A posteriori error estimates of two-grid finite volume element methods for nonlinear elliptic problems

In this paper, we study the a posteriori error estimates of two-grid finite volume element method for second-order nonlinear elliptic equations. We derive the residual-based a posteriori error estimator and prove the computable upper and lower bounds on the error in $H^1$-norm. The a posteriori error estimator can be used to assess the accuracy of the two-grid finite volume element solutions in practical applications. Numerical examples are provided to illustrate the performance of the proposed estimator.     

A unified framework for the numerical solution of optimal control problems using pseudospectral methods

A unified framework is presented for the numerical solution of optimal control problems using collocation at Legendre-Gauss (LG), Legendre-Gauss-Radau (LGR), and Legendre-Gauss-Lobatto (LGL) points. It is shown that the LG and LGR differentiation matrices are rectangular and full rank whereas the LGL differentiation matrix is square and singular. Consequently, the LG and LGR schemes can be expressed equivalently in either differential or integral form, while the LGL differential and integral forms are not equivalent. Transformations are developed that relate the Lagrange multipliers of the discrete nonlinear programming problem to the costates of the continuous optimal control problem. The LG and LGR discrete costate systems are full rank while the LGL discrete costate system is rank-deficient. The LGL costate approximation is found to have an error that oscillates about the true solution and this error is shown by example to be due to the null space in the LGL discrete costate system. An example is considered to assess the accuracy and features of each collocation scheme.     

Identification methods in a unified framework

The paper derives a framework suitable to discuss the classical Koopmans-Levin (KL) and maximum likelihood (ML) algorithms to estimate parameters of errors-in-variables linear models in a unified way. Using the capability of the unified approach a new parameter estimation algorithm is presented offering flexibility to ensure acceptable variance in the estimated parameters. The developed algorithm is based on the application of Hankel matrices of variable size and can equally be considered as a generalized version of the KL method (GKL) or as a reduced version of the ML estimation. The methodology applied to derive the GKL algorithm is used to present a straightforward derivation of the subspace identification algorithm.     

A unified hyper-heuristic framework for solving bin packing problems

One- and two-dimensional packing and cutting problems occur in many commercial contexts, and it is often important to be able to get good-quality solutions quickly. Fairly simple deterministic heuristics are often used for this purpose, but such heuristics typically find excellent solutions for some problems and only mediocre ones for others. Trying several different heuristics on a problem adds to the cost. This paper describes a hyper-heuristic methodology that can generate a fast, deterministic algorithm capable of producing results comparable to that of using the best problem-specific heuristic, and sometimes even better, but without the cost of trying all the heuristics. The generated algorithm handles both one- and two-dimensional problems, including two-dimensional problems that involve irregular concave polygons. The approach is validated using a large set of 1417 such problems, including a new benchmark set of 480 problems that include concave polygons.     

A unified approach for stability analysis of a class oftime-varying nonlinear systems

It is shown that an integral with negative feedback from its output terminal through a strictly passive system is still strictly passive. By using successive negative feedback around integrals, the conditions of strict passivity and a unified approach for constructing strictly passive systems are obtained for time-varying nonlinear systems of different orders. A new concept of “dissipation factor” is defined in this paper. The stability of a class of time-varying nonlinear systems is examined by using this passivity analysis     

The monotone convergence of a class of parallel nonlinear relaxation methods for nonlinear complementarity problems

We set up a class of parallel nonlinear multisplitting AOR methods by directly multisplitting the nonlinear mapping involved in the nonlinear complementarity problems. The different choices of the relaxation parameters can yield all the known and a lot of new relaxation methods, as well as a lot of new relaxed parallel nonlinear multisplitting methods for solving the nonlinear complementarity problems. The two-sided approximation properties and the influences on the convergence rates from the relaxation parameters about our new methods are shown, and sufficient conditions guaranteeing the methods to converge globally are discussed. Finally, a lot of numerical results show that our new methods are feasible and efficient.     

SubSpace Projection: A unified framework for a class of partition-based dimension reduction techniques

Similarity search in high dimensional space is a nontrivial problem due to the so-called curse of dimensionality. Recent techniques such as Piecewise Aggregate Approximation (PAA), Segmented Means (SMEAN) and Mean-Standard deviation (MS) prove to be very effective in reducing data dimensionality by partitioning dimensions into subsets and extracting aggregate values from each dimension subset. These partition-based techniques have many advantages including very efficient multi-phased approximation while being simple to implement. They, however, are not adaptive to the different characteristics of data in diverse applications.We propose SubSpace Projection (SSP) as a unified framework for these partition-based techniques. SSP projects data onto subspaces and computes a fixed number of salient features with respect to a reference vector. A study of the relationships between query selectivity and the corresponding space partitioning schemes uncovers indicators that can be used to predict the performance of the partitioning configuration. Accordingly, we design a greedy algorithm to efficiently determine a good partitioning of the data dimensions. The results of our extensive experiments indicate that the proposed method consistently outperforms state-of-the-art techniques.     

A full discrete two-grid finite-volume method for a nonlinear parabolic problem

A fully discrete two-grid finite-volume method (FVM) for a nonlinear parabolic problem is studied in this paper. This method involves solving a nonlinear parabolic equation on coarse mesh space and a linearized parabolic equation on fine grid. Both L ² and H ¹ norm error estimates of the standard FVM for the nonlinear parabolic problem are derived. Compared with the standard FVM, the two-level method is of the same order as the one-level method in the H ¹-norm as long as the mesh sizes satisfy h=𝒪(H ^3/2). However, the two-level method involves much less work than the standard method. Numerical results are provided to demonstrate the effectiveness of our algorithm.     

A unified framework for consistency check

Constraint Satisfaction Problem (CSP) involves finding values for variables to satisfy a set of constraints. Consistency check is the key technique in solving this class of problems. Past research has developed many algorithms for such a purpose, e.g., node consistency, are consistency, generalized node and arc consistency, specific methods for checking specific constraints, etc. In this article, an attempt is made to unify these algorithms into a common framework. This framework consists of two parts. the first part is a generic consistency check algorithm, which allows and encourages each individual constraint to be checked by its specific consistency methods. Such an approach provides a direct way of practical implementation of the CSP model for real problem-solving. the second part is a general schema for describing the handling of each type of constraint. the schema characterizes various issues of constraint handling in constraint satisfaction, and provides a common language for expressing, discussing, and exchanging constraint handling techniques. © 1995 John Wiley & Sons, Inc.     

A unified framework for multimodal retrieval

In this paper, a unified framework for multimodal content retrieval is presented. The proposed framework supports retrieval of rich media objects as unified sets of different modalities (image, audio, 3D, video and text) by efficiently combining all monomodal heterogeneous similarities to a global one according to an automatic weighting scheme. Then, a multimodal space is constructed to capture the semantic correlations among multiple modalities. In contrast to existing techniques, the proposed method is also able to handle external multimodal queries, by embedding them to the already constructed multimodal space, following a space mapping procedure of a submanifold analysis. In our experiments with five real multimodal datasets, we show the superiority of the proposed approach against competitive methods.     

A unified framework for heterogeneous patterns

Knowledge patterns, such as association rules, clusters or decision trees, can be defined as concise and relevant information that can be extracted, stored, analyzed, and manipulated by knowledge workers in order to drive and specialize business decision processes. In this paper we deal with data mining patterns. The ability to manipulate different types of patterns under a unified environment is becoming a fundamental issue for any ‘intelligent’ and data-intensive application. However, approaches proposed so far for pattern management usually deal with specific and predefined types of patterns and mainly concern pattern extraction and exchange issues. Issues concerning the integrated, advanced management of heterogeneous patterns are in general not (or marginally) taken into account.     

A unified framework for structure identification

We propose a general framework for structure identification, as defined by Dechter and Pearl. It is based on the notion of prime implicate, and handles Horn, bijunctive and affine, as well as Horn-renamable formulas, for which, to our knowledge, no polynomial algorithm has been proposed before. This framework, although quite general, gives good complexity results, and in particular we get for Horn formulas the same running time and better output size than the algorithms previously known.     

Numerical methods for a class of nonlinear integro-differential equations

In a previous article (Glowinski, J. Math. Anal. Appl. 41, 67–96, 1973) the first author discussed several methods for the numerical solution of nonlinear equations of the integro-differential type with periodic boundary conditions. In this article we discuss an alternative methodology largely based on the Strang’s symmetrized operator-splitting scheme. Several numerical experiments suggest that the new method is robust and accurate. It is also easier to implement than the various methods discussed by Glowinski in J. Math. Anal. Appl. 41, 67–96 (1973).     

A unified derivation of quasi-Newton methods for solving non-sparse and sparse nonlinear equations

An augmented quasi-Newton equation is solved by using theW-V generalized inverse to give a unified derivation of the known quasi-Newton methods for solving systems of nonlinear algebraic equations. This approach makes it possible to get new formulas for sparse and non-sparse systems, and also to determine what norms of the update matrices are minimized when several useful quasi-Newton update formulas are derived.     

A unified framework for managing Web-based services

The emergence of Web technologies enables a variety of Web-based service applications, which can be examined from business process integration, supply chain management, and knowledge management perspectives. To categorize existing Web-based services while foreseeing potential new types, a unified view is needed to represent the structures and processes of Web-based services. This paper proposes a general framework to identify essential structures and operations of Web-based services, and then models these components. We articulate the framework with Web technologies, such as Web service and semantic Web, multi-agent and peer-to-peer, and Web information retrieval and mining. Two comprehensive examples in insurance and knowledge services are used to elaborate the use of Web-based service framework in fulfilling business processes. This study synthesizes essential structures and processes of Web-based services to build a framework for researchers and practitioners to develop Web-based services and techniques.