Discrete Space Research Articles

Algebraically stable SDIRK methods with controllable numerical dissipation for first/second-order time-dependent problems

In this paper, a family of four-stage singly diagonally implicit Runge-Kutta methods are proposed to solve first-/second-order time-dependent problems, exhibiting the following numerical properties: fourth-order accuracy in time, unconditional stability, controllable numerical dissipation, and adaptive time step selection. The BN-stability condition is employed as a constraint to optimize parameters in the Butcher table, having significant benefits, and hence is recommended for nonlinear dynamics problems in contrast to existing methods. Numerical examples involving both first- and second-order linear/nonlinear dynamics problems validate the proposed method, and numerical results reveal that the proposed methods are free from the order reduction phenomenon when applied to nonlinear dynamics problems. The performance of adaptive time-stepping using the embedded scheme is further illustrated by the phase-field modeling problem. Additionally, the advantages and disadvantages of three-stage third-order accurate algebraically stable methods are discussed. The proposed high-order time integration can be readily integrated into high-order spatial discretization methods, such as the high-order spectral element method employed in this paper, to obtain high-order discretization in space and time dimensions.

Many-option collective decision making: discrete collective estimation in large decision spaces

AbstractCollective consensus forming in spatially distributed systems is a challenging task. In previous literature, multi-option consensus-forming scenarios, with the number of options being smaller or equal to the number of agents, have been well studied. However, many well-performing decision-making strategies on a few options suffer from scalability when the number of options increases, especially for many-option scenarios with significantly more options than agents. In this paper, we investigate the viabilities of discrete decision-making strategies with ranked voting (RV) and belief fusion (DBBS) decision mechanisms in many-option scenarios with large decision spaces compared to the number of agents. We test the investigated strategies on an expanded discrete collective estimation scenario where the decision space can be expanded using two factors: a higher number of environmental features and/or finer decision space discretization. We have used a continuous collective consensus forming strategy, linear consensus protocol (LCP), as a baseline. Our experimental results have shown that, although susceptible to environmental influences, discrete decision-making strategies can reliably outperform those of LCP in terms of error and convergence time at the tested sizes of decision space. We have also shown that the two factors that lead to the expansion of the decision space have different impacts on performances for both RV and DBBS strategies, due to differences in the correlations between the discrete options. When facing a higher number of features, both discrete strategies experience a smaller error and a significant increase in decision time, while a finer decision space discretization has a negative influence on all considered metrics.

Optimal control of an evolutionary variational–hemivariational inequality model with new results in numerical analysis and control of the dynamic frictional contact problem

In this paper, we are concerned with numerical analysis on optimal control of a dynamic contact model. Our work builds on some existing results of this model. We derive a weak formulation, and present an approximate problem for the weak formulation by discretization in space. We prove that the approximate problem has a unique solution, which converges to the unique solution of the original problem. The existence of optimal solutions to the control problem is established based on a known continuous dependence result. We consider the numerical solutions of the optimal control problem and prove a convergence result of numerical solutions.

On the product of almost discrete Grothendieck spaces

A topological space X is called almost discrete, if it has precisely one nonisolated point. In this paper, we get that for a countable product X=∏Xi of almost discrete spaces Xi the space Cp(X) of all continuous real-valued functions with the topology of pointwise convergence is a μ-space if, and only if, X is a weak q-space if, and only if, t(X)=ω if, and only if, X is functionally generated by the family of all its countable subspaces.This result makes it possible to solve Archangel'skii's problem on the product of Grothendieck spaces. It is proved that in the model of ZFC, obtained by adding one Cohen real, there are Grothendieck spaces X and Y such that X×Y is not weakly Grothendieck space. In (PFA): the product of any countable family almost discrete Grothendieck spaces is a Grothendieck space.

Deep learning-aided topology design of metasurfaces for Rayleigh waves

Abstract Metasurfaces can effectively attenuate Rayleigh waves propagating in soil, safeguarding structures from ambient vibrations or earthquakes. However, there remains a lack of efficient approaches for designing metasurfaces that can isolate Rayleigh waves within desired frequency ranges under different site conditions. This study presents a deep learning (DL)-based topology optimization method for designing metasurfaces isolating Rayleigh waves in a target frequency range, which has potential applications for surface wave control. The proposed DL model employs a variational autoencoder to transform high-dimensional and discrete topologies into low-dimensional and continuous latent vectors, reducing the design difficulty. On this basis, the conditional tandem neural network is constructed to optimize latent vectors under different soil conditions, improving the design efficiency and verifying the universality of the method. The design reliability of the DL model is validated through 100 tests with determination coefficients more than 0.99. In addition, the generations of different metasurfaces for the same target are explored, providing designers with more choices. The insulation capabilities of the designed metasurfaces against Metro-induced Rayleigh waves are demonstrated from the time- and frequency-domain responses. The presented DL-aided design method provides a novel insight into the customization of metasurfaces for manipulating Rayleigh waves.

Optimizing large-scale hydrogen storage: A novel hybrid genetic algorithm approach for efficient pipeline network design

As hydrogen production technologies continue to advance, efficient and high-capacity hydrogen storage solutions become increasingly crucial. To address the complex optimization challenges in the design of large-scale hydrogen storage pipeline networks, the study introduces a mixed integer nonlinear optimization model inspired by hydrogen pipeline design optimization theory. The model not only focuses on optimizing network design parameters but also aims to minimize investment costs. To achieve this, the study proposes an innovative hybrid genetic algorithm that combines the Modified Feasible Directions Method (MFDM) and Genetic Algorithm Theory (GAT), specifically targeting the optimization of pipeline diameter's discrete distribution challenges. Comparative analyses with SUMT and GA algorithms demonstrate the superiority of the approach. In continuous space, the algorithm achieves a lower convergence cost of 232.4437 × 104 Yuan, while in discrete space, the cost further decreases to 212.1636 × 104 Yuan. Additionally, the study delves into the sensitivity of the HGA-MFDM algorithm to ambient temperature and wellhead temperature in hydrogen storage facilities. The analysis reveals that wellhead temperature has a more significant impact on pipeline network investment, whereas ambient temperature exhibits a more pronounced effect on iteration curves. The findings provide valuable insights for the precise adjustment and optimization of hydrogen storage pipeline networks in practical applications.

Artificial Protozoa Optimizer (APO): A novel bio-inspired metaheuristic algorithm for engineering optimization

This study proposes a novel artificial protozoa optimizer (APO) that is inspired by protozoa in nature. The APO mimics the survival mechanisms of protozoa by simulating their foraging, dormancy, and reproductive behaviors. The APO was mathematically modeled and implemented to perform the optimization processes of metaheuristic algorithms. The performance of the APO was verified via experimental simulations and compared with 32 state-of-the-art algorithms. Wilcoxon signed-rank test was performed for pairwise comparisons of the proposed APO with the state-of-the-art algorithms, and Friedman test was used for multiple comparisons. First, the APO was tested using 12 functions of the 2022 IEEE Congress on Evolutionary Computation benchmark. Considering practicality, the proposed APO was used to solve five popular engineering design problems in a continuous space with constraints. Moreover, the APO was applied to solve a multilevel image segmentation task in a discrete space with constraints. The experiments confirmed that the APO could provide highly competitive results for optimization problems. The source codes of Artificial Protozoa Optimizer are publicly available at https://seyedalimirjalili.com/projects and https://ww2.mathworks.cn/matlabcentral/fileexchange/162656-artificial-protozoa-optimizer.

Learning to Branch: Generalization Guarantees and Limits of Data-Independent Discretization

Tree search algorithms, such as branch-and-bound, are the most widely used tools for solving combinatorial and non-convex problems. For example, they are the foremost method for solving (mixed) integer programs and constraint satisfaction problems. Tree search algorithms come with a variety of tunable parameters that are notoriously challenging to tune by hand. A growing body of research has demonstrated the power of using a data-driven approach to automatically optimize the parameters of tree search algorithms. These techniques use a training set of integer programs sampled from an application-specific instance distribution to find a parameter setting that has strong average performance over the training set. However, with too few samples, a parameter setting may have strong average performance on the training set but poor expected performance on future integer programs from the same application. Our main contribution is to provide the first sample complexity guarantees for tree search parameter tuning. These guarantees bound the number of samples sufficient to ensure that the average performance of tree search over the samples nearly matches its future expected performance on the unknown instance distribution. In particular, the parameters we analyze weight scoring rules used for variable selection. Proving these guarantees is challenging because tree size is a volatile function of these parameters: we prove that, for any discretization (uniform or not) of the parameter space, there exists a distribution over integer programs such that every parameter setting in the discretization results in a tree with exponential expected size, yet there exist parameter settings between the discretized points that result in trees of constant size. In addition, we provide data-dependent guarantees that depend on the volatility of these tree-size functions: our guarantees improve if the tree-size functions can be well approximated by simpler functions. Finally, via experiments, we illustrate that learning an optimal weighting of scoring rules reduces tree size.

Forest fire spreading: A nonlinear stochastic model continuous in space and time

AbstractForest fire spreading is a complex phenomenon characterized by a stochastic behavior. Nowadays, the enormous quantity of georeferenced data and the availability of powerful techniques for their analysis can provide a very careful picture of forest fires opening the way to more realistic models. We propose a stochastic spreading model continuous in space and time that has the potentiality to use such data in their full power. The state of the forest fire is described by the subprobability densities of the green trees and of the trees on fire that can be estimated thanks to data coming from satellites and earth detectors. The fire dynamics is encoded into a kernel function that can take into account wind conditions, land slope, spotting phenomena, and so on, bringing to a system of integrodifferential equations whose solutions provide the evolution in time of the subprobability densities. That makes the model complementary to models based on cellular automata that furnish single instantiations of the stochastic phenomenon. Moreover, stochastic models based on cellular automata can be derived from the present model by space and time discretization. Existence and uniqueness of the solutions is proved by using Banach's fixed‐point theorem. The asymptotic behavior of the model is analyzed as well. By specifying a particular structure for the kernel, we obtain numerical simulations of the fire spreading under different conditions. For example, in the case of a forest fire evolving toward a river, the simulations show that the probability density of the trees on fire is different from zero beyond the river due to the spotting phenomenon. The kernel could be slightly modified to include firefighters interventions and weather changes.

Viscosity solutions approach to finite-horizon continuous-time Markov decision process

This paper investigates the optimal control problems for the finite-horizon continuous-time Markov decision processes with delay-dependent control policies. We develop compactification methods in decision processes and show that the existence of optimal policies. Subsequently, through the dynamic programming principle of the delay-dependent control policies, the differential-difference Hamilton-Jacobi-Bellman (HJB) equation in the setting of discrete space is established. Under certain conditions, we give the comparison principle and further prove that the value function is the unique viscosity solution to this HJB equation. Based on this, we show that among the class of delay-dependent control policies, there is an optimal one which is Markovian.

Rediscovery of Numerical Lüscher’s Formula from the Neural Network

Abstract We present that by predicting the spectrum in discrete space from the phase shift in continuous space, the neural network can remarkably reproduce the numerical Lüscher’s formula to a high precision. The model-independent property of the Lüscher’s formula is naturally realized by the generalizability of the neural network. This exhibits the great potential of the neural network to extract model-independent relation between model-dependent quantities, and this data-driven approach could greatly facilitate the discovery of the physical principles underneath the intricate data.

On the stability and convergence of discontinuous Galerkin schemes for incompressible flows

Abstract The property that the velocity $\textbf{u}$ belongs to $L^{\infty }(0,T;L^{2}(\varOmega )^{d})$ is an essential requirement in the definition of energy solutions of models for incompressible fluids. It is, therefore, highly desirable that the solutions produced by discretization methods are uniformly stable in the $L^{\infty }(0,T;L^{2}(\varOmega )^{d})$-norm. In this work, we establish that this is indeed the case for discontinuous Galerkin (DG) discretizations (in time and space) of non-Newtonian models with $p$-structure, assuming that $p\geq \frac{3d+2}{d+2}$; the time discretization is equivalent to the RadauIIA Implicit Runge–Kutta method. We also prove (weak) convergence of the numerical scheme to the weak solution of the system; this type of convergence result for schemes based on quadrature seems to be new. As an auxiliary result, we also derive Gagliardo–Nirenberg-type inequalities on DG spaces, which might be of independent interest.

Assessing and Improving the Robustness of Bayesian Evidential Learning in One Dimension for Inverting Time-Domain Electromagnetic Data: Introducing a New Threshold Procedure

Understanding the subsurface is of prime importance for many geological and hydrogeological applications. Geophysical methods offer an economical alternative for investigating the subsurface compared to costly borehole investigations. However, geophysical results are commonly obtained through deterministic inversion of data whose solution is non-unique. Alternatively, stochastic inversions investigate the full uncertainty range of the obtained models, yet are computationally more expensive. In this research, we investigate the robustness of the recently introduced Bayesian evidential learning in one dimension (BEL1D) for the stochastic inversion of time-domain electromagnetic data (TDEM). First, we analyse the impact of the accuracy of the numerical forward solver on the posterior distribution, and derive a compromise between accuracy and computational time. We also introduce a threshold-rejection method based on the data misfit after the first iteration, circumventing the need for further BEL1D iterations. Moreover, we analyse the impact of the prior-model space on the results. We apply the new BEL1D with a threshold approach on field data collected in the Luy River catchment (Vietnam) to delineate saltwater intrusions. Our results show that the proper selection of time and space discretization is essential for limiting the computational cost while maintaining the accuracy of the posterior estimation. The selection of the prior distribution has a direct impact on fitting the observed data and is crucial for a realistic uncertainty quantification. The application of BEL1D for stochastic TDEM inversion is an efficient approach, as it allows us to estimate the uncertainty at a limited cost.

AoI minimization of ambient backscatter-assisted EH-CRN with cooperative spectrum sensing

To evaluate the timeliness and freshness of information from energy-constrained devices, we minimize the weighted average age of information (AoI) of the ambient backscatter-assisted energy harvesting cognitive radio network (AB-EH-CRN), where secondary transmitters (STs) follow the non-orthogonal multiple access (NOMA) strategy to transmit status updates. The secondary receiver (SR) schedules the selected actions for STs to perform spectrum access, and receives status updates from them. We formulate the AoI minimization problem where the action selection and transmission power of STs are considered as variables. To exploit spectrum resources in the AB-EH-CRN with unknown signal-to-noise ratios (SNRs) at the STs, we propose the deep neural network (DNN)-based cooperative spectrum sensing (CSS), and adopt an energy threshold approach for STs to determine the energy allocation between spectrum sensing (SS) and packet transmission. Moreover, we address the AoI minimization problem by the proposed hybrid action and energy penalty deep deterministic policy gradient (HAEP-DDPG) algorithm, which adapts to the hybrid continuous and discrete action spaces. Simulation results validate the advantage of the proposed HAEP-DDPG algorithm compared with comparison schemes in terms of AoI.

Well-posedness and Tikhonov regularization of an inverse source problem for a parabolic equation with an integral constraint

Abstract We investigate a time-dependent inverse source problem for a parabolic partial differential equation with an integral constraint and subject to Neumann boundary conditions in a domain of R d \mathbb{R}^{d} , d ≥ 1 d\geq 1 . We prove the well-posedness as well as higher regularity of solutions in Hölder spaces. We then develop and implement an algorithm that we use to approximate solutions of the inverse problem by means of a finite element discretization in space. Due to instability in inverse problems, we apply Tikhonov regularization combined with the discrepancy principle for selecting the regularization parameter in order to obtain a stable reconstruction. Our numerical results show that the proposed scheme is an accurate technique for approximating solutions of this inverse problem.

Error analysis of second-order local time integration methods for discontinuous Galerkin discretizations of linear wave equations

This paper is dedicated to the full discretization of linear wave equations, where the space discretization is carried out with a discontinuous Galerkin method on spatial meshes which are locally refined or have a large wave speed on only a small part of the mesh. Such small local structures lead to a strong Courant–Friedrichs–Lewy (CFL) condition in explicit time integration schemes causing a severe loss in efficiency. For these problems, various local time-stepping schemes have been proposed in the literature in the last years and have been shown to be very efficient. Here, we construct a quite general class of local time integration methods preserving a perturbed energy and containing local time-stepping and locally implicit methods as special cases. For these two variants we prove stability and optimal convergence rates in space and time. Numerical results confirm the stability behavior and show the proved convergence rates.

An Improved Machine Learning Model for Pure Component Property Estimation

Information on the physicochemical properties of chemical species is an important prerequisite when performing tasks such as process design and product design. However, the lack of extensive data and high experimental costs hinder the development of prediction techniques for these properties. Moreover, accuracy and predictive capabilities still limit the scope and applicability of most property estimation methods. This paper proposes a new Gaussian process-based modeling framework that aims to manage a discrete and high-dimensional input space related to molecular structure representation with the group-contribution approach. A warping function is used to map discrete input into a continuous domain in order to adjust the correlation between different compounds. Prior selection techniques, including prior elicitation and prior predictive checking, are also applied during the building procedure to provide the model with more information from previous research findings. The framework is assessed using datasets of varying sizes for 20 pure component properties. For 18 out of the 20 pure component properties, the new models are found to give improved accuracy and predictive power in comparison with other published models, with and without machine learning.

Joint magnetic resonance imaging artifacts and noise reduction on discrete shape space of images

Joint magnetic resonance imaging artifacts and noise reduction on discrete shape space of images

A novel structure learning method of Bayesian networks based on the neighboring complete node ordering search

Learning the optimal structure of a Bayesian network (BN) from observational data has received considerable research attention. In most structure learning methods based on scoring and searching, the discrete space of BNs is used, and the scale of the network grows exponentially with the number of nodes n, that is, O(n!2Cn2). Any existing scoring function in the BN space has multiple peaks. Thus, when using heuristic algorithms, local optimum problems may occur while searching in this space. As the search process involves only a small part of the space, the final learning result is not always ideal. In this study, the scoring and searching task is implemented in the complete node ordering space, and a novel neighbor operation is proposed for improving learning accuracy. This technique reduces the search space to O(n!) and facilitates optimized learning of a structure. Furthermore, techniques for efficiency optimization are proposed, which address the problems of high computational complexity and repetitive calculation of family scores.

Multi-Attribute Decision-Making Method in Preventive Maintenance of Asphalt Pavement Based on Optimized Triangular Fuzzy Number

By choosing the right pavement maintenance plan, we can reduce resource utilization, reduce environmental pollution, and extend road life, which is important for improving social sustainability. At present, the selection of road maintenance programs mostly adopts multiple attribute decision-making (MADA), in particular, the analytic hierarchy process (AHP) is often used. However, this method needs to use expert scoring data, which leads to strong subjectivity and poor reliability. Therefore, it reduces the science of road maintenance scheme selection. In order to reduce the subjectivity of the score and obtain a more suitable road maintenance scheme, this paper applies a multi-criteria decision-making method that characterizes attribute information by triangular fuzzy numbers (TFN) in the discrete decision space. Firstly, we invite experts to score the importance of the selection of pavement preventive maintenance technical solutions with respect to the indicators affecting the selection of solutions. Secondly, the two indicators of similarity and reliability are used to quantitatively evaluate the indicators and programs, respectively. Finally, we compare the weighted programs according to the overall possibility degree of each program. In actual cases, the overall possibility degree of each scheme obtained by this method is 1.0002–0.0477, and the optimal solution is fog sealing technology. The decision-making model applied in this paper can be considered in multiple dimensions, which can scientifically reduce the subjectivity of expert scoring. The best maintenance plan can also be quickly obtained through the simple calculation method in this paper.