Optimal Estimates Research Articles

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 Method for Correction of Dynamic Errors When Measuring Flat Surfaces.

This paper presents a new method for correction of dynamic errors occurring when measuring flat surfaces in the presence of mechanical effects. Mechanical effects cause inertial forces and moments that affect the moving components of measuring instruments, thereby causing dynamic errors. The study proposes a mathematical model, on the basis of which algorithms for correction of dynamic errors can be developed. The basic concept of the model is based on determining the optimal estimate in the current coordinate point on the basis of the theoretical model of the measured surface and the information from the measurement that contains errors caused by internal and external factors. Based on this model, an algorithm for real-time data processing has been developed. The algorithm works in "predictor-corrector" mode at each step of which the best estimate is obtained. The estimate is based on minimizing the variance of a random component in which the main values are formed from the accumulated statistical data of the error of the model and the measurement error. This paper presents the results of experimental studies, carried out with simulations of mechanical effects in four modes. The results confirm the high efficiency of the algorithm for high-accuracy measurement of flat surfaces in the presence of mechanical effects.

Asymptotics of spectral functions of lower energy forms on weakly 1-complete manifolds

Let M be a weakly 1-complete manifold. We show that the optimal fundamental estimate holds true on M with mild conditions, then we establish the weak Morse inequalities for lower energy forms on M. We also study the case for q-convex manifolds.

3MSP2: Sequential picking planning for multi-fruit congregated tomato harvesting in multi-clusters environment based on multi-views

Fast and effective picking planning from candidate fruits is important to improve the efficiency and success rate of robotic selective harvesting. However, it is a challenge for tomato picking decisions to simultaneously meet low- damage and high-performance requirements in multi-fruits congregation and multi-clusters environments. For such robotic application scenario, a novel sequential picking planning method is proposed for tomatoes harvesting sequence and picking posture estimation combined eye-on-shoulder and eye-in-hand views. First, the detection and localization of multiple tomatoes in both views are achieved by Yolov5. Then, a cluster Optimal Sequence Planning method was developed by us based on the view of eye-on-shoulder. The method combined the density clustering idea and the shortest movement path to automatically classify multiple neighboring tomatoes into candidate picking clusters and make decisions on the optimal picking order of the clusters. Finally, we developed a tomato Optimal Sequential Grasping method based on the view of eye-in-hand. The method combines collision-free constraints and a shortest time objective to determine the optimal grasping posture estimate and the optimal post-grasping observation position estimate for the target tomato. In indoor picking experiments, the proposed method improves the collision-free picking success rate of picking by 17.2 % and 36.1 %, and the picking efficiency by 25 % and 41.4 %, respectively, compared with the picking planning methods based on random traversal and scanning traversal. In the picking experiments in industrial tomato greenhouse, the collision-free picking success rate reaches 70.9 %, and the average cycle time for each picking task from target-recognizing to success-picking is 8.6 s. This approach offers a promising solution for picking planning of harvesting robot to lower damages and improve success rate, efficiency in the congregated fruit environment.

Riesz transform and Hardy spaces related to elliptic operators having Robin boundary conditions on Lipschitz domains with their applications to optimal endpoint regularity estimates

Riesz transform and Hardy spaces related to elliptic operators having Robin boundary conditions on Lipschitz domains with their applications to optimal endpoint regularity estimates

Utilization of genetic algorithm in tuning the hyper-parameters of hybrid NN-based side-slip angle estimators

This paper proposes a solution to enhance and compare different neural network (NN)-based side-slip angle estimators. The feed-forward neural networks (FFNNs), recurrent neural networks, long short-term memory units (LSTMs), and gated recurrent units are investigated. However, there is a lack in the selection criteria of the architectures’ hyper-parameters. Therefore, the genetic algorithm is integrated with the NN-based estimators to find the optimal hyper-parameters for the studied architectures. The tuned hyper-parameters in this work include the number of neurons, number of layers, activation function, optimizer type, and learning rate. The objective function of the optimization problem is minimizing the root-mean-square error (RMSE) on multiple testing data. The optimal models are further included in the design of a hybrid NN estimator with Kalman filter. In the hybrid estimators, the optimal NN estimators are used as virtual sensors to correct the prediction of the side-slip angle resulting from the mathematical lateral vehicle model. Eventually, the performance of the best selected model is evaluated in terms of different metrics; mean RMSE, mean error variance, mean training time, and mean estimation time. LSTMs are found to achieve the lowest mean RMSE while being tested on highly generalized data yielding the highest training and estimation time. However, FFNNs achieve the lowest RMSE while being tested on low generalized data and the lowest training and estimation time. Meanwhile, it is observed that the hybrid estimators achieved lower RMSE with great enhancement compared to the non-hybridized ones proving the effectiveness of the proposed approach and increasing the side-slip estimation generalization ability in unknown environments with high uncertainties, which are not covered by the training dataset for the NNs estimators.

A type of multigrid method for semilinear elliptic problems based on symmetric interior penalty discontinuous Galerkin method

AbstractThis article introduces a new kind of multigrid approach for semilinear elliptic problems, which is based on the symmetric interior penalty discontinuous Galerkin (SIPDG) method. We first give an optimal error estimate of the SIPDG method for the problem. Then, we design a type of multigrid method, which is called the multilevel correction method, and derive a priori error estimates. The primary idea of this method is to take the solution of the semilinear problem and utilize it to establish a sequence of solutions for associated linear boundary value problem on discontinuous finite element spaces and a newly defined low dimensional augmented subspace. Lastly, numerical experiments are offered to confirm the suggested method's precision and effectiveness.

Uncertainty propagation from probe spacing to Fourier 3-probe straightness measurement

Reliable and precise straightness profile measurements are crucial for manufacturing ultra-precision components and are capable of further enhancing their accuracy. The Fourier three-probe (F3P) straightness measurement allows for precise assessment of the workpiece profile on the machine by eliminating the harmful influence of the error motion of the sliding table. However, the probe spacing uncertainty deteriorates the measurement accuracy remarkably; and, the affecting mechanism behind this phenomenon has not yet been studied in detail. In this context, this paper thoroughly investigated the propagation of the probe spacing uncertainty in the F3P measurement. First, the influence of the probe spacing deviation is analyzed. Next, by calculating the partial differential of Laplace transform of the workpiece profile, we algebraically deduce the probe spacing uncertainty propagation law, especially in the harmonic domain. Subsequently, Monte Carlo simulations are carried out to confirm the derived propagation law. To reduce uncertainty propagation, a hybrid approach is presented: (I) F3P measurements are carried out under changing probe spacings to produce several sets of Fourier coefficients; (II) optimal harmonic estimates are selected individually according to the harmonic uncertainty. Finally, simulations and experimental measurements are performed for verification.

Uniform and optimal error estimates of a nested Picard integrator for the nonlinear Schrödinger equation with wave operator

AbstractWe propose a second‐order nested Picard iterative integrator sine pseudospectral (NPI‐SP) method for the nonlinear Schrödinger equation with wave operator (NLSW) involving a parameter and carry out rigorous error estimates. Actually, the equation propagates wave with wavelength in time, while the amplitude of the leading oscillation is for well‐prepared initial data, and for ill‐prepared initial data, respectively. Based on the exponential integrator and nested Picard iteration, the uniformly accurate (w.r.t. ) NPI‐SP scheme is proposed with the optimal uniform error bounds at in time and spectral accuracy in space for both well‐prepared and ill‐prepared data in ‐norm. This result significantly improves the error bounds of the finite difference methods and exponential wave integrator for the NLSW. Error estimates are rigorously carried out and numerical examples are provided to confirm the theoretical analysis.

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.

Application-based evaluation of multi-basin hydrological models

Hydrological models are generally calibrated and validated using a suite of well-known statistical metrics, which sometimes lack clear connection and tailoring to the local users’ need and therefore limits the evaluation, especially in the case of global climate services. Therefore, in this study, two types of application-based evaluation metrics are introduced, addressing (i) temporal matching of quantile extremes and (ii) relative bias in flow signatures, which supplements commonly used model performance assessment metrics. The introduced metrics are compared to conventional statistical metrics, at seven case study areas across the world, with three model settings representing different datasets and calibrations, generated from the global hydrological model World-Wide HYPE (WW HYPE). The results suggest that different performance results can occur when comparing application-based metrics to conventional ones. This implies that different evaluation metrics reveal models’ capability in various aspects, supporting their application under the corresponding circumstances. Finally, these metrics enable us to propose two model applicability scenarios: generally applicable models and conditionally applicable models. For instance, the WW HYPE with global dataset and local calibration can yield optimal estimates concerning the timing of quantile extremes and temporal variations in flow signatures, despite its suboptimal performance in conventional evaluation metrics. Therefore, it may be considered as a conditionally applicable model which can be used in regions with limited local datasets, supplying reliable information for decision-makers in formulating strategic plans for water resources management.

Interpreting PPV and NPV of Diagnostic Tests with Uncertain Prevalence.

Medical decision-making is often uncertain. The positive predictive value (PPV) and negative predictive value (NPV) are conditional probabilities characterizing diagnostic tests and assessing diagnostic interventions in clinical medicine and epidemiology. The PPV is the probability that a patient has a specified disease, given a positive test result for that disease. The NPV is the probability that a patient does not have the disease, given a negative test result for that disease. Both values depend on disease incidence or prevalence, which may be highly uncertain for unfamiliar diseases, epidemics, etc. Probability distributions for this uncertainty are usually unavailable. We develop a non-probabilistic method for interpreting PPV and NPV with uncertain prevalence. Uncertainty in PPV and NPV is managed with the non-probabilistic concept of robustness in info-gap theory. Robustness of PPV or NPV estimates is the greatest uncertainty (in prevalence) at which the estimate's error is acceptable. Four properties are demonstrated. Zeroing: best estimates of PPV or NPV have no robustness to uncertain prevalence; best estimates are unreliable for interpreting diagnostic tests. Trade-off: robustness increases as error increases; this trade-off identifies robustly reliable error in PPV or NPV. Preference reversal: sometimes sub-optimal PPV or NPV estimates are more robust to uncertain incidence or prevalence than optimal estimates, motivating reversal of preference from the putative optimum to the sub-optimal estimate. Trade-off between specificity and robustness to uncertainty: the robustness increases as test-specificity decreases. These four properties underlie the interpretation of PPV and NPV. The PPV and NPV assess diagnostic tests, but are sensitive to lack of knowledge that generates non-probabilistic uncertain prevalence and must be supplemented with robustness analysis. When uncertainties abound, as with unfamiliar diseases, assessing robustness is critical to avoiding erroneous decisions.

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.

Regularization techniques and inverse approaches in 3D Traction Force Microscopy

The conception of inverse methods in the context of Traction Force Microscopy (TFM) is influenced by multiple factors, such as nonlinear effects, dimensionality (2D/3D), and regularization/constraints, amongst others. Solving the inverse problem often requires the inversion of a matrix, and the presence of noise in the measured displacements can lead to unrealistic reconstructed tractions. To address this issue, Tikhonov regularization is commonly used, penalizing high norm values of computed tractions. The aim of this study is to compare the performance of different inverse methodologies (including some original variations) in 3D TFM, considering constraint imposition and regularization along the formulation. The different methodologies are numerically elaborated within a novel combined Newton–Raphson/Finite Element Method scheme that provides converged solutions in few iterations. The impact of constraint imposition and regularization in traction reconstruction is evaluated in terms of accuracy, efficiency (CPU time), and inherent characteristics of the methodology. Results show that, applying regularization and constraints (based on the fulfillment of fundamental principles) provides the best traction reconstruction, while simultaneously ensuring an optimum estimate of the maximum traction, at the cost of high CPU time and low efficiency. Moreover, regularization-based methods introduce the challenge of calibrating the regularization parameter, usually done under subjective criteria. On the other hand, non-regularized but constrained methods may represent a nice compromise between accuracy and efficiency, while avoiding pre-processing and calibration of such regularization parameter. It is emphasized the importance of considering not only traction reconstruction quality but also the efficiency and complexity of implementation (intrusiveness) when selecting an appropriate inverse method for TFM analysis.

Adaptive minimax estimation of service time distribution in the Mt/G/∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_t/G/\\infty $$\\end{document} queue from departure data

This article deals with the problem of estimating the service time distribution of the Mt/G/∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_t/G/\\infty $$\\end{document} queue from observation of the departure epochs. We develop minimax optimal estimators of G and study behavior of the minimax pointwise risk over a suitable family of service time distribution functions. In addition, we address the problem of adaptive estimation and propose a data–driven estimation procedure that adapts to unknown smoothness of the service time distribution function G. Lastly, a numerical study is presented to illustrate practical performance of the developed adaptive procedure.

Optimal measurement-based cost gradient estimate for feedback real-time optimization

This work presents a simple and efficient way of estimating the steady-state cost gradient Ju based on available uncertain measurements y. The main motivation is to control Ju to zero in order to minimize the economic cost J. For this purpose, it is shown that the optimal cost gradient estimate for unconstrained operation is simply Jˆu=H(ym−y∗) where H is a constant matrix, ym is the vector of measurements and y∗ is their nominally unconstrained optimal value. The derivation of the optimal H-matrix is based on existing methods for self-optimizing control and therefore the result is exact for a convex quadratic economic cost J with linear constraints and measurements. The optimality holds locally in other cases. For the constrained case, the unconstrained gradient estimate Jˆu should be multiplied by the nullspace of the active constraints and the resulting “reduced gradient” controlled to zero.

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.

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.

Implication of Different Data Split Ratios on the Performance of Models in Price Prediction of Used Vehicles Using Regression Analysis

Introduction; Artificial intelligence (AI) and Machine Learning have become buzzwords lately due to technological changes and data quality testing, especially in shape and finish analysis. Lots of research has been conducted for linear regression algorithms to predict the price in different sectors for share stock, rental properties, prices of used cars etc. This study provides suitable data split ratio for optimum cost estimation based on linear regression model. In present days there is an increasing demand for having own car for every middle class family therefore this have given opportunity to motor vehicle business to offer wide range of used vehicle for re-sale especially companies like Maruti Suzuki, Tata motors & Mahendra motors in Indian motor vehicle industries. Therefore, it is important to know the current value of your car before spending your hard-earned money on any item. Objective; The objective of this paper is finding appropriate value of cars in Metropolitans or even in state capitals. Features like model, mileage, AC, seating capacities, fuel type automatic will be taken into account when doing this. This estimate is designed to help customers find the right options to suit their needs. Methods; We have used a linear regression model to estimate the value of the respective car. Results; For doing this price prediction in this paper using liner regression we have tried to find the optimum accuracy of model by varying data split ratio for training and test data set and concluded with the result that 80/20 ratio is the best ratio with optimum model accuracy for business domain analysis with labelled data set. Conclusion; The findings underscore the importance of careful consideration when selecting a data split ratio for price prediction models in the used vehicle market. The insights gleaned from this study can inform future research and contribute to the development of more accurate and reliable regression models in similar domains.

Optimal Estimators of Cross-Partial Derivatives and Surrogates of Functions

Computing cross-partial derivatives using fewer model runs is relevant in modeling, such as stochastic approximation, derivative-based ANOVA, exploring complex models, and active subspaces. This paper introduces surrogates of all the cross-partial derivatives of functions by evaluating such functions at N randomized points and using a set of L constraints. Randomized points rely on independent, central, and symmetric variables. The associated estimators, based on NL model runs, reach the optimal rates of convergence (i.e., O(N−1)), and the biases of our approximations do not suffer from the curse of dimensionality for a wide class of functions. Such results are used for (i) computing the main and upper bounds of sensitivity indices, and (ii) deriving emulators of simulators or surrogates of functions thanks to the derivative-based ANOVA. Simulations are presented to show the accuracy of our emulators and estimators of sensitivity indices. The plug-in estimates of indices using the U-statistics of one sample are numerically much stable.