Optimal Estimates Research Articles

Using Lean-and-Green Supersaturated Poly-Factorial Mini Datasets to Profile Energy Consumption Performance for an Apartment Unit

The Renovation Wave for Europe initiative aspires to materialize the progressive greening of 85–95% of the continental older building stock as part of the European Green Deal objectives to reduce emissions and energy use. To realistically predict the energy performance even for a single apartment building is a difficult problem. This is because an apartment unit is inherently a customized construction which is subject to year-round occupant use. We use a standardized energy consumption response approach to accelerate the setting-up of the problem in pertinent energy engineering terms. Nationally instituted Energy Performance Certification databases provide validated energy consumption information by taking into account an apartment unit’s specific shell characteristics along with its installed electromechanical system configuration. Such a pre-engineered framework facilitates the effect evaluation of any proposed modifications on the energy performance of a building. Treating a vast building stock requires a mass-customization approach. Therefore, a lean-and-green, industrial-level problem-solving strategy is pursued. The TEE-KENAK Energy Certification database platform is used to parametrize a real standalone apartment. A supersaturated mini dataset was planned and collected to screen as many as 24 controlling factors, which included apartment shell layout details in association with the electromechanical systems arrangements. Main effects plots, best-subsets partial least squares, and entropic (Shannon) mutual information predictions—supplemented with optimal shrinkage estimations—formed the recommended profiler toolset. Four leading modifications were found to be statistically significant: (1) the thermal insulation of the roof, (2) the gas-sourced heating systems, (3) the automatic control category type ‘A’, and (4) the thermal insulation of the walls. The optimal profiling delivered an energy consumption projection of 110.4 kWh/m2 (energy status ‘B’) for the apartment—an almost 20% reduction in energy consumption while also achieving upgrading from the original ‘C’ energy status. The proposed approach may aid energy engineers to make general empirical screening predictions in an expedient manner by simultaneously considering the apartment unit’s structural configuration as well as its installed electromechanical systems arrangement.

Two-grid fully discrete finite element algorithms on temporal graded meshes for nonlinear multi-term time-fractional diffusion equations with variable coefficient

In this paper, we are concerned with two-grid fully discrete finite element algorithms for solving the nonlinear multi-term time-fractional diffusion equation with a variable coefficient. Taking into account the weak singularity of the solution at the initial time t=0, we employ graded meshes in time, which is an effective method to compensate for the lack of smoothness. A fully discrete implicit finite element scheme is established with the L1 method on temporally graded meshes; the stability and optimal error estimates are theoretically proved. To improve the computational efficiency, we propose a two-grid algorithm based on the fully discrete scheme, and then the second algorithm is constructed to further improve the optimal convergence order. Similar properties as standard FEM are presented for the two algorithms. A significant fact is that this technique reduces the computation time without loss of numerical precision. Several numerical examples are provided to support our theoretical findings and verify the performance of the proposed algorithms.

A stabilized divergence-free virtual element scheme for the nematic liquid crystal flows

In this work, we propose and analyze a stabilized divergence-free virtual element method (VEM) for approximating the hydrodynamics system of nematic liquid crystal flows. By adding appropriate stabilization term so that the nonlinear potential function can be treated explicitly, and using the implicit-explicit (IMEX) approach to handle the nonlinear coupling terms, we develop a linear and energy-stable fully discrete virtual element scheme. We further prove that the fully discrete scheme is uniquely solvable and energy stable in discrete sense, moreover, we obtain the optimal error estimates rigorously. Finally, numerical examples are provided to demonstrate the accuracy, stability and efficiency of the proposed scheme.

Optimal estimation of the binned mask-free power spectrum, bispectrum, and trispectrum on the full sky: Scalar edition

We derive optimal estimators for the two-, three-, and four-point correlators of statistically isotropic scalar fields defined on the sphere, such as the cosmic microwave background temperature fluctuations, allowing for arbitrary (linear) masking and inpainting schemes. In each case, we give the optimal unwindowed estimator (obtained via a maximum-likelihood prescription, with an associated Fisher deconvolution matrix), and an idealized form, and pay close attention to their efficient computation. For the trispectrum, we include both parity-even and parity-odd contributions, as allowed by symmetry. The estimators can include arbitrary weighting of the data (and remain unbiased), but are shown to be optimal in the limit of inverse-covariance weighting and Gaussian statistics. The normalization of the estimators is computed via Monte Carlo methods, with the rate-limiting steps (involving spherical harmonic transforms) scaling linearly with the number of bins. An accompanying code package, polybin, implements these estimators in python, and we demonstrate the estimators' efficacy via a suite of validation tests.

Optimal estimates for transmission problems including relative conductivities with different signs

We study the gradient and higher order derivative estimates for the transmission problem in the presence of closely located inclusions. We show that in two dimensions, when relative conductivities of circular inclusions have different signs, the gradient and higher order derivatives are bounded independent of ε, the distance between the inclusions. We also show that for general smooth strictly convex inclusions, when one inclusion is an insulator and the other one is a perfect conductor, the derivatives of any order is bounded independent of ε in any dimensions n≥2.

Localized pointwise error estimates for hybrid finite element methods

We present localized pointwise error estimates for a numerical method designed for accurate and efficient approximations for the flux variables. The method can be considered as a reduced version of the mixed finite element method. The method uses much less degrees of freedom compared to the mixed finite element methods while produces approximation solutions for the flux variable of the same accuracy. In this paper, we present optimal pointwise error estimates for the method, and they show local dependence of the error at a point and weaker dependence on global norms.

Image plane wavefront sensing and Kalman filtering for automated alignment of off-axis aspherics.

Automated alignment can significantly increase optical system precision and flexibility, and reduce time and labor costs in system setup and maintenance. We present an automated alignment technique on a double off-axis parabolic mirror system, which poses challenging alignment problems due to the mirrors' high sensitivity to aberrations, rotational asymmetry, and non-orthogonality in stage adjustments. In our methodology, we use focal plane wavefront sensing to eliminate the non-common path error, increase optical throughput, and reduce cost and system complexity. We incorporate model-based optimal estimation and control to better handle the nonlinearity, model uncertainty, and noise. Using either an iterated extended Kalman filter or a square-root unscented Kalman filter as the optimal nonlinear misalignment state estimator, we are able to consistently reduce the linear misalignment from around 1mm to <5µm and the angular misalignment from around 500 to <6a r c s e c in simulation, achieving a final wavefront error of <5∗10-5 waves in the field of view when tested at wavelength 635nm. We discover a multi-state coupling effect, which implies that different misalignment states have compensating effects on system measurements, thus interfering with the estimator's observation of misalignment state changes. We further investigate the coupling's effects on alignment quality through observability analysis.

A method for separating unmodeled errors in GNSS based on the constraint of the prior variance of unit weight

In global navigation satellite system (GNSS), unmodeled errors critically affect the accuracy and reliability of positioning solutions. When the unmodeled errors are significant in the mathematical model, they are mainly processed by choosing adjustment models with additional systematic error parameters or semiparametric estimation. However, many existing methods require knowledge of prior information on unmodeled errors; otherwise, achieving better processing results is difficult. To address this problem, this study proposes a GNSS unmodeled error separation method that does not rely on prior information on unmodeled errors. This method is based on the constraint of prior variance of unit weight. First, the method effectively separates the effect of unmodeled errors in the residuals under this constraint. Second, the initial estimate of the unmodeled errors in the observation domain is used as a virtual observation. Thus, the optimal estimate and variance of the unmodeled errors are obtained. Finally, the observations are effectively corrected by combining the concepts of mean shift and variance inflation. Multiple experiments were performed in this study. The results show that the proposed method can effectively weaken the impact of unmodeled errors on the float and fixed solutions of GNSS positioning, regardless of whether unmodeled errors exist in a single observation or multiple observations.

$$hp$$-Optimal Interior Penalty Discontinuous Galerkin Methods for the Biharmonic Problem

We prove $$hp$$ -optimal error estimates for interior penalty discontinuous Galerkin methods (IPDG) for the biharmonic problem with homogeneous essential boundary conditions. We consider tensor product-type meshes in two and three dimensions, and triangular meshes in two dimensions. An essential ingredient in the analysis is the construction of a global $$H^2$$ piecewise polynomial approximants with $$hp$$ -optimal approximation properties over the given meshes. The $$hp$$ -optimality is also discussed for $$\mathcal C^0$$ -IPDG in two and three dimensions, and the stream formulation of the Stokes problem in two dimensions. Numerical experiments validate the theoretical predictions and reveal that $$p$$ -suboptimality occurs in presence of singular essential boundary conditions.

Genetic least square estimation approach to wind power curve modelling and wind power prediction

Wind power curve (WPC) is an important index of wind turbines, and it plays an important role in wind power prediction and condition monitoring of wind turbines. Motivated by model parameter estimation of logistic function in WPC modelling, aimed at the problem of selecting initial value of model parameter estimation and local optimum result, based on the combination of genetic algorithm and least square estimation method, a genetic least square estimation (GLSE) method of parameter estimation is proposed, and the global optimum estimation result can be obtained. Six evaluation indices including the root mean square error, the coefficient of determination R2, the mean absolute error, the mean absolute percentage error, the improved Akaike information criterion and the Bayesian information criterion are used to select the optimal power curve model in the different candidate models, and avoid the model’s over-fitting. Finally, to predict the annual energy production and output power of wind turbines, a two-component Weibull mixture distribution wind speed model and five-parameter logistic function power curve model are applied in a wind farm of Jiangsu Province, China. The results show that the GLSE approach proposed in this paper is feasible and effective in WPC modelling and wind power prediction, which can improve the accuracy of model parameter estimation, and five-parameter logistic function can be preferred compared with high-order polynomial and four-parameter logistic function when the fitting accuracy is close.

Optimal error estimates of an IPDG scheme for the resistive magnetic induction equation

Optimal error estimates of an IPDG scheme for the resistive magnetic induction equation

Large time behavior and optimal decay estimate for solutions to the generalized Kadomtsev–Petviashvili–Burgers equation in 2D

We consider the Cauchy problem for the generalized Kadomtsev–Petviashvili–Burgers equation in 2D. This is one of the nonlinear dispersive-dissipative type equations, which has a spatial anisotropic dissipative term. Under some suitable regularity assumptions on the initial data u0, especially the condition ∂x−1u0∈L1(R2), it is known that the solution to this problem decays at the rate of t−74 in the L∞-sense. In this paper, we investigate the more detailed large time behavior of the solution and construct the approximate formula for the solution at t→∞. Moreover, we obtain a lower bound of the L∞-norm of the solution and prove that the decay rate t−74 of the solution given in the previous work to be optimal.

Analysis of a new kind of elastic-rigid bilateral obstacle problem

This paper is devoted to a study of a new kind of bilateral obstacle problem. The obstacles are made of rigid bodies covered by soft layers which are deformable and allow penetration. A model of an elastic rigid bilateral obstacle problem is established, and three equivalent descriptions are derived: the energy form, the variational inequality form, and the differential equation form. We prove the solution existence and uniqueness of this model and provide an error estimate of the numerical solutions. The optimal order error estimate for the linear finite element method is derived under the proper regularity assumption. A penalty method is introduced to solve the finite element approximation problem, and the convergence results are obtained when the penalty parameter tends to infinity. Several numerical examples are reported, and the results are in good agreement with the theoretical analysis.

Continuous interior penalty stabilization for divergence-free finite element methods

Abstract In this paper, we propose, analyze and test numerically a pressure-robust stabilized finite element for a linearized problem in incompressible fluid mechanics, namely, the steady Oseen equation with low viscosity. Stabilization terms are defined by jumps of different combinations of derivatives for the convective term over the element faces of the triangulation of the domain. With the help of these stabilizing terms, and the fact the finite element space is assumed to provide a point-wise divergence-free velocity, an $\mathcal O\big(h^{k+\frac 12}\big)$ error estimate in the $L^2$-norm is proved for the method (in the convection-dominated regime), and optimal order estimates in the remaining norms of the error. Numerical results supporting the theoretical findings are provided.

Optimal error estimate of an accurate second-order scheme for Volterra integrodifferential equations with tempered multi-term kernels

Optimal error estimate of an accurate second-order scheme for Volterra integrodifferential equations with tempered multi-term kernels

A modified weak Galerkin method for (curl)-elliptic problem

In this paper, we design and analyze a modified weak Galerkin (MWG) finite element method for H ( curl ) -elliptic problem. We first introduce a new discrete weak curl operator and the MWG finite element space. The modified weak Galerkin method does not require the penalty parameter by comparing with traditional DG methods. We prove an optimal error estimate in energy norm. At last, we provide the numerical results to confirm these theoretical results.

An effective second-order scheme for the nonstationary incompressible magnetohydrodynamics equations

We are proposing the Crank-Nicolson/Adams-Bashforth (CN/AB) algorithm for the nonstationary magnetohydrodynamic (MHD) equations. For time discretization, the time derivative terms are approximated by a first-order Euler-backward scheme, an implicit second-order Crank-Nicolson scheme acting on linear terms, and an explicit Adams-Bashforth scheme dealing with nonlinear terms. We use the finite element method for spatial discretization. Theoretical studies show that the proposed algorithm is almost unconditionally stabilized with τ≤c. The optimal error estimate of the algorithm is obtained by a parabolic argument mathematical trick. Finally, the theoretical results are verified by numerical experiments.

Model-Based Event-Triggered Dynamic Pursuing and Surrounding Control for a Multi-Robotic Fish System

This paper investigates the event-triggered-based pursuing and surrounding control of a multi-robotic fish system. A distributed surrounding control framework is put forward to deploy the whole system and enclose the dynamic evader into a convex hull. In particular, distributed kinematics-based motion predictions are hold for common agents in the multi-robotic fish system in addition to the evader. Based on the optimal state estimations, an innovative model-based event-triggered mechanism is tailored rather than frequent interactions to evaluate the importance of the messages. Afterwards, a two-stage position-based surrounding control method is provided. By arbitrating the trade-off between the control performance and interaction frequency, the event-triggered function is regulated to delicately execute the surrounding control task while sharply slashing the interactions. Finally, adequate simulations and experiments are conducted to ensure the effectiveness and robustness of the proposed method. Therefore, whether for communication energy conservation or attack defense, the proposed method shows its significance for future marine assignments.

Optimal error estimate of discontinuous Galerkin methods for advection-diffusion-reaction problems with low regularity

In this paper, we present a class of discontinuous Galerkin finite element methods for advection-diffusion-reaction problems and establish a priori error estimates when the solution is only in H1+s(Ω) with s∈(0,1/2].

Guaranteeing Differential Privacy in Distributed Fusion Estimation

With the increasing openness of networked sensor systems, information leakage has become an ever-critical issue due to the malicious eavesdropping attacks on privately available data. The distributed fusion estimates (DFEs), which perform well by combining local state estimates (LSEs) supplied by multiple sensors, imperatively require privacy-preserving. Notice that the eavesdropper can acquire DFEs by wiretapping fusion information from FC and weighting LSEs from sensors. For countering such dual eavesdropping, we introduce the idea called differential privacy from database literature. Basically, the publicly released estimates (PREs), which gather statistical information about DFEs, are considered to be disturbed with stochastic noises. The lower bounds of noise covariances can be designed with the privacy parameters and estimation systems. Furthermore, we disperse the location of noise insertion from PREs to LSEs. In this case, the eavesdropper cannot obtain optimal estimate even if it fuses local data by using original criterion, while the differential privacy is still satisfied. Finally, a target tracking system with multi-sensor observation is given to verify the effectiveness of the proposed methods.