Discrete Problem Research Articles

Best-in-class modeling: A novel strategy to discover constitutive models for soft matter systems

The ability to automatically discover interpretable mathematical models from data could forever change how we model soft matter systems. For convex discovery problems with a unique global minimum, model discovery is well-established. It uses a classical top-down approach that first calculates a dense parameter vector, and then sparsifies the vector by gradually removing terms. For non-convex discovery problems with multiple local minima, this strategy is infeasible since the initial parameter vector is generally non-unique. Here we propose a novel bottom-up approach that starts with a sparse single-term vector, and then densifies the vector by systematically adding terms. Along the way, we discover models of gradually increasing complexity, a strategy that we call best-in-class modeling. To identify and select successful candidate terms, we reverse-engineer a library of sixteen functional building blocks that integrate a century of knowledge in material modeling with recent trends in machine learning and artificial intelligence. Yet, instead of solving the NP hard discrete combinatorial problem with 216=65,536 possible combinations of terms, best-in-class modeling starts with the best one-term model and iteratively repeats adding terms, until the objective function meets a user-defined convergence criterion. Strikingly, for most practical purposes, we achieve good convergence with only one or two terms. We illustrate the best-in-class one- and two-term models for a variety of soft matter systems including rubber, brain, artificial meat, skin, and arteries. Our discovered models display distinct and unexpected features for each family of materials, and suggest that best-in-class modeling is an efficient, robust, and easy-to-use strategy to discover the mechanical signatures of traditional and unconventional soft materials. We anticipate that our technology will generalize naturally to other classes of natural and man made soft matter with applications in artificial organs, stretchable electronics, soft robotics, and artificial meat.

A maximum entropy deep reinforcement learning method for sequential well placement optimization using multi-discrete action spaces

Well placement optimization is a crucial method to solve the planar conflicts in reservoir development, mainly to determine the optimal well locations and drilling sequence to maximize the economic benefits of reservoir development. However, the current well placement optimization methods face the problems of high-dimensional discretization of optimization variables and lack of effective incentives for policy exploration, which make it challenging to improve the global optimization ability (the ability to jump out of locally optimal solutions in time and continuously search for better solutions in the whole optimization process) and real-time adjustability of the well placement optimization methods under limited numerical simulation times. In this paper, we propose a new sequential well placement optimization method, based on the Discrete Soft Actor-Critic Algorithm (DSAC), which incorporates the maximum entropy mechanism to formulate well placement and drilling sequencing schemes more efficiently and maximize the net present value (NPV) over the entire life cycle of reservoir development. Specifically, the method models the well placement optimization problem as a Markov Decision Process (MDP) and achieves sequential well placement optimization by training a Deep Reinforcement Learning (DRL) agent that maps reservoir states to a stochastic policy of well placement variables as well as evaluates the value function of the current policy. The DRL agent can determine the optimal infill well location in real-time based on the reservoir state at different times during the development process, thus obtaining the optimal drilling sequence. The proposed method in this paper has two innovations. First, by reconstructing the large-scale discrete action space of well placement optimization variables into multi-discrete action spaces, and with the maximum entropy mechanism, policy exploration is encouraged to improve the global optimization capability. Second, the trained policy can swiftly adapt the subsequent well placement scheme for a specific state of the target reservoir without the requirement to initiate training from scratch, which can realize the offline application of the trained policy and has better real-time adjustability. To verify the effectiveness of the proposed method, it is tested in 2D and 3D reservoir models. The results show that DSAC not only outperforms the gradient-based optimization method, classical evolutionary algorithms, and existing reinforcement learning proximal policy optimization (PPO) method in terms of global optimization ability but also shows better real-time adjustability of the trained policy when applied offline.

Revealing the Correlation Between Lyapunov Exponent and Modulus of an n-Dimensional Nondegenerate Hyperchaotic Map

For their good randomness and long iteration periods, chaotic maps have been widely used in cryptography. Recently, we have revealed the correlation between Lyapunov exponent and sequence randomness of multidimensional chaotic maps based on modular operation. Since the modular operation can realize the boundedness of chaotic state points, it is important to further reveal the deterministic correlation between Lyapunov exponent and modulus. First, we constructed an [Formula: see text]-dimensional nondegenerate hyperchaotic map model with the desired Lyapunov exponents. Then, we gave the existence and uniqueness proof of quadrature rectangle decomposition theorem and revealed the correlation between Lyapunov exponent and modulus. The novelty lies in that (1) in order to realize the irreversibility of the iterative processes of chaotic maps, we constructed a chaotic map based on modular exponentiation, and its inverse function is the discrete logarithm problem; and (2) we reveal for the first time the correlation between Lyapunov exponent and modulus, and give the lower bound of the modulus of the nondegenerate chaotic map. In addition, to verify the effectiveness of the scheme, we constructed four-dimensional and five-dimensional chaotic maps, respectively, and analyzed their dynamical behaviors, and the results revealed that there exist linear or nonlinear correlation between Lyapunov exponent and modulus.

Effective boundary conditions for second-order homogenization

Using matched asymptotic expansions, we derive an equivalent bar model for a periodic, one-dimensional lattice made up of linear elastic springs connecting both nearest and next-nearest neighbors. We obtain a strain-gradient model with effective boundary conditions accounting for the boundary layers forming at the endpoints. It is accurate to second order in the scale separation parameter ɛ≪1, as shown by a comparison with the solution to the discrete lattice problem. The homogenized modulus associated with the gradient effect (gradient stiffness) is found negative, as is often the case in second-order homogenization. Negative gradient stiffnesses are widely viewed as paradoxical as they can induce short-wavelength oscillations in the homogenized solution. In the one-dimensional lattice, the asymptotically correct boundary conditions are shown to suppress the oscillations, thereby restoring consistency. By contrast, most of the existing work on second-order homogenization makes use of postulated boundary conditions which, we argue, not only ruin the order of the approximation but are also the root cause of the undesirable oscillations.

A metaheuristic-based comparative structure for solving discrete space mechanical engineering problem

A metaheuristic-based comparative structure for solving discrete space mechanical engineering problem

Discrete Structural Design Synthesis: A Hierarchical-Inspired Deep Reinforcement Learning Approach Considering Topological and Parametric Actions

Abstract Structural design synthesis considering discrete elements can be formulated as a sequential decision process solved using deep reinforcement learning, as shown in prior work. By modeling structural design synthesis as a Markov decision process (MDP), the states correspond to specific structural designs, the discrete actions correspond to specific design alterations, and the rewards are related to the improvement in the altered design’s performance with respect to the design objective and specified constraints. Here, the MDP action definition is extended by integrating parametric design grammars that further enable the design agent to not only alter a given structural design’s topology, but also its element parameters. In considering topological and parametric actions, both the dimensionality of the state and action space and the diversity of the action types available to the agent in each state significantly increase, making the overall MDP learning task more challenging. Hence, this paper also addresses discrete design synthesis problems with large state and action spaces by significantly extending the network architecture. Specifically, a hierarchical-inspired deep neural network architecture is developed to allow the agent to learn the type of action, topological or parametric, to apply, thus reducing the complexity of possible action choices in a given state. This extended framework is applied to the design synthesis of planar structures considering both discrete elements and cross-sectional areas, and it is observed to adeptly learn policies that synthesize high performing design solutions.

Research on the Human–Robot Collaborative Disassembly Line Balancing of Spent Lithium Batteries with a Human Factor Load

The disassembly of spent lithium batteries is a prerequisite for efficient product recycling, the first link in remanufacturing, and its operational form has gradually changed from traditional manual disassembly to robot-assisted human–robot cooperative disassembly. Robots exhibit robust load-bearing capacity and perform stable repetitive tasks, while humans possess subjective experiences and tacit knowledge. It makes the disassembly activity more adaptable and ergonomic. However, existing human–robot collaborative disassembly studies have neglected to account for time-varying human conditions, such as safety, cognitive behavior, workload, and human pose shifts. Firstly, in order to overcome the limitations of existing research, we propose a model for balancing human–robot collaborative disassembly lines that take into consideration the load factor related to human involvement. This entails the development of a multi-objective mathematical model aimed at minimizing both the cycle time of the disassembly line and its associated costs while also aiming to reduce the integrated smoothing exponent. Secondly, we propose a modified multi-objective fruit fly optimization algorithm. The proposed algorithm combines chaos theory and the global cooperation mechanism to improve the performance of the algorithm. We add Gaussian mutation and crowding distance to efficiently solve the discrete optimization problem. Finally, we demonstrate the effectiveness and sensitivity of the improved multi-objective fruit fly optimization algorithm by solving and analyzing an example of Mercedes battery pack disassembly.

Base of exponent representation matters -- more efficient reduction of discrete logarithm problem and elliptic curve discrete logarithm problem to the QUBO problem

This paper presents further improvements in the transformation of the Discrete Logarithm Problem (DLP) and Elliptic Curve Discrete Logarithm Problem (ECDLP) over prime fields to the Quadratic Unconstrained Binary Optimization (QUBO) problem. This is significant from a cryptanalysis standpoint, as QUBO problems may be solved using quantum annealers, and the fewer variables the resulting QUBO problem has, the less time is expected to obtain a solution.The main idea presented in the paper is allowing the representation of the exponent in different bases than the typically used base 2 (binary representation). It is shown that in such cases, the reduction of the discrete logarithm problem over the prime field \(\mathbb{F}_p\) to the QUBO problem may be obtained using approximately \(1.89n^2\) logical variables for \(n\) being the bitlength of prime \(p\), instead of the \(2n^2\) which was previously the best-known reduction method. The paper provides a practical example using the given method to solve the discrete logarithm problem over the prime field \(\mathbb{F}_{47}\). Similarly, for the elliptic curve discrete logarithm problem over the prime field \(\mathbb{F}_p\), allowing the representation of the exponent in different bases than typically used base two results in a lower number of required logical variables for \(n\) being the bitlength of prime \(p\), from \(3n^3\) to \(\frac{6n^3}{\log_{2}\left(\frac{3}{4}n\right)}\) logical variables, in the case of Edwards curves.

Conclusion: The State and Local–Legal Governance in the Early American Republic

Abstract: This essay comments on four new articles on key institutions of local and legal governance in the history of the early American republic: e.g., nightwatchmen, overseers of the poor, justices of the peace, and sheriffs. It attempts to place these contributions in wider historical and historiographical context by noting recent developments in the history of the American state, municipal governance, the history of law, the history of administration, and the problem of local administrative discretion.

A fast solvable operator-splitting scheme for time-dependent advection diffusion equation

It is well known that the implicit central difference discretization for unsteady advection diffusion equation (ADE) suffers from being time-consuming to solve when the advection term dominates. In this paper, we propose an operator-splitting scheme for the unsteady ADE, in which the ADE is firstly discretized by Crank-Nicolson (CN) scheme in time and central difference scheme in space; and then the discrete advection-diffusion problem is split as advection sub-problem and diffusion sub-problem at each time-level. The significance of the new scheme is that these sub-problems can be fast and directly solved within a linearithmic complexity (a linear-times-logarithm complexity) by means of fast sine transforms (FSTs). In particular, the complexity is independent of the dominance of the advection term. Theoretically, we show that proposed scheme is unconditionally stable and of second-order convergence in time and space. Numerical results are reported to show the efficiency of the proposed scheme.

Satellite navigation signal quality monitoring algorithm based on DBO-SVM

Satellite navigation signals can provide users with accurate and continuous time and position information. Since the satellite navigation signal is susceptible to interference from terrain and other electromagnetic signals during transmission. In this study, to improve the accuracy of the traditional satellite navigation signal quality monitoring algorithm and to solve the problem of discretization in the monitoring results, the satellite navigation signal quality monitoring algorithm based on the Dung Beetle Optimization assisted Support Vector Machine (DBO-SVM) is proposed. The algorithm uses four feature quantities (correlation loss and symmetry of correlation peaks in the correlation domain, signal power in the frequency domain, eye diagram opening in the modulation domain) to classify the quality of satellite navigation signals, and optimizing the parameters of the SVM through the application of the dung beetles’ foraging mechanism. Experimental validation was conducted by simulating satellite navigation signal data, and the results indicate that the DBO-SVM algorithm is more accurate than the traditional SVM algorithm, with an accuracy rate of over 98%.

Efficient Authentication Steps for Vehicle Transmission in Ad-Hoc Networks

Vehicular ad-hoc networks (VANETs) are underactive development, thanks in part to recent advances in wireless communication and networking technologies. The most fundamental part of VANET is to enable message authentications between vehicles and roadside units. Message authentication using proxy vehicles has been proposed to reduce the computational overhead of roadside units significantly. In this message authentication scheme, a proxy vehicle that verifies multiple messages at the same time improves roadside units’ efficiency. In this paper first, we show that the only proxy-based authentication scheme (PBAS) presented for this goal by Liu et al. cannot guarantee message authenticity and also it is not resistant to impersonation and modification attacks and false acceptance of batched invalid signatures. Next, we propose a new identity-based message authentication scheme using proxy vehicles (ID-MAP). Then guarantee that it can satisfy the message authentication requirements, existential unforgeability of underlying signatures against adaptability chosen message, and identity attack is proved under the Elliptic Curve discrete logarithm problem (ECDLP) in the random oracle model. It should be highlighted that ID MAP not only is more efficient than PBAS since it is pairing-free and identity based and also it does not use map-to-point hash functions, also, it satisfies the security and privacy requirements of VANETs. Furthermore, analysis shows that the required time to verify 3000 messages in ID-MAP is reduced by 76% compared to that PBAS.

The Regularized Block GMERR Method and Its Simpler Version for Solving Large-Scale Linear Discrete Ill-Posed Problems

The Regularized Block GMERR Method and Its Simpler Version for Solving Large-Scale Linear Discrete Ill-Posed Problems

Improved Snake Optimizer Using Sobol Sequential Nonlinear Factors and Different Learning Strategies and Its Applications

The Snake Optimizer (SO) is an advanced metaheuristic algorithm for solving complicated real-world optimization problems. However, despite its advantages, the SO faces certain challenges, such as susceptibility to local optima and suboptimal convergence performance in cases involving discretized, high-dimensional, and multi-constraint problems. To address these problems, this paper presents an improved version of the SO, known as the Snake Optimizer using Sobol sequential nonlinear factors and different learning strategies (SNDSO). Firstly, using Sobol sequences to generate better distributed initial populations helps to locate the global optimum solution faster. Secondly, the use of nonlinear factors based on the inverse tangent function to control the exploration and exploitation phases effectively improves the exploitation capability of the algorithm. Finally, introducing learning strategies improves the population diversity and reduces the probability of the algorithm falling into the local optimum trap. The effectiveness of the proposed SNDSO in solving discretized, high-dimensional, and multi-constraint problems is validated through a series of experiments. The performance of the SNDSO in tackling high-dimensional numerical optimization problems is first confirmed by using the Congress on Evolutionary Computation (CEC) 2015 and CEC2017 test sets. Then, twelve feature selection problems are used to evaluate the effectiveness of the SNDSO in discretized scenarios. Finally, five real-world technical multi-constraint optimization problems are employed to evaluate the performance of the SNDSO in high-dimensional and multi-constraint domains. The experiments show that the SNDSO effectively overcomes the challenges of discretization, high dimensionality, and multi-constraint problems and outperforms superior algorithms.

Operator-splitting finite element method for solving the radiative transfer equation

AbstractAn operator-splitting finite element scheme for the time-dependent radiative transfer equation is presented in this paper. The streamline upwind Petrov-Galerkin finite element method and discontinuous Galerkin finite element method are used for the spatial-angular discretization of the radiative transfer equation, whereas the backward Euler scheme is used for temporal discretization. Error analysis of the proposed numerical scheme for the fully discrete radiative transfer equation is presented. The stability and convergence estimates for the fully discrete problem are derived. Moreover, an operator-splitting algorithm for the numerical simulation of high-dimensional equations is also presented. The validity of the derived estimates and implementation is illustrated with suitable numerical experiments.

Quantum Annealing for Computer Vision minimization problems

Computer Vision (CV) labeling problems play a pivotal role in low-level vision. For decades, it has been known that these problems can be elegantly formulated as discrete energy-minimization problems derived from probabilistic graphical models such as Markov Random Fields (MRFs). Despite recent advances in MRF inference algorithms (such as graph-cut and message-passing methods), the resulting energy-minimization problems are generally viewed as intractable. The emergence of quantum computations, which offer the potential for faster solutions to certain problems than classical methods, has led to an increased interest in utilizing quantum properties to overcome intractable problems. Recently, there has also been a growing interest in Quantum Computer Vision (QCV), hoping to provide a credible alternative/assistant to deep learning solutions. This study investigates a new Quantum Annealing-based inference algorithm for CV discrete energy minimization problems. Our contribution is focused on Stereo Matching as a significant CV labeling problem. As a proof of concept, we also use a hybrid quantum–classical solver provided by D-Wave System to compare our results with the best classical inference algorithms in the literature. Our results show that Quantum Annealing can yield promising results for Stereo Matching problems, with improved accuracy on certain stereo images and competitive performance on others.

Weighted least squares collocation methods

We consider overdetermined collocation methods and propose a weighted least squares approach to derive a numerical solution. The discrete problem requires the evaluation of the Jacobian of the vector field which, however, appears in a O(h) term, h being the stepsize. We show that, by neglecting this infinitesimal term, the resulting scheme becomes a low-rank Runge–Kutta method. Among the possible choices of the weights distribution, we analyze the one based on the quadrature formula underlying the collocation conditions. A few numerical illustrations are included to better elucidate the potential of the method.

Constrained Device Performance Benchmarking with the Implementation of Post-Quantum Cryptography

Advances in quantum computers may pose a significant threat to existing public-key encryption methods, which are crucial to the current infrastructure of cyber security. Both RSA and ECDSA, the two most widely used security algorithms today, may be (in principle) solved by the Shor algorithm in polynomial time due to its ability to efficiently solve the discrete logarithm problem, potentially making present infrastructures insecure against a quantum attack. The National Institute of Standards and Technology (NIST) reacted with the post-quantum cryptography (PQC) standardization process to develop and optimize a series of post-quantum algorithms (PQAs) based on difficult mathematical problems that are not susceptible to being solved by Shor’s algorithm. Whilst high-powered computers can run these PQAs efficiently, further work is needed to investigate and benchmark the performance of these algorithms on lower-powered (constrained) devices and the ease with which they may be integrated into existing protocols such as TLS. This paper provides quantitative benchmark and handshake performance data for the most recently selected PQAs from NIST, tested on a Raspberry Pi 4 device to simulate today’s IoT (Internet of Things) devices, and provides quantitative comparisons with previous benchmarking data on a range of constrained systems. CRYSTALS-Kyber and CRYSTALS-Dilithium are shown to be the most efficient PQAs in the key encapsulation and signature algorithms, respectively, with Falcon providing the optimal TLS handshake size.

A Secure Key Exchange Protocol and a Public Key Cryptosystem for Healthcare Systems

Diffie-Hellman (DH) is the reason for the existence of a solution to the key distribution problem, where two parties can mutually set up a shared secret key over an insecure channel without any previous communication. In this study, a novel key exchange protocol based on the difficulty of the Discrete Logarithm Problem with Factor Problem (DLPFP) utilizing matrices in non-commutative semigroup over semiring is proposed. The security and complexity analyses of the protocol are discussed. Also, an ElGamal cryptosystem based on a group of invertible matrices and DLPFP over a semiring to encrypt the Sensitive Health Information (SHI) in healthcare systems is suggested.

Second order discrete boundary value problem with the $(p_1(k); p_2(k))$-Laplacian

In this paper we investigate existence and non-existence of solutions for a Dirichlet boundary value problem involving the $(p_1(k), p_2(k))$-Laplacian operator when variational methods are applied to obtain the results.