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We introduce and study the bounded word problem and the precise word problem for groups given by means of generators and defining relations. For example, for every finitely presented group, the bounded word problem is in

This study presents a new analytic approach to optimal coordination of directional overcurrent relays. This approach is based on the selection of optimum pickup current and time dial setting, in order to obtain minimum operating time for the relays, while satisfying various coordination and boundary constraints. Based on the new optimal relay setting procedure, an iterative numerical solution is proposed. The proposed numerical algorithm converges to the global optimum values, which are independent of initial values and the order of relay setting. The proposed method is applied to three different test systems. The new method is compared with some previously proposed analytic and evolutionary approaches. The results demonstrate the advantages of the proposed method over the previous works.

In this paper, the numerical algorithms for solution of pore volume and surface diffusion model of adsorption systems are constructed and investigated. The approximation of PDEs is done by using the finite volume method for space derivatives and ODE15s solvers for numerical integration in time. The analysis of adaptive in time integration algorithms is presented. The main aim of this work is to analyze the sensitivity of the solution with respect to the main parameters of the mathematical model. Such a control analysis is done for a linearized and normalized mathematical model. The obtained results are compared with simulations done for a full nonlinear mathematical model.

This volume corresponds to the Banff International Research Station Workshop on Randomization, Relaxation, and Complexity, held from February 28-March 5, 2010 in Banff, Ontario, Canada. This volume contains a sample of advanced algorithmic techniques underpinning the solution of systems of polynomial equations. The papers are written by leading experts in algorithmic algebraic geometry and touch upon core topics such as homotopy methods for approximating complex solutions, robust floating point methods for clusters of roots, and speed-ups for counting real solutions. Vital related topics such as circuit complexity, random polynomials over local fields, tropical geometry, and the theory of fewnomials, amoebae, and coamoebae are treated as well. Recent advances on Smale's 17th Problem, which deals with numerical algorithms that approximate a single complex solution in average-case polynomial time, are also surveyed.

Purpose The purpose of this paper is to suggest a general numerical algorithm for nonlinear problems by the variational iteration method (VIM). Design/methodology/approach Firstly, the Laplace transform technique is used to reconstruct the variational iteration algorithm-II. Secondly, its convergence is strictly proved. Thirdly, the numerical steps for the algorithm is given. Finally, some examples are given to show the solution process and the effectiveness of the method. Findings No variational theory is needed to construct the numerical algorithm, and the incorporation of the Laplace method into the VIM makes the solution process much simpler. Originality/value A universal iteration formulation is suggested for nonlinear problems. The VIM cleans up the numerical road to differential equations.

In this article, we present simple and robust numerical methods for two-dimensional geometrical shape optimization problems, in the context of viscous flows driven by the stationary Navier-Stokes equations at low Reynolds number. The salient features of our algorithm are exposed with an educational purpose; in particular, the numerical resolution of the nonlinear stationary Navier-Stokes system, the Hadamard boundary variation method for calculating the sensitivity of the minimized function of the domain, and the mesh update strategy are carefully described. Several pedagogical examples are discussed. The corresponding program is written in the FreeFem++ environment, and it is freely available. Its chief features—and notably the implementation details of the main steps of our algorithm—are carefully presented, so that it can easily be handled and elaborated upon to deal with different, or more complex physical situations.

As noted in Wikipedia, skin in the game refers to having ‘incurred risk by being involved in achieving a goal’, where ‘ skin is a synecdoche for the person involved, and game is the metaphor for actions on the field of play under discussion’. For exascale applications under development in the US Department of Energy Exascale Computing Project, nothing could be more apt, with the skin being exascale applications and the game being delivering comprehensive science-based computational applications that effectively exploit exascale high-performance computing technologies to provide breakthrough modelling and simulation and data science solutions. These solutions will yield high-confidence insights and answers to the most critical problems and challenges for the USA in scientific discovery, national security, energy assurance, economic competitiveness and advanced healthcare. This article is part of a discussion meeting issue ‘Numerical algorithms for high-performance computational science’.

We introduce the feature rejection algorithm for meshing (FRAM) to generate a high quality conforming Delaunay triangulation of a three-dimensional discrete fracture network (DFN). The geometric features (fractures, fracture intersections, spaces between fracture intersections, etc.) that must be resolved in a stochastically generated DFN typically span a wide range of spatial scales and make the efficient automated generation of high-quality meshes a challenge. To deal with these challenges, many previous approaches often deformed the DFN to align its features with a mesh through various techniques including redefining lines of intersection as stair step functions and distorting the fracture edges. In contrast, FRAM generates networks on which high-quality meshes occur automatically by constraining the generation of the network. The cornerstone of FRAM is prescribing a minimum length scale and then restricting the generation of the network to only create features of that size and larger. The process is fully automated, meaning no adjustments of the mesh are performed, and the meshing of the individual fractures has been parallelized. Reported mesh statistics show that the computational meshes generated using FRAM are of high quality. Furthermore, the method does not require solving additional systems of linear equations which is needed if a nonconforming mesh is used. Details of the FRAM approach are provided, including its mathematical underpinnings and an algorithm for its implementation. We demonstrate the method's applicability by generating a DFN with similar statistics as the fractured granite at the Forsmark site in Sweden containing 6 700 fractures, and also solve the fully saturated Darcy flow equations to show the quality of the flow solutions which result from computation on the mesh.

The challenging limitations of prohibitive computation and Frequency Principle remain significantly difficult for deep learning methods to effectively resolve multi-scale problems. In this work, a higher-order multi-scale deep Ritz method (HOMS-DRM) is developed to address this issue and effectively compute thermal transfer equation of composite materials with highly oscillatory, discontinuous and high-contrast coefficients. In the computational framework of HOMS-DRM, higher-order multi-scale modeling is first employed to overcome limitations of prohibitive computation and Frequency Principle when direct deep learning simulation. Then, improved deep Ritz method is designed to high-accuracy and mesh-free simulation for lower-order and higher-order microscopic unit cell functions, and macroscopic homogenized equations of multi-scale composites, which are then assembled into higher-order multi-scale solutions for multi-scale thermal transfer problems by using automatic differentiation technology. Besides, corresponding numerical algorithm of HOMS-DRM is developed for implementing high-accuracy multi-scale simulation in periodic composite medium. Moreover, the theoretical convergence of the proposed HOMS-DRM is rigorously demonstrated under appropriate assumptions. Finally, 2D and 3D numerical experiments including high-contrast composite materials are presented to validate the computational performance of HOMS-DRM.

Abstract-Data interpolation is a common challenge in both scientific research and engineering. Multidimensional interpolation, Which encompasses one-dimensional interpolation as a subset, covers a broader range of problems. Notably, interpolation issues in dimensions higher than three can be reduced to a two-dimensional framework through dimensionality reduction, making two-dimensional interpolation a representative paradigm. In this paper, a mathematical model is presented consideration of dimensionality reduction to derive the two-dimensional interpolation polynomial for the tabular function f(x, y). This model is further employed to analyze the interpolation error and determine the remainder term of the polynomial, which is then used to evaluate computational results and perform error estimation. Finally, an engineering example of two-dimensional interpolation is given. While the number of interpolation points is optimized, the proposed algorithm of two-dimensional interpolation yields accurate interpolation outcomes with reduced the complexity of computations.