Error Analysis Of Approximations Research Articles

Numerical Study of Multi-Term Time-Fractional Sub-Diffusion Equation Using Hybrid L1 Scheme with Quintic Hermite Splines

Anomalous diffusion of particles has been described by the time-fractional reaction–diffusion equation. A hybrid formulation of numerical technique is proposed to solve the time-fractional-order reaction–diffusion (FRD) equation numerically. The technique comprises the semi-discretization of the time variable using an L1 finite-difference scheme and space discretization using the quintic Hermite spline collocation method. The hybrid technique reduces the problem to an iterative scheme of an algebraic system of equations. The stability analysis of the proposed numerical scheme and the optimal error bounds for the approximate solution are also studied. A comparative study of the obtained results and an error analysis of approximation show the efficiency, accuracy, and effectiveness of the technique.

Variable Structure Controller for Energy Savings in an Underwater Sensor Platform.

This paper introduces a new variable structure controller designed for depth control of an autonomous underwater sensor platform equipped with a variable buoyancy module. To that end, the prototype linear model is presented, and a finite element-based method is used to estimate one of its parameters, the hull deformation due to pressure. To manage potential internal disturbances like hull deformation or external disturbances like weight changes, a disturbance observer is developed. An analysis of the observer steady-state estimation error in relation to input disturbances and system parameter uncertainties is developed. The locations of the observer poles according to its parameters are also identified. The variable structure controller is developed, keeping energy savings in mind. The proposed controller engages when system dynamics are unfavorable, causing the vehicle to deviate from the desired reference, and disengages when dynamics are favorable, guiding the vehicle toward the target reference. A detailed analysis determines the necessary switching control actions to ensure the system reaches the desired reference. Finally, simulations are run to compare the proposed controller's performance with that of PID-based controllers recently developed in the literature, assessing dynamic response and energy consumption under various operating conditions. Both the VBM- and propeller-actuated vehicles were evaluated. The results demonstrate that the proposed controller achieves an average energy consumption reduction of 22% compared to the next most efficient PID-based controller for the VBM-actuated vehicle, though with some impact on control performance.

Virtual element method for solving a viscoelastic contact problem with long memory

In this paper, we use the virtual element method to solve a history-dependent hemivariational inequality arising in contact problems. The contact problem concerns the deformation of a viscoelastic body with long memory, subjected to a contact condition with non-monotone normal compliance and unilateral constraints. A fully discrete scheme based on the trapezoidal rule for the discretization of the time integration and the virtual element method for the spatial discretization are analyzed. We provide a unified priori error analysis for both internal and external approximations. For the linear virtual element method, we obtain the optimal order error estimate. Finally, three numerical examples are reported, providing numerical evidence of the theoretically predicted optimal convergence orders.

A mean flow velocity estimation scheme based on flow field characteristics and distribution point optimization for carbon monitoring in coal-fired power plants: A numerical study

CO2 online monitoring provides a method to obtain carbon emissions by directly quantifying the flue gas flow rate and CO2 concentration. However, significant flow rate measurement errors arise due to the complexity of turbulence in large-diameter stacks. To address the issue, this study utilizes turbulence numerical simulation combined with mean velocity estimation error analysis to investigate the flow field of typical stacks in coal-fired power plants. The simulation results demonstrated that the evolution of turbulence exaggerates velocities. The flow field exhibits stable turbulence characteristics in a single-introduced stack, while in the double-introduced stack, the flow field dynamically evolves due to asymmetric collisions and abrupt expansions. Furthermore, the effects of load and temperature on the estimation error of flow velocity are established. The optimum monitoring height, the ideal chord angle and the stepped number of deployment points based on the log-linear method are determined. Within the single-introduced stack, a correction factor is introduced and a corresponding formula is established that links estimation errors with monitoring heights. This resulted in a dramatic reduction of the maximum estimation error from 14% to 0.3%. Finally, the systematic and simplified optimization scheme for estimating mean flow velocity is summarised, which provides a powerful reference for flow rate measurement for online CO2 monitoring in coal-fired power plants.

State of health estimation embedded with hardware accelerator based on long short-term memory combined with Bayesian optimization considering extracted health indicator in charging conditions

Recently, the importance of renewable energy has increased as fossil fuel use is regulated worldwide to protect the environment. Accordingly, the use of electric vehicles (EVs) that use a battery as a power source is increasing and estimating the battery state for an efficient and safe operation of EV is a very significant task. This study proposes a method to estimate the state-of-health (SOH) in which batteries are a major state indicator. A health indicator (HI) capable of determining the state of the battery is derived from the charging part by considering the stopping situation in which there is a slight variation in the battery characteristics of the EV. As an integrity indicator, the partial capacity according to the voltage range set by the user can be considered, and the correlation between the SOH and HI is analyzed for validity verification. To learn the derived HI, a long short-term memory (LSTM) model, a type of deep-neural network, is designed, and model tuning was performed through Bayesian optimization. The absolute error of SOH estimation for the proposed LSTM model is 1.1 %, compared and validated against recurrent neural network (RNN) (1.7 %) and gated recurrent unit (GRU) (1.85 %). After validating the performance of the LSTM algorithm, an analysis of SOH estimation error (1 %) based on LSTM was conducted following pack-level experiments to assess scalability. Finally, for evaluation of operation within a hardware accelerator (HA) for integration into EVs, NVIDIA Jetson Nano was utilized. Upon mounting the algorithm on Jetson Nano, the mean absolute error (MAE) was 0.6581, mean absolute percentage error (MAPE) was 0.7209, and root mean square error (RMSE) was 0.7133, demonstrating excellent estimation performance.

Weak error analysis for a nonlinear SPDE approximation of the Dean–Kawasaki equation

We consider a nonlinear SPDE approximation of the Dean–Kawasaki equation for independent particles. Our approximation satisfies the physical constraints of the particle system, i.e. its solution is a probability measure for all times (preservation of positivity and mass conservation). Using a duality argument, we prove that the weak error between particle system and nonlinear SPDE is of the order N-1-1/(d/2+1)logN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N^{-1-1/(d/2+1)}\\log N$$\\end{document}. Along the way we show well-posedness, a comparison principle, and an entropy estimate for a class of nonlinear regularized Dean–Kawasaki equations with Itô noise.

Error analysis for the finite element approximations of Dirichlet parabolic boundary control problem with measure data

This article investigates the convergence analysis of finite element method for Dirichlet boundary control problem governed by parabolic equation with measure data. The presence of measure data in the source term and Dirichlet boundary control lead the solution of the state equation to have low regularity, which creates a challenging task for the error analysis. We prove the existence, uniqueness and regularity results of the solution to the control problem. For the discretization of the state and adjoint variables, we employ continuous piecewise linear polynomials while the control variable is approximated by piecewise constant functions. The backward Euler technique provides the foundation for the temporal discretization. For the completely discrete optimal control problem, a priori error bounds for the state variable in the L2(0,T;L2(Ω))-norm and for the control variable in the L2(0,T;L2(∂Ω))-norm are derived. Numerical experiment is performed to validate our theoretical analysis.

Online regularized learning algorithm for functional data

In recent years, functional linear models have attracted growing attention in statistics and machine learning for recovering the slope function or its functional predictor. This paper considers online regularized learning algorithm for functional linear models in a reproducing kernel Hilbert space. It provides convergence analysis of excess prediction error and estimation error with polynomially decaying step-size and constant step-size, respectively. Fast convergence rates can be derived via a capacity dependent analysis. Introducing an explicit regularization term extends the saturation boundary of unregularized online learning algorithms with polynomially decaying step-size and achieves fast convergence rates of estimation error without capacity assumption. In contrast, the latter remains an open problem for the unregularized online learning algorithm with decaying step-size. This paper also demonstrates competitive convergence rates of both prediction error and estimation error with constant step-size compared to existing literature.

Expected Policy Gradient for Network Aggregative Markov Games in Continuous Space.

In this article, we investigate the Nash-seeking problem of a set of agents, playing an infinite network aggregative Markov game. In particular, we focus on a noncooperative framework where each agent selfishly aims at maximizing its long-term average reward without having explicit information on the model of the environment dynamics and its own reward function. The main contribution of this article is to develop a continuous multiagent reinforcement learning (MARL) algorithm for the Nash-seeking problem in infinite dynamic games with convergence guarantee. To this end, we propose an actor-critic MARL algorithm based on expected policy gradient (EPG) with two general function approximators to estimate the value function and the Nash policy of the agents. We consider continuous state and action spaces and adopt a newly proposed EPG to alleviate the variance of the gradient approximation. Based on such formulation and under some conventional assumptions (e.g., using linear function approximators), we prove that the policies of the agents converge to the unique Nash equilibrium (NE) of the game. Furthermore, an estimation error analysis is conducted to investigate the effects of the error arising from function approximation. As a case study, the framework is applied on a cloud radio access network (C-RAN) by modeling the remote radio heads (RRHs) as the agents and the congestion of baseband units (BBUs) as the dynamics of the environment.

Truncation Error Analysis for an Accurate Nonlocal Approximation to Manifold Poisson Models with Dirichlet Boundary

Truncation Error Analysis for an Accurate Nonlocal Approximation to Manifold Poisson Models with Dirichlet Boundary

Moving-Target Circumnavigation Using Adaptive Neural Anti-Synchronization Control via Distance-Only Measurements.

In this work, we investigate the unknown moving-target circumnavigation problem in GPS-denied environments. A minimum of two tasking agents is excepted to circumnavigate the target cooperatively and symmetrically without prior knowledge of its position and velocity in order to achieve optimal sensor coverage persistently for the target. To achieve this goal, we develop a novel adaptive neural anti-synchronization (AS) controller. Based on relative distance-only measurements between the target and two tasking agents, a neural network is used to approximate the displacement of the target such that the position of the target can be estimated accurately and in real time. On this basis, a target position estimator is designed by considering whether all agents are in the same coordinate system. Furthermore, an exponential forgetting factor and a new information utilization factor are introduced to improve the accuracy of the aforementioned estimator. Rigorous convergence analysis of position estimation errors and AS error shows that the closed-loop system is globally exponentially bounded by the designed estimator and controller. Both numerical and simulation experiments are conducted to demonstrate the correctness and effectiveness of the proposed method.

Error analysis for deep neural network approximations of parametric hyperbolic conservation laws

We derive rigorous bounds on the error resulting from the approximation of the solution of parametric hyperbolic scalar conservation laws with ReLU neural networks. We show that the approximation error can be made as small as desired with ReLU neural networks that overcome the curse of dimensionality. In addition, we provide an explicit upper bound on the generalization error in terms of the training error, number of training samples and the neural network size. The theoretical results are illustrated by numerical experiments.

Convergent finite element methods for the perfect conductivity problem with close-to-touching inclusions

Abstract In the perfect conductivity problem (i.e., the conductivity problem with perfectly conducting inclusions), the gradient of the electric field is often very large in a narrow region between two inclusions and blows up as the distance between the inclusions tends to zero. The rigorous error analysis for the computation of such perfect conductivity problems with close-to-touching inclusions of general geometry still remains open in three dimensions. We address this problem by establishing new asymptotic estimates for the second-order partial derivatives of the solution with explicit dependence on the distance $\varepsilon $ between the inclusions, and use the asymptotic estimates to design a class of graded meshes and finite element spaces to solve the perfect conductivity problem with possibly close-to-touching inclusions. In particular, we propose a special finite element basis function that resolves the asymptotic singularity of the solution by making the interpolation error bounded in $W^{1,\infty }$ in a neighborhood of the close-to-touching point, even though the solution itself is blowing up in $W^{1,\infty }$. This is crucial in the error analysis for the numerical approximations. We prove that the proposed method yields optimal-order convergence in the $H^1$ norm, uniformly with respect to the distance $\varepsilon $ between the inclusions, in both two and three dimensions for general convex smooth inclusions, which are possibly close-to-touching. Numerical experiments are presented to support the theoretical analysis and to illustrate the convergence of the proposed method for different shapes of inclusions in both two- and three-dimensional domains.

The use of physics-informed neural network approach to image restoration via nonlinear PDE tools

In this work, we propose a physics-informed solution for a blending PDE model with application in image denoising and edge-preserving. The blending model contains isotropic diffusion (ID) model and the modified Perona-Malik model introduced by Catte et al. Then we solve this model applying a physics-informed approach. Solving the new blending model is done using DeepXDE, which is a deep learning library for solving differential equations, by considering the image as input. Error analysis for neural network approximations of the new PDE model is shown. Applying PSNR criteria for some images, it can be seen that the proposed method can compete with solving denoising PDE models using finite difference method (FDM).

Low-Complexity Wideband Interference Mitigation for UWB ToA Estimation.

Reliable time of arrival (ToA) estimation in dense multipath (DM) environments is a difficult task, especially when strong interference is present. The increasing number of multiple services in a shared spectrum comes with the demand for interference mitigation techniques. Multiple receiver elements, even in low-energy devices, allow for interference mitigation by processing coherent signals, but computational complexity has to be kept at a minimum. We propose a low-complexity, linearly constrained minimum variance (LCMV) interference mitigation approach in combination with a detection-based ToA estimator. The performance of the method within a realistic multipath and interference environment is evaluated based on measurements and simulations. A statistical analysis of the ToA estimation error is provided in terms of the mean absolute error (MAE), and the results are compared to those of a band-stop filter-based interference blocking approach. While the focus is on receivers with only two elements, an extension to multiple elements is discussed as well. Results show that the influence of strong interference can be drastically reduced, even when the interference bandwidth exceeds 60% of the signal bandwidth. Moreover, the algorithm is robust to uncertainties in the angle of arrival (AoA) of the desired signal. Based on these results, the proposed mitigation method is well suited when the interference bandwidth is large and when computational power is a critical resource.

Error analysis for the numerical approximation of the harmonic map heat flow with nodal constraints

Abstract An error estimate for a canonical discretization of the harmonic map heat flow into spheres is derived. The numerical scheme uses standard finite elements with a nodal treatment of linearized unit-length constraints. The analysis is based on elementary approximation results and only uses the discrete weak formulation.

Optimal \(\boldsymbol{{L^2}}\) Error Estimates of Unconditionally Stable Finite Element Schemes for the Cahn–Hilliard–Navier–Stokes System

The paper is concerned with the analysis of a popular convex-splitting finite element method (FEM) for the Cahn–Hilliard–Navier–Stokes system, which has been widely used in practice. Since the method is based on a combined approximation to multiple variables involved in the system, the approximation to one of the variables may seriously affect the accuracy for others. Optimal-order error analysis for such combined approximations is challenging. The previous works failed to present optimal error analysis in -norm due to the weakness of the traditional approach. Here we first present an optimal error estimate in -norm for the convex-splitting FEMs. We also show that optimal error estimates in the traditional (interpolation) sense may not always hold for all components in the coupled system due to the nature of the pollution/influence from lower-order approximations. Our analysis is based on two newly introduced elliptic quasi-projections and the superconvergence of negative norm estimates for the corresponding projection errors. Numerical examples are also presented to illustrate our theoretical results. More important is that our approach can be extended to many other FEMs and other strongly coupled phase field models to obtain optimal error estimates.

Analysis and Compensation of Position Estimation Error for Sensorless Reduced DC-Link Capacitance IPMSM Drives

To extend the lifetime, improve the power density and reduce the system cost, the film capacitor is applied to replace the large-volume electrolytic capacitor in motor drives. As the dc-link voltage fluctuates periodically, the rotor position estimation error and operation stability are necessary to be concerned in the position sensorless control methods. A cross-decoupling complex filtering-based position estimation method is investigated for the reduced capacitance interior permanent magnet synchronous motor drive system. Harmonic components of the extended back electromotive force (EEMF) caused by the dc-link voltage fluctuation are analyzed to evaluate the additional rotor position error. A cross-decoupling complex filter realized by the multiple first-order complex filters with different center frequencies is designed to improve the position estimation precision. Harmonics of the EEMF are suppressed and the position error can be eliminated. The estimation precision and operation ability are improved. Experimental results are performed to verify the effectiveness of the proposed method.

Disturbance Utilization-Based Tracking Control for the Fixed-Wing UAV With Disturbance Estimation

This paper investigates the disturbance utilization-based attitude control for the fixed-wing unmanned aerial vehicle (UAV). The disturbance utilization condition (DUC) is formed based on the Lyapunov function analysis to retain the disturbance that benefits the stability of closed-loop systems. To reduce the sign-misjudgment of disturbance coupling terms induced by the DUC, the disturbance estimation error analysis auxiliary system (DEEAAS) is designed based on the disturbance observer (DO). Combined with the DEEAAS and the DO, the boundaries of the disturbance estimation error (DEE) are derived. Subsequently, the composite DUC is proposed based on the above boundaries and the given thresholds to replace the basic DUC. Then, the boundedness of the attitude tracking errors of the fixed-wing UAV can be ensured by the controller designed with the composite DUC. And combined with the derived boundaries of the DEE and the given thresholds, the adaptive DO and the adaptive DEEAAS are also designed to avoid the use of big parameters. In addition, sufficient conditions that stabilize the attitude closed-loop system of fixed-wing UAVs equipped with the disturbance utilization-based controller (DUBC) are given. Finally, the numerical simulation for the fixed-wing UAV illustrates the effectiveness of the proposed DUBC.

Theoretical and Algorithmic Study of Inverses of Arbitrary High-Dimensional Multi-Input Multi-Output Linear-Time-Invariant Systems

Nowadays, systems need to be built and/or characterized to handle exceptional circumstances or adapt to a world itself more complex. A typical phenomenon can often be found that systems are required to accommodate high-dimensional inputs and outputs. Although the theories and methods for inverting a single-input single-output (SISO) linear-time-invariant (LTI) system have been well established, the generalized framework (consisting of theories and algorithms) for extending to arbitrary high-dimensional multi-input multi-output (MIMO) scenarios is still unsubstantial in the existing literature as this extension is far from trivial. In this work, we would like to develop such a new framework for governing the inversion of arbitrary high-dimensional discrete-time MIMO LTI systems, where any individual transfer function from a certain input to a certain output may have the infinite-impulse-response (IIR) characteristics. We propose two new inversion algorithms to invert the transfer-function tensors (TFTs) of arbitrary MIMO LTI systems. The pertinent computational complexities are also investigated for our proposed two TFT-inversion algorithms. The approximation of the inverse of an arbitrary TFT by a finite-impulse-response (FIR) TFT is studied and the corresponding approximation-error analysis is derived as well. Finally, numerical evaluations are presented to study the computational complexities with respect to different TFT dimensions and ranks.