Class Of Problems Research Articles

Solve Riemann–Liouville boundary value problems using collocation boundary value methods with the graded mesh

The paper explores numerical approaches to a class of fractional boundary value problems that can be converted into Volterra integral equations with multiple singular kernels. We propose a collocation boundary value method with a graded mesh for the resulting weakly singular integral equations. Moreover, we demonstrate the local convergence property of the presented method using the Gronwall’s inequality. In addition, we assess the stability of the algorithm by examining the L2-norm of the approximate solution. It is found that the proposed collocation method exhibits both fast convergence order and good stability. The numerical experiments provide rigorous validation of the theoretical results.

Inference of Probabilistic Programs with Moment-Matching Gaussian Mixtures

Computing the posterior distribution of a probabilistic program is a hard task for which no one-fit-for-all solution exists. We propose Gaussian Semantics, which approximates the exact probabilistic semantics of a bounded program by means of Gaussian mixtures. It is parametrized by a map that associates each program location with the moment order to be matched in the approximation. We provide two main contributions. The first is a universal approximation theorem stating that, under mild conditions, Gaussian Semantics can approximate the exact semantics arbitrarily closely. The second is an approximation that matches up to second-order moments analytically in face of the generally difficult problem of matching moments of Gaussian mixtures with arbitrary moment order. We test our second-order Gaussian approximation (SOGA) on a number of case studies from the literature. We show that it can provide accurate estimates in models not supported by other approximation methods or when exact symbolic techniques fail because of complex expressions or non-simplified integrals. On two notable classes of problems, namely collaborative filtering and programs involving mixtures of continuous and discrete distributions, we show that SOGA significantly outperforms alternative techniques in terms of accuracy and computational time.

On [formula omitted]-stable locally checkable problems parameterized by mim-width

In this paper we continue the study of locally checkable problems under the framework introduced by Bonomo-Braberman and Gonzalez in 2020, by focusing on graphs of bounded mim-width. We study which restrictions on a locally checkable problem are necessary in order to be able to solve it efficiently on graphs of bounded mim-width. To this end, we introduce the concept of d-stability of a check function. The related locally checkable problems contain large classes of problems, among which we can mention, for example, LCVP problems. We give an algorithm showing that these problems are XP when parameterized by the mim-width of a given binary decomposition tree of the input graph , that is, that they can be solved in polynomial time given a binary decomposition tree of bounded mim-width. We explore the relation between d-stable locally checkable problems and the recently introduced DN logic (Bergougnoux, Dreier and Jaffke, 2022), and show that both frameworks model the same family of problems. We include a list of concrete examples of d-stable locally checkable problems whose complexity on graphs of bounded mim-width was open so far.

Enhanced bi-directional dynamic programming algorithm for the resource constrained shortest path problem

In this paper we propose an enhanced bi-directional dynamic programming algorithm for the Resource Constrained Shortest Path, Righini and Salani (2006). In particular we study the join procedure, one of the time consuming components of the algorithm, and propose a dominance based join that uses binary search and Pareto frontier exploration. The proposed method can be used in both node-join and arc-join procedures. The computational experiments are performed on two classes of problems both describing complete graphs with negative cost cycles. On the first set, with a single resource consumption, the proposed procedure slightly improves the overall computational time with savings up to 8.4% for the B class and just 0.4% for the most time consuming class M. On the second set, with multiple resources consumption, the proposed procedure exhibits a remarkable improvement with an average speedup of 39.2%.

Modelling the user’s information needs based on the assessment of pragmatic informativeness of data

The purpose of this study is to describe the model of the user›s information needs when working with a geographic information system, as well as the technology of forming this model. The model is based on quantifying the pragmatic informativeness of the data output to the user. The main methods used in the work are methods of system analysis, ontological design, set-theoretic description of models and systems, as well as methods of calculating the semantic proximity of concepts. As a result of the work, a model of user›s information needs and the technology of its formation and description are proposed. The proposed model and technology allow to increase the efficiency of user interaction with geographic information system and can be used in solving such classes of problems as resolving semantic ambiguity when working with geodata, taking into account the context when performing geovisualisation procedures, and others.

A Lyapunov Type Inequality for a Class of Fractional Boundary Value Problems with Riemann-Liouville Derivative

A Lyapunov Type Inequality for a Class of Fractional Boundary Value Problems with Riemann-Liouville Derivative

Cooperative Eco-Driving Control of Connected Multi-Vehicles With Spatio-Temporal Constraints

In this article, we propose a novel time-energy optimal control approach with applications in cooperative eco- driving of connected and automated vehicles (CAVs) in urban traffic networks. Safely approaching and departing signalized intersections requires the satisfaction of both spatial equality constraints determined by intersection locations and temporal inequality constraints in compliance with the green light phases. To generate time- and energy-optimal trajectories, the optimal crossing times at intersections are firstly treated as characteristic time constraints, which makes the problem tractable. Then the di- rect multiple shooting method and time transformation technique are applied to find a numerical solution. The contribution of this article is twofold. The first one is the development of a novel time- and energy-optimal control approach that ensures a trade- off between minimizing energy and time for a general class of optimal control problems with multiple characteristic times. The second contribution is the application of the proposed method to the challenging problem of multi-CAVs’ cooperative eco-driving control, in which multiple vehicles must simultaneously minimize travel time and energy consumption in the presence of spatio- temporal constraints. Simulation analysis over real-world urban route scenarios shows that the proposed eco-driving control strategy can save up to 8.2% of energy or reduce up to 6.7% of travel time compared to a baseline method. Furthermore, hardware-in-the-loop (HiL) experimental results indicate that the proposed strategy can be implemented in real-time.

State Estimation and Detectability of Networked Discrete Event Systems With Multi-Channel Communication Networks

In this paper, we extend existing results on detectability of Discrete Event Systems (DES) to Networked Discrete Event Systems (NDES), a class of DES whose communication between the plant and the agent (supervisor, diagnoser, prognoser, etc) is carried out through a network that can have several channels. We assume that each channel is subject to different communication delays, which may cause changes in the order of the event observations with respect to their actual occurrences in the system automaton, and also that event transmission may be lost in the communication channels causing loss of observations. We follow a previously presented approach to construct an equivalent nondeterministic untimed automaton, and, based on this model, we present a method for the state estimation of NDES. We extend existing D-detectability definitions of DES to NDES subject to delays and loss of observation, referred here to as strong networked D-detectability, weak networked D-detectability, strong periodic networked D-detectability, weak periodic networked D-detectability. Finally, necessary and sufficient conditions for the networked D-detectability notions proposed here are proposed and algorithms for their verification are developed. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —State estimation is an important problem that appear in several practical applications, e.g. industrial systems, aircraft control and business management. Of current interest is the problem of distinguishing in real time if the system is not in pairs of states, the so-called D-detectability. In practical terms, it is not necessary to actually know the current state of the system but if the system may not be in two or more conflicting states. With the use of communication networks, additional features such as communication delay must also be taken into account in the process of performing D-detectability. In this paper we consider discrete event systems, a class of systems used to model a wide class of practical engineering problems whose dynamic evolution is driven by event occurrences, that are embedded in a communication network; we will refer to these systems as networked discrete event systems. In this regard, we generalize existing notions of D-detectability, and we call them networked D-detectability. Algorithms for networked D-detectability verification and a method to perform online state estimation are given in the paper.

A Numerical Solution of a Coupling System of Conformable Time-Derivative Two-Dimensional Burgers’ Equations

In this paper, we deal with a numerical solution of a coupling system of fractional conformable time-derivative two-dimensional (2D) Burgers’ equations. The presence of both the fractional time derivative and the nonlinear terms in this system of equations makes solving it more difficult. Firstly, we use the Cole-Hopf transformation in order to reduce the coupling system of equations to a conformable time-derivative 2D heat equation for which the numerical solution is calculated by the explicit and implicit schemes. Secondly, we calculate the numerical solution of the proposed system by using both the obtained solution of the conformable time-derivative heat equation and the inverse Cole-Hopf transformation. This approach shows its efficiency to deal with this class of fractional nonlinear problems. Some numerical experiments are displayed to consolidate our approach

Optimal strategy for Bayesian two-armed bandit problem with an arched reward function

In the classical two-armed bandit (TAB) problems, the goal of maximizing the total return specifies the optimal decision rule and fosters the development of abundant practicable methods, such as $ \epsilon $-greedy and upper confidence bound for TAB procedures. However, the existence of an arched reward function for the total return breaks the previously updated algorithms, like traditional myopic strategy, and prompts the generation of novel computation and theory for maximizing its expectation. Here we develop a special class of Bayesian two-armed bandit problems by imposing a prior probability on which arm has a greater expectation of the returns and propose a myopic strategy to specify the decision rule. We prove that the proposed myopic strategy is optimal under the structure of the arched reward function by exploring the dynamic programming principle. This article also demonstrates that arched functions are extensions of the monotone functions including the classical linear functions. Meanwhile, we establish a corresponding law of large numbers for our myopic strategy. Simulation studies pose supportive evidence that the newly proposed strategy performs well.

Characteristics of convergence and stability of some methods for inverting the Laplace transform

The problem of inversion of the integral Laplace transform, which belongs to the class of ill-posed problems, is considered. Integral equations are reduced to ill-conditioned systems of linear algebraic equations, in which the unknowns are either the coefficients of the series expansion in terms of special functions, or the approximate values of the desired original at a number of points. Various inversion methods are considered and their characteristics of accuracy and stability are indicated, which must be known when choosing an inversion method for solving applied problems. Quadrature inversion formulas are constructed, which are adapted for the inversion of long-term and slow processes of linear viscoelasticity. A method of deformation of the integration contour in the Riemann-Mellin inversion formula is proposed, which leads the problem to the calculation of certain integrals and allows obtaining error estimates.

Efficient approximate algorithms for a class of dynamic lot size problems under product substitution

Efficient approximate algorithms for a class of dynamic lot size problems under product substitution

On existence results for a class of biharmonic elliptic problems without (AR) condition

&lt;abstract&gt;&lt;p&gt;In this paper, we study the following biharmonic elliptic equation in $ \mathbb{R}^{N} $:&lt;/p&gt; &lt;p&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ \Delta^{2}\psi-\Delta \psi+P(x)\psi = g(x, \psi), \ \ x\in\mathbb{R}^{N}, $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt; &lt;p&gt;where $ g $ and $ P $ are periodic in $ x_{1}, \cdots, x_{N} $, $ g(x, \psi) $ is subcritical and odd in $ \psi $. Without assuming the Ambrosetti-Rabinowitz condition, we prove the existence of infinitely many geometrically distinct solutions for this equation, and the existence of ground state solutions is established as well.&lt;/p&gt;&lt;/abstract&gt;

On a class of Lyapunov's inequality involving $ \lambda $-Hilfer Hadamard fractional derivative

&lt;abstract&gt;&lt;p&gt;In this paper, we presented and proved a general Lyapunov's inequality for a class of fractional boundary problems (FBPs) involving a new fractional derivative, named $ \lambda $-Hilfer. We proved a criterion of existence which extended that of Lyapunov concerning the ordinary case. We used this criterion to solve the fractional differential equation (FDE) subject to the Dirichlet boundary conditions. In order to do so, we invoked some properties and essential results of $ \lambda $-Hilfer fractional boundary value problem (HFBVP). This result also retrieved all previous Lyapunov-type inequalities for different types of boundary conditions as mixed. The order that we considered here only focused on $ 1 &amp;lt; r\leq 2 $. General Hartman-Wintner-type inequalities were also investigated. We presented an example in order to provide an application of this result.&lt;/p&gt;&lt;/abstract&gt;

Traffic network analysis via multidimensional split variational inequality problem with multiple output sets

&lt;abstract&gt;&lt;p&gt;In this study, we introduce and investigate a new class of split inverse problems, comprising a multidimensional parameter of evolution, which we call the multidimensional split variational inequality problem with multiple output sets. To demonstrate its applicability, we formulate the equilibrium flow of multidimensional traffic network models for an arbitrary number of locations. We define a multidimensional split Wardrop condition with multiple output sets and establish its equivalence with the formulated equilibrium flow of multidimensional traffic network models. We then establish the existence and uniqueness of equilibria for our proposed model. In addition, we propose a method for solving the introduced problem. We then validate our results using some numerical experiments.&lt;/p&gt;&lt;/abstract&gt;

Global optimization algorithm for a class of linear ratios optimization problem

&lt;abstract&gt;&lt;p&gt;We presented an image space branch-and-bound algorithm for globally minimizing the sum of linear ratios problem. In the algorithm, a new linearizing technique was proposed for deriving the linear relaxation problem. An image space region reduction technique was constructed for improving the convergence speed of the algorithm. Moreover, by analyzing the computational complexity of the algorithm, the maximum iterations of the algorithm were estimated, and numerical experimental results showed the potential computing benefits of the algorithm. Finally, a practical application problem in education investment was solved to verify the usefulness of the proposed algorithm.&lt;/p&gt;&lt;/abstract&gt;

Convergence properties of stochastic proximal subgradient method in solving a class of composite optimization problems with cardinality regularizer

Convergence properties of stochastic proximal subgradient method in solving a class of composite optimization problems with cardinality regularizer

On determining and breaking the gauge class in inverse problems for reaction-diffusion equations

Abstract We investigate an inverse boundary value problem of determination of a nonlinear law for reaction-diffusion processes, which are modeled by general form semilinear parabolic equations. We do not assume that any solutions to these equations are known a priori, in which case the problem has a well-known gauge symmetry. We determine, under additional assumptions, the semilinear term up to this symmetry in a time-dependent anisotropic case modeled on Riemannian manifolds, and for partial data measurements on ${\mathbb R}^n$ . Moreover, we present cases where it is possible to exploit the nonlinear interaction to break the gauge symmetry. This leads to full determination results of the nonlinear term. As an application, we show that it is possible to give a full resolution to classes of inverse source problems of determining a source term and nonlinear terms simultaneously. This is in strict contrast to inverse source problems for corresponding linear equations, which always have the gauge symmetry. We also consider a Carleman estimate with boundary terms based on intrinsic properties of parabolic equations.

HYPER-DIFFERENTIAL SENSITIVITY ANALYSIS FOR NONLINEAR BAYESIAN INVERSE PROBLEMS

We consider hyper-differential sensitivity analysis (HDSA) of nonlinear Bayesian inverse problems governed by partial differential equations (PDEs) with infinite-dimensional parameters. In previous works, HDSA has been used to assess the sensitivity of the solution of deterministic inverse problems to additional model uncertainties and also different types of measurement data. In the present work, we extend HDSA to the class of Bayesian inverse problems governed by PDEs. The focus is on assessing the sensitivity of certain key quantities derived from the posterior distribution. Specifically, we focus on analyzing the sensitivity of the MAP point and the Bayes risk and make full use of the information embedded in the Bayesian inverse problem. After establishing our mathematical framework for HDSA of Bayesian inverse problems, we present a detailed computational approach for computing the proposed HDSA indices. We examine the effectiveness of the proposed approach on an inverse problem governed by a PDE modeling heat conduction.

Stochastic maximum principle for a time-changed mean field game

This work studies a class of time-changed mean field game problems by the stochastic maximum principle, and establishes a duality principle for the derived forward backward stochastic differential equation. As an application, we provide an explicit solution for the mean of the subdiffusion under the framework of the linear-quadratic structure.