Optimal Error Research Articles

A rotational velocity-correction projection method for the Micropolar Navier-Stokes equations

In this paper, we introduce a velocity correction projection method for the Micropolar Navier-Stokes Equations. The velocity correction method are adopted to approximate the time derivative term, stability analysis and error estimation of the first-order semi-discrete scheme are proved. At the same time, the optimal error estimate using the technique of dual norm are obtained. In this way, the divergence free of the velocity u can be conserved. Finally, the numerical results show the method has an optimal convergence order. The numerical results are consistent with our theoretical analysis, and our method is effective.

OEDG: Oscillation-eliminating discontinuous Galerkin method for hyperbolic conservation laws

Suppressing spurious oscillations is crucial for designing reliable high-order numerical schemes for hyperbolic conservation laws, yet it has been a challenge actively investigated over the past several decades. This paper proposes a novel, robust, and efficient oscillation-eliminating discontinuous Galerkin (OEDG) method on general meshes, motivated by the damping technique (see J. Lu, Y. Liu, and C. W. Shu [SIAM J. Numer. Anal. 59 (2021), pp. 1299–1324]). The OEDG method incorporates an oscillation-eliminating (OE) procedure after each Runge–Kutta stage, and it is devised by alternately evolving the conventional semidiscrete discontinuous Galerkin (DG) scheme and a damping equation. A novel damping operator is carefully designed to possess both scale-invariant and evolution-invariant properties. We rigorously prove the optimal error estimates of the fully discrete OEDG method for smooth solutions of linear scalar conservation laws. This might be the first generic fully discrete error estimate for nonlinear DG schemes with an automatic oscillation control mechanism. The OEDG method exhibits many notable advantages. It effectively eliminates spurious oscillations for challenging problems spanning various scales and wave speeds, without necessitating problem-specific parameters for all the tested cases. It also obviates the need for characteristic decomposition in hyperbolic systems. Furthermore, it retains the key properties of the conventional DG method, such as local conservation, optimal convergence rates, and superconvergence. Moreover, the OEDG method maintains stability under the normal Courant–Friedrichs–Lewy (CFL) condition, even in the presence of strong shocks associated with highly stiff damping terms. The OE procedure is nonintrusive, facilitating seamless integration into existing DG codes as an independent module. Its implementation is straightforward and efficient, involving only simple multiplications of modal coefficients by scalars. The OEDG approach provides new insights into the damping mechanism for oscillation control. It reveals the role of the damping operator as a modal filter, establishing close relations between the damping technique and spectral viscosity techniques. Extensive numerical results validate the theoretical analysis and confirm the effectiveness and advantages of the OEDG method.

Research on CNN-LSTM-Attention based surface temperature prediction model for fermented grains

The prediction of the temperature on the surface of the fermented grains during the steaming process holds significant importance in improving the alcohol yield and quality. To accurately forecast the temperature variations on the surface of the fermented grains, a predictive model based on CNN-LSTM-Attention is proposed. Leveraging convolutional neural networks, latent features of the temperature data on the fermented grains surface are extracted. These extracted feature vectors, representing a time series, are then inputted into a long short-term memory network to further capture the temporal characteristics of the sequence. Finally, employing an attention mechanism, the influential features are highlighted to achieve precise prediction of the fermented grains surface temperature. Historical temperature data from the steaming process of the fermented grains is utilized, and experimental comparisons are conducted with other neural network prediction models. The results demonstrate that the CNN-LSTM-Attention model achieves optimal root mean square error of 1.282 and average absolute error of 0.647, surpassing other models. The experimental findings substantiate the superior accuracy of the CNN-LSTM-Attention model in forecasting the changing trend of the fermented grains surface temperature.

Prediction of titanium burn-off and untimate titanium content in electroslag process

In this study, we investigate the burning behavior of titanium during the electroslag remelting (ESR) process and its impact on the titanium content at the endpoint using machine learning. Initially, a comprehensive database was established by collecting data from literature and experiments, encompassing slag system composition, smelting temperature, and material composition content. Subsequently, six machine learning algorithms, including random forest and Bayesian regression, were employed to model the burning loss behavior of titanium. The random forest model, which exhibited optimal mean square error (MSE) performance, was utilized to generate partial dependence plots. These plots, in conjunction with experimental observations and existing studies, facilitated the analysis of key factors influencing titanium burn loss. Furthermore, the same six machine learning models were applied to predict the endpoint titanium content. The Bayesian regression model demonstrated superior performance in terms of R2 and MSE, leading to the derivation of an empirical formula for predicting endpoint titanium content. This empirical formula was subsequently validated, refined, and optimized using a thermodynamic model based on the theory of molecular ion coexistence. The final prediction formula achieved an error margin of 0.123%.

Estimation of Hamiltonian Parameters from Thermal States.

We upper bound and lower bound the optimal precision with which one can estimate an unknown Hamiltonian parameter via measurements of Gibbs thermal states with a known temperature. The bounds depend on the uncertainty in the Hamiltonian term that contains the parameter and on the term's degree of noncommutativity with the full Hamiltonian: higher uncertainty and commuting operators lead to better precision. We apply the bounds to show that there exist entangled thermal states such that the parameter can be estimated with an error that decreases faster than 1/sqrt[n], beating the standard quantum limit. This result governs Hamiltonians where an unknown scalar parameter (e.g., a component of a magnetic field) is coupled locally and identically to n qubit sensors. In the high-temperature regime, our bounds allow for pinpointing the optimal estimation error, up to a constant prefactor. Our bounds generalize to joint estimations of multiple parameters. In this setting, we recover the high-temperature sample scaling derived previously via techniques based on quantum state discrimination and coding theory. In an application, we show that noncommuting conserved quantities hinder the estimation of chemical potentials.

A discontinuous Galerkin method for the Brinkman–Darcy-transport problem

In this paper, a weighted discontinuous Galerkin finite element method and an upwind format are presented to solve the coupled Brinkman–Darcy flow and transport model. The flow in a highly permeable region of the model is governed by the Brinkman equations, and the percolation in the porous media is controlled by the Darcy equations, which are coupled by three interface conditions. The permeability coefficients in this model are strongly nonhomogeneous, anisotropic, and discontinuous; the transport equation is a convection-dominated problem; and the velocity field in the porous media domain does not satisfy the divergence-free condition. A weighted discontinuous Galerkin finite element method is used to solve the complex permeability coefficient problem and an upwind scheme is used to solve the convection dominated problem. The interface conditions can be naturally incorporated into the discrete formulation without introducing additional variables. Optimal error estimates are obtained for the semi-discretization and the full discretization with the backward Euler scheme in suitable energy norm. A series of numerical experiments are provided to illustrate the proposed method, including testing the convergence and accuracy of various types of meshes; simulating fluid flow in complex porous media such as obstacles, layered media, and curved interfaces; and studying the coupled flow behavior of surface-subsurface flows with contaminant transport.

Small Errors in Random Zeroth-Order Optimization Are Imaginary

Small Errors in Random Zeroth-Order Optimization Are Imaginary

Optimal error estimates of a non-uniform IMEX-L1 finite element method for time fractional PDEs and PIDEs

Stability and optimal convergence analysis of a non-uniform implicit-explicit L1 finite element method (IMEX-L1-FEM) is studied for a class of time-fractional linear partial differential/integro-differential equations with non-self-adjoint elliptic part having (space-time) variable coefficients. The proposed scheme is based on a combination of an IMEX-L1 method on graded mesh in the temporal direction and a finite element method in the spatial direction. With the help of a discrete fractional Grönwall inequality, global almost optimal error estimates in L2- and H1-norms are derived for the problem with initial data u0∈H01(Ω)∩H2(Ω). The novelty of our approach is based on managing the interaction of the L1 approximation of the fractional derivative and the time discrete elliptic operator to derive the optimal estimate in H1-norm directly. Furthermore, a super convergence result is established when the elliptic operator is self-adjoint with time and space varying coefficients, and as a consequence, an L∞ error estimate is obtained for 2D problems that too with the initial condition is in H01(Ω)∩H2(Ω). All results proved in this paper are valid uniformly as α→1−, where α is the order of the Caputo fractional derivative. Numerical experiments are presented to validate our theoretical findings.

Green method for 17-hydroxyprogesterone extraction and determination using PDMS stir bar sorptive extraction coupled with HPLC: optimization by response surface methodology

Quantifying small amounts of the 17-hydroxyprogesterone in various matrix is crucial for different purposes. In this study, a commercial polydimethylsiloxane stir bar was used to extract hormone from water and urine samples. Analysis was performed by high-performance liquid chromatography using a UV detector. The response surface methodology was used to optimize the desorption and extraction steps, with predicted optimal point relative errors of 1.25% and 6.40%, respectively. The optimized method was validated with a linear range of 1.21–1000.00 for aqueous and 2.43–2000.00 ng mL–1 for urine samples. The coefficient of determination was 0.9998 and 0.9967, and the detection limit of the proposed method was obtained to be 0.40 and 0.80 ng mL–1 for aqueous and urine samples, respectively. The recovery percentage and relative standard deviation within a day and between three days after the addition of three different concentration levels of the standard to the control sample were 87–103% and 0.4–3.6% for aqueous and 87.5–101% and 0.1–5.2% for urine samples, respectively. The results show that the proposed method can be appropriate and cost-effective for extracting and analyzing this hormone. In addition, using three different tools, the greenness of the proposed method was proven.

Spectral codes of real symmetric operators for error correction

In this paper, we introduce a novel class of real number codes termed as [Formula: see text]-spectral codes, derived from real symmetric operators. Our focus is on addressing the challenges associated with error correction in communication systems utilizing real number codes. Our study initiates with a meticulous definition and analysis of the fundamental properties of [Formula: see text]-spectral codes. These properties serve as the foundation for comprehending the distinctive characteristics that [Formula: see text]-spectral codes offer, ensuring the integrity and reliability of transmitted data. Our exploration extends to the construction of maximum distance separable (MDS) [Formula: see text]-spectral codes, which boast the maximum error correction capability. As a culmination of our research, we present an efficient decoding technique tailored for [Formula: see text]-spectral codes. Leveraging an MDS [Formula: see text]-spectral code and our decoding technique, we achieve optimal error correction and ensure good numerical stability. This innovative approach opens new avenues for enhancing the performance of real-number codes in practical applications.

Time-Stepping Error Estimates of Linearized Grünwald–Letnikov Difference Schemes for Strongly Nonlinear Time-Fractional Parabolic Problems

A fully discrete scheme is proposed for numerically solving the strongly nonlinear time-fractional parabolic problems. Time discretization is achieved by using the Grünwald–Letnikov (G–L) method and some linearized techniques, and spatial discretization is achieved by using the standard second-order central difference scheme. Through a Grönwall-type inequality and some complementary discrete kernels, the optimal time-stepping error estimates of the proposed scheme are obtained. Finally, several numerical examples are given to confirm the theoretical results.

Low regularity error analysis for an H(div)-conforming discontinuous Galerkin approximation of Stokes problem

In this paper, we derive an improved error estimate for the H(div)-conforming discontinuous Galerkin (DG) approximation of the Stokes equations, assuming only minimal regularity on the exact solution. The estimate relies on both a priori and a posteriori analysis, and thus is called a medius error analysis. More precisely, we proved an optimal order error estimate under the assumption (u,p)∈H1+s(Ω)×Hs(Ω) with any s∈(0,1]. Extension to the standard interior penalty DG methods is also explored. Finally, numerical results are provided to verify our theoretical findings.

A novel discontinuous Galerkin projection scheme for the hydrodynamics of nematic liquid crystals

This paper is focused on the numerical approximations for the hydrodynamic model of nematic liquid crystals. Under the framework of a splitting projection method, we propose a novel interior penalty discontinuous Galerkin (DG) method for solving the coupled system, which is employed by combining the scalar auxiliary variables (SAV) approach, implicit-explicit (IMEX) treatments and a rotational pressure-correction method. One prominent feature of the developed scheme here is by introducing an additional stabilization term artificially in liquid crystal equation to balance the explicit treatment for the coupling term, so that the computations of vector field from velocity field are decoupled. Hence a linear, fully decoupled and unconditionally energy-stable DG scheme can be achieved in a fully discrete manner. We show that the resulting scheme is uniquely solvable and unconditional energy stable, and the optimal error estimates of the proposed scheme are further proved theoretically. Finally, numerical results are carried out to demonstrate the accuracy and energy stability of our scheme.

Error estimates and numerical simulations of a thermoviscoelastic contact problem with damage and long memory

This paper aims to investigate a thermal frictional contact model with damage and long memory effects. We consider a deformable body made of viscoelastic material and assume the process to be dynamic. The material is expected to adhere to the Kelvin–Voigt constitutive law, with damage and thermal effects incorporated. The variational formulation of the model results in a coupled system comprising a history-dependent hemivariational inequality governing the displacement field, a parabolic variational inequality describing the damage field and an evolution equation for the temperature field. In the analysis of this system, we initially introduce a fully discrete scheme, and then concentrate on deriving error estimates of numerical solutions. An optimal order error estimate is attained under some appropriate solution regularity assumptions. At the tail of this manuscript, numerical simulations are provided for the contact problem to validate the theoretical results.

An energy-stable variable-step L1 scheme for time-fractional Navier–Stokes equations

We propose a structure-preserving scheme and its error analysis for time-fractional Navier–Stokes equations (TFNSEs) with periodic boundary conditions. The equations are first rewritten as an equivalent system by eliminating the pressure explicitly. Then, the spatial and temporal discretization are done by the Fourier spectral method and variable-step L1 scheme, respectively. It is proved that the fully-discrete scheme is energy-stable and divergence-free. The energy is an asymptotically compatible one since it recovers the classical energy when α→1. Moreover, optimal error estimates are presented very technically by the obtained boundedness of the numerical solutions and some Sobolev inequalities. To our knowledge, they are the first results of the construction and analysis of structure-preserving schemes for TFNSEs. Several interesting numerical examples are given to confirm the theoretical results at last.

Two-grid methods for nonlinear pseudo-parabolic integro-differential equations by finite element method

In this paper, two efficient two-grid finite element algorithms are proposed for solving two-dimensional nonlinear pseudo-parabolic integro-differential equations. Firstly, we obtain optimal error estimates of the fully discrete finite element method using a temporal-spatial error splitting technique in H1 and Lp norms. Then the two-grid technique to improve computation efficiency of the proposed finite element method. Error estimates in H1 and Lp norms of two-grid solutions are presented. Theoretical analysis shows that the two-grid algorithms maintain asymptotically optimal accuracy. Finally, numerical examples are provided to support our theoretical results and demonstrate the effectiveness of these methods.

Efficient Byzantine-robust distributed inference with regularization: A trade-off between compression and adversary

In large-scale distributed learning, the direct application of traditional inference is often not feasible, because it may contain multiple themes, such as communication costs, privacy issues, and Byzantine failures. Nowadays, the internet is vulnerable to attacks, and Byzantine failures frequently occur. For copying with Byzantine failures, the paper develops two Byzantine-robust distributed learning algorithms under a framework of communication-efficient surrogate likelihood. In our algorithms, we adopt the δ-approximate compressors, including sign-based operator and topk sparsification, to improve communication efficiency, and an unsophisticated thresholding of local gradient norms to guard against Byzantine failures. For accelerating convergence and achieving an optimal statistical error rate, error feedback is exploited in the second algorithm. The two algorithms are robust to arbitrary adversaries, although Byzantine workers don't adhere to the mandated compression mechanism. We explicitly establish statistical error rates, which imply that our algorithms don't sacrifice the quality of learning, and attain the order-optimal under some settings. In addition, we provide a trade-off between compression and adversary in the presence of Byzantine worker machines. Extensive numerical experiments validate our theoretical results and demonstrate a good performance of our algorithms.

An efficient weak Galerkin FEM for third-order singularly perturbed convection-diffusion differential equations on layer-adapted meshes

In this article, we study the weak Galerkin finite element method to solve a class of a third order singularly perturbed convection-diffusion differential equations. Using some knowledge on the exact solution, we prove a robust uniform convergence of order O(N−(k−1/2)) on the layer-adapted meshes including Bakhvalov-Shishkin type, and Bakhvalov-type and almost optimal uniform error estimates of order O((N−1ln⁡N)(k−1/2)) on Shishkin-type mesh with respect to the perturbation parameter in the energy norm using high-order piecewise discontinuous polynomials of degree k. Here N is the number mesh intervals. We conduct numerical examples to support our theoretical results.

Precise error bounds for numerical approximations of fractional HJB equations

Abstract We prove precise rates of convergence for monotone approximation schemes of fractional and nonlocal Hamilton–Jacobi–Bellman equations. We consider diffusion-corrected difference-quadrature schemes from the literature and new approximations based on powers of discrete Laplacians, approximations that are (formally) fractional order and second-order methods. It is well known in numerical analysis that convergence rates depend on the regularity of solutions, and here we consider cases with varying solution regularity: (i) strongly degenerate problems with Lipschitz solutions and (ii) weakly nondegenerate problems where we show that solutions have bounded fractional derivatives of order $\sigma \in (1,2)$. Our main results are optimal error estimates with convergence rates that capture precisely both the fractional order of the schemes and the fractional regularity of the solutions. For strongly degenerate equations, these rates improve earlier results. For weakly nondegenerate problems of order greater than one, the results are new. Here we show improved rates compared to the strongly degenerate case, rates that are always better than $\mathcal{O}\big (h^{\frac{1}{2}}\big )$.

Kinetic Langevin MCMC sampling without gradient Lipschitz continuity - the strongly convex case

In this article we consider sampling from log concave distributions in Hamiltonian setting, without assuming that the objective gradient is globally Lipschitz. We propose two algorithms based on monotone polygonal (tamed) Euler schemes, to sample from a target measure, and provide non-asymptotic 2-Wasserstein distance bounds between the law of the process of each algorithm and the target measure. Finally, we apply these results to bound the excess risk optimization error of the associated optimization problem.