Problem decomposition in distributed problem-solving systems

Distributed Problem Solving (DPS) is defined as the cooperative solution of problems by a decentralized and loosely coupled collection of problem solvers (agents), each of them knowing how to execute only some of the necessary tasks. This approach considers the problem-solving process as occurring in three phases: problem decomposition, subproblem solution, and answer synthesis. In the problem decomposition phase, one has to determine which tasks will be executed by each agent and when. One of the key research questions in the problem decomposition process is how to decompose a problem in order to minimize the cost of resources needed for its solution. In this article, we construct mathematical programming models in order to describe the decomposition process under the above criterion, study its complexity, and present exact and heuristic algorithms for its solution. Our work was motivated by the operation of an actual system that can be considered as a distributed problem solver for the assessment of irrigation projects design.     

基于细分曲面的三维服装柔性实体模拟

提出一种基于细分曲面的三维服装柔性实体模拟算法，该算法将整个模拟过程分为两个阶段：首先利用四点细分曲面造型方法生成三维服装刚性曲面，然后在刚性曲面基础上通过引入织物的物理模型来模拟三维服装柔性曲面，通过物理和几何模拟方法有机结合，算法有效解决了复杂衣片间的缝合问题，较大地提高了模拟的计算效率，同时，也提出了一种基于细分曲面层次数据结构的碰撞检测算法，有效提高了模拟速度，提出的算法已全部在所开发的三维虚拟服装试衣系统中得以实现，实验结果表明：该算法具有模拟效率高、交互性强和易于计算机实现等优点。     

有约束竞争选址问题的降阶回溯算法

有约束竞争选址问题是组合优化中一个经典的NP-hard问题,现有算法研究该问题时或是无法求得最优解或是求解速度慢.针对现有算法的缺点,首先在这个经典问题的基础上进行修改,构建了一个新的数学模型;接着对该模型的数学性质进行研究,并在数学性质的基础上提出了上下界算法和降阶子算法对问题进行降阶,达到了缩减问题搜索解空间的目的,降阶的过程中既有单个的降阶,也有成批的降阶;然后在前面的基础上设计了一个回溯子算法来求解问题的最优解;最后通过两个示例分析更清楚地阐述该算法的原理,结果证明该算法可以较快求得最优解.     

Solution validation technique for optimal shakedown design problems

Modern computer technology allows conjoining shakedown theory, optimization and ever stricter standardized design requirements in a single mathematical problem formulation. However it raises a question of reliability: easily achieved solution should not be taken for granted but should be adequately assessed. This paper focuses on the physical validation technique for optimal shakedown design problem solution in the aspect of Melan theorem (statics) and residual deformation compatibility (kinematics). For that purpose Rosen gradient projection method is used. Optimization problem of bending circular, symmetric plate at shakedown, which is subjected by a variable repeated load, is considered for illustration of the validation technique.     

带法向约束的圆平均非线性细分曲线设计

目的 对采样设备获取的测量数据进行拟合,可实现原模型的重建及功能恢复。但有些情况下,获取的数据点不仅包含位置信息,还包含法向量信息。针对这一问题,本文提出了基于圆平均的双参数4点binary非线性细分法与单参数3点ternary插值非线性细分法。方法 首先将线性细分法改写为点的重复binary线性平均,然后用圆平均代替相应的线性平均,最后用加权测地线平均计算的法向量作为新插入顶点的法向量。基于圆平均的双参数4点binary细分法的每一次细分过程可分为偏移步与张力步。基于圆平均的单参数3点ternary细分法的每一次细分过程可分为左插步、插值步与右插步。结果 对于本文方法的收敛性与C¹连续性条件给出了理论证明;数值实验表明,与相应的线性细分相比,本文方法生成的曲线更光滑且具有圆的再生力,可以较好地实现3个封闭曲线重建。结论 本文方法可以在带法向量的初始控制顶点较少的情况下,较好地实现带法向约束的离散点集的曲线重建问题。     

Subdivision Schemes for Thin Plate Splines

Thin plate splines are a well known entity of geometric design. They are defined as the minimizer of a variational problem whose differential operators approximate a simple notion of bending energy. Therefore, thin plate splines approximate surfaces with minimal bending energy and they are widely considered as the standard "fair" surface model. Such surfaces are desired for many modeling and design applications.
Traditionally, the way to construct such surfaces is to solve the associated variational problem using finite elements or by using analytic solutions based on radial basis functions. This paper presents a novel approach for defining and computing thin plate splines using subdivision methods. We present two methods for the construction of thin plate splines based on subdivision: A globally supported subdivision scheme which exactly minimizes the energy functional as well as a family of strictly local subdivision schemes which only utilize a small, finite number of distinct subdivision rules and approximately solve the variational problem. A tradeoff between the accuracy of the approximation and the locality of the subdivision scheme is used to pick a particular member of this family of subdivision schemes.
Later, we show applications of these approximating subdivision schemes to scattered data interpolation and the design of fair surfaces. In particular we suggest an efficient methodology for finding control points for the local subdivision scheme that will lead to an interpolating limit surface and demonstrate how the schemes can be used for the effective and efficient design of fair surfaces.     

Solution of the laminated plate discrete design problem

The design problem of laminated plate possessing the required stiffnesses is analyzed under the condition that a given finite set of materials may be used to manufacture the plate. The mathematical formulation of the problem represents the convex combinations problem with the condition that the solution belongs to a finite set. This problem is called a discrete convex combinations problem. The algorithm for constructing the general solution (the set of all possible solutions) of the problem is proposed. The algorithm can be used for a numerical solution of the problem.     

A systematic knowledge-based method for design of transformable product

A transformable product can perform different functions or change functionality by changing its physical structure. It is formed by integration of different components whose states can be transformed each other. However, there is a lack of systematic methods to guide design of the transformable product. In order to improve the design efficiency of transformable products, a large number of products are studied in this research to build a case base of transformation parameters and transformation principles for the design knowledge. A systematic design process is proposed to apply the design knowledge. The transformation design problem is first mapped from the problem domain to the knowledge domain expressed by transformation parameters. A general solution is then obtained in the knowledge domain. A multi-classification support vector machine is used to train a model of the transformation recommendation based on transformation parameters. Finally, the general solution is mapped into the problem domain for the specific solution using the analogy design. The effectiveness of the method is verified in the design of a self-propelled boom sprayer.     

n-Dimensional multiresolution representation of subdivision meshes with arbitrary topology

We present a new model for the representation of n-dimensional multiresolution meshes. It provides a robust topological representation of arbitrary meshes that are combined in closely interlinked levels of resolution. The proposed combinatorial model is formalized through the mathematical model of combinatorial maps allowing us to give a general formulation, in any dimensions, of the topological subdivision process that is a key issue to robustly and soundly define mesh hierarchies. It fully supports multiresolution edition what allows the implementation of most mesh processing algorithms – like filtering or compression – for n-dimensional meshes with arbitrary topologies.We illustrate this model, in dimension 3, with an new truly multiresolution representation of subdivision volumes. It allows us to extend classical subdivision schemes to arbitrary polyhedrons and to handle adaptive subdivision with an elegant solution to compliance issues. We propose an implementation of this model as an effective and relatively inexpensive data structure.     

Hash functions and triangular mesh reconstruction

Some applications use data formats (e.g. STL file format), where a set of triangles is used to represent the surface of a 3D object and it is necessary to reconstruct the triangular mesh with adjacency information. It is a lengthy process for large data sets as the time complexity of this process is O(N log N), where N is number of triangles. Triangular mesh reconstruction is a general problem and relevant algorithms can be used in GIS and DTM systems as well as in CAD/CAM systems. Many algorithms rely on space subdivision techniques while hash functions offer a more effective solution to the reconstruction problem. Hash data structures are widely used throughout the field of computer science. The hash table can be used to speed up the process of triangular mesh reconstruction but the speed strongly depends on hash function properties. Nevertheless the design or selection of the hash function for data sets with unknown properties is a serious problem. This paper describes a new hash function, presents the properties obtained for large data sets, and discusses validity of the reconstructed surface. Experimental results proved theoretical considerations and advantages of hash function use for mesh reconstruction.     

Construction of minimal subdivision surface with a given boundary

The fascinating characters of minimal surface make it to be widely used in shape design. While the flexibility and high quality of subdivision surface make it a powerful mathematical tool for shape representation. In this paper, we construct minimal subdivision surfaces with given boundaries using the mean curvature flow, a second order geometric partial differential equation. This equation is solved by a finite element method where the finite element space is spanned by the limit functions of an extended Loop’s subdivision scheme proposed by Biermann et al. Using this extended Loop’s subdivision scheme we can treat a surface with boundary, thereby construct the perfect minimal subdivision surfaces with any topology of the control mesh and any shaped boundaries.     

基于推理的组卷数学建模及其应用

针对多目标约束空间优化的组卷问题,采用目标参数降维和推理技术相结合的方法,提出一种推理匹配组卷数学模型,该模型具有动态规则库设计免维护、推理抽题满足用户主动参与组卷等特点,能够提高试卷的实用性,并可有效避免后续遗传算法组卷早熟现象。     

Translational polygon containment and minimal enclosure using mathematical programming

A new algorithm is given for the two-dimensional translational containment problem: find translations for k polygons which place them inside a polygonal container without overlapping. Both the polygons and the container can be non-convex. The algorithm is based on mathematical programming principles. The containment algorithm is generalized to solve minimal enclosure problems: find the minimal area enclosing square or minimal area enclosing rectangle for k translating polygons. The containment algorithm consists of new algorithms for restriction , evaluation , and subdivision of two-dimensional configuration spaces. Restriction establishes lower bounds through relaxation and the solution of linear programs. Evaluation establishes upper bounds by finding potential solutions. Subdivision branches, when necessary, by introducing a cutting plane. The algorithm shows that either the upper bound of the objective (overlap) is exactly zero or the lower bound is greater than zero. Hence, it gives an exact solution to the containment problem. Experiments show that new containment algorithm clearly outperforms purely geometric containment algorithms. For data sets from the apparel industry, it can solve containment for up to ten non-convex polygons in practice. An implementation of the algorithm given in this paper has been licensed by Gerber Garment Technologies, the largest provider of textile CAD/CAM software in the US, and they are incorporating it into an existing CAD/CAM software product.     

Data base design constrained by traffic load estimates

This paper discusses a mathematical model for finding an effective design of a logical data base. It is argued that a logical data base should not only integrate multiple users' views of information. It should also reflect the data base activity of each individual application and be based on performance tradeoffs among the different applications. Our model shows how this activity can be mapped into activity within the global information structure which describes the integrated view of the information. This is an aid for subsequent logical record definition and schema design.

The model is based on knowledge of a large number of input parameters, some of which have greater impact on the final data base solution than others. Sensitivity analysis of the system's behaviour with respect to changes in parameters help focus on the most critical parameters. This reduces the efforts devoted to quantification of parameters, so that default values or very rough estimates are sufficient for the least critical parameters.     

On the selection of secondary indices in relational databases

An important problem in the physical design of databases is the selection of secondary indices. In general, this problem cannot be solved in an optimal way due to the complexity of the selection process. Often use is made of heuristics such as the well-known ADD and DROP algorithms. In this paper it will be shown that frequently used cost functions can be classified as super- or submodular functions. For these functions several mathematical properties have been derived which reduce the complexity of the index selection problem. These properties will be used to develop a tool for physical database design and also give a mathematical foundation for the success of the before-mentioned ADD and DROP algorithms.     

Toward a Meta-Model for Computational Engineering

Geometric modeling and finite element analysis have matured in recent decades. Both methods are used extensively in engineering design. However, the link between geometric modeling, physical modeling and finite element analysis is currently cumbersome, error-prone, and ad-hoc. Topological domain modeling provides the missing link. In this paper, we propose a combined topological modeling and finite element modeling method that allows not only topological modeling, but also promotes geometric and physical modeling, by providing a topological base space for the definition of finite element meshes, fields, and the definition and solution of boundary value problems. We call the method the Constructive Topological Domain Method (CTDM). In this method, Primitive Topological Domains (PTDs), each possessing a natural coordinate space, are combined in multiple n-dimensional Cartesian coordinate spaces, called charts, using generalizations of Boolean set operations, to create Constructed Topological Domains (CTDs) capable of acting as the base spaces of fiber bundles. The charts are glued together to create an atlas, within which the CTD is defined. The fiber of the bundle may describe, in addition to geometry, physical fields like density, stress, and temperature. Finite element meshes may be defined upon each of the PTDs from which the CTD is constructed, enabling the definition and solution of boundary value problems, thus avoiding the difficult and messy problem of creating a single finite element mesh to represent the entire CTD. A modified finite element method, to handle the individually meshed PTDs, is described. The boundary conditions may be specified as analytical or as finite element-based fields upon each of the PTDs. The CTDM appears to be a promising approach to robust mathematical and computational modeling of physical objects. Simple examples are presented. ID="A1"Correspondance and offprint requests to: W. Gerstle, Department of Civil Engineering, University of New Mexico, Albuquerque, NM 87131, USA. E-mail: gerstle@unm.edu     

A mixed integer linear program and tabu search approach for the complementary edge covering problem

In this paper we study a complementary edge covering problem (CECP) as a variant of the total edge covering problem (TECP) which has application in the area of facility location. Unlike TECP, the partial cover of an edge through vertices is allowed in CECP such that in a feasible solution the entire edge will be covered. We propose a new mixed integer linear formulation for the problem. A number of size reduction rules are proposed which speed up getting optimal solution(s). Since the CECP is NP-Hard, a solution method based on tabu search is designed to search for optimal or near-optimal solutions. Computational experiments are carried out to evaluate effectiveness of the proposed mathematical formulation and the modified tabu search algorithm. Results justify that the proposed mathematical model can solves problems with up to 40 vertices and 456 edges optimally. Our computational efforts signify that the proposed tabu search is very effective and can find high quality solutions for larger problems in reasonable amount of time.     

Stochastic robustness metric and its use for static resource allocations

This research investigates the problem of robust static resource allocation for distributed computing systems operating under imposed Quality of Service (QoS) constraints. Often, such systems are expected to function in a physical environment replete with uncertainty, which causes the amount of processing required to fluctuate substantially over time. Determining a resource allocation that accounts for this uncertainty in a way that can provide a probabilistic guarantee that a given level of QoS is achieved is an important research problem. The stochastic robustness metric proposed in this research is based on a mathematical model where the relationship between uncertainty in system parameters and its impact on system performance are described stochastically.The utility of the established metric is then exploited in the design of optimization techniques based on greedy and iterative approaches that address the problem of resource allocation in a large class of distributed systems operating on periodically updated data sets. The performance results are presented for a simulated environment that replicates a heterogeneous cluster-based radar data processing center. A mathematical performance lower bound is presented for comparison analysis of the heuristic results. The lower bound is derived based on a relaxation of the Integer Linear Programming formulation for a given resource allocation problem.