Optimal convergence in finite element semidiscrete error analysis of the Doyle–Fuller–Newman model beyond one dimension with a novel projection operator

For linear elastostatics, the Lagrange multiplier to couple the displacement (i.e., Dirichlet) condition is well known in mathematics community, but the Lagrange multiplier to couple the traction (i.e., Neumann) condition is popular for elasticity problems by the Trefftz method in engineering community, which is called the Hybrid Trefftz method (HTM). However, there has not been any analysis for these Lagrange multipliers to couple the traction condition so far. New error analysis of the HTM for elasticity problems is explored in this paper, to derive error bounds with the optimal convergence rates. Numerical experiments are reported to support this analysis. The error analysis of the HTM for linear elastostatics is the main aim of this paper. In this paper, the collocation Trefftz method (CTM) without a multiplier is also introduced, accompanied with error analysis. Numerical comparisons are made for HTM and CTM using fundamental solutions (FS) and particular solutions (PS). The error analysis and numerical computations show that the accuracy of the HTM is equivalent to that of the CTM, but the stability of the CTM is good. For elasticity and other complicated problems, the simplicity of algorithms and programming grants the CTM a remarkable advantage. More numerical comparisons show that using PS is more efficient than using FS in both HTM and CTM. However, since the optimal convergence rates are the most important criterion in evaluation of numerical methods, the global performance of the HTM is as good as that of the CTM. The comparisons of HTM and CTM using FS and PS are the next aim of this article. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011

In this paper we consider the problem of determining optimal convergence rates of Galerkin approximations to infinite dimensional operator Riccati equations (OREs). Optimal rates are obtained for a class of abstract distributed parameter systems evolving in an infinite dimensional Hilbert space. These general results are then applied to systems modeled by partial differential equations that generate compact and analytic semigroups. The estimates apply to distributed control and observation of classical parabolic equations and to certain vibration problems with sufficiently strong damping. The ORE is formulated as an equivalent operator‐valued Bochner integral equation and the Brezzi–Rappaz–Raviart theorem is used to obtain convergence rates. First we establish smoothing property and bounds for the solutions of the infinite dimensional ORE. Then it is shown that, under sui\ assumptions on the coefficients and domain geometry, the hp‐finite element approximations of the classical solution converges on the order of . Furthermore, these optimal error bounds are shown to hold for the functional gains that define observer and control gain operators. We provide numerical examples that corroborate the theoretical convergence rates.

The finite volume method and application in combinations

Nonparametric deconvolution problems require one to recover an unknown density when the data are contaminated with errors. Optimal global rates of convergence are found under the weighted Lp‐loss (1 ≤ p ≤ ∞). It appears that the optimal rates of convergence are extremely low for supersmooth error distributions. To resolve this difficulty, we examine how high the noise level can be for deconvolution to be feasible, and for the deconvolution estimate to be as good as the ordinary density estimate. It is shown that if the noise level is not too high, nonparametric Gaussian deconvolution can still be practical. Several simulation studies are also presented.

We systematically study the optimal linear convergence rates for several relaxed alternating projection methods and the generalized Douglas-Rachford splitting methods for finding the projection on the intersection of two subspaces. Our analysis is based on a study on the linear convergence rates of the powers of matrices. We show that the optimal linear convergence rate of powers of matrices is attained if and only if all subdominant eigenvalues of the matrix are semisimple. For the convenience of computation, a nonlinear approach to the partially relaxed alternating projection method with at least the same optimal convergence rate is also provided. Numerical experiments validate our convergence analysis

The modeling of lithium-ion battery is an important element to the management of batteries in industrial applications. Various models have been studied and investigated, ranging from simple to complex. The second-order equivalent circuit model was studied and investigated since the dynamic behavior of the battery is fully characterized. The simulation model was built in Matlab Simulink using the Kirchhoff Laws principle in mathematical equations, while the battery's internal parameters were identified by using the BTS4000 (Battery tester) device. To estimate the full state of charge (SOC), the initial state of charge (SOC0) must be identified or measured. Hence, this paper seeks for the SOC estimation by using experimental terminal voltage data and SOC with Matlab lookup table. Then, the simulated terminal voltage, as well as the SOC of the battery are compared and validated against measured data. The maximum relative error of 0.015 V and 2% for terminal voltage and SOC respectively shows that the proposed model is accurate and relevant based on the error analysis.

Many well-known first-order gradient methods have been extended to cope with large-scale composite problems, which often arise as a regularized empirical risk minimization in machine learning. However, their optimal convergence is attained only in terms of the weighted average of past iterative solutions. How to make the individual convergence of stochastic gradient descent (SGD) optimal, especially for strongly convex problems has now become a challenging problem in the machine learning community. On the other hand, Nesterov's recent weighted averaging strategy succeeds in achieving the optimal individual convergence of dual averaging (DA) but it fails in the basic mirror descent (MD). In this paper, a new primal averaging (PA) gradient operation step is presented, in which the gradient evaluation is imposed on the weighted average of all past iterative solutions. We prove that simply modifying the gradient operation step in MD by PA strategy suffices to recover the optimal individual rate for general convex problems. Along this line, the optimal individual rate of convergence for strongly convex problems can also be achieved by imposing the strong convexity on the gradient operation step. Furthermore, we extend PA-MD to solve regularized nonsmooth learning problems in the stochastic setting, which reveals that PA strategy is a simple yet effective extra step toward the optimal individual convergence of SGD. Several real experiments on sparse learning and SVM problems verify the correctness of our theoretical analysis.

We consider the wavelet-based estimators of mean regression function with long memory moving average errors and investigate their asymptotic rates of convergence based on thresholding of empirical wavelet coefficients. We show that these estimators achieve nearly optimal minimax convergence rates within a logarithmic term over a large range of Besov function classes \(B^{s}_{p,q}\). Therefore, in the presence of long memory non-Gaussian moving average noise, wavelet estimators still achieve nearly optimal convergence rates and provide explicitly the extraordinary local adaptability. The theory is illustrated with some numerical examples.

Problem. This article addresses the challenge of enhancing the environmental friendliness and energy efficiency of vehicles. It does so by conducting a comparative analysis and identifying ways to improve the electrical models of lithium-ion batteries used in electric vehicles. The study includes an examination of well-known electrical models of lithium-ion rechargeable batteries, such as the Rint model, the RC model, the Thevenin model, and the PNGV model. It identifies key characteristics of lithium-ion batteries in electric vehicles, including state of charge, mass, actual voltage, energy required for recharging, among others. The study also explores models of battery degradation, focusing on capacity reduction and the increase in active resistance. It substantiates directions for improving electrical models of lithium-ion batteries in electric vehicles by considering changes in capacity, internal resistance, polarization resistance, and both calendar and cyclic degradation. Goal. The aim of this work is to enhance the environmental friendliness and energy efficiency of vehicles through a comparative analysis and by determining ways to improve the electrical models of lithium-ion batteries in electric vehicles. Methodology. Our approach to achieving this goal involves using electrical models of lithium-ion batteries in electric vehicles, which describe various parameters such as state of charge, actual voltage during charge/discharge processes, and energy required for recharging. The study encompasses an investigation into the degradation of electric vehicle batteries, including their use in Vehicle to Grid (V2G) technology. Results. The analysis of electrical models of lithium-ion batteries in electric vehicles, aiming to increase their accuracy, considers the following aspects: changes in internal resistance and polarization resistance; capacity variation; and battery degradation. The change in internal resistance and polarization resistance should be considered based on two factors: the state of charge of the battery and the degree of its degradation. While the first factor is relevant primarily when the battery is deeply discharged (SoC<30%), the second factor must be considered at any state of charge. Capacity changes should be accounted for based on calendar and cyclic degradation. It has been determined that the primary causes of degradation in electric vehicle batteries are calendar aging (service life) and aging due to charge/discharge cycles. Contrarily, it is argued that using Vehicle to Grid (V2G) technology can reduce battery degradation by 10%. Originality. The results of this study provide a comprehensive understanding of the electrical models of lithium-ion batteries in electric vehicles and contribute to the improvement of existing models. Practical value. This research enhances the accuracy of current electrical models of lithium-ion batteries in electric vehicles by considering the variable nature of internal resistance and capacity during vehicle operation. It may be valuable in assessing the residual parameters of electric vehicle batteries during their secondary use, such as in the residential sector for solar energy support. The findings can be recommended to scientific and technical professionals involved in developing energy storage systems for electric vehicles.

We develop and analyze a new hybridizable discontinuous Galerkin (HDG) method for solving third-order Korteweg-de Vries type equations. The approximate solutions are defined by a discrete version of a characterization of the exact solution in terms of the solutions to local problems on each element which are patched together through transmission conditions on element interfaces. We prove that the semi-discrete scheme is stable with proper choices of stabilization function in the numerical traces. For the linearized equation, we carry out error analysis and show that the approximations to the exact solution and its derivatives have optimal convergence rates. In numerical experiments, we use an implicit scheme for time discretization and the Newton-Raphson method for solving systems of nonlinear equations, and observe optimal convergence rates for both the linear and the nonlinear third-order equations.

Nitsche’s method for linear Kirchhoff–Love shells: Formulation, error analysis, and verification

Moving least-square reproducing kernel methods (I) Methodology and convergence

Models built with deep neural network (DNN) can handle complicated real-world data extremely well, seemingly without suffering from the curse of dimensionality or the non-convex optimization. To contribute to the theoretical understanding of deep learning, this work studies the nonparametric perspective of DNNs by considering the following questions: (1) What is the underlying estimation problem and what are the most appropriate data assumptions? (2) What is the corresponding optimal convergence rate and does the curse of dimensionality occur? (3) Is the optimal rate achievable for DNN estimators and is there any optimization guarantee? These questions are investigated on two of the most fundamental problems --- regression and classification. Specifically, statistical optimality of DNN estimators is established under various settings with special focuses on the curse of dimensionality and optimization guarantee.In the classic binary classification problem, statistical optimal convergence rates that suffer less from the curse of dimensionality are established under two settings:(1) Under the smooth boundary assumption, I show that DNN classifiers with proper architectures can benefit from the compositional smoothness structure underlying the high dimensional data in the sense that the optimal convergence rates only depend on some effective dimension d*, potentially much smaller than the data dimension d. (2) Under a novel teacher-student framework that assumes the Bayes classifier to be expressed as ReLU neural networks, I obtain a dimension-free rate of convergence O(n^{-2/3}) for DNN classifiers, which is also proven optimal. The optimization of DNN is highly complicated with generally no algorithmic guarantee on finding the global minimizer. To this end, I turn to the recently proposed neural tangent kernel (NTK) literature where the similarity between overparametrized DNN trained by gradient descent (GD) and kernel methods are established. Specifically, through a comprehensive analysis of L2-regularized GD trajectories, I prove that for overparametrized one-hidden-layer ReLU neural networks with L2 regularization, the output from GD is close to that from the kernel ridge regression with the corresponding NTK and optimal rate of L2 estimation error can be achieved.

Schock (Ref. 1) considered a general a posteriori parameter choice strategy for the Tikhonov regularization of the ill-posed operator equationTx=y which provides nearly the optimal rate of convergence if the minimal-norm least-squares solution $$\hat x$$ belongs to the range of the operator (T * T) v , o<v≤1. Recently, Nair (Ref. 2) improved the result of Schock and also provided the optimal rate ifv=1. In this note, we further improve the result and show in particular that the optimal rate can be achieved for 1/2≤v≤1.

This paper is concerned with the large time behaviors of smooth solutions to the Cauchy problem of the one dimensional bipolar Euler-Poisson equations with the time dependent critical overdamping. We show that in this critical overdamping case the bipolar Euler-Poisson system admits a unique global smooth solution that asymptotically converges to the nonlinear diffusion wave. In particular, the optimal convergence rate in logarithmic form is derived when the initial perturbations are L 2 sense by using the technical time-weighted energy method.