Error Estimates Research Articles

Strong error bounds for Trotter and strang-splittings and their implications for quantum chemistry

Efficient error estimates for the Trotter product formula are central in quantum computing, mathematical physics, and numerical simulations. However, the Trotter error's dependency on the input state and its application to unbounded operators remains unclear. Here, we present a general theory for error estimation, including higher-order product formulas, with explicit input state dependency. Our approach overcomes two limitations of the existing operator-norm estimates in the literature. First, previous bounds are too pessimistic as they quantify the worst-case scenario. Second, previous bounds become trivial for unbounded operators and cannot be applied to a wide class of Trotter scenarios, including atomic and molecular Hamiltonians. Our method enables analytical treatment of Trotter errors in chemistry simulations, illustrated through a case study on the hydrogen atom. Our findings reveal the following: (i) for states with fat-tailed energy distribution, such as low-angular-momentum states of the hydrogen atom, the Trotter error scales worse than expected (sublinearly) in the number of Trotter steps; (ii) certain states do not admit an advantage in the scaling from higher-order Trotterization and, thus, the higher-order Trotter hierarchy breaks down for these states, including the hydrogen atom's ground state; (iii) the scaling of higher-order Trotter bounds might depend on the order of the Hamiltonians in the Trotter product for states with fat-tailed energy distribution. Physically, the enlarged Trotter error is caused by the atom's ionization due to the Trotter dynamics. Mathematically, we find that certain domain conditions are not satisfied by some states so higher moments of the potential and kinetic energies diverge. Our analytical error analysis agrees with numerical simulations, indicating that we can estimate the state-dependent Trotter error scaling genuinely. Published by the American Physical Society 2024

The hyperbolic multi-term time fractional integro-differential equation with generalized Caputo derivative and error estimate in $$L_{p,\gamma ,\upsilon }$$ space

The hyperbolic multi-term time fractional integro-differential equation with generalized Caputo derivative and error estimate in $$L_{p,\gamma ,\upsilon }$$ space

Error Analysis of the Vector Penalty-Projection Methods for the Time-Dependent Stokes Equations with Open Boundary Conditions

Abstract We present in this paper a rigorous error analysis of the vector penalty-projection method for solving the time-dependent incompressible Stokes equations with open boundary conditions on part of the boundary. First, we prove the stability of the scheme. Then we provide an error analysis for the second-order vector penalty-projection method which shows that the convergence rate of the error on the velocity and the pressure is of order 2 in l ∞ ⁢ ( L 2 ⁢ ( Ω ) ) l^{\infty}(\mathbf{L}^{2}(\Omega)) and l 2 ⁢ ( L 2 ⁢ ( Ω ) ) l^{2}(L^{2}(\Omega)) respectively. In addition, it is shown that the splitting errors of the method varies as O ⁢ ( ε ) \mathcal{O}(\varepsilon) , where 𝜀 is a penalty parameter chosen as small as desired. Several numerical tests in agreement with the theoretical results are presented. To the best of our knowledge, this paper provides the first rigorous proof of optimal error estimates for second-order splitting schemes with open boundary conditions.

Deep Learning‐Based Anomaly Diagnosis Method for Capacitive Voltage Transformer Measurement Error in PIoT

ABSTRACTIn the era of power Internet of Things (PIoT), the accuracy of capacitive voltage transformers (CVTs) is crucial for maintaining the reliability of voltage measurement and protection systems in smart grids, thereby contributing to overall grid stability and efficiency. Accurate and timely detection of anomalies in CVT measurement errors is essential for preventing equipment failures, reducing maintenance costs, and improving overall system reliability. However, existing anomaly diagnosis methods often rely on statistical analysis and rule‐based approaches, which have limitations in capturing complex patterns and adapting to evolving anomaly types. This paper proposes a novel deep learning‐based anomaly diagnosis method for CVT measurement error in PIoT, called LSTM‐CVT. The proposed method leverages a long short‐term memory (LSTM) neural network architecture with three key strategies: bidirectional temporal dependency capture, hierarchical feature learning, and joint anomaly diagnosis and error estimation. The experimental results demonstrate the superior performance of LSTM‐CVT compared to state‐of‐the‐art baseline methods.

The velocity tracking problem for Navier–Stokes equations with pointwise-integral control constraints in time-space

In this work, we study the velocity tracking control problem of the evolutionary two-dimensional Navier–Stokes system. The regularizing Tikhonov term is not involved in the cost functional and pointwise-integral control constraints in time-space are considered. First- and second-order optimality conditions are established using a suitable extended cone for the sufficient ones. We also address the numerical approximation of the control problem, proving the convergence of the discrete problems and establishing error estimates between the optimal discrete and continuous states.

International intercomparison of Radon 222 activity concentration calibration facilities

The Coalition of International Radon Associations (COIRA) organised an inter-comparison of Rn-222 (radon) activity concentrations reported by calibration laboratories. A set of three AlphaGUARDs were used as transfer reference instruments against which to compare reported Rn-222 activity concentrations. Rn-222 activity concentration calibration facilities (sometimes termed chambers) from seven countries (Australia, US, Czech Republic, Spain, England, Sweden, Canada), and three continents participated in this project. The objective of the study was to provide information useful to calibration chamber operators and public health officials in the improvement of measurement and control systems, the maintenance of performance standards for measurements, and regulatory requirements for calibrations. This work builds upon and expands previous interlaboratory comparisons and provides data for estimating and using calibration uncertainty values as part of overall field error estimates and limits. A simple proportional difference between laboratories is presented here, calculated as the average for N hours of each hour’s difference between the laboratory’s concentration and the average of the three reference instruments. This percent difference ranged from less than 0.5% to just less than 8%. This work demonstrates that the ANSI/AARST standards limit of 8% for the estimated unexpanded (one sigma) individual calibration estimated uncertainty for continuous radon monitor calibration facilities in the US is achievable. However, given the few standards regarding calibration of Rn-222 activity concentration measurement instruments that themselves are often used to calibrate secondary and tertiary Rn-222 calibration facilities, there is a great need for continued interlaboratory comparisons to harmonise and document the calibration of Rn-222 activity concentrations.

A demonstration of the effect of fringe-rate filtering in the hydrogen epoch of reionization array delay power spectrum pipeline

Abstract Radio interferometers targeting the 21cm brightness temperature fluctuations at high redshift are subject to systematic effects that operate over a range of different timescales. These can be isolated by designing appropriate Fourier filters that operate in fringe-rate (FR) space, the Fourier pair of local sidereal time (LST). Applications of FR filtering include separating effects that are correlated with the rotating sky vs. those relative to the ground, down-weighting emission in the primary beam sidelobes, and suppressing noise. FR filtering causes the noise contributions to the visibility data to become correlated in time however, making interpretation of subsequent averaging and error estimation steps more subtle. In this paper, we describe fringe rate filters that are implemented using discrete prolate spheroidal sequences, and designed for two different purposes – beam sidelobe/horizon suppression (the ‘mainlobe’ filter), and ground-locked systematics removal (the ‘notch’ filter). We apply these to simulated data, and study how their properties affect visibilities and power spectra generated from the simulations. Included is an introduction to fringe-rate filtering and a demonstration of fringe-rate filters applied to simple situations to aid understanding.

Approximation of optimal feedback controls for stochastic reaction-diffusion equations

In this paper we present a method to approximate optimal feedback controls for stochastic reaction-diffusion equations. We derive two approximation results providing the theoretical foundation of our approach and allowing for explicit error estimates. The approximation of optimal feedback controls by neural networks is discussed as an explicit application of our method. We illustrate our findings in the case of a linear quadratic control problem with a numerical example.

Fast standard error estimation for joint models of longitudinal and time-to-event data based on stochastic EM algorithms.

Maximum likelihood inference can often become computationally intensive when performing joint modeling of longitudinal and time-to-event data, due to the intractable integrals in the joint likelihood function. The computational challenges escalate further when modeling HIV-1 viral load data, owing to the nonlinear trajectories and the presence of left-censored data resulting from the assay's lower limit of quantification. In this paper, for a joint model comprising a nonlinear mixed-effect model and a Cox Proportional Hazards model, we develop a computationally efficient Stochastic EM (StEM) algorithm for parameter estimation. Furthermore, we propose a novel technique for fast standard error estimation, which directly estimates standard errors from the results of StEM iterations and is broadly applicable to various joint modeling settings, such as those containing generalized linear mixed-effect models, parametric survival models, or joint models with more than two submodels. We evaluate the performance of the proposed methods through simulation studies and apply them to HIV-1 viral load data from six AIDS Clinical Trials Group studies to characterize viral rebound trajectories following the interruption of antiretroviral therapy (ART), accounting for the informative duration of off-ART periods.

Local error analysis of L1 scheme for time-fractional diffusion equation on a star-shaped pipe network

Abstract The numerical analysis for differential equations on networks has become a significant issue in theory and diverse fields of applications. Nevertheless, solving time-fractional diffusion problem on metric graphs has been less studied so far, as one of the major challenging tasks of this problem is the weak singularity of solution at initial moment. In order to overcome this difficulty, a new L1-finite difference method considering the weak singular solution at initial time is proposed in this paper. Specifically, we utilize this method on temporal graded meshes and spacial uniform meshes, which has a new treatment at the junction node of metric graph by employing Taylor expansion method, Neumann-Kirchhoff and continuity conditions. Over the whole star graph, the optimal error estimate of this fully discrete scheme at each time step is given. Also, the convergence analysis for a discrete scheme that preserves the Neumann-Kirchhoff condition at each time level is demonstrated. Finally, numerical results show the effectiveness of proposed full-discrete scheme, which can be applied to star graphs and even more general graphs with multiple cross points.

Quadratic constraint consistency in the projection-free approximation of harmonic maps and bending isometries

We devise a projection-free iterative scheme for the approximation of harmonic maps that provides a second-order accuracy of the constraint violation and is unconditionally energy stable. A corresponding error estimate is valid under a mild but necessary discrete regularity condition. The method is based on the application of a BDF2 scheme and the considered problem serves as a model for partial differential equations with holonomic constraint. The performance of the method is illustrated via the computation of stationary harmonic maps and bending isometries.

Forecast Errors and Welfare Conclusions Based on the Flyvbjerg Database

Abstract The so-called Flyvbjerg database is the largest source of data on the performance of major investment projects. It has generated influential analyses of the magnitude of and reasons for cost overruns and demand shortfalls in major projects. Those analyses have demonstrated, among other things, the systematic presence of large forecast errors in both construction costs and in user demand in the first year of operation. They have also linked those results to the social welfare consequences of the underlying projects, suggesting that the large and systematic forecast errors are indicative of welfare destruction. Given how influential those analyses have been, this paper examines the link between the database, empirical analyses thereof, and social benefit–cost analysis (BCA). To that end, both the measurement of variables in the database and the estimation of forecast errors are contrasted against BCA. The conditions for the estimated forecast errors to approximate those obtained from a BCA are spelled out, and the scope for drawing welfare conclusions based on those estimates is discussed. Furthermore, numerical simulations are presented to explore whether the estimated forecast errors do indeed imply likely welfare destruction. The simulations suggest that as large as the forecast errors are, welfare destruction is no foregone conclusion.

Local discontinuous Galerkin methods with implicit–explicit BDF time marching for Newell–Whitehead–Segel equations

The Newell–Whitehead–Segel type equations with time-dependent Dirichlet boundary conditions are solved by the local discontinuous Galerkin (LDG) method coupled with the implicit–explicit backward difference formulas (IMEX-BDF). With a suitable setting of numerical fluxes and by the aid of the multiplier technique and the a priori error assumption technique, the optimal error estimate for the corresponding fully discrete LDG-IMEX-BDF schemes is obtained by energy analysis, under the condition τ ≤ C h 1 / s , where h and τ are mesh size and time step, respectively, the positive constant C is independent of h, and s = 1 , … , 5 is the order of the IMEX-BDF method. Numerical experiments are also presented to verify the accuracy of the considered schemes.

Reparameterization of the mW model to accurately predict the experimental phase diagram of methane hydrate.

Due to their high computational efficiency, the coarse-grained water models are of particular importance for practical molecular simulations of gas hydrates. In these models, the mW model is successfully used to study many thermodynamics and dynamics of methane hydrate. Yet, despite several decades of intense research, the mW model is still found to overestimate the melting temperature of methane hydrate. We here employ the minimum mean squared error estimation to revisit the key parameter of the mW model, which determines the strength of the tetrahedral angle of the water system. Relying on the free energy calculations, we first estimate the chemical potentials of water in the liquid phase for temperatures at which methane hydrate forms. We then turn to the mean squared error to describe the chemical potential deviation between the mW model and the TIP4P/ice model (the latter could reproduce the experimental phase diagram of methane hydrate). By minimizing the mean squared error, we finally have an optimized parameter for the mW model. In this part, we also discuss the pressure effect on such reparameterization procedure. Moreover, relying on the direct coexistence method, the melting temperature determined using the reparameterized mW model is found to be consistent with the experimental data. This strategy provides a means to improve the coarse-grained model to match the experimental observations for temperatures in the range of interest.

Estimation of direction and zero errors of satellite laser terminals in low-light conditions based on machine learning

In the assembly, launch, and on-orbit operation of satellite optical communication terminals, small deviations are difficult to avoid, which can lead to pointing errors and challenges to the establishment of optical communication links. To estimate the pointing errors of on-orbit satellite terminals, a calibration algorithm is developed based on lunar surface imagery. First, a feature extraction algorithm for low-light images is employed to process consecutive frames of low-light images to obtain a lunar surface feature map. Then, by combining the feature map and error estimation model, predictions of direction errors and zero errors were achieved. The ground validation results demonstrate the effectiveness and feasibility of the proposed on-orbit error estimation algorithm under low-signal-to-noise-ratio conditions.

Is Binary Grading Better Than WHO System for Grading Epithelial Dysplasia? A Systematic Review and Meta-Analysis.

This meta-analysis summarizes the current evidence on the intra- and inter-observer agreement between WHO and the binary grading systems used to assess epithelial dysplasia (ED). A systematic search for observational studies that compared the level of agreement among pathologists between WHO and binary grading systems for ED was conducted using three databases: Medline, Scopus, and EBSCOhost. For the meta-analysis, summary estimations of kappa value (κ) and standard error (SE) were utilized. The pooled analysis of observations by 46 pathologists from a total of eight studies showed better interobserver agreement in the interpretation of ED for the binary system (κ = 0.31; 95% confidence interval [CI], 0.23-0.40) in comparison with the WHO (κ = 0.14; 95% CI, 0.10-0.19). The intra-observer agreement was reported only by five studies and was also found to be higher for the binary system (κ = 0.44; 95% CI, 0.31-0.57) compared to the WHO (κ = 0.25; 95% CI, 0.11-0.39). Our results validate that the binary system has better overall intra-observer and interobserver agreement than the WHO system. Further studies with larger cohorts are mandatory before clinically relevant conclusions are drawn, as evidence remains inadequate.

FDI, industrialisation and environmental quality in SSA—the role of institutional quality towards environmental sustainability

In light of the quest to achieve economic development without compromising environmental quality, we empirically examine whether institutional quality (INSQY) can help moderate the possible harmful effects of foreign direct investments (FDI) and industrialisation on environmental quality in sub-Saharan Africa (SSA). We utilise the Driscoll and Kraay standard error estimation technique on a panel of 45 SSA countries from 2000 to 2019. The results indicate that FDI and industrialisation generally have a significant harmful effect on the environment. Our findings reveal that INSQY directly promotes environmental quality. Notably, the results confirm that INSQY plays a stimulating role in mitigating the adverse effects of FDI and industrialisation on environmental quality. The results further validate the environmental Kuznets curve (EKC) hypothesis in SSA. These findings contribute to environmental sustainability literature and offer policymakers insights on how INSQY can enhance environmental quality. Our empirical results are also robust to different estimation techniques, such as the two-stage least squares. We recommend SSA leaders strengthen institutional capacities, enforce environmental regulations, and implement strict policies to ensure environmental quality while promoting industrialisation and FDI inflows.

Stopping criteria for nonlinear variational problems: iterative approach

AbstractIn this paper, we present and study a stopping criteria based on a priori error estimate for nonlinear variational problems. We show that for nonlinear variational problems, the a priori error estimate is divided into two sub errors namely: the discretization error and the linearization error. We then used that representation to devise a stopping criteria that help to reduce the computational time. The methodology is applied to three problems and the numerical results demonstrate the superiority of the new approach.

Small-sample Bayesian error estimation for ergodic, chaotic systems of ordinary differential equations

The discretization of chaotic systems introduces statistical errors in addition to discretization errors into the estimation of quantities of interest. In order to efficiently arrive at estimates of quantities of interest, these two forms of error should be balanced; however, simulations are run without knowledge of the true/asymptotic outputs of interest or their error behaviors. In this work, we develop a framework for error modeling and identification using small-sample Bayesian inference that allows approximation of the optimal balance between sampling time and discretization precision without the computation of high-cost libraries of reference solutions. The result enables the possibility of running chaotic and turbulent simulations in a way that minimizes the total error between sampling and discretization without prior knowledge of the error behavior of the system.

Investigating Model Form Error Estimation for Sparse Data

Computational simulations of dynamical systems involve the use of mathematical models and algorithms to mimic and analyze complex real-world phenomena. By leveraging computational power, simulations enable researchers to explore and understand systems that are otherwise challenging to study experimentally. They offer a cost-effective and efficient means to predict and analyze the behavior of physical, biological, and social systems. However, model form error arises in computational simulations from simplifications, assumptions, and limitations inherent in the mathematical model formulation. Several methods for addressing model form error have been proposed in the literature, but their robustness in the face of challenges inherent to real-world systems has not been thoroughly investigated. In this work, a data assimilation-based approach for model form error estimation is investigated in the presence of sparse observation data. Including physics-based domain knowledge to improve estimation performance is also explored. The Lotka-Volterra equations are employed as a simple computational simulation for demonstration.