Continuous Approximation Research Articles

Continuity approximation in hybrid Bayesian networks structure learning

Continuity approximation in hybrid Bayesian networks structure learning

Regularization of the Cauchy problem for matrix factorizations of the Helmholtz equation in an unbounded domain

<p>In this paper, a regularized solution to the Cauchy problem for matrix factorization of the Helmholtz equation in a three-dimensional unbounded domain is constructed explicitly based on the Carleman matrix. When solving applied problems, in addition to an approximate solution, the derivative of the approximate solution is found. It is assumed that the solution to the problem exists and is continuously differentiable in a closed domain with precisely specified Cauchy data. An explicit formula for continuing the solution and its derivative is established, as well as a regularization formula for the case when, under the specified conditions, instead of the original Cauchy data, their continuous approximations with a specified error in the uniform metric are given. As a result, the stability of the solution to the Cauchy problem in the classical sense is estimated.</p>

Step-by-step time discrete Physics-Informed Neural Networks with application to a sustainability PDE model

The use of Artificial Neural Networks (ANNs) has spread massively in several research fields. Among the various applications, ANNs have been exploited for the solution of Partial Differential Equations (PDEs). In this context, the so-called Physics-Informed Neural Networks (PINNs) are considered, i.e. neural networks generally constructed in such a way as to compute a continuous approximation in time and space of the exact solution of a PDE.In this manuscript, we propose a new step-by-step approach that allows to define PINNs capable of providing numerical solutions of PDEs that are discrete in time and continuous in space. This is done by establishing connections between the network outputs and the numerical approximations computed by a classical one-stage method for stiff Initial Value Problems (IVPs). Links are also highlighted between the step-by-step PINNs derived here, and the time discrete models based on Runge–Kutta (RK) methods proposed so far in literature. To evaluate the efficiency of the new approach, we build such PINNs to solve a nonlinear diffusion-reaction PDE model describing the process of production of renewable energy through dye-sensitized solar cells. The numerical experiments show that not only the new step-by-step PINNs are able to well reproduce the model solution, but also highlight that the proposed approach can constitute an improvement over existing continuous and time discrete models.

Resource optimization of multi-UAV assisted communication system based on user scheduling

In order to improve the transmission rate of the multi-user mobile communication downlink wireless transmission system, a resource optimization method for multi-unmanned aerial vehicle (UAV) assisted communication system based on user scheduling and trajectory optimization is proposed. The proposed method establishes an optimization problem with the maximization of the total multi-user throughput as the criterion under the constraints of user scheduling, total energy consumption of UAVs and user service quality requirements. In order to solve this non-convex problem, the original non-convex problem is decomposed into three easy-to-handle non-convex sub-problems by using the block coordinate descent method (BCD). The sub-problems are transformed and solved by introducing slack variables, first-order Taylor expressions, continuous convex approximation (SCA) and other methods, and then alternately iteratively optimized to obtain an approximate suboptimal solution to the original non-convex problem. Simulation results show that the proposed algorithm can effectively improve the total system throughput and has good convergence in both single and multi-UAV communication systems.

A continuous approximation model for the electric vehicle fleet sizing problem

A continuous approximation model for the electric vehicle fleet sizing problem

From operational to strategic modelling: A continuous multi-scale approach for last-mile analysis

Supply chain planning requires decision-making at all levels, especially in the new normal of intervened supply networks. The integration of strategic network design and operational routing decisions has been widely studied in the literature as the location-routing problem (LRP). However, the LRP does not consider the differences in the planning horizon of each sub-problem, nor have geospatial elements been included in the distribution of demand. This paper aims to redesign the conceptual modelling of the last mile to inform strategic network design problems. A continuous multi-scale approach (CMA) is proposed by taking elements from the districting problem (DP) and continuous approximation (CA). This approach includes stochastic demand in the analysis and the effects of time windows and failed deliveries. The validation of CMA in a case of rural parcel deliveries in Belgium shows an estimate of the distance travelled similar to traditional routing algorithms in scenarios with high demand density. Likewise, the effects of time windows on the spatial configuration of multi-scale districting are explored. This approach provides insights for decision-making in strategic and tactical planning, such as sonification, differentiated services to satisfy consumer preferences, and fleet management. The limitations of the CMA lie in its implementation in purely operational scenarios since it does not offer detailed routing information. Subsequent research aims to exploit the potential of strategic last-mile modelling with effective integration into network design problems.

Robust [formula omitted] stabilization for systems with uncertain input time-delay

This paper presents a robust H∞ stabilization approach for actively controlled systems that include uncertain input time-delay. The approach comprises two essential steps: (1) converting a controlled system with uncertain time-delay to have only deterministic time-delay through a time-scale transformation; and (2) applying the Chebyshev spectral continuous time approximation (CHsCTA) method and defining appropriate extension matrices to transform the system so there are no time-delays. A robust H∞ control design is implemented on the transformed system, which facilitates the derivation of a robust H∞ controller. An iterative algorithm is proposed for computing the feedback gain matrix, which overcomes the challenges posed by the nonlinear coupling terms when solving the matrix inequalities. Case studies validate the effectiveness of the proposed control algorithm, which, through comparison with existing control algorithms, is shown to be less conservative.

The pressure-wired Stokes element: a mesh-robust version of the Scott–Vogelius element

The Scott–Vogelius finite element pair for the numerical discretization of the stationary Stokes equation in 2D is a popular element which is based on a continuous velocity approximation of polynomial order k and a discontinuous pressure approximation of order k-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k-1$$\\end{document}. It employs a “singular distance” (measured by some geometric mesh quantity Θz≥0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\Theta \\left( \ extbf{z}\\right) \\ge 0$$\\end{document} for triangle vertices z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{z}$$\\end{document}) and imposes a local side condition on the pressure space associated to vertices z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{z}$$\\end{document} with Θz=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Theta \\left( \ extbf{z}\\right) =0$$\\end{document}. The method is inf-sup stable for any fixed regular triangulation and k≥4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k\\ge 4$$\\end{document}. However, the inf-sup constant deteriorates if the triangulation contains nearly singular vertices 0<Θz≪1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<\\Theta \\left( \ extbf{z}\\right) \\ll 1$$\\end{document}. In this paper, we introduce a very simple parameter-dependent modification of the Scott–Vogelius element with a mesh-robust inf-sup constant. To this end, we provide sharp two-sided bounds for the inf-sup constant with an optimal dependence on the “singular distance”. We characterise the critical pressures to guarantee that the effect on the divergence-free condition for the discrete velocity is negligibly small, for which we provide numerical evidence.

Crack propagation arrest by the Joule heating in micro/nano-sized structures

The Joule heating technique is utilized to arrest the crack propagation so to protect some important structures being further damaged. In this approach there are created compression stresses at the crack tip vicinity due to a specific temperature distribution induced by the electric current. The Joule heating is amulti-physical problem which is applied here to micro/nano-sized structures. The size-effects are considered for description of both the elastic and the thermal fields. The heat conduction can’t be well described by classical Fourier’s law in nano-sized structures. A novel gradient theory is developed in such structures adopting the size effect of heat conduction. Besides, the strain gradients and double stresses are introduced into the constitutive laws. Then, the governing equations are given by partial differential equations with order of derivatives being higher than in classical theory. The principle of virtual work is the base of weak formulation and derivation of the discretized finite element equations for ageneral 2d boundary value problem with cracks. The collocation mixed FEM with large strain gradients is developed for numerical solution, where the C0 continuous approximation is applied independently to displacements and strains and also to temperature and temperature gradients. Kinematic constraints between strains and displacements are satisfied by the collocation method at selected interior points. The collocation mixed FEM is applied to solve 2D crack problems with Joule heating.

Study of Power Equivalent Continuous Approximation Based on the Recent Consensus Recommendations for Brain Tumor Imaging with Pulsed Chemical Exchange Saturation Transfer at 3T.

The quantitative analysis of pulsed-chemical exchange saturation transfer (CEST) using a full model-based method is computationally challenging, as it involves dealing with varying RF values in pulsed saturation. A power equivalent continuous approximation of B1 power was usually applied to accelerate the analysis. In line with recent consensus recommendations from the CEST community for pulsed-CEST at 3T, particularly recommending a high RF saturation power (B1 = 2.0 µT) for the clinical application in brain tumors, this technical note investigated the feasibility of using average power (AP) as the continuous approximation. The simulated results revealed excellent performance of the AP continuous approximation in low saturation power scenarios, but discrepancies were observed in the z-spectra for the high saturation power cases. Cautions should be taken, or it may lead to inaccurate fitted parameters, and the difference can be more than 10% in the high saturation power cases.

Optimal fare and headway for a demand adaptive paired-line hybrid transit system in a rectangular area with elastic demand

Demand adaptive paired-line hybrid transit systems that integrate fixed- and flex-route transit have emerged in the last decade and attracted increasing attention because of their potential to improve accessibility for passengers. To facilitate the operation of such a hybrid transit system, this study develops a model to determine the optimal fare and headways associated with fixed- and flex-route transit along a rectangular corridor. Compared with existing literature, the novelty of this study lies in designing the fare structure while simultaneously considering demand elasticity and passenger behaviour. A continuous approximation modelling approach is employed to derive the agency’s and travellers’ cost components. Using these, a nonlinear programming optimisation model is formulated to minimise the total user cost subject to the agency’s nonnegative revenue constraints and passengers’ route choice behaviour, which is characterised as a path-size logit model. Numerical experiments are performed using a stylised network to examine the properties of the model, in which the solution is obtained by combining a brute force method that enumerates headway and fare combinations and an iterative method that determines equilibrated passenger choices. The results show that as potential demand density increases, fare and headway fluctuate and drop, while the percentage of passengers choosing to ride on flex-route transit increases. In addition, there may be an optimal maximum offset distance, which is defined as the width of the corridor to be covered by the transit system, when the potential demand density is low, leading to minimum user cost and maximum travel demand within the service area.

Robust secure resource optimization for active STAR-RIS systems

In this paper, we investigate the robust secure resource optimization for the active simultaneously transmitting and reflecting reconfigurable intelligent surface (STAR-RIS) system under the imperfect eavesdroppers channel state information. Considering the fairness, a max–min secure rate optimization problem is formulated based on several practical constraints. To deal with the original non-convex problem, an alternative iteration algorithm is proposed. First, the original problem is decomposed into two non-convex sub-problems. Next, the continuous convex approximation and S-procedure techniques are applied to deal with the non-convex and uncertainty constraints, respectively. Then, the first-order Taylor expansion formula is utilized to approximate the convex difference form of the objective function. Finally, two sub-problems are transformed into convex ones and alternately solved until convergence. Simulation results show that the secure rate of the proposed scheme is higher than the conventional schemes.

In-Plane Vibrations of Elastic Lattice Plates and Their Continuous Approximations

This paper presents an analytical study on the in-plane vibrations of a rectangular elastic lattice plate. The plane lattice is modelled considering central and angular interactions. The lattice difference equations are shown to coincide with a spatial finite difference scheme of the corresponding continuous plate. The considered lattice converges to a 2D linear isotropic elastic continuum at the asymptotic limit for a sufficiently small lattice spacing. This continuum has a free Poisson’s ratio, which must be lower than that foreseen by the rare-constant theory, to preserve the definite positiveness of the associated discrete energy. Exact solutions for the in-plane eigenfrequencies and modes are analytically derived for the discrete plate. The stiffness characterising the lattice interactions at the boundary is corrected to preserve the symmetry properties of the discrete displacement field. Two classes of constraints are considered, i.e., sliding supports at the nodes, one normal and the other parallel to the boundary. For both boundary conditions, a single equation for the eigenfrequency spectrum is derived, with two families of eigenmodes. Such behaviour of the lattice plate is like that of the continuous plate, the eigenfrequency spectrum of which has been given by Rayleigh. The convergence of the spectrum of the lattice plate towards the spectrum of the continuous plate from below is confirmed. Two continuous size-dependent plate models, considering the strain gradient elasticity and non-local elasticity, respectively, are built from the lattice difference equations and are shown to approximate the plane lattice accurately.

Assessing the viability of parking slot utilization as transhipment points for parcel carriers: a case study in Barcelona

AbstractThe measures proposed in the sustainable and smart mobility strategy developed by the European commission are crucial for achieving the environmental objectives for 2030. This strategy includes measures specifically aimed at reducing the impact of e-commerce distribution. One widely adopted strategy for mitigating the impact of e-commerce distribution is the establishment of urban consolidation centres (UCCs). While this infrastructure quickly demonstrates its environmental success in urban areas, achieving long-term economic sustainability proves challenging. In this paper, we introduce a methodology aimed at incentivizing parcel carriers to shift towards an economically feasible and environmentally sustainable distribution mode within urban areas by leveraging their own transhipment space. This methodology includes a mathematical program for determining the quantity and locations of parking slots, taking into account the number of parcels and transportation vehicles used. It also utilizes the Continuous Approximation technique to assess the transportation and environmental costs of the proposed solutions. The proposed methodology is implemented for the city of Barcelona, considering three different sizes of parcel carriers: the carrier handling the largest volume of parcels, the carrier handling the lowest volume of parcels, and an intermediate case. The vehicles considered for delivery from the transhipment points include vans, cargo bikes, and trolleys for distribution on foot. The results indicate that a significant reduction in both transport and environmental costs is achieved when using zero-emissions vehicles for delivery from transhipment points for any of the parcel carriers considered.

Reliability-based G1 continuous arc spline approximation

This paper introduces an algorithm for approximating a set of data points with G1 continuous arcs, leveraging covariance data associated with the points. Prior approaches to arc spline approximation typically assumed equal contribution from all data points, resulting in potential algorithmic instability when outliers are present. To address this challenge, we propose a robust method for arc spline approximation, taking into account the 2D covariance of each data point. Beginning with the definition of models and parameters for single-arc approximation, we extend the framework to support multiple-arc approximation for broader applicability. Finally, we validate the proposed algorithm using both synthetic noisy data and real-world data collected through vehicle experiments conducted in Sejong City, South Korea.

W0_sample = np.random.normal(0,1)?

In this note, we explore the distribution of vacuum expectation values of the superpotential W0 in explicit Type IIB flux compactifications. We show that the distribution can be approximated universally across geometries by a two-dimensional Gaussian with a model-dependent standard deviation. We identify this behaviour in 20 Calabi-Yau orientifold compactifications with between two and five complex structure moduli by constructing a total of O(107) flux vacua. We observe a characteristic scaling behaviour of the width σ of our distributions with respect to the D3-charge contributions Nflux from fluxes which can be approximated by σ∼Nflux. This W0 distribution implies that locating small values of |W0| as a preferred regime associated with classes of string theory solutions typically featuring hierarchies, simplifies to the basic statement of finding small Euclidean norms of normally distributed values. We also identify small modifications to this Gaussian behaviour in our samples, which might be seen as indications for the breakdown of the continuous flux approximation commonly used in the context of statistical analyses of the flux landscape.

Length Scale Insensitive Phase-Field Fracture Methodology for Brittle and Ductile Materials

We present length scale insensitive phase-field fracture models for brittle and ductile fracture to address the deficiencies of widely implemented models which over-estimate crack dissipation. An approach is proposed to attain a regularization length scale insensitive mechanical response, which considers a continuous approximation of a crack boundary with a function of infinite support. This is in contrast to previous approaches to attain a length scale insensitive response which exploited approximations of the crack boundary with functions of finite support. The choice of these functions with finite support creates implementational challenges which are avoided in the presented quasi-brittle and ductile fracture models with infinite support of the phase-field. The capability of these models is demonstrated in several benchmarks with attention given to the sensitivity of the structural response with respect to the choice of regularization length scale. The models are validated against a three-point bending test on a concrete specimen and an in-plane shear test on a steel specimen.

Modeling and optimization of expected travel time for multi‐aisle AS/RSs with two‐class‐based storage policy

AbstractClass‐based storage policy with optimized contour‐shaped class boundary can significantly improve storage system's performance. Surprisingly, this policy has not been explored in widely used multi‐aisle automated storage and retrieval systems (MA‐AS/RSs), which use one storage and retrieval machine to serve multiple aisles with the help of an aisle‐transfer technique. This paper investigates the two‐class‐based storage policy with contour‐shaped class boundary in MA‐AS/RSs that use a transfer car for aisle transfer. The aim is to optimize the system dimensions and class boundary by minimizing system's expected travel time. Based on the approximation of the MA‐AS/RS with a continuous cube and the proposed hierarchical procedure, analytical expected travel time expressions for systems with any dimensions and class boundary are calculated. In addition, based on several proved properties, closed‐form optimal system dimensions and class boundary are derived. Numerical results show the accuracy of our continuous cubic approximation is sufficient. By measuring the performance using the average of expected travel time over all tested systems with various dimensions, we find that (1) class‐based policy with our optimal class boundary can respectively improve the performance by at least 40%, 10%, and 50% compared to three previous policies in the case of 20/80 ABC curve; and (2) system with our optimal dimensions can improve the performance by about 20%–30%. Several managerial insights for warehouse practitioners are presented.

A responsive closed-loop supply chain network design under demand uncertainty

A closed-loop supply chain (CLSC) is important in developing a green industrial supply chain and a sustainable economy. This study addresses the challenge of designing a closed-loop supply chain network (CLSCN) under demand uncertainty by integrating forward and reverse logistics. A continuous approximation (CA) approach is proposed to solve a responsive CLSCN model. An inventory-sharing strategy is applied in the continuous approximation (CA) approach to further solve uncertain demand in a CLSCN. We conduct sensitivity analysis to demonstrate how the optimal design characteristics and the responsive supply chain network (SCN) design depend on various scenarios. A responsive CLSCN's network density is higher than a responsive SCN's. The changes in network density of the distribution center (DC) are not significantly affected by increased risks of uncertainties. The proposed model has demonstrated substantial cost savings compared to the traditional responsive CLSCN model.

Optimizing service design for the intercity demand responsive transit system: Model, algorithm, and comparative analysis

Intercity Demand Responsive Transit (IDRT) systems, providing on-demand “door-to-door” services for intercity travelers, have been recently implemented in a few cities and gained popularity. To devise a theoretical tool for designing efficient IDRT systems, this study proposes a continuous approximation (CA) model to describe the operation metrics of the IDRT. In a two-city system, an optimization model is proposed to study the zoning strategies in the main city, the service area in the satellite city, and headways. The objective is to minimize the average system cost, including the costs of both the operators and the passengers. A two-stage partition algorithm is proposed to generate a near-optimal solution by combining the greedy algorithm and sensitivity-based coordination optimization method. Subsequently, we develop a direct solution algorithm to establish a lower bound for the optimal solution, facilitating a comparison with the near-optimal solution. Throughout all sensitivity analyses, the gaps between the near-optimal solution and the lower-bound solution remain below 3 %. The comparison results demonstrate that the computation time and solution quality of the two-stage partition algorithm are considered acceptable. Simulation is performed to test the validity of the CA model, and the errors in the average system costs are less than 1.16 %. Sensitivity analyses are carried out to evaluate the impacts of several key parameters (e.g., demand densities and line-haul distances) on the optimization results. By comparison with Intercity Fixed Route Transit (IFRT), it can be seen that the choice of IDRT depends on passengers' preference for door-to-door transportation. This study may serve as a valuable reference for the practical implementation of IDRT.