Approximation Scheme Research Articles

Novel adaptive control approach to fractal fractional order deforestation model and its impact on soil erosion.

Deforestation exerts profound ramifications on soil quality and biodiversity, thereby exerting substantial economic repercussions. The depletion of organic matter and structural integrity of soil following tree removal for agricultural purposes underscores the severity of this issue. In elucidating the soil pollution stemming from deforestation, this research employs a sophisticated five-compartment SDIFR model integrating fractal dimension and fractional order dynamics. The rigorous analysis, including the application of Picard Lindelof's fixed point theorem, establishes the existence and uniqueness of explicit solutions. Furthermore, the examination of local and global stability sheds light on the system's behavior, delineating between pollution-free equilibrium and pollution-extinct equilibrium states. To regulate system behavior, an adaptive control framework grounded in fractal fractional order is proposed, leveraging the Adams-Bashforth numerical approximation scheme for implementation. Through numerical simulations, the study underscores the pivotal role of parameters, thus substantiating the significance of the proposed model in comprehensively addressing the complexities of soil pollution induced by deforestation.

ANALYSIS OF OPTIMAL PORTFOLIO ON FINITE AND SMALL-TIME HORIZONS FOR A STOCHASTIC VOLATILITY MODEL WITH MULTIPLE CORRELATED ASSETS

In this paper, we consider the portfolio optimization problem in a financial market where the underlying stochastic volatility model is driven by [Formula: see text]-dimensional Brownian motions. At first, we derive a Hamilton–Jacobi–Bellman equation including the correlations among the standard Brownian motions. We use an approximation method for the optimization of portfolios. With such approximation, the value function is analyzed using the first-order terms of expansion of the utility function in the powers of time to the horizon. The error of this approximation is controlled using the second-order terms of expansion of the utility function. It is also shown that the one-dimensional version of this analysis corresponds to a known result in the literature. We also generate a close-to-optimal portfolio near the time to horizon using the first-order approximation of the utility function. It is shown that the error is controlled by the square of the time to the horizon. Finally, we provide an approximation scheme to the value function for all times and generate a close-to-optimal portfolio.

Parameter identification of shallow water waves using the generalized equal width equation and physics-informed neural networks: a conservative approximation scheme

Parameter identification of shallow water waves using the generalized equal width equation and physics-informed neural networks: a conservative approximation scheme

Fixed-Ratio Approximation Algorithm for the Minimum Cost Cover of a Digraph by Bounded Number of Cycles

We consider polynomial time approximation for the minimum cost cycle cover problem of an edge-weighted digraph, where feasible covers are restricted to have at most k disjoint cycles. In the literature this problem is referred to as Min-k-SCCP. The problem is closely related to classic Traveling Salesman Problem (TSP) and Vehicle Routing Problem (VRP) and has many important applications in algorithms design and operations research. Unlike its unconstrained variant, the Min-k-SCCP is strongly NP-hard even on undirected graphs and remains intractable in very specific settings. For any metric, the problem can be approximated in polynomial time within ratio 2, while in fixed-dimensional Euclidean spaces it admits Polynomial Time Approximation Schemes (PTAS). In the same time, approximation of the more general asymmetric Min-k-SCCP still remains weakly studied. In this paper, we propose the first fixed-ratio approximation algorithm for this problem, which extends the recent breakthrough Svensson-Tarnawski-Vegh and Traub-Vygen results for the Asymmetric Traveling Salesman Problem.

Slope of the Delocalization Function Is Proportional to Analytical Hardness.

Conceptual Density Functional Theory (CDFT) has been extended beyond its traditional role in elucidating chemical reactivity to the development of density functional theory methods, e.g., the investigation of the delocalization error. This delocalization error causes the dependence of the energy on the number of electrons (N) to deviate from its exact piecewise linear behavior, an error which is the basis of many well-known limitations of commonly used density-functional approximations (DFAs). Following our previous work on the analytical hardness η± for pure functionals, we extend its application to hybrid and range-separated functionals. A comparison is made between the analytical hardness and the slope of the delocalization function introduced by Hait and Head-Gordon. Our results show that there is a linear relationship between its slope and the analytical hardness. An approximate scheme is presented to construct the energy vs N curve without fractional occupation number calculations. The extension to densities is discussed.

Deterministic fault-tolerant connectivity labeling scheme

AbstractThe f-fault-tolerant connectivity labeling (f-FTC labeling) is a scheme of assigning each vertex and edge with a small-size label such that one can determine the connectivity of two vertices s and t under the presence of at most f faulty edges only from the labels of s, t, and the faulty edges. This paper presents a new deterministic f-FTC labeling scheme attaining $$O(f^2 \textrm{polylog}(n))$$ O ( f 2 polylog ( n ) ) -bit label size and a polynomial construction time, which settles the open problem left by Dory and Parter (in: Proceedings of the 2021 ACM symposium on principles of distributed computing (PODC), pp 445–455, 2021). The key ingredient of our construction is to develop a deterministic counterpart of the graph sketch technique by Ahn et al. (in: Proceedings of the 31st ACM SIGMOD-SIGACT-SIGAI symposium on principles of database systems (PODS), pp 5–14, 2012), via some natural connection with the theory of error-correcting codes. This technique removes one major obstacle in de-randomizing the Dory–Parter scheme. The whole scheme is obtained by combining this technique with a new deterministic graph sparsification algorithm derived from the seminal $$\epsilon $$ ϵ -net theory, which is also of independent interest. As byproducts, our result deduces the first deterministic fault-tolerant approximate distance labeling scheme with a non-trivial performance guarantee and an improved deterministic fault-tolerant compact routing. The authors believe that our new technique is potentially useful in the future exploration of more efficient FTC labeling schemes and other related applications based on graph sketches.

Ground state solutions for the Hamilton–Choquard elliptic system with critical exponential growth

In this paper, we study the following Hamilton–Choquard type elliptic system: − Δ u + u = ( I α ∗ F ( v ) ) f ( v ) , x ∈ R 2 , − Δ v + v = ( I β ∗ F ( u ) ) f ( u ) , x ∈ R 2 , where I α and I β are Riesz potentials, f : R → R possessing critical exponential growth at infinity and F ( t ) = ∫ 0 t f ( s ) d s. Without the classic Ambrosetti–Rabinowitz condition and strictly monotonic condition on f, we will investigate the existence of ground state solution for the above system. The strongly indefinite characteristic of the system, combined with the convolution terms and critical exponential growth, makes such problem interesting and challenging to study. With the help of a proper auxiliary system, we employ an approximation scheme and the non-Nehari manifold method to control the minimax levels by a fine threshold, and succeed in restoring the compactness for the critical problem. Existence of a ground state solution is finally established by the concentration compactness argument and some detailed estimates.

Science in the Language of the Heart: Ukrainian Language and Literature in the New Ukrainiian School as a Driver of STEM Development

The article outlines the pathways for integrating STEM disciplines into the planning of Ukrainian language and literature lessons in secondary education institutions within the framework of the New Ukrainian School. It substantiates the growing need for interdisciplinary approaches that foster the development of core competencies while considering the advantages, potential risks, and limitations of the proposed methods. The emphasis is placed on the importance of combining humanities with natural and mathematical sciences to enhance students' understanding of a holistic worldview, illustrating the need to avoid focusing solely on one aspect of development (so-called «humanitarians» and «technocrats»). The article offers theoretical and practical recommendations for developing STEM projects that engage students in creative literary exploration through applying scientific principles and doctrines. The effective combination of STEM education principles with the field of philology enables the creation of a dynamic and effective educational environment where students perceive the interconnectedness of knowledge, develop a well-rounded intellect, and receive foundational preparation for higher education. Methods of linguistic analysis of scientific texts are discussed, demonstrating how language skills can be enhanced through scientific discourse. To illustrate the material, approximate schemes for planning integrated lessons with specific examples from both linguistic-literary and scientific perspectives are provided. To improve understanding and engagement of modern students, emphasis is placed on using digital tools and platforms that ensure integration, including online resources, interactive modeling, multimedia content and artificial intelligence. The necessity for alignment among educators of various disciplines is evident to ensure a fair assessment of proficiency in the state language and scientific literacy, which is a crucial issue during wartime when language matters become increasingly prominent, alongside scientific and mathematical literacy, for building an intellectually developed country.

Barrier-critic-disturbance approximate optimal control of nonzero-sum differential games for modular robot manipulators

In this paper, for addressing the safe control problem of modular robot manipulators (MRMs) system with uncertain disturbances, an approximate optimal control scheme of nonzero-sum (NZS) differential games is proposed based on the control barrier function (CBF). The dynamic model of the manipulator system integrates joint subsystems through the utilization of joint torque feedback (JTF) technique, incorporating interconnected dynamic coupling (IDC) effects. By integrating the cost functions relevant to each player with the CBF, the evolution of system states is ensured to remain within the safe region. Subsequently, the optimal tracking control problem for the MRM system is reformulated as an NZS game involving multiple joint subsystems. Based on the adaptive dynamic programming (ADP) algorithm, a cost function approximator for solving Hamilton–Jacobi (HJ) equation using only critic neural networks (NN) is established, which promotes the feasible derivation of the approximate optimal control strategy. The Lyapunov theory is utilized to demonstrate that the tracking error is uniformly ultimately bounded (UUB). Utilizing the CBF’s state constraint mechanism prevents the robot from deviating from the safe region, and the application of the NZS game approach ensures that the subsystems of the MRM reach a Nash equilibrium. The proposed control method effectively addresses the problem of safe and approximate optimal control of MRM system under uncertain disturbances. Finally, the effectiveness and superiority of the proposed method are verified through simulations and experiments.

An optimal ansatz space for moving least squares approximation on spheres

We revisit the moving least squares (MLS) approximation scheme on the sphere Sd-1⊂Rd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb S^{d-1} \\subset {\\mathbb R}^d$$\\end{document}, where d>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d>1$$\\end{document}. It is well known that using the spherical harmonics up to degree L∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L \\in {\\mathbb N}$$\\end{document} as ansatz space yields for functions in CL+1(Sd-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {C}^{L+1}(\\mathbb S^{d-1})$$\\end{document} the approximation order OhL+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}\\left( h^{L+1} \\right) $$\\end{document}, where h denotes the fill distance of the sampling nodes. In this paper, we show that the dimension of the ansatz space can be almost halved, by including only spherical harmonics of even or odd degrees up to L, while preserving the same order of approximation. Numerical experiments indicate that using the reduced ansatz space is essential to ensure the numerical stability of the MLS approximation scheme as h→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h \\rightarrow 0$$\\end{document}. Finally, we compare our approach with an MLS approximation scheme that uses polynomials on the tangent space of the sphere as ansatz space.

An Improved Scheme for the Finite Difference Approximation of the Advective Term in the Heat or Solute Transport Equations

AbstractTransport equations are widely used to describe the evolution of scalar quantities subject to advection, dispersion and, possibly, reactions. Numerical methods are required to solve these equations in applications, adopting either the advective or conservative formulations. Conservative formulations are usually preferred in practice because they conserve mass. Advective formulations do not, but have received more mathematical attention and are required for Lagrangian solution methods. To obtain an advective formulation that conserves mass, we subtract the discretized fluid flow equation, multiplied by concentration, from the conservative form of the transport equation. The resulting scheme not only conserves mass, but is also elegant in that it can be interpreted as averaging the advective term at cell interfaces, instead of approximating it at cell centers as in traditional centered schemes. The two schemes are identical when fluid velocity is constant, and both have second-order convergence, but the truncation errors are slightly different. We argue that the error terms appearing in the proposed scheme actually imply an improved representation of subgrid spreading/contraction and acceleration/deceleration caused by variable velocity. We compare the proposed and traditional schemes on several problems with variable velocity caused by recharge, discharge or evaporation, including two newly developed analytical solutions. The proposed method yields results that are slightly, but consistently, better than the traditional scheme, while always conserving mass (i.e., mass at the end equals mass at the beginning plus inputs minus outputs), which the traditional centered finite differences scheme does not. We conclude that this scheme should be preferred in finite difference solutions of transport.

Deciphering competing interactions of Kitaev-Heisenberg-Γ system in clusters: II. Dynamics of Majorana fermions

We perform a systematic and exact study of Majorana fermion dynamics in the Kitaev-Heisenberg-Γ model in a few finite-size clusters increasing in size up to twelve sites. We employ exact Jordan-Wigner transformations to evaluate certain measures of Majorana fermion correlation functions, which effectively capture matter and gauge Majorana fermion dynamics in different parameter regimes. An external magnetic field is shown to produce a profound effect on gauge fermion dynamics. Depending on certain non-zero choices of other non-Kitaev interactions, it can stabilise it to its non-interacting Kitaev limit. For all the parameter regimes, gauge fermions are seen to have slower dynamics, which could help build approximate decoupling schemes for appropriate mean-field theory. The probability of Majorana fermions returning to their original starting site shows that the Kitaev model in small clusters can be used as a test bed for the quantum speed limit.

Tail index estimation for tail adversarial stable time series with an application to high‐dimensional tail clustering

For stationary time series with regularly varying marginal distributions, an important problem is to estimate the associated tail index which characterizes the power‐law behavior of the tail distribution. For this, various results have been developed for independent data and certain types of dependent data. In this article, we consider the problem of tail index estimation under a recently proposed notion of serial tail dependence called the tail adversarial stability. Using the technique of adversarial innovation coupling and a martingale approximation scheme, we establish the consistency and central limit theorem of the tail index estimator for a general class of tail dependent time series. Based on the asymptotic normal distribution from the obtained central limit theorem, we further consider an application to cluster a large number of regularly varying time series based on their tail indices by using a robust mixture algorithm. The results are illustrated using numerical examples including Monte Carlo simulations and a real data analysis.

Exponential series approximation of the SIR epidemiological model

IntroductionThe SIR (Susceptible-Infected-Recovered) model is one of the simplest and most widely used frameworks for understanding epidemic outbreaks.MethodsA second-order dynamical system for the R variable is formulated using an infinite exponential series expansion, and a recursion relation is established between the series coefficients. A numerical approximation scheme for the R variable is also developed.ResultsThe proposed numerical method is compared to a double exponential (DE) nonlinear approximate analytic solution, which reveals two coupled timescales: a relaxation timescale, determined by the ratio of the model’s time constants, and an excitation timescale, dictated by the population size. The DE solution is applied to estimate model parameters for a well-known epidemiological dataset—the boarding school flu outbreak.DiscussionFrom a theoretical standpoint, the primary contribution of this work is the derivation of an infinite exponential, Dirichlet, series for the model variables. Truncating the series yields a finite approximation, known as a Prony series, which can be interpreted as a sequence of coupled exponential relaxation processes, each with a distinct timescale. This apparent complexity can be approximated well by the DE solution, which appears to be of main practical interest.

Near-Optimal Adaptive Policies for Serving Stochastically Departing Customers

In, “Near-Optimal Adaptive Policies for Serving Stochastically Departing Customers,” Segev considers a multistage stochastic optimization problem originally introduced by Cygan et al. [Cygan M, Englert M, Gupta A, Mucha M, Sankowski P (2013) Catch them if you can: How to serve impatient users. Proc. 4th Innovations Theoretical Comput. Sci. Conf., 485–494], studying how a single server should prioritize stochastically departing customers. In this setting, the objective is to determine an adaptive service policy that maximizes the expected total reward collected along a discrete planning horizon, in the presence of customers who are independently departing between one stage and the next with known stationary probabilities. The paper’s main contribution resides in proposing a quasi-polynomial-time approximation scheme for serving impatient customers. Specifically, letting n be the number of underlying customers, our algorithm identifies in [Formula: see text] time a service policy whose expected reward is within factor [Formula: see text] of the optimal adaptive reward. The method for deriving this approximation scheme synthesizes various stochastic analyses in order to investigate how the adaptive optimum is affected by alterations to several instance parameters, including the reward values, the departure probabilities, and the collection of customers itself.

Packing squares independently

Given a set of squares and a strip with bounded width and infinite height, we consider a square strip packaging problem, which we call the square independent packing problem (SIPP), to minimize the strip height so that all the squares are packed into independent cells separated by horizontal and vertical partitions. For the SIPP, we first investigate efficient solution representations and propose a compact representation that reduces the search space from Ω(n!) to O(2n), with n the number of given squares, while guaranteeing that there exists a solution representation that corresponds to an optimal solution. Based on the solution representation, we show that the problem is NP-hard. To solve the SIPP, we propose a dynamic programming method that can be extended to a fully polynomial-time approximation scheme (FPTAS). We also propose three mathematical programming formulations based on different solution representations and confirm their performance through computational experiments with a mathematical programming solver. Finally, we discuss several extensions that are relevant to practical applications.

Overcoming the curse of dimensionality in the numerical approximation of high-dimensional semilinear elliptic partial differential equations

Recently, so-called full-history recursive multilevel Picard (MLP) approximation schemes have been introduced and shown to overcome the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations (PDEs) with Lipschitz nonlinearities. The key contribution of this article is to introduce and analyze a new variant of MLP approximation schemes for certain semilinear elliptic PDEs with Lipschitz nonlinearities and to prove that the proposed approximation schemes overcome the curse of dimensionality in the numerical approximation of such semilinear elliptic PDEs.

ANALYSIS OF THE REGULATORY AND LEGAL BASE FOR THE FORMATION OF GENERAL REQUIREMENTS FOR TEST SLOTS FOR THE CHECKING OF DEMINING MACHINES (HUMANITARIAN) BY FUNCTIONAL PURPOSE

The relevance of the chosen topic is determined primarily by the need to organize testing and certification activities, the introduction of international demining standards and NATO standards in humanitarian demining. To assess the suitability of demining machines under the same controlled conditions. This makes it possible to evaluate products under the same conditions and determine the most productive samples. This actualizes the need to define general requirements for test sites for testing mechanized means of humanitarian demining by functional purpose. It is worth improving approaches to the organization and functioning of test sites based on the existing international experience and new domestic developments. In the course of the conducted research, it was established that there is an urgent need to organize demining of territories in Ukraine, both for military tasks and for humanitarian demining tasks. It has been established that the lack of general requirements for test sites for conducting tests and assessing the conformity of demining means approved as a normative legal act is an urgent problem that needs immediate resolution. During the research, a classification of demining tools was carried out, which are divided into machines for detonation, destruction, removal of mines; machines for soil preparation; machines for detecting sources of danger, have different types of working equipment (excavator or bulldozer type) and different types of attached equipment: roller trawls, milling-rotor trawls, knife trawls, chain trawls, fighting trawls, mulchers for soil treatment. In accordance with the set goal, the article developed recommendations for determining general requirements for test areas of demining machines equipment, substantiated the quantitative and qualitative indicators of objects and elements of test areas for conducting certification tests of mechanical (humanitarian) demining equipment according to their functional purpose, which are based on the practices of conducting such works in Ukraine and other countries. The necessity of forming requirements for test areas for assessing the survivability and productivity of demining means is substantiated. In particular, with the definition of requirements for test sites and related infrastructure. Recommendations are given for the placement of the main and additional elements on the test plots. Recommendations were provided and an approximate scheme for the placement of the main objects and elements of test areas for the organization of certification tests was provided, which will allow to ensure the performance of testing and certification tasks at the level established by state and international standards. In general, the substantiation of the requirements for test sites and the adoption of general requirements for test sites at the level of a normative document will allow normalizing the processes of testing samples of mechanized demining equipment and will allow the implementation of mine action tasks in Ukraine at a new, organizationally and practically improved level.

Algorithms for the thief orienteering problem on directed acyclic graphs

We consider the scenario of routing an agent called a thief through a weighted graph G=(V,E) from a start vertex s to an end vertex t. A set I of items each with weight wi and profit pi is distributed among V∖{s,t}. The thief, who has a knapsack of capacity W, must follow a simple path from s to t within a given time T while packing in the knapsack a set of items taken from the vertices along the path of total weight at most W and maximum profit. The travel time across an edge depends on the edge length and current knapsack load.The thief orienteering problem (ThOP) is a generalization of the orienteering problem, the longest path problem, and the 0-1 knapsack problem. We prove that there exists no approximation algorithm for ThOP with constant approximation ratio unless P=NP, and we present a polynomial-time approximation scheme (PTAS) for a relaxed version of ThOP when G is directed and acyclic that produces solutions that use time at most T(1+ϵ) for any constant ϵ>0. We also present a fully polynomial-time approximation scheme (FPTAS) for ThOP on arbitrary undirected graphs where the travel time depends only on the lengths of the edges and T is the length of a shortest path from s to t plus a constant K. Finally, we present a FPTAS for a restricted version of the problem where the input graph is a clique.

Two-phase flows through porous media described by a Cahn–Hilliard–Brinkman model with dynamic boundary conditions

We investigate a new diffuse-interface model that describes creeping two-phase flows (i.e., flows exhibiting a low Reynolds number), especially flows that permeate a porous medium. The system of equations consists of a Brinkman equation for the volume averaged velocity field and a convective Cahn–Hilliard equation with dynamic boundary conditions for the phase field, which describes the location of the two fluids within the domain. The dynamic boundary conditions are incorporated to model the interaction of the fluids with the wall of the container more precisely. In particular, they allow for a dynamic evolution of the contact angle between the interface separating the fluids and the boundary, and for a convection-induced motion of the corresponding contact line. For our model, we first prove the existence of global-in-time weak solutions in the case where regular potentials are used in the Cahn–Hilliard subsystem. In this case, we can further show the uniqueness of the weak solution under suitable additional assumptions. We further prove the existence of weak solutions in the case of singular potentials. Therefore, we regularize such singular potentials by a Moreau–Yosida approximation, such that the results for regular potentials can be applied, and eventually pass to the limit in this approximation scheme.