First-order Problem Research Articles

A New Methodology for the Development of Efficient Multistep Methods for First-Order Initial Value Problems with Oscillating Solutions V: The Case of the Open Newton–Cotes Differential Formulae

A New Methodology for the Development of Efficient Multistep Methods for First-Order Initial Value Problems with Oscillating Solutions V: The Case of the Open Newton–Cotes Differential Formulae

An Effective Discontinuous Galerkin Method for Solving Acoustic Wave Equations in Heterogeneous Media

Abstract A discontinuous Galerkin (DG) method is introduced for solving 2D and 3D first-order velocity-pressure acoustic wave propagation problems in heterogeneous media. First, acoustic equations are reformulated into a first-order hyperbolic system, which can be integrated into the framework of the DG method. Then, we introduce an effective numerical flux in DG spatial discretization, which preserves physical laws on both sides of interfaces with discontinuous material parameters. It is compared with both the classic Lax-Friedrichs flux and the exact upwind flux, with the latter derived from solving the Riemann problem and satisfying the Rankine-Hugoniot condition. The comparison reveals that while our flux is formally similar to the classic local Lax-Friedrichs flux, its numerical behavior is analogous to the exact upwind flux. The primary advantage of this method lies in its simplicity and straightforward computation yet can maintain physical continuity conditions near interfaces with discontinuous material parameters. Several numerical experiments of acoustic wave propagation in 2D and 3D heterogeneous media are carried out, illustrating the effectiveness of this method.

Investigations into Proof Structures

We introduce and elaborate a novel formalism for the manipulation and analysis of proofs as objects in a global manner. In this first approach the formalism is restricted to first-order problems characterized by condensed detachment. It is applied in an exemplary manner to a coherent and comprehensive formal reconstruction and analysis of historical proofs of a widely-studied problem due to Łukasiewicz. The underlying approach opens the door towards new systematic ways of generating lemmas in the course of proof search to the effects of reducing the search effort and finding shorter proofs. Among the numerous reported experiments along this line, a proof of Łukasiewicz ’s problem was automatically discovered that is much shorter than any proof found before by man or machine.

Rate of Convergence for First-Order Singular Perturbation Problems: Hamilton–Jacobi–Isaacs Equations and Mean Field Games of Acceleration

Rate of Convergence for First-Order Singular Perturbation Problems: Hamilton–Jacobi–Isaacs Equations and Mean Field Games of Acceleration

Invariant neural architecture for learning term synthesis in instantiation proving

The development of strong CDCL-based propositional (SAT) solvers has greatly advanced several areas of automated reasoning (AR). One of the directions in AR is therefore to make use of SAT solvers in expressive formalisms such as first-order logic, for which large corpora of general mathematical problems exist today. This is possible due to Herbrand's theorem, which allows reduction of first-order problems to propositional problems by instantiation. The core challenge is synthesizing the appropriate instances from the typically infinite Herbrand universe.In this work, we develop a machine learning system targeting this task, addressing its combinatorial and invariance properties. In particular, we develop a GNN2RNN architecture based on a graph neural network (GNN) that learns from problems and their solutions independently of many symmetries and symbol names (addressing the abundance of Skolems), combined with a recurrent neural network (RNN) that proposes for each clause its instantiations. The architecture is then combined with an efficient ground solver and, starting with zero knowledge, iteratively trained on a large corpus of mathematical problems. We show that the system is capable of solving many problems by such educated guessing, finding proofs for 32.12% of the training set. The final trained system solves 19.74% of the unseen test data on its own. We also observe that the trained system finds solutions that the iProver and CVC5 systems did not find.

Reduced Element for Adaptive Finite Element Analysis of First-Order IVP with Built-in Error Estimator in Maximum Norm

This paper proposes a novel yet simple approach to the adaptive finite element (FE) analysis of the first-order Initial Value Problems (IVPs) in the maximum norm by introducing the reduced element technique. In the present approach, the FE solution uh of the conventional Galerkin element of degree m + 1 is decomposed into two parts: a reduced solution uRh from the reduced element of degree m obtained by ignoring the highest degree term of uh, and a built-in point-wise error estimator εRh directly given by the ignored term. Since the end node solutions of the reduced element are inherited from the full order element, it gains O(h2m+2) accuracy and achieves a nodal/element accuracy ratio as high as two, which greatly enhances its adaptive capability regarding solving IVPs on long time domains. The related error analysis is addressed and a complete adaptivity algorithm is given. Typical numerical examples of both linear and nonlinear IVPs of both single and systems of equations are presented to show the validity and effectiveness of the proposed approach.

A New Methodology for the Development of Efficient Multistep Methods for First-Order Initial Value Problems with Oscillating Solutions: III the Role of the Derivative of the Phase Lag and the Derivative of the Amplification Factor

A New Methodology for the Development of Efficient Multistep Methods for First-Order Initial Value Problems with Oscillating Solutions: III the Role of the Derivative of the Phase Lag and the Derivative of the Amplification Factor

A novel A-stable method of order of accuracy three, based on optimization for solving initial value problems

This paper presents a novel A-stable nonlinear method of order of accuracy three, based on optimization for solving the first-order initial value problems. The A-stability is proven and the consistency of order three is investigated as well. The convergence of the method is established under certain assumptions. Additionally, a second-order method is derived based on this approach, and a variable step size formulation is implemented using the second and third-order methods to enhance efficiency. Numerical examples provided in the paper demonstrate the validity of the theory and underscore the advantages of the method compared to three other A-stable methods of order three, including an implicit RK method, a Rosenbrock method, and an explicit (nonstandard) method.

New computational methods using seventh-derivative type for the solution of first-order initial value problems

This study uses finite power series as the basis function and interpolation and collocation techniques to study a class of implicit block methods of a seventh-derivative type. Discrete schemes are implicit two-point block methods that are obtained by selecting collocation points carefully and unevenly in order to improve the stability of the methods through testing. Nevertheless, in contrast to other current numerical equations, these methods require seventh-derivative functions. The novel techniques are identified, examined, and shown to be A-stable and convergent. Newton Raphson’s approach is used to accomplish method implementation. Trials demonstrated the effectiveness and precision of the derived equations in terms of computational time and absolute errors on a variety of first-order initial value issues, such as first-order, second-order linear differential systems and SIR model. When compared to similar methods that are currently in the literature, the suggested methods produce better numerical results.

Perturbations of non-autonomous second-order abstract Cauchy problems

In this paper we present time-dependent perturbations of second-order non-autonomous abstract Cauchy problems associated to a family of operators with constant domain. We make use of the equivalence to a first-order non-autonomous abstract Cauchy problem in a product space, which we elaborate in full detail. As an application we provide a perturbed non-autonomous wave equation.

Effect of distribution of fines on evolution of cooperation in spatial public goods game

Abstract In the realm of public goods game, punishment, as a potent tool, stands out for fostering cooperation. While it effectively addresses the first-order free-rider problem, the associated costs can be substantial. Punishers incur expenses in imposing sanctions, while defectors face fines. Unfortunately, these monetary elements seemingly vanish into thin air, representing a loss to the system itself. However, by virtue of the redistribution of fines to cooperators and punishers, not only can we mitigate this loss, but the rewards for these cooperative individuals can be enhanced. Based upon this premise, this paper introduces a fine distribution mechanism to the traditional pool punishment model. Under identical parameter settings, by conducting a comparative experiment with the conventional punishment model, the paper aims to investigate the impact of fine distribution on the evolution of cooperation in spatial public goods game. And the experimental results clearly demonstrate that, in instances where the punishment cost is prohibitively high, the cooperative strategies of the traditional pool punishment model may completely collapse. However, the model enriched with fine distribution manages to sustain a considerable number of cooperative strategies, thus highlighting its effectiveness in promoting and preserving cooperation, even in the face of substantial punishment cost.

The Role of Convection in the Limit Shape of the Critical Front Profile for Born-Infeld Diffusion Models

AbstractIn this paper, we deal with models with Born-Infeld (or relativistic) type diffusion and monostable reaction, investigating the effect of the introduction of a convection term on the limit shape of the critical front profile for vanishing diffusion. We first provide an estimate of the critical speed and then, through a careful analysis of an equivalent first-order problem, we show that different convection terms may lead either to a complete sharpening of the limit profile or to its complete regularization, presenting some related numerical simulations.

Global AI governance: barriers and pathways forward

Abstract Recent advances in artificial intelligence (AI), combined with a proliferation in use, have led to a newfound emphasis on strengthening the global governance of AI. In this article, we assess the prospects for stronger global AI governance and consider potential pathways forward. We map the nascent landscape of international regimes focused on AI governance and conclude that a governance deficit remains due to the inadequacy of existing initiatives, gaps in the landscape and difficulties reaching agreement over more appropriate mechanisms. First-order cooperation problems stemming from interstate competition and second-order cooperation problems from dysfunctional international institutions problematize overcoming current deficiencies in global AI governance. In light of these cooperation problems, we evaluate two pathways for strengthening global AI governance: (a) developing new centralized international AI institution(s); and (b) strengthening coordination between, and capacities of, existing institutions. We argue that strengthening the existing weak ‘regime complex’ of international institutions is the more politically legitimate and viable path forward. Improving coordination between, and capacities of, existing international institutions governing AI would support inclusive and mutually reinforcing policy change that can mitigate a range of risks associated with these technologies. As first steps for strengthening the weak AI regime complex, we recommend that the OECD is foregrounded as a centre of expert AI knowledge so that it can facilitate peer pressure among states and harmonization of policies. We also encourage scrutiny of different nodes in the regime complex to ensure that they are fulfilling appropriate functions based on their democratic mandates.

Relation-Theoretic Nonlinear Almost Contractions with an Application to Boundary Value Problems

This article investigates certain fixed-point results enjoying nonlinear almost contraction conditions in the setup of relational metric space. Some examples are constructed in order to indicate the profitability of our results. As a practical use of our findings, we demonstrate the existence of a unique solution to a specific first-order boundary value problem.

Efficient Multistep Algorithms for First-Order IVPs with Oscillating Solutions: II Implicit and Predictor–Corrector Algorithms

This research introduces a fresh methodology for creating efficient numerical algorithms to solve first-order Initial Value Problems (IVPs). The study delves into the theoretical foundations of these methods and demonstrates their application to the Adams–Moulton technique in a five-step process. We focus on developing amplification-fitted algorithms with minimal phase-lagor phase-lag equal to zero (phase-fitted). The request of amplification-fitted (zero dissipation) is to ensure behavior like symmetric multistep methods (symmetric multistep methods are methods with zero dissipation). Additionally, the stability of the innovative algorithms is examined. Comparisons between our new algorithm and traditional methods reveal its superior performance. Numerical tests corroborate that our approach is considerably more effective than standard methods for solving IVPs, especially those with oscillatory solutions.

Reinforcement Learning and Data-Generation for Syntax-Guided Synthesis

Program synthesis is the task of automatically generating code based on a specification. In Syntax-Guided Synthesis (SyGuS) this specification is a combination of a syntactic template and a logical formula, and the result is guaranteed to satisfy both. We present a reinforcement-learning guided algorithm for SyGuS which uses Monte-Carlo Tree Search (MCTS) to search the space of candidate solutions. Our algorithm learns policy and value functions which, combined with the upper confidence bound for trees, allow it to balance exploration and exploitation. A common challenge in applying machine learning approaches to syntax-guided synthesis is the scarcity of training data. To address this, we present a method for automatically generating training data for SyGuS based on anti-unification of existing first-order satisfiability problems, which we use to train our MCTS policy. We implement and evaluate this setup and demonstrate that learned policy and value improve the synthesis performance over a baseline by over 26 percentage points in the training and testing sets. Our tool outperforms state-of-the-art tool cvc5 on the training set and performs comparably in terms of the total number of problems solved on the testing set (solving 23% of the benchmarks on which cvc5 fails). We make our data set publicly available, to enable further application of machine learning methods to the SyGuS problem.

New Computational Methods Using Seventh Derivative Type for the Solution of First Order Initial Value Problems

In this research, a class of implicit block methods of a seventh derivative type are examined through interpolation and collocation techniques using finite power series as the basis function. The discrete schemes, which are implicit two-point block methods, are obtained by carefully and unevenly choose collocation points that ensure better methods’ stability via test. However, these schemes require seventh derivative functions unlike other existing numerical formulae. The new methods are found, investigated and proven to be convergent and A-stable. The implementation of methods is achieved by using Newton Raphson’s method. Experiments show the efficiency and accuracy of the developed formulae on different class of first-order initial value problems, including SIR, growth models and Prothero-Robinson oscillatory problem and with comparison to such existing methods. In addition, it is observed that uneven and positioning of collocation points greatly influence the efficiency and accuracy of numerical methods.

A One-Step Hybrid Obrechkoff-Type Block Method for First-Order Initial Value Problems in Ordinary Differential Equations

In this paper, we propose a one-step hybrid block method of the Obrechkoff-type block method for solving initial value problems in ordinary differential equations using Taylor series expansion. The new method was implemented with some selected initial value problems to determine the efficiency and accuracy of the method. The convergence and stability properties were also examined. It was discovered that the new method compared favorably with other existing methods in the selected literature.

A Decomposition Approach to Multiagent Systems With Bernoulli Packet Loss

In this paper, we extend the decomposable systems framework to multi-agent systems with Bernoulli distributed packet loss with uniform probability. The proposed sufficient analysis conditions for mean-square stability and $H_2$-performance -- which are expressed in the form of linear matrix inequalities -- scale linearly with increased network size and thus allow to analyse even very large-scale multi-agent systems. A numerical example demonstrates the potential of the approach by application to a first-order consensus problem.

A unified immersed finite element error analysis for one-dimensional interface problems

It has been known that the traditional scaling argument cannot be directly applied to the error analysis of immersed finite elements (IFE) because, in general, the spaces on the reference element associated with the IFE spaces on different interface elements via the standard affine mapping are not the same. By analyzing a mapping from the involved Sobolev space to the IFE space, this article is able to extend the scaling argument framework to the error estimation for the approximation capability of a class of IFE spaces in one spatial dimension. As demonstrations of the versatility of this unified error analysis framework, the manuscript applies the proposed scaling argument to obtain optimal IFE error estimates for a typical first-order linear hyperbolic interface problem, a second-order elliptic interface problem, and the fourth-order Euler-Bernoulli beam interface problem, respectively.