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.     

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.     

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.     

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 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.     

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)].     

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.     

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.     

Sufficiency conditions under which a lagrangian is an ordinary divergence

An expression is derived for the variation of Lagrangians which are such that the set of admissible variables of variation is star-shaped. If such a Lagrangian leads to identically vanishing Euler-Lagrange expressions then it is shown that under suitable circumstances the Lagrangian in question must be an ordinary divergence. Furthermore, an expression is given for the ‘vector’ field which appears in this ordinary divergence.     

Proximal-like contraction methods for monotone variational inequalities in a unified framework I: Effective quadruplet and primary methods

Approximate proximal point algorithms (abbreviated as APPAs) are classical approaches for convex optimization problems and monotone variational inequalities. To solve the subproblems of these algorithms, the projection method takes the iteration in form of u ^k+1=P _Ω [u ^k −α _k d ^k ]. Interestingly, many of them can be paired such that [(u)\tilde]^k = P_\varOmega[u^k - b_kF(v^k)] = P_\varOmega[[(u)\tilde]^k - (d₂^k - G d₁^k)]\tilde{u}^{k} = P_{\varOmega}[u^{k} - \beta_{k}F(v^{k})] = P_{\varOmega}[\tilde {u}^{k} - (d_{2}^{k} - G d_{1}^{k})], where inf {β _k }>0 and G is a symmetric positive definite matrix. In other words, this projection equation offers a pair of directions, i.e., d₁^kd_{1}^{k} and d₂^kd_{2}^{k} for each step. In this paper, for various APPAs we present a unified framework involving the above equations. Unified characterization is investigated for the contraction and convergence properties under the framework. This shows some essential views behind various outlooks. To study and pair various APPAs for different types of variational inequalities, we thus construct the above equations in different expressions according to the framework. Based on our constructed frameworks, it is interesting to see that, by choosing one of the directions (d₁^kd_{1}^{k} and d₂^kd_{2}^{k}) those studied proximal-like methods always utilize the unit step size namely α _k ≡1.     

A unified derivation for fast estimation algorithms by the conjugate direction method

Fast estimation algorithms inspired by the classical method of Levinson have been developed in the areas of time series analysis, system identification, and signal processing. This paper provides a unified derivation for the Levinson-Durbin-Whittle-Wiggins-Robinson, fast recursive least squares (RLS), ladder (or lattice), and fast Cholesky algorithms as special cases of the conjugate direction method (CDM). This gives a novel derivation and interpretation for all these methods.     

Network science faces the challenge and opportunity:exploring ``Network of Networks" and its unified theoretical framework

In the era of big data, network science is facing new challenges and opportunities. This review article focuses on discussing one of the hottest subjects of network science - ``network of networks" (NON). The main features, several typical examples and the main progress for NON are outlined, including the epidemic spreading in multilayer coupled networks. Finally the most challenging tasks for NON are proposed.     

Interest rate futures: estimation of volatility parameters in an arbitrage-free framework

Hedging interest rate exposures using interest rate futures contracts requires some knowledge of the volatility function of the interest rates. Use of historical data as well as interest rate options like caps and swaptions to estimate this volatility function have been proposed in the literature. In this paper the interest rate futures price is modelled within an arbitrage-free framework for a volatility function which includes a stochastic variable, the instantaneous spot interest rate. The resulting system is expressed in a state space form which is solved using an extended Kalman filter. The residual diagnostics indicate suitability of the model and the bootstrap resampling technique is used to obtain small sample properties of the parameters of the volatility function.