for elliptical singularly perturbed problems with mixed shifts in two dimensions

In this work, we embarked on the research of linear two-dimensional elliptic singularly perturbed problems having positive and negative shifts in both directions of the reaction terms, whose solution exhibits characteristic boundary layers. The study of this class of problems was started in the mid-eighties, but all the studies were restricted to 1D. The Taylor series expansion is used to estimate decelerated terms of the problem and the emerging problem is discretized using the fitted mesh finite difference method to establish parameter uniform error estimates. The effect of positive and negative shifts on the behaviour of the solution is explained by executing numerical experiments on two test examples.

Diffractive optical element design based on vector diffraction theory and improved PSO-SA algorithm

The existence of the beam vector effect requires rigorous electromagnetic computational methods for the optimization of subwavelength diffractive optical elements (DOEs). To design DOEs with high performance, we proposed an optimization strategy combining a global optimization algorithm with the finite difference time domain (FDTD) method. First, scalar initial solutions are obtained by the Gerchberg–Saxton algorithm. Then, regarding the diffraction efficiency, the uniformity error and figure of merit are calculated by FDTD as evaluation functions, and the optimization of the initial solutions is implemented by improved particle swarm optimization and continues with the simulated annealing algorithm. After several numerical simulations, the analysis results confirm the robustness of the algorithm and the reduction of the computational cost. The experimental data demonstrate that the correlation of diffraction energy orders between the simulated design and fabricated sample are above 85%. This method efficiently improves the DOEs design.

A study of error diversity in robotic swarms for task partitioning in foraging tasks.

Often in swarm robotics, an assumption is made that all robots in the swarm behave the same and will have a similar (if not the same) error model. However, in reality, this is not the case, and this lack of uniformity in the error model, and other operations, can lead to various emergent behaviors. This paper considers the impact of the error model and compares robots in a swarm that operate using the same error model (uniform error) against each robot in the swarm having a different error model (thus introducing error diversity). Experiments are presented in the context of a foraging task. Simulation and physical experimental results show the importance of the error model and diversity in achieving the expected swarm behavior.

Analysis and improvement of the uniformity of magnetic field coil based on the cylindrical magnetic shield in atomic magnetometers

The magnetic field performance of the uniform coils will be affected by magnetic shields and produce errors in the atomic magnetometer. It is necessary to analyze the influencing factors and guide the coil design. Aiming at the above shortcomings, this paper analyzes the coupling effect of magnetic shield and coil and proposed a Lee-Whiting coil design method based on the linearly decreasing inertia weight particle swarm optimization (LDW-PSO) algorithm. Firstly, the magnetic field model under the ferromagnetic boundary is established, and the effects of the size, thickness and permeability of the magnetic shield on the magnitude and uniformity of the Lee-Whiting coil magnetic field are analyzed. The results show that the coupling coefficient of the coil changes monotonically with the change of the size of the magnetic shield, while the uniformity error has a minimum value with the change of the size of the shield. In addition, an improved Lee-Whiting coil is designed by combining the LDW-PSO algorithm, which improves the uniformity of the coil. Compared with the original coil, the measured uniformity error of the improved coil is reduced from 8.39 × 1 0 − 4 to 2.81 × 1 0 − 4 . The accuracy of the theoretical analysis is verified by experiments. This paper is of great significance to study the coupling and optimization design of the coil based on the magnetic shields in atomic magnetometers. • The influence of magnetic shield parameters on coil uniformity is analyzed. • The optimal coil size for the uniformity in magnetic shield is given by analysis and testing. • The improved Lee-Whiting coil is designed by intelligent optimization algorithm. • Improved Lee-Whiting coil overcomes the coupling effect between coil and magnetic shield. • The correctness of the analysis and the effectiveness of the design are proved by experiments.

Uniform Error Bound of an Exponential Wave Integrator for Long-Time Dynamics of the Nonlinear Schrödinger Equation with Wave Operator

Uniform Error Bound of an Exponential Wave Integrator for Long-Time Dynamics of the Nonlinear Schrödinger Equation with Wave Operator

Error estimations of a weak Galerkin finite element method for a linear system of $ \ell\geq 2 $ coupled singularly perturbed reaction-diffusion equations in the energy and balanced norms

<abstract><p>This paper introduces a weak Galerkin finite element method for a system of $ \ell\geq 2 $ coupled singularly perturbed reaction-diffusion problems. The proposed method is independent of parameter and uses piecewise discontinuous polynomials on interior of each element and constant on the boundary of each element. By the Schur complement technique, the interior unknowns can be locally efficiently eliminated from the resulting linear system, and the degrees of freedom of the proposed method are comparable with the classical FEM. It has been reported that the energy norm is not adequate for singularly perturbed reaction-diffusion problems since it can not efficiently reflect the behaviour of the boundary layer parts when the diffusion coefficient is very small. For the first time, the error estimates in the balanced norm has been presented for a system of coupled singularly perturbed problems when each equation has different parameter. Optimal and uniform error estimates have been established in the energy and balanced norm on an uniform Shishkin mesh. Finally, we carry out various numerical experiments to verify the theoretical findings.</p></abstract>

Stochastic Convergence Rates and Applications of Adaptive Quadrature in Bayesian Inference

We provide the first stochastic convergence rates for a family of adaptive quadrature rules used to normalize the posterior distribution in Bayesian models. Our results apply to the uniform relative error in the approximate posterior density, the coverage probabilities of approximate credible sets, and approximate moments and quantiles, therefore, guaranteeing fast asymptotic convergence of approximate summary statistics used in practice. The family of quadrature rules includes adaptive Gauss-Hermite quadrature, and we apply this rule in two challenging low-dimensional examples. Further, we demonstrate how adaptive quadrature can be used as a crucial component of a modern approximate Bayesian inference procedure for high-dimensional additive models. The method is implemented and made publicly available in the aghq package for the R language, available on CRAN. Supplementary materials for this article are available online.

Improved uniform error bounds of the time-splitting methods for the long-time (nonlinear) Schrödinger equation

We establish improved uniform error bounds for the time-splitting methods for the long-time dynamics of the Schrödinger equation with small potential and the nonlinear Schrödinger equation (NLSE) with weak nonlinearity. For the Schrödinger equation with small potential characterized by a dimensionless parameter ε ∈ ( 0 , 1 ] \varepsilon \in (0, 1] , we employ the unitary flow property of the (second-order) time-splitting Fourier pseudospectral (TSFP) method in L 2 L^2 -norm to prove a uniform error bound at time t ε = t / ε t_\varepsilon =t/\varepsilon as C ( t ) C ~ ( T ) ( h m + τ 2 ) C(t)\widetilde {C}(T)(h^m +\tau ^2) up to t ε ≤ T ε = T / ε t_\varepsilon \leq T_\varepsilon = T/\varepsilon for any T > 0 T>0 and uniformly for ε ∈ ( 0 , 1 ] \varepsilon \in (0,1] , while h h is the mesh size, τ \tau is the time step, m ≥ 2 m \ge 2 and C ~ ( T ) \tilde {C}(T) (the local error bound) depend on the regularity of the exact solution, and C ( t ) = C 0 + C 1 t C(t) =C_0+C_1t grows at most linearly with respect to t t with C 0 C_0 and C 1 C_1 two positive constants independent of T T , ε \varepsilon , h h and τ \tau . Then by introducing a new technique of regularity compensation oscillation (RCO) in which the high frequency modes are controlled by regularity and the low frequency modes are analyzed by phase cancellation and energy method, an improved uniform (w.r.t ε \varepsilon ) error bound at O ( h m − 1 + ε τ 2 ) O(h^{m-1} + \varepsilon \tau ^2) is established in H 1 H^1 -norm for the long-time dynamics up to the time at O ( 1 / ε ) O(1/\varepsilon ) of the Schrödinger equation with O ( ε ) O(\varepsilon ) -potential with m ≥ 3 m \geq 3 . Moreover, the RCO technique is extended to prove an improved uniform error bound at O ( h m − 1 + ε 2 τ 2 ) O(h^{m-1} + \varepsilon ^2\tau ^2) in H 1 H^1 -norm for the long-time dynamics up to the time at O ( 1 / ε 2 ) O(1/\varepsilon ^2) of the cubic NLSE with O ( ε 2 ) O(\varepsilon ^2) -nonlinearity strength. Extensions to the first-order and fourth-order time-splitting methods are discussed. Numerical results are reported to validate our error estimates and to demonstrate that they are sharp.

Layout and optimization of LED light source for indoor visible light communication

AbstractThis paper proposes a novel polygonal layout based on rectangular plane stereo space. The illumination and signal‐to‐noise ratio of polygonal layout and circular layout is simulated in a 5 m × 4 m × 3 m room; The simulation results show that: compared with the illuminance uniformity of the circular layout of 0.67, the illuminance uniformity of the polygonal layout reaches 0.81, which is improved by 20.90%. The position and power of the polygonal layout are further optimized respectively by particle swarm optimization (PSO), and the illuminance uniformity and signal‐to‐noise ratio quality factors (Q) of the polygonal layout can be increased from 0.81 and 19.31 to 0.88 and 37.38 after optimization, respectively. The illuminance uniformity increased by 8.64%, and the Q increased by 93.58%. The experimental system of polygonal layout optimized by PSO is built, by comparing the simulation and experimental results, the average illuminance error of the simulation and experiment is 10.58%, the illuminance uniformity error is 5.68%, and the illuminance distribution trend is consistent.

The error model based on the special Euclidean group SE(3) of the INS: Comparison and extension

In this paper, the error model based on the Lie group is investigated with main focus on the Special Euclidean group SE(3) based error model construction and filtering comparison for the inertial navigation system (INS). The conventional theory of the SE(3) based error model is only focused on the state error definition. Moreover, the attitude representation is usually used by Euler angle. In order to enrich the family of the SE(3) based error model of the INS, this paper is to establish a theoretical bridge that can connect the attitude estimation to the error model construction for the Lie group. For this theoretical bridge, the SE(3) based error model based on Rodrigues parameter and modified Rodrigues parameter is proposed to fill the gap of the family of the SE(3) based error model. Further, based on the integrity of the SE(3) based error model, a uniform error model is also proposed to unify the family of the SE(3) based error model using the attitude transformation and direction cosine matrix. The field tests are carried out to assess the comprehensive performance of the existing SE(3) based error model under different conditions for the INS.

NIPG Method on Shishkin Mesh for Singularly Perturbed Convection–Diffusion Problem with Discontinuous Source Term

In this paper, we provide numerical approaches for singularly perturbed convection–diffusion problem with a discontinuous source term. As a result of the discontinuity in the source term, the problem highlights an exponential-type boundary layer and a weak interior layer. To handle both the layers at the same time, we design a nonsymmetric discontinuous Galerkin finite element technique with interior penalties (NIPG). The problem is solved using the standard Shishkin mesh. In the DG norm, uniform error estimates are derived for the problem. To verify the theoretical estimates, numerical tests are carried out.

Self-powered acceleration sensors arrayed by swarm intelligence for table tennis umpiring system

Table tennis competition is voted as one of the most popular competitive sports. The referee umpires the competition mainly based on visual observation and experience, which may make misjudgments on competition results due to the referee’s subjective uncertainty or imprecision. In this work, a novel intelligent umpiring system based on arrayed self-powered acceleration sensor nodes was presented to enhance the competition accuracy. A sensor node array model was established to detect ball collision point on the table tennis table. This model clearly illuminated the working mechanism of the proposed umpiring system. And an improved particle swarm optimization (level-based competitive swarm optimization) was applied to optimize the arrayed sensor nodes distribution by redefining the representations and update rules of position and velocity. The optimized results showed that the number of sensors decreased from 58 to 51. Also, the reliability of the optimized nodes distribution of the table tennis umpiring system has been verified theoretically. The results revealed that our system achieved a precise detection of the ball collision point with uniform error distances below 3.5 mm. Besides, this research offered an in-depth study on intelligent umpiring system based on arrayed self-powered sensor nodes, which will improve the accuracy of the umpiring of table tennis competition.

A higher‐order hybrid spline difference method on adaptive mesh for solving singularly perturbed parabolic reaction–diffusion problems with robin‐boundary conditions

AbstractWe propose a hybrid numerical scheme to discretize a class of singularly perturbed parabolic reaction–diffusion problems with robin‐boundary conditions on an equidistributed grid. The hybrid difference scheme is developed by using a modified backward difference scheme in time, a combination of the cubic spline and exponential spline difference scheme in space. The proposed scheme uses a cubic spline difference scheme for the discretization of robin‐boundary conditions. For the time discretization of the problem, we use the standard uniform mesh while a layer adapted equidistributed grid is generated for the spatial discretization. By equidistributing a curvature‐based monitor function, the spatial adaptive grid is able to capture the presence of parabolic boundary layers without using any prior information about the solution. Parameter uniform error estimates are derived to illustrate an optimal convergence of first‐order in time and second‐order in space for the proposed discretization. The accuracy of the proposed scheme is confirmed by the numerical experiments that underpin the theoretical analysis.

Improved Uniform Error Bounds on Time-Splitting Methods for the Long-Time Dynamics of the Dirac Equation with Small Potentials

We establish improved uniform error bounds on time-splitting methods for the long-time dynamics of the Dirac equation with small electromagnetic potentials characterized by a dimensionless parameter $\varepsilon\in (0, 1]$ representing the amplitude of the potentials. We begin with a semi-discritization of the Dirac equation in time by a time-splitting method, and then followed by a full-discretization in space by the Fourier pseudospectral method. Employing the unitary flow property of the second-order time-splitting method for the Dirac equation, we prove uniform error bounds at $C(t)\tau^2$ and $C(t)(h^m+\tau^2)$ for the semi-discretization and full-discretization, respectively, for any time $t\in[0,T_\varepsilon]$ with $T_\varepsilon = T/\varepsilon$ for $T > 0$, which are uniformly for $\varepsilon \in (0, 1]$, where $\tau$ is the time step, $h$ is the mesh size, $m\geq 2$ depends on the regularity of the solution, and $C(t) = C_0 + C_1\varepsilon t\le C_0+C_1T$ grows at most linearly with respect to $t$ with $C_0\ge0$ and $C_1>0$ two constants independent of $t$, $h$, $\tau$ and $\varepsilon$. Then by adopting the regularity compensation oscillation (RCO) technique which controls the high frequency modes by the regularity of the solution and low frequency modes by phase cancellation and energy method, we establish improved uniform error bounds at $O(\varepsilon\tau^2)$ and $O(h^m +\varepsilon\tau^2)$ for the semi-discretization and full-discretization, respectively, up to the long-time $T_\varepsilon$. Numerical results are reported to confirm our error bounds and to demonstrate that they are sharp. Comparisons on the accuracy of different time discretizations for the Dirac equation are also provided.

Ride comfort improvement using robust multi-input multi-output fuzzy logic dynamic compensator

This article aims to improve the ride comfort of a vehicle keeping a satisfactory without road-holding performance by a novel adaptive control method that is insensitive to unknown system dynamics, model parameter changes, external disturbances and without any loss in suspension working space. With this purpose in mind, a new fuzzy integrated model-based adaptive control law for vehicle suspension systems is proposed in this study to improve the ride comfort and to ensure the robustness of system towards unknown model parameters and external disturbances. First, a model-based adaptive control law, which has the robust characteristics is presented. Afterwards, a multi-input multi-output fuzzy logic controller is designed to determine the controller gains dynamically. The stability of controller is ensured by Lyapunov Theory to achieve uniform boundedness error convergence. A 4 degree-of-freedom half-car model with active suspension system is used in this study to assess the performance of the controller. The results are compared among passive, model-based adaptive control law-controlled and novel fuzzy model-based adaptive control law-controlled systems. It has been concluded that fuzzy model-based adaptive control law further attenuates linear and angular motions of the vehicle increasing the ride comfort. The robustness is also verified for vehicle components having different possible parameter values. It is noteworthy that suspension working length returns to its initial position. Thus, the vehicle ride comfort is improved with no suspension working space loss. Finally, the economic feasibility of controllers has been checked in terms of energy consumption.

Grid Adaptive Bucketing Algorithm Based on Differential Privacy

With the popularity of location-based services, the restricted relationship between the availability of big location data and user’s privacy security has become a challenging issue. The spatial division is an effective measure for statistical location data characteristics. This paper proposes a grid adaptive bucketing algorithm based on differential privacy (GAB) to solve the problem that the existing differential privacy location data division method does not fully consider the distribution characteristics of spatial data points and noise superimposition. The algorithm first divides the mobile app location spatial data set into two layers of grids. It then performs adaptive binning judgment according to the square sum error value in the divided area. It puts areas with similar distributions into the same bucket, reducing the uniform assumption error and the noise error caused by many blank areas. Finally, a noise allocation strategy based on Hopkins statistic is adopted to achieve a reasonable noise allocation. The experimental results on two real large-scale location data sets, Checkin and Beijing, show that, compared with other classical algorithms based on differential privacy space partitioning, the GAB algorithm performs better in common evaluation indicators such as relative error and absolute error, which means GAB can obtain better range query effects and privacy protection effects.

Parallel transmission 2D RARE imaging at 7T with transmit field inhomogeneity mitigation and local SAR control

PurposeWe develop and test a parallel transmit (pTx) pulse design framework to mitigate transmit field inhomogeneity with control of local specific absorption rate (SAR) in 2D rapid acquisition with relaxation enhancement (RARE) imaging at 7T. MethodsWe design large flip angle RF pulses with explicit local SAR constraints by numerical simulation of the Bloch equations. Parallel computation and analytical expressions for the Jacobian and the Hessian matrices are employed to reduce pulse design time. The refocusing-excitation “spokes” pulse pairs are designed to satisfy the Carr-Purcell-Meiboom-Gill (CPMG) condition using a combined magnitude least squares-least squares approach. ResultsIn a simulated dataset, the proposed approach reduced peak local SAR by up to 56% for the same level of refocusing uniformity error and reduced refocusing uniformity error by up to 59% (from 32% to 7%) for the same level of peak local SAR compared to the circularly polarized birdcage mode of the pTx array. Using explicit local SAR constraints also reduced peak local SAR by up to 46% compared to an RF peak power constrained design. The excitation and refocusing uniformity error were reduced from 20%–33% to 4%–6% in single slice phantom experiments. Phantom experiments demonstrated good agreement between the simulated excitation and refocusing uniformity profiles and experimental image shading. ConclusionPTx-designed excitation and refocusing CPMG pulse pairs can mitigate transmit field inhomogeneity in the 2D RARE sequence. Moreover, local SAR can be decreased significantly using pTx, potentially leading to better slice coverage, enabling larger flip angles or faster imaging.

Improved Uniform Error Bounds on Time-Splitting Methods for Long-Time Dynamics of the Nonlinear Klein--Gordon Equation with Weak Nonlinearity

We establish improved uniform error bounds on a second-order Strang time-splitting method which is equivalent to an exponential wave integrator for the long-time dynamics of the nonlinear Klein--Gordon equation (NKGE) with weak cubic nonlinearity, whose strength is characterized by $\varepsilon^2$ with $0 < \varepsilon \leq 1$ a dimensionless parameter. Actually, when $0 < \varepsilon \ll 1$, the NKGE with $O(\varepsilon^2)$ nonlinearity and $O(1)$ initial data is equivalent to that with $O(1)$ nonlinearity and small initial data, the amplitude of which is at $O(\varepsilon)$. We begin with a semidiscretization of the NKGE by the second-order time-splitting method and derive a full-discretization by the Fourier spectral method in space. Employing the regularity compensation oscillation technique which controls the high frequency modes by the regularity of the exact solution and analyzes the low frequency modes by phase cancellation and energy method, we carry out the improved uniform error bounds at $O(\varepsilon^2\tau^2)$ and $O(h^m+\varepsilon^2\tau^2)$ for the second-order semidiscretization and full-discretization up to the long time $T_\varepsilon = T/\varepsilon^2$ with $T$ fixed, respectively. Extensions to higher-order time-splitting methods and the case of an oscillatory complex NKGE are also discussed. Finally, numerical results are provided to confirm the improved error bounds and to demonstrate that they are sharp.

Uniform Finite Element Error Estimates with Power-Type Asymptotic Constants for Unsteady Navier-Stokes Equations.

Uniform error estimates with power-type asymptotic constants of the finite element method for the unsteady Navier–Stokes equations are deduced in this paper. By introducing an iterative scheme and studying its convergence, we firstly derive that the solution of the Navier–Stokes equations is bounded by power-type constants, where we avoid applying the Gronwall lemma, which generates exponential-type factors. Then, the technique is extended to the error estimate of the long-time finite element approximation. The analyses show that, under some assumptions on the given data, the asymptotic constants in the finite element error estimates for the unsteady Navier–Stokes equations are uniformly power functions with respect to the initial data, the viscosity, and the body force for all time . Finally, some numerical examples are shown to verify the theoretical predictions.