A unified framework for a posteriori error estimation for the Stokes problem

In this paper, a unified framework for a posteriori error estimation for the Stokes problem is developed. It is based on $[H^1_0(\Omega )]^d$ -conforming velocity reconstruction and $\underline{\varvec{H}}(\mathrm{div},\Omega )$ -conforming, locally conservative flux (stress) reconstruction. It?gives guaranteed, fully computable global upper bounds as well as local lower bounds on the energy error. In order to apply this framework to a given numerical method, two simple conditions need to be checked. We show how to do this for various conforming and conforming stabilized finite element methods, the discontinuous Galerkin method, the Crouzeix–Raviart nonconforming finite element method, the mixed finite element method, and a general class of finite volume methods. The tools developed and used include a new simple equilibration on dual meshes and the solution of local Poisson-type Neumann problems by the mixed finite element method. Numerical experiments illustrate the theoretical developments.     

Distance matrices of a tree: Two more invariants,and in a unified framework

A classical result of Graham and Pollak (1971) states that the determinant of the distance matrix $D_T$ of any tree $T$ depends only on the number of edges of $T$. This and several other variants of $D_T$ have since been studied – including a $q$-version, a multiplicative version, and directed versions of these – and in all cases, $det\left(D_T\right)$ depends only on the edge-data.In this paper, we introduce a more general framework for bi-directed weighted trees that has not been studied to date; our work is significant for three reasons. First, our setting strictly generalizes – and unifies – all variants of $D_T$ studied to date (with coefficients in an arbitrary unital commutative ring) – including in Graham and Pollak (1971) above, as well as Graham and Lovász (1978), Yan and Yeh (2006), Yan and Yeh (2007), Sivasubramanian (2010), and others.Second, our results strictly improve on state-of-the-art for every variant of the distance matrix studied to date, even in the classical Graham–Pollak case. Here are three results for trees: (1) We compute the minors obtained by deleting arbitrary equinumerous sets of pendant nodes (in fact, more general sub-forests) from the rows and columns of $D_T$, and show these minors depend only on the edge-data and not the tree-structure. (2) We compute a second function of the distance matrix $D_T$: the sum of all its cofactors, termed $cof\left(D_T\right)$. We do so in our general setting and in stronger form, after deleting equinumerous pendant nodes (and more generally) as above – and show these quantities also depend only on the edge-data. (3) We compute in closed form the inverse of $D_T$, extending a result of Graham and Lovász (1978) and answering an open question of Bapat et al. (2006) in greater generality.Third, a new technique is to crucially use commutative algebra arguments – specifically, Zariski density – which to our knowledge are hitherto unused for such matrices/invariants, but are richly rewarding. We also explain why our setting is “most general”, in that for more general edgeweights, $det\left(D_T\right),cof\left(D_T\right)$ depend on the tree structure. In a sense, this completes the study of the invariants $det\left(D_T\right),cof\left(D_T\right)$ for distance matrices of trees $T$ with edge-data in a commutative ring.Our proofs use novel results for arbitrary bi-directed strongly connected graphs $G$: we prove a multiplicative analogue of an additive result by Graham et al. (1977), as well as a novel $q$-version thereof. In particular, we provide closed-form expressions for $det\left(D_G\right)$, $cof\left(D_G\right)$, and $D_G^{-1}$ in terms of their strong blocks. We then show how this subsumes the classical 1977 result, and provide sample applications to adding pendant trees and to cycle-clique graphs (including cactus/polycyclic graphs and hypertrees), subsuming variants in the literature. The final section introduces and computes a third – and novel – invariant for trees, as well as a parallel Graham–Hoffman–Hosoya type result for our “most general” distance matrix $D_T$.     

Recursive estimation: A unified approach to the identification estimation,and forecasting of hydrological systems

Mathematical models of hydrological and water-resource systems have been formulated in many different ways and with various levels of complexity. There are advantages to be gained, therefore, by trying to unify some of the more common models within a statistical framework which will allow for more objective methods of model calibration. In this paper, we consider the general class of linear, dynamic models, as applied to the characterisation of flow and dispersion behavior in rivers, and show how these can be unified within the context of recursive time-series analysis and estimation. This allows not only for more objective, data-based approaches to stochastic model structure identification, but also for improved statistical estimation and the development of both constant parameter and self-adaptive, Kalman-filter-based forecasting procedures. The unified approach presented in the paper is being applied successfully in other environmental areas, such as soil science, climatic data analysis, meterological forecasting, and plant physiology.     

Detecting and interpreting clusters of economic activity in rural areas using scan statistic and LISA under a unified framework

The primary aim of this paper is to expose the use and the value of spatial statistical analysis in business and especially in designing economic policies in rural areas. Specifically, we aim to present under a unified framework, the use of both point and area‐based methods, in order to analyze in‐depth economic data, as well as, to drive conclusions through interpreting the analysis results. The motivating problem is related to the establishment of women‐run enterprises in a rural area of Greece. Moreover, in this article, the spatial scan statistic is successfully applied to the spatial economic data at hand, in order to detect possible clusters of small women‐run enterprises in a rural mountainous and disadvantaged region of Greece. Then, it is combined with Geographical Information System based on Local Indicator of Spatial Autocorrelation scan statistic for further exploring and interpreting the spatial patterns. The rejection of the random establishment of women‐run enterprises and the interpretation of the clustering patterns are deemed necessary, in order to assist government in designing policies for rural development. Copyright © 2014 John Wiley & Sons, Ltd.     

MgNet: A unified framework of multigrid and convolutional neural network

We develop a unified model, known as MgNet, that simultaneously recovers some convolutional neural networks(CNN) for image classification and multigrid(MG) methods for solving discretized partial differential equations(PDEs). This model is based on close connections that we have observed and uncovered between the CNN and MG methodologies. For example, pooling operation and feature extraction in CNN correspond directly to restriction operation and iterative smoothers in MG, respectively. As the solution space is often the dual of the data space in PDEs, the analogous concept of feature space and data space(which are dual to each other) is introduced in CNN. With such connections and new concept in the unified model, the function of various convolution operations and pooling used in CNN can be better understood. As a result,modified CNN models(with fewer weights and hyperparameters) are developed that exhibit competitive and sometimes better performance in comparison with existing CNN models when applied to both CIFAR-10 and CIFAR-100 data sets.     

A unified framework for the discretization of nonlinear operator equations

A general theory for the discretization of non-linear operator equations is presented. A given operator with certain continuity and compactness properties is approximated by a sequence of operators acting in different spaces, usually finite dimensional. Connection maps, such as restriction and interpolation, relate the spaces. The abstract convergence theory is formulated in terms of metric spaces. Specializations and applications to differential and integral equations involve normed linear spaces. The case with the same setting for the original and approximate problems was treated in [1]. For typical problems, both types of discretization methods are available. They are related by means of the connection maps.     

A software framework based on a conceptual unified model for evolutionary multiobjective optimization: ParadisEO-MOEO

This paper presents a general-purpose software framework dedicated to the design and the implementation of evolutionary multiobjective optimization techniques: ParadisEO-MOEO. A concise overview of evolutionary algorithms for multiobjective optimization is given. A substantial number of methods has been proposed so far, and an attempt of conceptually unifying existing approaches is presented here. Based on a fine-grained decomposition and following the main issues of fitness assignment, diversity preservation and elitism, a conceptual model is proposed and is validated by regarding a number of state-of-the-art algorithms as simple variants of the same structure. This model is then incorporated into the ParadisEO-MOEO software framework. This framework has proven its validity and high flexibility by enabling the resolution of many academic, real-world and hard multiobjective optimization problems.     

A double projection algorithm for multi-valued variational inequalities and a unified framework of the method

In this paper, we propose a double projection algorithm for a generalized variational inequality with a multi-valued mapping. Under standard conditions, our method is proved to be globally convergent to a solution of the variational inequality problem. Moreover, we present a unified framework of projection-type methods for multi-valued variational inequalities. Preliminary computational experience is also reported.     

A unified framework for approximation of general telecommunication networks

In this paper, we develop a unified framework for approximation of the performance of general telecommunication networks based on a decomposition strategy. The method is an extension of the work presented in [15]. The algorithms assume finite buffer space at each switch, state-dependent arrival rates of data packets, and general service time distribution at the switches. Two methods of buffer space allocation at the switches, and two congestion control mechanisms are modeled. The proposed algorithms have been extensively tested against simulation values. The results show that the proposed framework yields robust, reliable and accurate estimates of network performance measures, such as throughput, number of packets in the system, and switch and link utilization. The computation time required is minimal. The unified framework presents a useful set of tools for telecommunication network designers in evaluating numerous network designs.     

The divergence of the image of a point sequence under a conformal mapping

We show that every point sequence contained in an open disk and converging towards its boundary can, by means of a conformal homeomorphism of this disk into the Riemann sphere, be carried over into a sequence whose limit set contains the whole continuum.     

Entropy differential metric, distance and divergence measures in probability spaces: A unified approach

The paper is devoted to metrization of probability spaces through the introduction of a quadratic differential metric in the parameter space of the probability distributions. For this purpose, a φ-entropy functional is defined on the probability space and its Hessian along a direction of the tangent space of the parameter space is taken as the metric. The distance between two probability distributions is computed as the geodesic distance induced by the metric. The paper also deals with three measures of divergence between probability distributions and their interrelationships.     

Possibility distributions: A unified representation of usual direct-probability-based parameter estimation methods

The paper presents a possibility theory based formulation of one-parameter estimation that unifies some usual direct probability formulations. Point and confidence interval estimation are expressed in a single theoretical formulation and incorporated into estimators of a generic form: a possibility distribution. New relationships between continuous possibility distribution and probability concepts are established. The notion of specificity ordering of a possibility distribution, corresponding to fuzzy subsets inclusion, is then used for comparing the efficiency of different estimators for the case of data points coming from a symmetric probability distribution. The usefulness of the approach is illustrated on common mean and median estimators from identical independent data sample of different size and of different common symmetric continuous probability distributions.     

A unified framework to prove multiplicative inequalities for the partition function

In this paper, we consider a certain class of inequalities for the partition function of the following form:$\prod _{i=1}^{T}p(n+s_i)\ge \prod _{i=1}^{T}p(n+r_i),$ which we call multiplicative inequalities. Given a multiplicative inequality with the condition that ${\sum }_{i=1}^T{s}_i^m\ne {\sum }_{i=1}^T{r}_i^m$ for at least one $m\ge 1$, we shall construct a unified framework so as to decide whether such a inequality holds or not. As a consequence, we will see that study of such inequalities has manifold applications. For example, one can retrieve log-concavity property, strong log-concavity, and the multiplicative inequality for $p(n)$ considered by Bessenrodt and Ono, to name a few. Furthermore, we obtain an asymptotic expansion for the finite difference of the logarithm of $p(n)$, denoted by ${(-1)}^{r-1}{\mathrm{\Delta }}^r\log p(n)$, which generalizes a result by Chen, Wang, and Xie.     

A unified framework for the construction of various algebraic multilevel preconditioning methods

1.IntroductionThediscretizationofmanysecondorderselfadjointellipticboundaryvalueproblemsbythefiniteelementmethodleadstolargesparsesystemsoflinearequationswithsymmetricpositivedefinite(SPD)coefficientmatrices.Fortheselinearsystems,algebraicmultilevelp...     

A unified approach to a posteriori error estimation using element residual methods

Summary This paper deals with the problem of obtaining numerical estimates of the accuracy of approximations to solutions of elliptic partial differential equations. It is shown that, by solving appropriate local residual type problems, one can obtain upper bounds on the error in the energy norm. Moreover, in the special case of adaptiveh-p finite element analysis, the estimator will also give a realistic estimate of the error. A key feature of this is the development of a systematic approach to the determination of boundary conditions for the local problems. The work extends and combines several existing methods to the case of fullh-p finite element approximation on possibly irregular meshes with, elements of non-uniform degree. As a special case, the analysis proves a conjecture made by Bank and Weiser [Some A Posteriori Error Estimators for Elliptic Partial Differential Equations, Math. Comput.44, 283–301 (1985)].     

A unified framework for the numerical solution of general quadratic matrix equations

^** Email: vassilios.tsachouridis{at}ieee.org^*** Email: N.karcanias{at}city.ac.uk^**** Email: ixp{at}le.ac.uk Algebraic quadratic equations are special cases of a singlegeneralized algebraic quadratic matrix equation (GQME). Thispaper focuses on the numerical solution of the GQME using probability-1homotopy methods. A synoptic review of these methods and theirapplication to algebraic matrix equations is provided as background.A large variety of analysis and design problems in systems andcontrol are reported as special cases of the presented frameworkand some of them are illustrated via numerical examples fromthe literature.     

Algorithms for nonnegative matrix and tensor factorizations: a unified view based on block coordinate descent framework

We review algorithms developed for nonnegative matrix factorization (NMF) and nonnegative tensor factorization (NTF) from a unified view based on the block coordinate descent (BCD) framework. NMF and NTF are low-rank approximation methods for matrices and tensors in which the low-rank factors are constrained to have only nonnegative elements. The nonnegativity constraints have been shown to enable natural interpretations and allow better solutions in numerous applications including text analysis, computer vision, and bioinformatics. However, the computation of NMF and NTF remains challenging and expensive due the constraints. Numerous algorithmic approaches have been proposed to efficiently compute NMF and NTF. The BCD framework in constrained non-linear optimization readily explains the theoretical convergence properties of several efficient NMF and NTF algorithms, which are consistent with experimental observations reported in literature. In addition, we discuss algorithms that do not fit in the BCD framework contrasting them from those based on the BCD framework. With insights acquired from the unified perspective, we also propose efficient algorithms for updating NMF when there is a small change in the reduced dimension or in the data. The effectiveness of the proposed updating algorithms are validated experimentally with synthetic and real-world data sets.     

Probabilistic temporal networks: A unified framework for reasoning with time and uncertainty

Complex real-world systems consist of collections of interacting processes/events. These processes change over time in response to both internal and external stimuli as well as to the passage of time itself. Many domains such as real-time systems diagnosis, story understanding, and financial forecasting require the capability to model complex systems under a unified framework to deal with both time and uncertainty. Current models for uncertainty and current models for time already provide rich languages to capture uncertainty and temporal information, respectively. Unfortunately, these semantics have made it extremely difficult to unify time and uncertainty in a way which cleanly and adequately models the problem domains at hand. Existing approaches suffer from significant trade offs between strong semantics for uncertainty and strong semantics for time. In this paper, we explore a new model, the Probabilistic Temporal Network (PTN), for representing temporal and atemporal information while fully embracing probabilistic semantics. The model allows representation of time constrained causality, of when and if events occur, and of the periodic and recurrent nature of processes.