Optimal Error Research Articles

Rank selection for non-negative matrix factorization.

Non-Negative Matrix Factorization (NMF) is a widely used dimension reduction method that factorizes a non-negative data matrix into two lower dimensional non-negative matrices: one is the basis or feature matrix which consists of the variables and the other is the coefficients matrix which is the projections of data points to the new basis. The features can be interpreted as sub-structures of the data. The number of sub-structures in the feature matrix is also called the rank. This parameter controls the model complexity and is the only tuning parameter for the NMF model. An appropriate rank will extract the key latent features while minimizing the noise from the original data. However due to the large amount of optimization error always present in the NMF computation, the rank selection has been a difficult problem. We develop a novel rank selection method based on hypothesis testing, using a deconvolved bootstrap distribution to assess the significance level accurately. Through simulations, we compare our method with a rank selection method based on hypothesis testing using bootstrap distribution without deconvolution and a method based on cross-validation; we demonstrate that our method is not only accurate at estimating the true ranks for NMF, especially when the features are hard to distinguish, but also efficient at computation. When applied to real microbiome data (eg, OTU data and functional metagenomic data), our method also shows the ability to extract interpretable subcommunities in the data.

A local energy-based discontinuous Galerkin method for fourth-order semilinear wave equations

Abstract This paper proposes an energy-based discontinuous Galerkin scheme for fourth-order semilinear wave equations, which we rewrite as a system of second-order spatial derivatives. Compared to the local discontinuous Galerkin methods, the proposed scheme uses fewer auxiliary variables and is more computationally efficient. We prove several properties of the scheme. For example, we show that the scheme is unconditionally stable and that it achieves optimal convergence in $L^2$ norm for both the solution and the auxiliary variables without imposing penalty terms. A key part of the proof of the stability and convergence analysis is the special choice of the test function for the auxiliary equation involving the time derivative of the displacement variable, which leads to a linear system for the time evolution of the unknowns. Then we can use standard mathematical techniques in discontinuous Galerkin methods to obtain stability and optimal error estimates. We also obtain energy dissipation and/or conservation of the scheme by choosing simple and mesh-independent interelement fluxes. Several numerical experiments are presented to illustrate and support our theoretical results.

An H1-Galerkin Space-Time Mixed Finite Element Method for Semilinear Convection–Diffusion–Reaction Equations

In this paper, the semilinear convection–diffusion–reaction equation is split into a lower-order system by introducing the auxiliary variable q=a(x)ux. An H1-Galerkin space-time mixed finite element method for the lower-order system is then constructed. The proposed method applies the finite element method to discretize the time and space directions simultaneously and does not require checking the Ladyzhenskaya–Babusˇka–Brezzi (LBB) compatibility constraints, which differs from the traditional mixed finite element method. The uniqueness of the approximate solutions u and q are proven. The L2(L2) norm optimal order error estimates of the approximate solution u and q are derived by introducing the space-time projection operator. The numerical experiment is presented to verify the theoretical results. Furthermore, by comparing with the classical H1-Galerkin mixed finite element scheme, the proposed scheme can easily improve computational accuracy and time convergence order by changing the basis function.

Distributed Adaptive Optimization Algorithm for Fractional High-Order Multiagent Systems Based on Event-Triggered Strategy and Input Quantization

This paper investigates the distributed optimization problem (DOP) for fractional high-order nonstrict-feedback multiagent systems (MASs) where each agent is multiple-input–multiple-output (MIMO) dynamic and contains uncertain dynamics. Based on the penalty-function method, the consensus constraint is eliminated and the global objective function is reconstructed. Different from the existing literatures, where the DOPs are addressed for linear MASs, this paper deals with the DOP through using radial basis function neural networks (RBFNNs) to approximate the unknown nonlinear functions for high-order MASs. To reduce transmitting and computational costs, event-triggered scheme and quantized control technology are combined to propose an adaptive backstepping neural network (NN) control protocol. By applying the Lyapunov stability theory, the optimal consensus error is proved to be bounded and all signals remain semi-global uniformly ultimately bounded. Simulation shows that all agents reach consensus and errors between agents’ outputs and the optimal solution is close to zero with low computational costs.

An improved pointwise error estimate of the LDG method for 1-d singularly perturbed reaction-diffusion problem

The local discontinuous Galerkin (LDG) method on a Shishkin mesh is studied for a one-dimensional singularly perturbed reaction-diffusion problem. Based on the discrete Green's function, we derive some improved pointwise error estimates in the regular and layer regions. For the LDG approximation to the solution itself, the convergence rate of the pointwise error in the regular region is sharp. For the LDG approximation to the derivative of the solution, the convergence rate of the pointwise error in the whole domain is optimal. We establish also optimal pointwise error estimates in the case that the regular component of the exact solution belongs to the finite element space. Numerical experiments are presented to validate the theoretical results.

On the Optimal Error Exponent of Type-Based Distributed Hypothesis Testing.

Distributed hypothesis testing (DHT) has emerged as a significant research area, but the information-theoretic optimality of coding strategies is often typically hard to address. This paper studies the DHT problems under the type-based setting, which is requested from the popular federated learning methods. Specifically, two communication models are considered: (i) DHT problem over noiseless channels, where each node observes i.i.d. samples and sends a one-dimensional statistic of observed samples to the decision center for decision making; and (ii) DHT problem over AWGN channels, where the distributed nodes are restricted to transmit functions of the empirical distributions of the observed data sequences due to practical computational constraints. For both of these problems, we present the optimal error exponent by providing both the achievability and converse results. In addition, we offer corresponding coding strategies and decision rules. Our results not only offer coding guidance for distributed systems, but also have the potential to be applied to more complex problems, enhancing the understanding and application of DHT in various domains.

Two‐grid finite element method on grade meshes for time‐fractional nonlinear Schrödinger equation

AbstractA two‐grid finite element method with nonuniform L1 scheme is developed for solving the time‐fractional nonlinear Schrödinger equation. The finite element solution in the ‐norm and ‐norm are proved bounded without any time‐step size conditions (dependent on spatial‐step size). Then, the optimal order error estimations of the two‐grid solution in the ‐norm are proved without any time‐step size conditions. Finally, the theoretical results are verified by numerical experiments.

Design and analysis of an exactly divergence-free hybridised discontinuous Galerkin method for incompressible flows on meshes with quadrilateral cells

We generalise a hybridised discontinuous Galerkin method for incompressible flow problems to non-affine cells, showing that with a suitable element mapping the generalised method preserves a key invariance property that eludes most methods, namely that any irrotational component of the prescribed force is exactly balanced by the pressure gradient and does not affect the velocity field. This invariance property can be preserved in the discrete problem if the incompressibility constraint is satisfied in a sufficiently strong sense. We derive sufficient conditions to guarantee discretely divergence-free functions are exactly divergence-free and give examples of divergence-free finite elements on meshes with triangular, quadrilateral, tetrahedral, or hexahedral cells generated by a (possibly non-affine) map from their respective reference cells. In the case of quadrilateral cells, we prove an optimal error estimate for the velocity field that does not depend on the pressure approximation. Our analysis is supported by numerical results.

Construction of Deep ReLU Nets for Spatially Sparse Learning.

Training an interpretable deep net to embody its theoretical advantages is difficult but extremely important in the community of machine learning. In this article, noticing the importance of spatial sparseness in signal and image processing, we develop a constructive approach to generate a deep net to capture the spatial sparseness feature. We conduct both theoretical analysis and numerical verifications to show the power of the constructive approach. Theoretically, we prove that the constructive approach can yield a deep net estimate that achieves the optimal generalization error bounds in the framework of learning theory. Numerically, we show that the constructive approach is essentially better than shallow learning in the sense that it provides better prediction accuracy with less training time.

Analytical Solution of Optimal Symbol Error Rate for Multilevel Pulse-Amplitude-Modulated Signals With One-Bit Quantization

Analytical Solution of Optimal Symbol Error Rate for Multilevel Pulse-Amplitude-Modulated Signals With One-Bit Quantization

Error estimates with low-order polynomial dependence for the fully-discrete finite element invariant energy quadratization scheme of the Allen–Cahn equation

In this paper, for the Allen–Cahn equation, we obtain the error estimate of the temporal semi-discrete scheme, and the fully-discrete finite element numerical scheme, both of which are based on the invariant energy quadratization (IEQ) time-marching strategy. We establish the relationship between the [Formula: see text]-error bound and the [Formula: see text]-stabilities of the numerical solution. Then, by converting the numerical schemes to a form compatible with the original format of the Allen–Cahn equation, using mathematical induction, the superconvergence property of nonlinear terms, and the spectrum argument, the optimal error estimates that only depends on the low-order polynomial degree of [Formula: see text] instead of [Formula: see text] for both of the semi and fully-discrete schemes are derived. Numerical experiment also validates our theoretical convergence analysis.

A heterogeneous ensemble architecture coupling model selection sorting and residual error iterative correction for crude oil price forecasting

Although crude oil price forecasts can provide policymakers, investors, consumers, and producers with scientific and reliable guidance, the nonlinearity and complexity of crude oil markets still pose serious challenges to this work. To this end, we propose a crude oil price ensemble prediction model that combines model selection sorting, multi-objective heterogeneous ensemble learning, and residual error iterative correction. First, a selection strategy based on the weighted information criterion is used to screen benchmark models in the model pool. Subsequently, candidate models are heterogeneously integrated using the multi-objective multi-verse optimization algorithm. VIseKriterijumska Optimizacija I Kompromisno Resenje, Technique for Order Preference by Similarity to Ideal Solution, and Grey Relational Analysis are utilized to select the optimal non-dominated solution of the Pareto surface. Finally, correction technology is designed to adaptively correct the residual error. Optimum error shrinkage and the number of corrections ensure maximum predictable component development. West Texas Intermediate and Europe Brent spot price data are utilized for experimental verification. Results show that the model can accurately describe crude oil price fluctuations, and has excellent robustness and stability. The forecasting method can be used as a reliable and promising energy economic decision algorithm.

11-th order of accuracy for numerical solution of 3-D Poisson equation with irregular interfaces on unfitted Cartesian meshes

For the first time the optimal local truncation error method (OLTEM) with 125-point stencils and unfitted Cartesian meshes has been developed in the general 3-D case for the Poisson equation for heterogeneous materials with smooth irregular interfaces. The 125-point stencils equations that are similar to those for quadratic finite elements are used for OLTEM. The interface conditions for OLTEM are imposed as constraints at a small number of interface points and do not require the introduction of additional unknowns, i.e., the sparse structure of global discrete equations of OLTEM is the same for homogeneous and heterogeneous materials. The stencils coefficients of OLTEM are calculated by the minimization of the local truncation error of the stencil equations. These derivations include the use of the Poisson equation for the relationship between the different spatial derivatives. Such a procedure provides the maximum possible accuracy of the discrete equations of OLTEM. In contrast to known numerical techniques with quadratic elements and third order of accuracy on conforming and unfitted meshes, OLTEM with the 125-point stencils provides 11-th order of accuracy, i.e., an extremely large increase in accuracy by 8 orders for similar stencils. The numerical results show that OLTEM yields much more accurate results than high-order finite elements with much wider stencils. The increased numerical accuracy of OLTEM leads to an extremely large increase in computational efficiency.Also, a new post-processing procedure with the 125-point stencil has been developed for the calculation of the spatial derivatives of the primary function. The post-processing procedure includes the minimization of the local truncation error and the use of the Poisson equation. It is demonstrated that the use of the partial differential equation (PDE) for the 125-point stencils improves the accuracy of the spatial derivatives by 6 orders compared to post-processing without the use of PDE as in existing numerical techniques. At an accuracy of 0.1% for the spatial derivatives, OLTEM reduces the number of degrees of freedom by 900−4⋅106 times compared to quadratic finite elements. The developed post-processing procedure can be easily extended to unstructured meshes and can be independently used with existing post-processing techniques (e.g., with finite elements).

An Improved Differential Evolution for Parameter Identification of Photovoltaic Models

Photovoltaic (PV) systems are crucial for converting solar energy into electricity. Optimization, control, and simulation for PV systems are important for effectively harnessing solar energy. The exactitude of associated model parameters is an important influencing factor in the performance of PV systems. However, PV model parameter extraction is challenging due to parameter variability resulting from the change in different environmental conditions and equipment factors. Existing parameter identification approaches usually struggle to calculate precise solutions. For this reason, this paper presents an improved differential evolution algorithm, which integrates a collaboration mechanism of dual mutation strategies and an orientation guidance mechanism, called DODE. This collaboration mechanism adaptively assigns mutation strategies to different individuals at different stages to balance exploration and exploitation capabilities. Moreover, an orientation guidance mechanism is proposed to use the information of the movement direction of the population centroid to guide the evolution of elite individuals, preventing them from being trapped in local optima and guiding the population towards a local search. To assess the effectiveness of DODE, comparison experiments were conducted on six different PV models, i.e., the single, double, and triple diode models, and three other commercial PV modules, against ten other excellent meta-heuristic algorithms. For these models, the proposed DODE outperformed other algorithms, with the separate optimal root mean square error values of 9.86021877891317 × 10−4, 9.82484851784979 × 10−4, 9.82484851784993 × 10−4, 2.42507486809489 × 10−3, 1.72981370994064 × 10−3, and 1.66006031250846 × 10−2. Additionally, results obtained from statistical analysis confirm the remarkable competitive superiorities of DODE on convergence rate, stability, and reliability compared with other methods for PV model parameter identification.

On Solution Regularity of the Two-Dimensional Radiation Transfer Equation and Its Implication on Numerical Convergence

In this paper, we deal with the differential properties of the scalar flux ϕ ( x ) defined over a two-dimensional bounded convex domain, as a solution to the integral radiation transfer equation. Estimates for the derivatives of ϕ ( x ) near the boundary of the domain are given based on Vainikko’s regularity theorem. The optimal pointwise error estimates in terms of the scalar flux are presented for the two classic finite difference methods: diamond difference (DD) and step difference (SD). Numerical results indicate the implication of the solution smoothness on the numerical convergence behavior.

Optimal error estimates of a SAV–FEM for the Cahn–Hilliard–Navier–Stokes model

We carry out the optimal error estimates of a scalar auxiliary variable (SAV) based the Euler semi-implicit finite element method for the Cahn–Hilliard–Navier–Stokes (CHNS) system in this paper. A SAV is introduced to reformulate the CHNS system into an equivalent system. We apply the Euler semi-implicit method and finite element method for the temporal and spatial discretizations to obtain a linear and unconditional energy stable scheme. The stability of numerical solutions under different norms and the optimal error estimates for our proposed numerical schemes are also presented. Finally, numerical examples are presented to demonstrate the accuracy of the proposed method, and the discrete energy is dissipated in numerical simulations.

Maximum norm error bounds for the full discretization of nonautonomous wave equations

Abstract In the present paper, we consider a specific class of nonautonomous wave equations on a smooth, bounded domain and their discretization in space by isoparametric finite elements and in time by the implicit Euler method. Building upon the work of Baker and Dougalis (1980, On the ${L}^{\infty }$-convergence of Galerkin approximations for second-order hyperbolic equations. Math. Comp., 34, 401–424), we prove optimal error bounds in the $W^{1,\infty } \times L^\infty $-norm for the semidiscretization in space and the full discretization. The key tool is the gain of integrability coming from the inverse of the discretized differential operator. For this, we have to pay with (discrete) time derivatives on the error in the $H^{1} \times L^2$-norm, which are reduced to estimates of the differentiated initial errors. To confirm our theoretical findings, we also present numerical experiments.

Error estimates of high-order compact finite difference schemes for the nonlinear abcd s systems

Abstract In this paper, some fourth-order compact finite difference schemes are derived and analyzed for the nonlinear $abcd$ Boussinesq systems. The optimal order error estimates for the semidiscrete compact finite difference schemes with different cases of dispersion coefficients $a,\ b,\ c,\ d$, are presented. The third-order and fourth-order linearized implicit multistep schemes are adopted for time discretization, and numerical experiments are conducted on the model problems. Numerical results show that the proposed schemes have high accuracy and are consistent with the theoretical analysis.

A weak Galerkin finite element method for nonlinear convection-diffusion equation

In this paper, a weak Galerkin (WG) finite element method for one-dimensional nonlinear convection-diffusion equation with Dirichlet boundary condition is developed. Based on a special variational form featuring two built-in parameters, the semi-discrete and fully discrete WG finite element schemes are proposed. The backward Euler method is utilized for time discretization. The WG finite element method adopts locally piecewise polynomials of degree k for the approximation of the primal variable in the interior of elements, and piecewise polynomials of degree k+1 for the weak derivatives. Theoretically, the optimal error estimates in both discrete H1 and standard L2 norms are derived. Numerical experiments are performed to demonstrate the effectiveness of the WG finite element approach and validate the theoretical findings.

A weak Galerkin finite element method for 1D semiconductor device simulation models

In this paper, we study a weak Galerkin (WG) finite element method for semiconductor device simulations. We consider the one-dimensional drift–diffusion (DD) and high-field (HF) models, which involves not only first derivative convection terms but also second derivative diffusion terms, as well as a coupled Poisson potential equation. The main difficulties in the analysis include the treatment of the nonlinearity and coupling of the models. The weak Galerkin finite element method adopts piecewise polynomials of degree k for the approximations of electron concentration and electric potential in the interior of elements, and piecewise polynomials of degree k+1 for the discrete weak derivative space. The optimal order error estimates in a discrete H1 norm and the standard L2 norm are derived. Numerical experiments are presented to illustrate our theoretical analysis. Moreover, numerical schemes also work out for the discontinuous diffusion coefficient problems.