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This paper presents a numerical comparison of finite-element methods resulting in local mass conservation at the element level for Poisson's problem, namely the primal hybrid and mixed methods. These formulations result in an indefinite system. Alternative formulations yielding a positive-definite system are obtained after hybridizing each method. The choice of approximation spaces yields methods with enhanced accuracy for the pressure variable, and results in systems with identical size and structure after static condensation. A regular pressure precision mixed formulation is also considered based on the classical RT k space. The simulations are accelerated using Open multi-processing (OMP) and Thread-Building Blocks (TBB) multithreading paradigms alongside either a coloring strategy or atomic operations ensuring a thread-safe execution. An additional parallel strategy is developed using C++ threads, which is based on the producer-consumer paradigm, and uses locks and semaphores as synchronization primitives. Numerical tests show the optimal parallel strategy for these finite-element formulations, and the computational performance of the methods are compared in terms of simulation time and approximation errors. Additional results are developed during the process. Numerical solvers often fail to find an accurate solution to the highly indefinite systems arising from finite-element formulations, and this paper documents a matrix ordering strategy to stabilize the resolution. A procedure to enable static condensation based on the introduction of piecewise constant functions that also fulfills Neumann's compatibility condition, and yet computes an average pressure per element is presented. • Numerical comparison of the mixed, primal hybrid, and other locally conservatives finite element methods. • Introduction of a twice hybridized primal hybrid method, yielding a positive definite matrix after static condensation. • Comparison of multi-threaded strategies and thread-safety mechanisms. • A numerical procedure to stabilize the linear system resolution for indefinite matrices is described. • A general procedure that enables static condensation and fulfills Neumann's compatibility condition. [ABSTRACT FROM AUTHOR]

To solve some scheduling problems of batch processes based on timed Petri net models, timed extended reachability graphs (TERGs) and approximated TERGs can be used. Such graphs abstract temporal specifications and represent parts of timed languages. By exploring the feasible trajectories in a TERG, optimal schedules can be obtained with respect to the makespans of batch processes that are modeled by timed Petri nets. Nevertheless, the rapid growth of the number of states in a TERG makes the problem intractable for large systems. In this paper, we improve the existing clustering TERG approach, and we make it suitable for large sized batch processes. We also enlarge a systematic approach to model batch processes with timed Petri nets. Finally, a comprehensive example of scheduling problem is studied for an archetypal chemical production plant in order to illustrate the efficiency of the proposed approach. • The scheduling issue of batch chemical processes is formalized with timed Petri nets. • A graph named ClusTERG approximates the state space through hierarchical clustering. • The approximation error of this ClusTERG is far less than other approximations. • Such an approach is used to obtain near-optimal schedules of a batch process. [ABSTRACT FROM AUTHOR]

Reconfigurable variable stiffness actuator (RVSA) has attracted increasing attention in robotics due to its safety, compliance, and robustness. However, the control of the RVSA is challenging due to nonlinear factors such as high-order nonlinear dynamic, model uncertainties, time-varying model parameters, and disturbances. In this paper, firstly, a lightweight RVSA structure with both passive and active nonlinear variable stiffness characteristic is developed. Secondly, a dynamic surface backstepping control method based on a radial basis neural network and disturbance observer (DSBC-RBFNN-DOB) is proposed to achieve position control of the lightweight RVSA with matched and unmatched uncertainties. To address solve the "complexity explosion" and noise problems in traditional backstepping control, the dynamic surface backstepping control (DSBC) method is used to design the controller. Then, a method based on radial basis neural network (RBFNN) and disturbance observer (DOB) are used to compensate for the matched and unmatched uncertainties in the link and motor. In this method, the matched uncertainties are compensated using RBFNN, and the DOB is integrated to compensate RBFNN approximation errors and unmatched uncertainties. Through Lyapunov stability analysis, the semi-global boundedness of the controller is proven. Finally, the proposed method is simulated and actually implemented, verifying the effectiveness of the method. Simulation and experimental results show that the root mean square error (RMSE) of the proposed method is only 0.97277° and 0.6418°, respectively. Compared with PID, DSBC, and DSBC-RBFNN, the error reduction percentages in simulation (experiment) are 85.6 % (88.9 %), 49.4 % (88.4 %) and 36.1 % (80.0 %) respectively. • A lightweight reconfigurable variable stiffness actuator with smaller mass and more compact structure is proposed.. • A dynamic surface backstepping control method based on neural network and disturbance observer is proposed. • A method of uncertainty and disturbance estimation combining neural network and disturbance observer is proposed. • Experiments and simulations were conducted under the condition of actively changing stiffness. [ABSTRACT FROM AUTHOR]

This paper proposes a Generalized Multilevel B-spline Approximation (GMBA) method, which addresses scattered data interpolation problems in image processing. Mathematically, the GMBA provides a better solution for the B-spline control lattice by superimposing identical level B-splines compared with traditional Multilevel B-spline Approximation (MBA). Specifically, the GMBA allows the spacing of next control lattice to be subdivided arbitrarily or remained unchanged, which is determined by a predefined spacing set or the current error level. These improvements bring higher approximation accuracy and more flexibility for algorithm design to avoid over-fitting. In this paper, basic GMBA algorithm and its refined algorithm are compiled for image processing. Finally, six relevant cases are involved to test the GMBA, including surface approximation, image enlargement, image completion, and Salt-and-Pepper (SAP) noise removal. The experimental results show that the GMBA has better performance than the MBA in surface approximation and image processing, performs comparatively fast with the best performance on more than half of the standard test images compared with traditional algorithms, and has partially better performance even than deep learning algorithms. The GMBA can effectively recover meaningful details in images contaminated with even extremely high SAP noise level (up to 99%). • We give a generalized multilevel B-spline approximation (GMBA) method. • GMBA allows the control spacing to be subdivided arbitrarily or remained unchanged. • GMBA offers higher approximation accuracy and greater flexibility than traditional MBA. • GMBA provides superior performance and less run time than many of the state-of-the-art methods for SAP noise removal. • GMBA can recovery meaningful detail at noise levels as high as 99% for SAP noise removal. [ABSTRACT FROM AUTHOR]

This work proposes a hybrid modeling framework based on recurrent neural networks (RNNs) and the finite element (FE) method to approximate model discrepancies in time-dependent, multi-fidelity problems, and use the trained hybrid models to perform bias correction of the low-fidelity models. The hybrid model uses FE basis functions as a spatial basis and RNNs for the approximation of the time dependencies of the FE basis' degrees of freedom. The training data sets consist of sparse, non-uniformly sampled snapshots of the discrepancy function, pre-computed from trajectory data of low- and high-fidelity dynamic FE models. To account for data sparsity and prevent overfitting, data upsampling and local weighting factors are employed, to instigate a trade-off between physically conforming model behavior and neural network regression. The proposed hybrid modeling methodology is showcased in three highly non-trivial engineering test-cases, all featuring transient FE models, namely, heat diffusion out of a heat sink, eddy-currents in a quadrupole magnet, and sound wave propagation in a cavity. The results show that the proposed hybrid model is capable of approximating model discrepancies to a high degree of accuracy and accordingly correct low-fidelity models. • A RNN/FE hybrid model is developed to learn model discrepancies of multi-fidelity FE simulations based on trajectory data. • The RNN captures the dynamics of the discrepancy function while spatial discrepancies are resolved by a FE basis. • Upsampling and Gaussian priors are employed to account for sparsity in the discrepancy data and prevent overfitting. • The model is employed as a correction operator for low-fidelity models in three non-trivial test cases. • The bias corrected models are significantly more accurate compared to the original low-fidelity models. [ABSTRACT FROM AUTHOR]

In this paper, we develop a method for the estimation of airgap spatial variation from the harmonics that appear in the stator's current under static eccentricity conditions. We move beyond the detection of the eccentricity fault and endeavour to quantify the severity of this fault. To achieve this, we formulate the problem as a mathematical inverse problem and attempt to solve it in the Bayesian framework using the so-called Bayesian approximation error approach. The solution takes the form of an algorithm capable of estimating the parameter(s) that characterize the spatial variation of the airgap length from noisy and limited data. Furthermore, it can quantify the uncertainty associated with the results as a byproduct of the Bayesian analysis approach employed in this paper. The results demonstrate that with each iteration of the algorithm, estimation accuracy improves, and the uncertainty of the estimated results reduces. • Bayesian approximation error analysis shows promise in monitoring electrical machines with eccentric rotors. • The paper applies Bayesian inversion for online estimation of airgap length in induction machines with eccentric rotors. • The study estimates static eccentricity ratio from limited data on harmonics of the machine line currents. • This paper gives an analytical method for calculating all inductances of induction machines under static eccentricity. [ABSTRACT FROM AUTHOR]

Different from the construction process of the traditional finite difference method, we derive a large class of high-accuracy methods for approximating derivatives. Since Taylor expansion is avoided, the requirement for function smoothness in the new methods is greatly reduced. We analyze the approximation errors of the proposed methods and compare their approximation effects. As a typical application, we use the proposed methods to solve Elliptic, Hyperbolic and Parabolic problems numerically. For time-dependent problems, a class of new fully implicit difference schemes and the fourth-order Runge-Kutta method are both used for discretization. When using fully implicit difference schemes, unconditional stability and convergence are proved for hyperbolic and parabolic problems with the periodic boundary conditions. A large number of numerical experiments are provided to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

The walking stability of a humanoid robot is a fundamental problem due to the complex nonlinear dynamic model of the robot's legs. This work introduces the performance tracking control for the humanoid NAO robot by using a Nonlinear Disturbance Observer (NDO)-based Fixed-time Terminal Sliding Mode (FTSM). The influence of uncertain external disturbance is considered while implementing the control strategy to improve the walking motion of the NAO robot. An NDO is adapted to estimate the uncertainties and external disturbances. A novel FTSM surface is proposed to drive the tracking errors to zero in fixed-time. The designed NDO-based FTSM control law achieves robustness while reducing the chattering phenomenon. The Lyapunov's stability theory is used to establish the fixed-time stability of the sliding surface and system states under the proposed control method. To validate the performance of the proposed NDO-based FTSM control, a real-time experiment was conducted on a humanoid NAO robot to demonstrate the improved tracking performance in the presence of the uncertain perturbation effect. The effectiveness of the proposed controller design is validated on a flat, upward inclined surface, and compared to another controller. • Novel FTSM control law can improve convergence speed to ensure tracking errors reach zero in a fixed time. • The NDO provides the estimate of an unknown disturbance in real-time, to minimize the approximation error. • Using the proposed control on the real NAO robots, stabilized motion is obtained. [ABSTRACT FROM AUTHOR]

An adaptive neural network energy-based control approach is proposed in this paper to address the issue of stabilizing a partial space elevator during cargo transportation. The method handles the challenges presented by external disturbances and model uncertainties. The dynamic model of the partial space elevator, with a typical multi-input and multi-output underactuated system, is presented. A control scheme using energy-based techniques is proposed to guarantee the stable configuration of the system, and auxiliary kinematic vector variables are designed to incorporate the uncontrollable variables to constitute controllable state subsets. Under these frameworks, adaptive control schemes are formulated using multi-layer neural networks to mitigate model uncertainty. Robust compensators are designed to effectively counteract the unknown boundaries of external disturbances and approximation errors. The stability of the closed control system is rigorously demonstrated through the application of the Lyapunov function and LaSalle's invariance principle. Numerical simulations demonstrate that the proposed control method is highly effective in suppressing oscillations during the transfer period of a partial space elevator. [ABSTRACT FROM AUTHOR]

This work presents the development of a state-input affine differential neural network that approximates a class of nonlinear systems with uncertain dynamics and perturbations. The feasibility of using differential neural networks is based on the approximation capabilities of artificial neural networks, assuring that the approximation error of the right-hand side of the system with uncertain dynamics is bounded. The dynamics of the free parameters of the differential neural network is obtained using the Lyapunov's second stability method, which provides the convergence of the identification error to an invariant set around the origin. The approximation capabilities of the proposed identifier are compared with a classical differential neural network, aiming to identify the dynamics of a Stewart platform, which is used as testing example. The results show that the cumulative norm of identification's error is smaller using the state-input affine differential neural network, which demonstrates better identification outcomes with the proposed modified identifier. An additional advantage of the state-input affine differential neural network is that it has a quasi-linear structure that could help formulate a controller for several systems, which is a further research interest. • Learning laws for State-input affine differential neural network (SIADNN) identifier. • Design based on the second stability method of Lyapunov. • The novel DNN identifier has more applications than the traditional DNN identifier. • Evaluation of performing the non-parametric identification of a Stewart platform. [ABSTRACT FROM AUTHOR]

The success of a multi-kilometre drive by a solar-powered rover at the lunar south pole depends upon careful planning in space and time due to highly dynamic solar illumination conditions. An additional challenge is that the rover may be subject to random faults that can temporarily delay long-range traverses. The majority of existing global spatiotemporal planners assume a deterministic rover-environment model and do not account for random faults. In this paper, we consider a random fault profile with a known, average spatial fault rate. We introduce a methodology to compute recovery policies that maximize the probability of survival of a solar-powered rover from different start states. A recovery policy defines a set of recourse actions to reach a safe location with sufficient battery energy remaining, given the local solar illumination conditions. We solve a stochastic reach-avoid problem using dynamic programming to find an optimal recovery policy. Our focus, in part, is on the implications of state space discretization, which is required in practical implementations. We propose a modified dynamic programming algorithm that conservatively accounts for approximation errors. To demonstrate the benefits of our approach, we compare against existing methods in scenarios where a solar-powered rover seeks to safely exit from permanently shadowed regions in the Cabeus area at the lunar south pole. We also highlight the relevance of our methodology for mission formulation and trade safety analysis by comparing different rover mobility models in simulated recovery drives from the LCROSS impact region. • The safety of a solar-powered rover depends on its location, energy and time of day. • Maximally safe behaviours depend on when and where delays occur during a traverse. • The probability of survival of a rover varies non-monotonically with time. • Min–max dynamic programming can conservatively account for the temporal risk ambiguity. [ABSTRACT FROM AUTHOR]

In this paper, we first design a residual-type error estimator for a modified weak Galerkin (MWG) method of 2D H (curl) elliptic problems. Then, we show that the indicator is reliable with respect to the approximation error measured in terms of a natural energy norm, the main ingredient of our approach is to translate the error between weak solution and MWG solution into the conforming part and nonconforming part. We also prove the efficiency of the error estimator by using standard bubble functions. Finally, we provide several experiments to verify the performance of the error estimator within both uniform meshes and adaptive meshes. [ABSTRACT FROM AUTHOR]

The free-float space manipulator (FFSM) control problem subject to coupling dynamics, uncertainties, and input saturation constraints is investigated in this paper. The screw-based dynamical model is established through the Lagrangian equation and Hamilton's variational principle such that the posture of the rigid body can be described uniquely and globally. The derived dynamics is then reduced to a decoupled one in the light of the conservation of the momentum. On this basis, a novel anti-windup finite-time sliding-mode control strategy is investigated to guarantee the tracking performance in the joint space of FFSM, where a learning-based adaptive neural network is designed to estimate the uncertainties. The approximation error is inhibited by a tanh-type slow-varying adaptive law which is integrated into an auxiliary system to compensate for input saturation nonlinearities. With the contributions above, the stability of the proposed nonsingular terminal sliding-mode (NTSM) controller is comprehensively proved. Comparative simulations based on a 6-DOF FFSM confirm the validity of the proposed control scheme. [ABSTRACT FROM AUTHOR]

In [1] , we proved that the inverse of the stiffness matrix of an h -version finite element method (FEM) applied to scalar second order elliptic boundary value problems can be approximated at an exponential rate in the block rank by H -matrices. Here, we improve on this result in multiple ways: (1) The class of meshes is significantly enlarged and includes certain exponentially graded meshes. (2) The dependence on the polynomial degree p of the discrete ansatz space is made explicit in our analysis. (3) The bound for the approximation error is sharpened, and (4) the proof is simplified. [ABSTRACT FROM AUTHOR]

This paper presents a new nonlinear robust attitude control strategy for the tilt trirotor unmanned aerial vehicle (UAV). Fixed-time convergence control of the UAV's attitude tracking errors under the effects of model uncertainties and unknown external disturbances is achieved by utilizing the proposed control design. Actor–critic (AC) structure based neural networks are trained only with the information of the UAV's inputs and outputs data, to handle the UAV's modeling uncertainties with bounded estimation error. Then a sliding-mode based fixed-time controller is designed to compensate the approximation error of the neural networks and the unknown external disturbances. Based on the Lyapunov stability theory, the stability analysis of the closed-loop system is presented. The performance of the presented nonlinear robust control strategy is validated through the real-time flight experiments. • Reinforcement learning networks are designed to compensate modeling uncertainties. • A fixed-time controller is proposed for approximation errors and disturbances. • Completed stability analysis of the closed-loop system is presented. • Real-time experiments are performed to validate the performance of the control law. [ABSTRACT FROM AUTHOR]

The low-order approximation emerges as a promising method for capturing the dynamic behavior of Printed Circuit Heat Exchangers (PCHEs) in sCO 2 Recompression Closed Brayton Cycles (RCBCs). However, two primary challenges hinder widespread adoption: concerns regarding feasibility and the determination of key parameters, such as the time constant. To address these challenges, this study first affirms the feasibility of approximating PCHE's dynamic behaviors with a low-order system. In specific, the validity of first-order approximation on the dynamics of high-temperature recuperator (HTR) with mild non-linear property variations is initially assumed. However, a discrepancy is noticed under various operating conditions by employing the same time constant after being compared with the reference model (SCOPE). Then, a novel concept of the optimal time constant, τ op , is introduced to underscore the influence of operating conditions on parameter accuracy. Through extensive parameter sweeping, the impact of operating conditions on approximation error is evaluated from two perspectives. First, utilizing τ op effectively reduces approximation errors to acceptable levels. The mean squared error remains below 2 ° C 2 for mass flow rate fluctuations below 32.5 kg/s, and below 4 ° C 2 for temperature variations below 40 ° C. Second, the optimal time constant is significantly influenced by mass flow rate, notably affected by geometry configuration, and minimally influenced by hot inlet temperature. Additionally, this study provides insights for design optimization and control strategy selection, as the findings highlight the trade-off between response speed and thermal performance, and the relatively rapid response of sCO 2 -PCHEs to temperature control, compared with mass flow rate control. [ABSTRACT FROM AUTHOR]

This paper proposes a fault-tolerant control scheme for tracking and controlling hypersonic vehicles with unknown dynamics, actuator failures, and unmeasurable states. The approach involves using a radial-based neural network to approximate the unknown dynamics and reconstruct the entire system model. Additionally, a neural network state observer is proposed to estimate the unmeasurable state of the system. To address the impact of actuator faults, a nonlinear observer is designed to estimate and compensate for the approximation error of the neural network system and fault values. Furthermore, a prescribed performance function is introduced to ensure both transient and steady-state performance of the system. The bounded stability of the closed-loop system is demonstrated through Lyapunov stability analysis. • Unmeasurable states, actuator faults, and state constraints in hypersonic flight vehicle control are investigated. • The neural network state observer realizes the estimation of the unmeasurable state of the system under fault conditions. • The nonlinear observer can estimate perturbations, and the prescribed performance function guarantees transient and steady-state performance. [ABSTRACT FROM AUTHOR]

Signed distance fields (SDFs) have emerged as an alternative shape representation for real-time collision detection and lighting effects. Computing these for complex models can be expensive, so one popular approach is to prepare an approximation via sampling and interpolation. Then, these may be rendered using sphere marching, which gets close to the surface quickly, but needs several iterations to converge to it. In this paper, we propose an alternative that computes the intersection of a given ray and the surface analytically at a narrow band. This may be combined with other enhancements like having variable error for the approximation depending on the distance to the surface and skipping regions that do not contain the surface to accelerate the outer band ray traversal while reducing the required memory. To achieve smoother representations with minimal computational cost, we propose a method for computing surface intersections and normals from separate interpolants. We evaluate all these to find the optimal combination improving the rendering performance and memory consumption of these SDF approximations. [Display omitted] • Enhancements for rendering SDF piecewise approximations. • Solving narrow band intersections using analytical methods. • Reducing SDF approximations for improving required memory and outer band ray traversal. [ABSTRACT FROM AUTHOR]

Recently, Cheval-Garabédian et al. (2023) published a paper in Ore Geology Reviews entitled "The La Lucette Sb-Au-(W) vein-deposit (Armorican Massif, France): new time and genetic constraints to decipher the 310–295 Ma Sb-Au metallogenic peak in the Variscan Belt." Studies on Sb mineralisation are highly welcomed, especially in France, where there is a lack of updated data hindering a comprehensive understanding of this mineral system. The paper provides new mineralogical, fluid inclusion and geochronological data from historical collection samples of La Lucette mine, a mining site no more accessible since at least 40 years (Serment, 1978). These data, together with structural data from literature, lead the authors to build a metallogenic model related to dextral shear zone during the Carboniferous-Permian transition. The authors further argue for a single and unique 310–295 Ma Sb-Au metallogenic peak in the European Variscan Belt, a late Variscan hydrothermal event coeval with post-thickening tectonics (Munoz et al., 1992; Bouchot et al., 1997, 2005). This assertion is not correct. In this comment, we highlight that (i) several errors and approximations question the validity of their model, and (ii) the authors omit certain information that contradicts their metallogenic model. In our view, their data do not fully support the existence of a generalised and unique Late Carboniferous Sb-Au metallogenic peak in the French Variscan Belt. Furthermore, the idea defended by the authors that ore deposits occurred during dextral strike-slip tectonics that affected the area may better support a possible long-lived history during Palaeozoic times. [ABSTRACT FROM AUTHOR]

Hybrid quantum mechanics/molecular mechanics (QM/MM) models play a pivotal role in molecular simulations. These models provide a balance between accuracy, surpassing pure MM models, and computational efficiency, offering advantages over pure QM models. Adaptive approaches have been developed to further improve this balance by allowing on-the-fly selection of the QM and MM subsystems as necessary. We propose a novel and robust adaptive QM/MM method for practical material defect simulations. To ensure mathematical consistency with the QM reference model, we employ machine-learning interatomic potentials (MLIPs) as the MM models (Chen et al., 2022 and Grigorev et al., 2023). Our adaptive QM/MM method utilizes a residual-based error estimator that provides both upper and lower bounds for the approximation error, thus indicating its reliability and efficiency. Furthermore, we introduce a novel adaptive algorithm capable of anisotropically updating the QM/MM partitions. This update is based on the proposed residual-based error estimator and involves solving a free interface motion problem, which is efficiently achieved using the fast marching method. We demonstrate the robustness of our approach via numerical tests on a range of crystalline defects comprising edge dislocations, cracks and di-interstitials. [ABSTRACT FROM AUTHOR]

Reconstructed temperature field will provide critical input for monitoring the aerodynamic heating on the thermal protection system (TPS). In this paper, a novel data-driven method is proposed that allows real-time reconstruction of the 3D transient non-uniform temperature field in a TPS ceramic tile. The measured temperature is obtained by embedding sparse sensors at different layers of the tile. The spatial feature bases and time projection coefficients of the non-uniform temperature fields are extracted using the multi-parameter proper orthogonal decomposition (POD) method. The Fully-connected Long Short-Term Memory (FC-LSTM) networks are used to establish the prediction relationship between the measurements and the full-field temperatures. The problem to estimate the projection coefficients of the spatial feature bases is solved by optimizing the loss function using Adam algorithm. The numerical examples show an approximation error less than 5% and the reconstruction time-consuming less than 1 s in 3D temperature field reconstruction (TFR). The proposed reconstruction method is robust at different measurement noise levels and tested point arrangements. [ABSTRACT FROM AUTHOR]

To address ambiguity arising from linguistic decision judgments, Guo and Zhao introduced a model of the well-known Multi-Criteria Decision-Making (MCDM) method, Best-Worst Method (BWM), using fuzzy sets, known as Fuzzy Best–Worst Method (FBWM). Despite its versatility, FBWM has limitations, such as criteria-weights being independent of fuzzy comparison values' shape and reliance on approximate fuzzy arithmetic operations. In response to these limitations, we introduce a novel model of Fuzzy Best–Worst Method based on α -cut intervals. In this model, the calculation of optimal weights involves formulating an optimization problem that utilizes exact fuzzy arithmetic operations defined in terms of α -cut intervals. This approach aims to optimize the entire shape of fuzzy comparison values simultaneously. However, it turns out that, although we have proven the existence of optimal solution for this problem, solving it proves to be challenging. To resolve this issue, we approximate optimal weight set(s) using triangular fuzzy weight sets and assess the Error of Approximation (EA). The approximate weight set with the desired EA is then defuzzified to obtain a crisp weight set. In situations where there are multiple approximate weight sets with the same EA, interval-weights are calculated. Following that, we establish the necessary conditions for a Fuzzy Pairwise Comparison System (FPCS) to be consistent. Using these conditions, we calculate a lower bound of the Consistency Index (CI), providing a measure of the accuracy of weights. Finally, we discuss numerical examples and a real-world scenario, ranking of risk factors in supply chain 4.0, to validate the effectiveness of the proposed model. • Introducing an α -cut interval based model of the best–worst method. • Approximation of optimal weights and estimation of error of approximation. • Addressing non-uniqueness of optimal weights by calculating interval-weights. • Introduction of a consistency ratio to assess the reliability of weights. • The proposed model outperforms existing models in a thorough comparison. [ABSTRACT FROM AUTHOR]

The generalized finite difference method (GFDM) is a typical meshless collocation method based on the Taylor series expansion and the moving least squares technique. In this paper, we first provide theoretical results of the meshless function approximation in the GFDM. Properties, stability and error estimation of the approximation are studied theoretically, and a stabilized approximation is proposed by revising the computational formulas of the original approximation. Then, we provide theoretical results consisting of error bound and condition number of the GFDM. Numerical results are finally provided to confirm these theoretical results. [ABSTRACT FROM AUTHOR]

This paper investigates the performance-guaranteed adaptive self-healing control for wastewater treatment processes (WWTPs), in which the non-ideal actuator, i.e., the actuator suffering from faults and constraints is considered. Firstly, an error conversion dynamic model based on a new prespecified-time performance function is presented to guarantee that the tracking error is quickly maintained in the user-defined area regardless of the occurrence of actuator faults and constraints. Secondly, a novel smooth approximation function is constructed to imitate the non-smooth characteristics of actuator constraints. Thirdly, the unknown bounds of non-ideal actuator parameters are estimated online in place of parameters themselves. Meanwhile, the lumped disturbances induced by the external disturbance and the fuzzy approximation error are suppressed via H ∞ control theory. In addition, the stability analysis is carried out by the Lyapunov stability criterion to guarantee that all closed-loop system signals are bounded. Finally, comparative simulation results demonstrate that the proposed method has the superiorities in terms of preferable dynamic tracking performance with fast transient convergence, strong robustness and performance recovery under non-ideal actuator. • Smooth approximation function is designed to simulate the constraint characteristic of WWTP actuator. • Error conversion dynamic model using prespecified-time performance function is constructed. • Unknown bounds of non-ideal actuator parameters are estimated online. • Performance-guaranteed adaptive self-healing controller is proposed. [ABSTRACT FROM AUTHOR]

Optimal a priori error bounds are theoretically derived, and numerically verified, for approximate solutions to the 2D homogeneous wave equation obtained by the spectral element method. To be precise, the spectral element method studied here takes advantage of the Gauss-Lobatto-Legendre quadrature, thus resulting in under-integrated elements but a diagonal mass matrix. The approximation error in H 1 is shown to be of order O (h p) with respect to the element size h and of order O (p − q) with respect to the degree p , where q is the spatial regularity of the solution. These results improve on past estimates in the L 2 norm, particularly with respect to h. Specific assumptions on the discretization by the spectral element method are the use of a triangulation by quadrilaterals constructed via affine transformations from a reference square element and of a second order discretization in time by the leap-frog scheme. • A priori error estimates are derived for the fully discrete wave equation using SEM. • Established sub-optimal h bounds in L2 are extended here to optimal H1 bounds. • Numerical tests show optimal rates even if GLL quadrature causes under-integration. • Theoretical and numerical rates of convergence agree with respect to h and p. [ABSTRACT FROM AUTHOR]

In this paper we analyze the error as well for the semi-discretization as the full discretization of a time-dependent convection-diffusion problem. We use for the discretization in space the local discontinuous Galerkin (LDG) method on a class of layer-adapted meshes including Shishkin-type and Bakhvalov-type meshes and the implicit θ -scheme in time. For piecewise tensor-product polynomials of degree k we obtain uniform or almost uniform error estimates with respect to space of order k + 1 / 2 in some energy norm and optimal error estimates with respect to time. Our analysis is based on careful approximation error estimates for the Ritz projection related to the stationary problem on the anisotropic meshes used. We discuss also improved estimates in the one-dimensional case and the use of a discontinuous Galerkin discretization in time. Numerical experiments are given to support our theoretical results. [ABSTRACT FROM AUTHOR]

In this article, we discuss goal-oriented a posteriori error estimation for the biharmonic plate bending problem. The error for a numerical approximation of a goal functional is represented by several computable estimators. One of these estimators is obtained using the dual-weighted residual method, which takes advantage of an equilibrated moment tensor. Then, an abstract unified framework for the goal-oriented a posteriori error estimation is derived based on the equilibrated moment tensor and the potential reconstruction that provides a guaranteed upper bound for the error of a numerical approximation for the goal functional. In particular, C 0 interior penalty and discontinuous Galerkin finite element methods are employed for the practical realisation of the estimators. Numerical experiments are performed to illustrate the effectivity of the estimators. [ABSTRACT FROM AUTHOR]

In this technical note, we present an adaptive fuzzy hierarchical sliding mode control method to deal with the control problem of under-actuated switched nonlinear systems. For the system under consideration, both the issues of unknown uncertain functions and aperiodically updating input are taken into account, which are of practical importance. A bounded time-varying function is employed to make a linear transformation of the control input, leading to a transformed system that can be applied to the control design. By introducing the so-called hierarchical structure, a top layer hierarchical sliding surface containing all the system states' information is obtained. Furthermore, by carrying out fuzzy logic systems' universal approximation, the problem caused by unknown system uncertainties is tackled. The approximation errors together with the measurement error resulted from the effects of the triggering event are lumped into a function, and its upper bound is estimated on-line. Based on these, the boundedness of all the signals are verified by combining the Lyapunov theory and projection algorithm. To testify the validity of our control scheme, a simulation example is carried out. [ABSTRACT FROM AUTHOR]

• Unique Augmentation Algorithm: Combines variational autoencoders with metric learning for time series. • Normalization of Irregularities: Normalizes heteroscedastic and non-stationary data effectively. • Advanced VAE Integration: Integrates GRU into VAE, enhanced by metric learning for better representation. • Improved Data Interpretability: Maintains temporal structure through multi-seasonal decomposition. • Superior Performance: Shows enhanced accuracy and fidelity in data-limited scenarios. In the field of time series classification, deep learning techniques have shown remarkable performance; however, their effectiveness is often compromised when confronted with challenges of insufficient data and class imbalance. To address this challenge, we propose an interpretable time series data augmentation algorithm integrating variational autoencoders (VAE) and metric learning. The core contribution of this algorithm is manifested in three aspects: First, it eliminates the heteroscedasticity and non-stationarity of the data, ensuring that the data satisfies the hypothesis of normal distribution in the potential space of the encoder, and effectively avoids the approximation error of the real data distribution; Secondly, the algorithm constructs a discriminant VAE potential space, suitable for data augmentation, with metric learning, ensuring that the hidden variable distribution accurately reflects the characteristics of the original data. Finally, this paper explores the multi-seasonal decomposition algorithm of time series to seamlessly integrate the structural features of the original time series in the generated data, thereby enhancing the interpretability of data generation. Through experimental verification on four multivariate time series data sets, including the electrical energy data set, the results demonstrate that the proposed algorithm outperforms existing methods in fidelity and prediction performance, exhibiting high stability and generalization ability, particularly in cases of limited data volume. The introduction of this algorithm not only contributes to enhancing the overall performance of time series classification models but also substantially reduces the cost of data collection and labeling, thereby demonstrating its significant value in practical applications. [ABSTRACT FROM AUTHOR]

High-dimensional dynamical systems often require computationally intensive physics-based simulations, making full physical space data assimilation impractical. Latent data assimilation methods perform assimilation in reduced-order latent space for efficiency but struggle with complex, nonlinear state-observation mappings. Recent solutions like Generalized Latent Data Assimilation (GLA) and Latent Space Data Assimilation (LSDA) address heterogeneous latent spaces by incorporating surrogate mapping functions but introduce computational costs and uncertainties. Furthermore, current algorithms that integrate data assimilation and deep learning still face limitations when it comes to handling non-explicit mapping functions. To address these challenges, this paper introduces a novel deep-learning-based data assimilation scheme, named Multi-domain Encoder–Decoder Latent Data Assimilation (MEDLA), capable of handling diverse data sources by sharing a common latent space. The proposed approach significantly reduces the computational burden since the complex mapping functions are mimicked by the multi-domain encoder–decoder neural network. It also enhances assimilation accuracy by minimizing interpolation and approximation errors. Extensive numerical experiments from three different test cases assess MEDLA's performance in high dimensional dynamical systems, benchmarking it against state-of-the-art latent data assimilation methods. The numerical results consistently underscore MEDLA's superiority in managing multi-scale observational data and tackling intricate, non-explicit mapping functions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the position value as a centrality measure to evaluate the relevance of the edges and players in a network, with the additional advantage that this value integrates the degree measure of each player in it. In fact, in the real world, it is particularly important to consider the natural influence of connections of a player in a network. Its applications were very limited in real-world situations due to the high computational complexity of exactly obtaining this value. With the aim of solving this problem we provide a method, based on sampling theory, to estimate the position value, which is analyzed in terms of the theoretical properties of the resulting estimator. Moreover, we establish specific statistical results for bounding the absolute error in this approximation. It is important to emphasize that this approach allows for obtaining rankings not only of the nodes but also of the edges of the network. To illustrate the advantages and interest of the proposed methodology, as well as the variety of problems that can be analyzed in this framework, we applied it in three very different settings, the suburban train network of Madrid in the year 2000, the Spanish national team in a match against Portugal, and the Zerkani network responsible for the terrorist attacks of Paris (2015) and Brussels (2016). • The position value gives a measure of the relevance of the links and players in a network. • We rank the members of networks using the position value. • Sampling techniques are used to solve the computational drawbacks in these settings. • As an application, we rank the Spanish national team and the Zerkani network. [ABSTRACT FROM AUTHOR]

Regarding the nonconvexity of the optimal power flow (OPF) problem, power industry typically employs convex approximation approaches to satisfy the computational demand. However, the accuracy performances of the convex OPF models are hard to be predicted in actual operating power systems. To control the maximum effect of the approximation error on power dispatch, it is important to solve and compare the worst-case approximation errors of these OPF models (i.e., the error bounds). In this paper, the error bound is represented by the maximum gap of the optimal solutions of the nonconvex and convex OPF models (i.e., the optimality gap) which is further equivalent to be the Hausdorff distance of the feasible regions of the two OPF models. Subsequently, an optimization model is proposed to generally solve the Hausdorff distance as well as the error bound. Case studies show that the proposed error bound is tight and effective when the power system is not seriously congested. The relaxation rate of the error bounds is less than 5% when congestion rate is less than 10%. It provides theoretical guidance for selecting the convex OPF model under the condition without historical operating data. • The error bound of convex OPF models is generally modeled based on Fenchel duality. • An approach for solving error bounds is proposed from Hausdorff distance calculation. • The accuracy and application of the error bound measurement are numerically presented. [ABSTRACT FROM AUTHOR]

High-order mesh optimization has many goals, such as improving smoothness, reducing approximation error, and improving mesh quality. The previous methods do not optimize these objectives together, resulting in suboptimal results. To this end, we propose a multi-objective optimization method for high-order meshes. Central to our algorithm is using the multi-objective genetic algorithm (MOGA) to adapt to the multiple optimization objectives. Specifically, we optimize each control point one by one, where the MOGA is applied. We demonstrate the feasibility and effectiveness of our method over various models. Compared to other state-of-the-art methods, our method achieves a favorable trade-off between multiple objectives. • We propose a multi-objective optimization method for high-order tetrahedral meshes. • Employ the multi-objective genetic algorithm to relocate control points to achieve a favorable trade-off among objectives. • Select the best individual via the weighted TOPSIS method, with weights determined by combining multi-objective problem. [ABSTRACT FROM AUTHOR]

During the actual flight of unmanned helicopters, there are widespread output constraints, system uncertainties and external disturbances, which bring significant threats to their safe flight. To keep the output constraints from violating predefined boundaries, the barrier Lyapunov function (BLF) is utilized in this paper. The system uncertainties are estimated by the fuzzy logic system (FLS). The compound disturbances composed of the external disturbances and the approximation errors caused by FLS are approximated by the fixed-time disturbance observers (FTDOs). According to the analysis of FTDOs and fixed-time stability theory, the observer errors and state errors converge to a region of the origin in fixed-time, and the appropriate parameters of FTDOs and controller are designed to ensure that the disturbance convergence time is faster than the one of the proposed controller, which further improves the stability of the system. Moreover, the convergence time of the FTDOs FTDOs and the proposed approach only depends on the design parameters under any initial conditions with constrained outputs. The performance of the proposed control approach is verified through the simulation results. [ABSTRACT FROM AUTHOR]

Flow direction algorithms have important application in attracting geomorphic features and topographic attributes, which serve as inputs for some hydrological and topographical models. Evaluating flow direction algorithms is of great significance and often conducted on synthetic surfaces instead of real digital elevation models for free of approximation errors. However, most widely-used synthetic surfaces are too simplified to represent complex topographical relief of real-world terrains. For this, this work applies a complex synthetic surface of modified Himmelblau's function (HF) to simulate sophisticated terrains encountered in real landscapes. HF surface is spatially smooth and continuous with four hilltops and one valley, where plan curvatures are clustered mainly from −0.1 to 0.1. In addition, a slope line-based discretization numerical (SLDN) approach is designed for obtaining numerical solution to theoretical total contributing area (TCA) on synthetic surfaces of non-integrable slope lines (e.g. HF surface). TCAs estimated by several flow direction algorithms are compared with SLDN-derived TCA quantitatively. Results indicate that the largest and smallest mean size errors are obtained by Random eight-node (Rho8) (i.e. 82.6 %) and Freeman multiple flow direction (FMFD) (i.e. 17.9 %), while the largest and smallest mean extent errors by Eight drainage directions (D8) (i.e. 128.2 %) and Eight drainage directions, least transversal deviation (D8-LTD) (i.e.55.4 %). Most mean errors are larger than 20.0 %, which may not be satisfactory in practice. This work can provide a reference for flow direction algorithms application in digital terrain analysis, and therefore improve accuracy of hydrological, geological and geomorphological models. [Display omitted] • Characteristics of the complex HF surface are comprehensively investigated. • An SLDN approach is used to obtain theoretical TCAs on complex synthetic surfaces. • D8-LTD and FMFD are respectively the best SFD and MFD algorithms on HF surface. [ABSTRACT FROM AUTHOR]

We present an improvement of the Multiscale Finite Element Method (MsFEM). MsFEM incorporates fine-scale features of a domain through special oscillatory coarse-scale trial functions and its essential step is an online computation of these special functions defined using local auxiliary problems. Our enhancement makes use of the dynamic condensation that transmits the construction of the oscillatory basis functions exclusively to the coarse mesh skeleton (element interfaces) where we defined special 1D eigenproblems. Consequently, both the approximation error and computational cost are reduced. Moreover, arbitrary coarse mesh element geometries, like polytopal ones, may be used without any additional modifications. We illustrate the accuracy and convergence of the improved method with higher order approximation through selected plane strain examples. [ABSTRACT FROM AUTHOR]

In this paper, we develop an efficient virtual element method to solve the two-dimension time fractional convection diffusion reaction equation involving the Caputo fractional derivative with non-smooth solutions in the time direction. The equation exhibits a weak singularity near the initial time. We first use the exponential transformation to eliminate the convection terms, and obtain the time fractional diffusion reaction equation. By utilizing properties of the energy projection operator and continuous Grönwall inequality, we derive estimates of the approximation error in the L 2 -norm and H 1 -norm under semi-discrete on polygonal meshes. Based on the L 1 scheme on graded mesh in time, the fully-discrete scheme is proved to be unconditionally stable and the optimal convergence results are derived with regards to the L 2 -norm and H 1 -norm. Finally, some numerical experiments are implemented to verify the theoretical results. [ABSTRACT FROM AUTHOR]

For strict-feedback nonlinear systems (SFNSs) with unknown control direction, this paper synthesizes an asymptotic tracking controller by a combination of the dynamic surface control (DSC) technique, the Nussbaum gain technique (NGT) and fuzzy logic systems (FLSs). The SFNSs under study feature unknown nonlinear uncertainties and external disturbances. By utilizing the DSC technique with nonlinear filters, the issue of 'differential explosion' is obviated, in which the adaptive laws are constructed to conquer the effect of unknown functions. The FLSs are exploited to cope with uncertainties without any prior conditions of the ideal weight vectors and the approximation errors. In addition, by introducing the NGT, the unknown control direction problem is solved. Compared with the existing results, the proposed design procedure is able to simultaneously overcome the 'differential explosion' and the unknown control direction problems, and asymptotic tracking is accomplished. At the end, a second-order numerical system and a more realistic Norrbin nonlinear mathematical model are applied to confirm the feasibility of the design procedure. • The asymptotic tracking control problem is considered for a class of uncertain strict-feedback nonlinear systems with unknown control direction. • The proposed design procedure is able to simultaneously overcome the 'differential explosion' and the unknown control direction problems. • The norms of the ideal weight vectors of fuzzy logic systems are taken as the part of estimation parameters, which significantly reduces the calculation amount. • Only the desired trajectory itself is required to be known without knowledge of its higher derivatives, which is different from traditional dynamic surface control results. [ABSTRACT FROM AUTHOR]

The problem of finite-time attitude-tracking control (ATC) for a rigid spacecraft subject to inertial uncertainties, external disturbances, actuator saturations and faults is addressed. First, a fast nonsingular terminal sliding mode (FNTSM) manifold is constructed to improve robustness. Second, the fuzzy logic system (FLS) is integrated into the manifold derivative to deal with the lumped unknown function. The specific assumptions about uncertainties in most of the existing achievements are no longer needed. Combining the FNTSM and fuzzy approximation techniques, an enhanced fault-tolerant control scheme is developed. Compared to almost all finite-time ATC results based on FLS or neural network, a new Proof line is proposed to prove that the approximation errors are finite-time stable instead of asymptotically stable. Therefore, the attitude controller presented herein guarantees the real finite-time stability in a complete sense. Finally, numerical experiments are conducted to verify the effectiveness of the solution, and comparisons with related works are displayed. • Assumptions on uncertainties in the existing achievements are abolished. • Finite-time stability of fuzzy approximation errors is proved via a new Proof line. • Norm approximation is exploited to reduce the parameters to be estimated online. • The attitude controller guarantees the finite-time stability in a complete sense. [ABSTRACT FROM AUTHOR]

• A new approach for greedy offline analysis of reduced basis method. • A smart stopping criterion for the greedy approach. • A Discrete Fracture Network Reduced Basis model. • Residual-based a posteriori error estimator for greedy offline analysis. In classic Reduced Basis (RB) framework, we propose a new technique for the offline greedy error analysis which relies on a residual-based a posteriori error estimator. This approach is as an alternative to classical a posteriori RB estimators, avoiding a discrete inf-sup lower bound estimate. We try to use less common ingredients of the RB framework to retrieve a better approximation of the RB error, such as the estimation of the distance between the continuous solution and the reduced one. In particular we focus on the application of the reduction model for the flow simulations in underground fractured media, in which high accurate simulations suffer for the complexity of the domain geometry. Finally, some numerical tests are assessed to confirm the viability and the efficacy of the technique proposed. [ABSTRACT FROM AUTHOR]

The goal of this paper is to introduce and study a new class of fractional differential hemivariational inequality formulated by an evolutionary hemivariational inequality and a fractional differential equation in Banach spaces. By employing the Rothe method and the surjectivity result, we derive the existence of unique solution for such a problem under some mild conditions. Moreover, we use the fully discrete scheme to approximate the fractional differential hemivariational inequality and provide an error estimate for the approximation. Finally, the main results are applied to obtain the unique solvability as well as the numerical analysis for a viscoelastic frictional contact problem with adhesion. [ABSTRACT FROM AUTHOR]

The aim of this work is to show an abstract framework to analyze the numerical approximation for a family of linear degenerate parabolic mixed equations by using a finite element method in space and a Backward-Euler scheme in time. We consider sufficient conditions to prove that the fully-discrete problem has a unique solution and prove quasi-optimal error estimates for the approximation. Furthermore, we show that mixed finite element formulations arising from dynamics fluids (time-dependent Stokes problem) and from electromagnetic applications (eddy current models) , can be analyzed as applications of the developed theory. Finally, we include numerical tests to illustrate the performance of the method and confirm the theoretical results. [ABSTRACT FROM AUTHOR]

We present the application of the Discontinuous Petrov–Galerkin (DPG) methodology for the mixed Multiscale Finite Element Method (MsFEM). The MsFEM upscaling technique relies on incorporating fine-scale features through special, in a sense optimized for approximability, trial functions while the DPG methodology allows for the selection of the optimal test functions to provide stability of the FEM approximation. The special trial functions are computed online by the solution of local boundary value problems. We improved this process using the static condensation that restricted the construction of the functions to the coarse mesh skeleton (element interfaces) only. We have verified by numerical tests that the proposed improvement of MsFEM reduced both the approximation error and computational cost. Moreover, it simplified the algorithm significantly. The key component of this prolongation construction is our novel method for prolongation of both traction and displacement vectors on element edges of arbitrary shape. The proposed prolongation operator may be also used in the multigrid solver for direct analysis of composites with varying material parameters, using an arbitrary well-posed functional setting with or without the DPG methodology. [ABSTRACT FROM AUTHOR]

This work realizes the adaptive neural disturbance rejection for the leader–follower cooperative synchronization of surface ships with model perturbations and ocean disturbances without leader velocity measurements. The virtual ship alleviates the requirements on leader ship's velocities such that the information requirements are only position and heading on the leader ship. The adaptive neural networks approximate model perturbations. The robustifying term attenuates neural network approximation errors. The adaptive neural network-based disturbance observer achieves the disturbance rejection which is integrated with the dynamic surface control technique. The supply ship synchronization control system is ensured to be practical stable. The synchronization control realizes the ship's cooperative synchronization navigation. Simulations with comparisons validate the synchronization scheme. • Virtual ship alleviates the requirements on the leader ship velocities. • Adaptive neural disturbance observer achieves disturbance rejection. • Robustifying term attenuates the neural approximation errors. [ABSTRACT FROM AUTHOR]

The paper is concerned with guaranteed a posteriori error estimates for a class of evolutionary problems related to poroelastic media governed by the quasi-static linear Biot equations. The system is decoupled by employing the fixed-stress split scheme, which leads to an iteratively solved semi-discrete system. The error bounds are derived by combining a posteriori estimates for contractive mappings with functional type error control for elliptic partial differential equations. The estimates are applicable to any approximation in the admissible functional space and are independent of the discretization method. They are fully computable, do not contain mesh-dependent constants, and provide reliable global estimates of the error measured in the energy norm. Moreover, they suggest efficient error indicators for the distribution of local errors and can be used in adaptive procedures. [ABSTRACT FROM AUTHOR]

To address the issue of low accuracy resulting from approximation errors in conventional interval finite element methods, this study presents an ɛ -accelerated bivariate dimension-reduction interval finite element method. The proposed method aims to accurately and efficiently predict the static response of structures with high dimensionality and large uncertainty parameters. We first approximate the structural interval equilibrium equation using a bivariate dimension-reduction method. An explicit expression for the sequence of displacements is derived by approximating the inverse of the structural interval stiffness matrix using the Neumann series. Adjoint-based sensitivity analysis can effectively avoid the response interval expansion caused by parameter coupling. For improving the convergence speed and computational accuracy, the displacement response sequence is accelerated by a vector ɛ acceleration algorithm and an adaptive strategy. Finally, three engineering structures with uncertain-but-bounded parameters are analyzed with the proposed method. Compared with conventional interval analysis methods, the proposed method effectively balances computational accuracy and efficiency, making it more suitable for non-linear and large uncertainty problems with multiple interval parameters. The study shows that ɛ -accelerated bivariate dimension-reduction interval finite element method has a promising application. • An interval finite element method based on bivariate dimension-reduction is proposed. • Explicit sequences and errors for the bounds of displacements were derived. • Dependencies between parameters are addressed by adjoint-based sensitivity analysis. • Convergence speed and accuracy are improved by the adaptive ɛ algorithm. • Structural response intervals are efficiently and accurately predicted. [ABSTRACT FROM AUTHOR]