Smooth Problems Research Articles

Statistical and computational efficiency for smooth tensor estimation with unknown permutations

We consider the problem of structured tensor denoising in the presence of unknown permutations. Such data problems arise commonly in recommendation systems, neuroimaging, community detection, and multiway comparison applications. Here, we develop a general family of smooth tensor models up to arbitrary index permutations; the model incorporates the popular tensor block models and Lipschitz hypergraphon models as special cases. We show that a constrained least-squares estimator in the block-wise polynomial family achieves the minimax error bound. A phase transition phenomenon is revealed with respect to the smoothness threshold needed for optimal recovery. In particular, we find that a polynomial of degree up to ( m − 2 ) ( m + 1 ) / 2 is sufficient for accurate recovery of order-m tensors, whereas higher degrees exhibit no further benefits. This phenomenon reveals the intrinsic distinction for smooth tensor estimation problems with and without unknown permutations. Furthermore, we provide an efficient polynomial-time Borda count algorithm that provably achieves the optimal rate under monotonicity assumptions. The efficacy of our procedure is demonstrated through both simulations and Chicago crime data analysis.

Research on Autonomous Underwater Vehicle Path Optimization Using a Field Theory-Guided A* Algorithm

Autonomous Underwater Vehicles (AUVs) have become indispensable tools in the fields of ocean exploration, resource exploitation, and environmental monitoring. Path planning and obstacle avoidance are crucial to improve the operational capabilities of AUVs. However, most algorithms focus only on macro-global or micro-local path planning and rarely address both problems simultaneously. This study extends the classical A* algorithm by integrating field theory principles. The resulting Field Theory Augmented A* (FT-A*) algorithm combines the constraints in the AUV’s dynamics and the threats posed by obstacles to ensure a safe navigation distance. The paths planned by the FT-A* algorithm were subsequently re-optimised in conjunction with Dubins curves, taking into account path smoothness and redundant node problems. Simulation experiments confirm that the improved algorithm can effectively help AUVs navigate safely around obstacles, which greatly improves navigation safety and increases the arithmetic power and navigation efficiency. The proposed FT-A* algorithm provides a robust solution for underwater path planning and demonstrates great practical value for AUV operation in complex marine environments.

Smooth Sigmoid Surrogate (SSS): An Alternative to Greedy Search in Decision Trees

Greedy search (GS) or exhaustive search plays a crucial role in decision trees and their various extensions. We introduce an alternative splitting method called smooth sigmoid surrogate (SSS) in which the indicator threshold function used in GS is approximated by a smooth sigmoid function. This approach allows for parametric smoothing or regularization of the erratic and discrete GS process, making it more effective in identifying the true cutoff point, particularly in the presence of weak signals, as well as less prone to the inherent end-cut preference problem. Additionally, SSS provides a convenient means of evaluating the best split by referencing a parametric nonlinear model. Moreover, in many variants of recursive partitioning, SSS can be reformulated as a one-dimensional smooth optimization problem, rendering it computationally more efficient than GS. Extensive simulation studies and real data examples are provided to evaluate and demonstrate its effectiveness.

A New Hybrid Descent Algorithm for Large-Scale Nonconvex Optimization and Application to Some Image Restoration Problems

Conjugate gradient methods are widely used and attractive for large-scale unconstrained smooth optimization problems, with simple computation, low memory requirements, and interesting theoretical information on the features of curvature. Based on the strongly convergent property of the Dai–Yuan method and attractive numerical performance of the Hestenes–Stiefel method, a new hybrid descent conjugate gradient method is proposed in this paper. The proposed method satisfies the sufficient descent property independent of the accuracy of the line search strategies. Under the standard conditions, the trust region property and the global convergence are established, respectively. Numerical results of 61 problems with 9 large-scale dimensions and 46 ill-conditioned matrix problems reveal that the proposed method is more effective, robust, and reliable than the other methods. Additionally, the hybrid method also demonstrates reliable results for some image restoration problems.

A symmetric interior-penalty discontinuous Galerkin isogeometric analysis spatial discretization of the self-adjoint angular flux form of the neutron transport equation

This paper presents the first application of a symmetric interior-penalty discontinuous Galerkin isogeometric analysis (SIP-DG-IGA) spatial discretization to the self-adjoint angular flux (SAAF) form of the multi-group neutron transport equation. The penalty parameters are determined, for general element types, from a mathematically rigorous coercivity analysis of the bilinear form. The proposed scheme produces a compact spatial discretization stencil. It also yields symmetric positive-definite (SPD) matrices, which can be efficiently solved using pre-conditioned conjugate gradient (PCG) solution algorithms. The proposed discretization scheme is verified using the method of manufactured solutions (MMS) and several nuclear reactor physics benchmark verification test cases. For sufficiently smooth elliptic problems, the proposed spatial discretization can exploit higher-order continuity, or k-refinement, of the NURBS basis to consistently yield greater numerical accuracy per degree of freedom (DoF) than standard h-refinement. Since this is a discontinuous scheme, it can also accurately model significant changes in the neutron scalar flux that may occur near the material interfaces of heterogeneous problems.

Supersonic turbulence simulations with GPU-based high-order Discontinuous Galerkin hydrodynamics

ABSTRACT We investigate the numerical performance of a Discontinuous Galerkin (DG) hydrodynamics implementation when applied to the problem of driven, isothermal supersonic turbulence. While the high-order element-based spectral approach of DG is known to efficiently produce accurate results for smooth problems (exponential convergence with expansion order), physical discontinuities in solutions, like shocks, prove challenging and may significantly diminish DG’s applicability to practical astrophysical applications. We consider whether DG is able to retain its accuracy and stability for highly supersonic turbulence, characterized by a network of shocks. We find that our new implementation, which regularizes shocks at subcell resolution with artificial viscosity, still performs well compared to standard second-order schemes for moderately high-Mach number turbulence, provided we also employ an additional projection of the primitive variables on to the polynomial basis to regularize the extrapolated values at cell interfaces. However, the accuracy advantage of DG diminishes significantly in the highly supersonic regime. Nevertheless, in turbulence simulations with a wide dynamic range that start with supersonic Mach numbers and can resolve the sonic point, the low-numerical dissipation of DG schemes still proves advantageous in the subsonic regime. Our results thus support the practical applicability of DG schemes for demanding astrophysical problems that involve strong shocks and turbulence, such as star formation in the interstellar medium. We also discuss the substantial computational cost of DG when going to high order, which needs to be weighted against the resulting accuracy gain. For problems containing shocks, this favours the use of comparatively low DG order.

Two-Step Quadratic Programming for Physically Meaningful Smoothing of Longitudinal Vehicle Trajectories

Longitudinal vehicle trajectories suffer from errors and noise because of detection and extraction techniques, challenging their applications. Existing smoothing methods either lack physical meaning or cannot ensure solution existence and uniqueness. To address this, we propose a two-step quadratic programming method that aligns smoothed speeds and higher-order derivatives with physical laws, drivers’ behaviors, and vehicle characteristics. Unlike the well-known smoothing splines method, which minimizes a weighted sum of discrepancy and roughness in a single quadratic programming problem, our method incorporates prior knowledge of position errors into two sequential quadratic programming problems. Step 1 solves half-smoothed positions by minimizing the discrepancy between them and raw positions, subject to physically meaningful bounds on speeds and higher-order derivatives of half-smoothed positions. Step 2 solves smoothed positions by minimizing the roughness while maintaining physically meaningful bounds and allowing the deviations from raw data of smoothed positions by at most those of the half-smoothed positions and prior position errors. The second step’s coefficient matrix is not positive definite, necessitating the matching of the first few smoothed positions with corresponding half-smoothed ones, with equality constraints equaling the highest order of derivatives. We establish the solution existence and uniqueness for both problems, ensuring their well-defined nature. Numerical experiments using Next Generation Simulation (NGSIM) data demonstrate that a third-order derivative constraint yields an efficient method and produces smoothed trajectories comparable with manually re-extracted ones, consistent with the minimum jerk principle for human movements. Comparisons with an existing approach and application to the Highway Drone data set further validate our method’s efficacy. Notably, our method is a postprocessing smoothing technique based on trajectory data and is not intended for systematic errors. Future work will extend this method to lateral vehicle trajectories and trajectory prediction and planning for both human-driven and automated vehicles. This approach also holds potential for broader smoothing problems with known average error in raw data. History: This paper has been accepted for the Transportation Science Special Issue on ISTTT25 Conference. Funding: The authors extend their gratitude to the Pacific Southwest Region University Transportation Center and the University of California Institute of Transportation Studies (UC ITS) Statewide Transportation Research Program (STRP) for their valuable financial support.

果园环境中巡检机器人的 SLAM 和路径规划方法研究

Orchard robots play a crucial role in agricultural production. Autonomous navigation serves as the foundation for orchard robots and eco-unmanned farms. Accurate sensing and localization are prerequisites for achieving autonomous navigation. However, current vision-based navigation solutions are sensitive to environmental factors, such as light, weather, and background, which can affect positioning accuracy. Therefore, they are unsuitable for outdoor navigation applications. LIDAR provides accurate distance measurements and is suitable for a wide range of environments. Its immunity to interference is not affected by light, colour, weather, or other factors, making it suitable for low objects and complex orchard scenes. Therefore, LiDAR navigation is more suitable for orchard environments. In complex orchard environments, tree branches and foliage can cause Global Positioning System (GNSS) accuracy to degrade, resulting in signal loss. Therefore, the major challenge that needs to be addressed is generating navigation paths and locating the position of orchard robots. In this paper, an improved method for Simultaneous Localization and Mapping (SLAM) and A-star algorithm is proposed. The SLAM and path planning method designed in this study effectively solves the problems of insufficient smoothness and large curvature fluctuation of the path planned in the complex orchard environment, and improves the detection efficiency of the robot. The experimental results indicate that the method can consistently and accurately fulfil the robot's detection needs in intricate orchard environments.

Global solutions of approximation problems in Krein spaces

Three approximation problems in Krein spaces are studied, namely the indefinite weighted least squares problem and the related problems of indefinite abstract splines and smoothing. In each case, we analyse the existence of a linear and continuous operator that maps each data point to its solution and when the associated operator problem considering the J-trace has a solution.

Improved GNN based on Graph-Transformer: A new framework for rolling mill bearing fault diagnosis

The structure of the rolling mill system is complex and the operating conditions are changeable. Therefore, the interdependence between the data needs to be fully considered in the fault diagnosis of the rolling mill. Although graph neural network (GNN) is a powerful architecture based on non-Euclidean spatial data, the current method is difficult to represent the long-range dependence of rolling mill fault vibration signals. Simply increasing the depth of GNN is not enough to expand the receptive field of the model, because the larger GNN model may have the problem of gradient disappearance or transition smoothing. In order to solve the above problems, an improved graph neural network based on Graph-Transformer is proposed to diagnose the health status of rolling mill. This method first performs sliding maximum sampling on the spectrum of the original vibration signal to improve the frequency resolution and reduce the feature dimension. Second, the relationship between fault features is characterized by constructing affinity graph. Finally, the long-range dependency between paired features is learned through the readout module and the self-attention mechanism in Graph-Transformer and the diagnostic results are output by the classifier. The experimental results on the rolling mill platform show that this method can not only adapt to the changing working conditions of the rolling mill but also achieve excellent performance in the case of sample imbalance and strong noise.

Upwind Summation-by-parts Finite Differences: error Estimates and WENO methodology

High order upwind summation-by-parts finite difference operators have recently been developed. When combined with the simultaneous approximation term method to impose boundary conditions, the method converges faster than using traditional summation-by-parts operators. We prove the convergence rate by the normal mode analysis for such methods for a class of hyperbolic partial differential equations. Our analysis shows that the penalty parameter for imposing boundary conditions affects the convergence rate for stable methods. In addition, to solve problems with discontinuous data, we extend the method to also have the weighted essentially nonoscillatory property. The overall method is stable, achieves high order accuracy for smooth problems, and is capable of solving problems with discontinuities.

A new spectral conjugate subgradient method with application in computed tomography image reconstruction

A new spectral conjugate subgradient method is presented to solve nonsmooth unconstrained optimization problems. The method combines the spectral conjugate gradient method for smooth problems with the spectral subgradient method for nonsmooth problems. We study the effect of two different choices of line search, as well as three formulas for determining the conjugate directions. In addition to numerical experiments with standard nonsmooth test problems, we also apply the method to several image reconstruction problems in computed tomography, using total variation regularization. Performance profiles are used to compare the performance of the algorithm using different line search strategies and conjugate directions to that of the original spectral subgradient method. Our results show that the spectral conjugate subgradient algorithm outperforms the original spectral subgradient method, and that the use of the Polak–Ribière formula for conjugate directions provides the best and most robust performance.

Analysis of Automotive Smoothness Driven by Wheel Hub Motors with Different Configuration under Dual Excitation

Abstract In response to the smoothness issue of wheel hub motor driven vehicles, electromagnetic vertical excitation was added on the basis of road excitation, and the noise source of a three-phase 8-pole 48-slot permanent magnet motor was investigated in the frequency domain; A 1/4 vehicle 2-DOF vertical vibration system model with three suspension configurations under dual excitation was established, and the vehicle smoothness problem was analyzed from a time domain perspective. The results show that the overall trend of the response values of each evaluation indicator under dual coupling excitation is correspondingly higher than that under single excitation. Under dual excitation, the suspension configuration of absorber type exhibits the smallest maximum variation in the vertical acceleration response of the vehicle over time compared to the distributed and isolated suspension configurations.

Improving path planning for mobile robots in complex orchard environments: the continuous bidirectional Quick-RRT* algorithm.

Efficient obstacle-avoidance path planning is critical for orchards with numerous irregular obstacles. This paper presents a continuous bidirectional Quick-RRT* (CBQ-RRT*) algorithm based on the bidirectional RRT (Bi-RRT) and Quick-RRT* algorithms and proposes an expansion cost function that evaluates path smoothness and length to overcome the limitations of the Quick-RRT* algorithm for non-holonomic mobile robot applications. To improve the zigzag between dual trees caused by the dual-tree expansion of the Bi-RRT algorithm, CBQ-RRT* proposes the CreateConnectNode optimization method, which effectively solves the path smoothness problem at the junction of dual trees. Simulations conducted on the ROS platform showed that the CBQ-RRT* outperformed the unidirectional Quick-RRT* in terms of efficiency for various orchard layouts and terrain conditions. Compared to Bi-RRT*, CBQ-RRT* reduced the average path length and maximum heading angle by 8.5% and 21.7%, respectively. In addition, field tests confirmed the superior performance of the CBQ-RRT*, as evidenced by an average maximum path lateral error of 0.334m, a significant improvement over Bi-RRT* and Quick-RRT*. These improvements demonstrate the effectiveness of the CBQ-RRT* in complex orchard environments.

A fully conservative and shift-invariant formulation for Galerkin discretizations of incompressible variable density flow

This paper introduces a formulation of the variable density incompressible Navier-Stokes equations by modifying the nonlinear terms in a consistent way. For Galerkin discretizations, the formulation leads to favorable discrete conservation properties without the divergence-free constraint being strongly enforced. In addition, the formulation is shown to make the density field invariant to global shifts. The effect of viscous regularizations on conservation properties is also investigated. Numerical tests validate the theory developed in this work. The new formulation shows superior performance compared to other formulations from the literature, both in terms of accuracy for smooth problems and in terms of robustness.

Research on traffic flow prediction method based on adaptive multi-channel graph convolutional neural networks

In order to address the issues of predefined adjacency matrices inadequately representing information in road networks, insufficiently capturing spatial dependencies of traffic networks, and the potential problem of excessive smoothing or neglecting initial node information as the layers of graph convolutional neural networks increase, thus affecting traffic prediction performance, this paper proposes a prediction model based on Adaptive Multi-channel Graph Convolutional Neural Networks (AMGCN). The model utilizes an adaptive adjacency matrix to automatically learn implicit graph structures from data, introduces a mixed skip propagation graph convolutional neural network model, which retains the original node states and selectively acquires outputs of convolutional layers, thus avoiding the loss of node initial states and comprehensively capturing spatial correlations of traffic flow. Finally, the output is fed into Long Short-Term Memory networks to capture temporal correlations. Comparative experiments on two real datasets validate the effectiveness of the proposed model.

Complex Environment Based on Improved A* Algorithm Research on Path Planning of Inspection Robots

The proposed research aims to accomplish an improved A* algorithm for mobile robots in complex environments. In this novel algorithm, the guidance of environment information is added to the evaluation function to enhance the adaptability of the algorithm in complex environments. Additionally, to solve the problem of path smoothness, the optimal selection rules for child nodes and the bidirectional optimization strategy for path smoothing are introduced to reduce redundant nodes, which effectively makes the search space smaller and the path smoother. The simulation experiments show that, compared with the colony algorithm and Dijkstra algorithms, the proposed algorithm has significantly improved performance. Compared with the A* algorithm, the average planning time is reduced by 17.2%, the average path length is reduced by 2.05%, the average turning point is reduced by 49.4%, and the average turning Angle is reduced by 75.5%. The improved A* algorithm reduces the search space by 61.5% on average. The simulation results show that the effectiveness and adaptability of the improved A* algorithm in complex environments are verified by multi-scale mapping and multi-obstacle environment simulation experiments.

Design and Test of Disturbed Fertilizer Strip-Ejection Device with Vertical Pendulum Bar Based on Discrete Element Method

Fertilizer can improve the yield of crops per unit area, and uniform fertilizer discharge can improve the fertilizer utilization rate. Therefore, it is meaningful to improve the performance of fertilizer-discharge devices in order to improve the modernization level of crop field fertilizer management. To address the problems of operational smoothness, stability and poor uniformity of fertilizer discharge, and other difficult problems encountered with strip fertilizer-discharge devices, this study designs a disturbed fertilizer strip-discharge device with a vertical pendulum. The main factors affecting the performance of fertilizer discharge were the wedge angle of the push-disturbing main pendulum bar (PMPB), the inclination angle of the aided-stirring pendulum pick (APP), the flow gap of the pendulum bar (FGPB), and the operation frequency of the swing-rod combination (SRC). The discrete element method (DEM) was used to establish a simulation model of the fertilizer device to explore the influence of the main factors on the performance of fertilizer discharge, with the coefficient of variation (CV) of fertilizer discharge uniformity and fertilizer discharge accuracy (FDA) used as the evaluation indices. The results show that the factors affecting the CV of fertilizer discharge uniformity and FDA were, in order of priority, the operation frequency of the SRC, the FGPB, the wedge angle of the PMPB, and the inclination angle of the APP. The optimal parameters after rounding were as follows: the wedge angle of the PMPB was 45°, the inclination angle of the APP was 46°, the operation frequency of the SRC was 188 times/min, and the FGPB was 4.5 mm. At this point, the model predicted that the CV of fertilizer discharge uniformity would be 10.53%, and that the FDA would be 3.19%. Using the optimal parameters for bench test verification, it was found that the wedge angle of the PMPB was 45°, the inclination angle of the APP was 46°, the operation frequency of the SRC was 188 times/min, the FGPB was 4.5 mm, the CV of the uniformity of the fertilizer discharge was 11.06%, and the FDA was 3.51%. In the test, the fertilizer-discharge device was stable and had good adaptability to different fertilizers. The results of this study can provide a theoretical reference for the development of precision strip-fertilizer application devices.

A structure preserving numerical method for the ideal compressible MHD system

We introduce a novel structure-preserving method in order to approximate the compressible ideal Magnetohydrodynamics (MHD) equations. This technique addresses the MHD equations using a non-divergence formulation, where the contributions of the magnetic field to the momentum and total mechanical energy are treated as source terms. Our approach uses the Marchuk-Strang splitting technique and involves three distinct components: a compressible Euler solver, a source-system solver, and an update procedure for the total mechanical energy. The scheme allows for significant freedom on the choice of Euler's equation solver, while the magnetic field is discretized using a curl-conforming finite element space, yielding exact preservation of the involution constraints. We prove that the method preserves invariant domain properties, including positivity of density, positivity of internal energy, and the minimum principle of the specific entropy. If the scheme used to solve Euler's equation conserves total energy, then the resulting MHD scheme can be proven to preserve total energy. Similarly, if the scheme used to solve Euler's equation is entropy-stable, then the resulting MHD scheme is entropy stable as well. In our approach, the CFL condition does not depend on magnetosonic wave-speeds, but only on the usual maximum wavespeed from Euler's system. To validate the effectiveness of our method, we solve a variety of ideal MHD problems, showing that the method is capable of delivering second-order accuracy in space for smooth problems, while also offering unconditional robustness in the shock hydrodynamics regime as well.

Generalized Alternating Projections on Manifolds and Convex Sets

In this paper, we extend the previous convergence results for the generalized alternating projection method applied to subspaces in [arXiv:1703.10547] to hold also for smooth manifolds. We show that the algorithm locally behaves similarly in the subspace and manifold settings and that the same rates are obtained. We also present convergence rate results for when the algorithm is applied to non-empty, closed, and convex sets. The results are based on a finite identification property that implies that the algorithm after an initial identification phase solves a smooth manifold feasibility problem. Therefore, the rates in this paper hold asymptotically for problems in which this identification property is satisfied. We present a few examples where this is the case and also a counter example for when this is not.