a domain decomposition method based on weighted interior penalties for advection diffusion reaction problems we propose a domain decomposition method for advection diffusion reaction equations based on nitsche s transmission conditions the advection dominated case is stabilized using a continuous interior penalty approach based on the jumps in the gradient over element boundaries we prove the convergence of the finite element solutions of the discrete problem to the exact solution and propose a parallelizable iterative method the convergence of the resulting domain decomposition method is proved and this result holds true uniformly with respect to the diffusion parameter the numerical scheme that we propose here can thus be applied straightforwardly to diffusion dominated advection dominated and hyperbolic problems some numerical examples are presented in different flow regimes showing the influence of the stabilization parameter on the performance of the iterative method and we compare our method with some other domain decomposition techniques for advection diffusion equations

on self similar singular solutions of the complex ginzburg landau equation we address the open problem of existence of singularities for the complex ginzburg landau equation using a combination of rigorous results and numerical computations we describe a countable family of self similar singularities our analysis includes the supercritical nonlinear schrodinger equation as a special case we also consider the problem of stability of these singularities 2001 john wiley sons inc

on t norms based measures of specificity this paper gives a general expression for families of measures of specificity of a fuzzy set or a possibility distribution based on three t norms and a negation other known measures of specificity are particular cases of this expression and new examples are provided

visual assessment of clustering tendency for rectangular dissimilarity matrices we have an m times n matrix d and assume that its entries correspond to pair wise dissimilarities between m row objects or and n column objects oc which taken together as a union comprise a set o of n m n objects this paper develops a new visual approach that applies to four different cluster assessment problems associated with o the problems are the assessment of cluster tendency pi amongst the row objects or p2 amongst the column objects oc p3 amongst the union of the row and column objects or u oc and p4 amongst the union of the row and column objects that contain at least one object of each type co clusters the basis of the method is to regard d as a subset of known values that is part of a larger unknown n times n dissimilarity matrix and then impute the missing values from d this results in estimates for three square matrices dr dc druc that can be visually assessed for clustering tendency using the previous vat or svat algorithms the output from assessment of druc ultimately leads to a rectangular covat image which exhibits clustering tendencies in d five examples are given to illustrate the new method two important points i because vat is scalable by svat to data sets of arbitrary size and because covat depends explicitly and only on vat this new approach is immediately scalable to say the scovat model which works for even very large unloadable data sets without alteration and ii vat svat and covat are autonomous parameter free models no hidden values are needed to make them work

on quantification of different facets of uncertainty abstract with a brief introduction to three major types of uncertainties randomness nonspecificity and fuzziness we discuss various attempts to quantify them we also discuss several attempts to quantify total uncertainty in a system we then talk about some new facets of uncertainty like higher order fuzzy entropy hybrid entropy and conflict in a body of evidence in conclusion we indicate some other aspects of uncertainty that need to be modeled and quantified

discontinuous galerkin finite element heterogeneous multiscale method for elliptic problems with multiple scales an analysis of a multiscale symmetric interior penalty discontinuous galerkin finite element method for the numerical discretization of elliptic problems with multiple scales is proposed this new method first described in a abdulle c r acad sci paris ser i 346 2008 is based on numerical homogenization it allows to significantly reduce the computational cost of a fine scale discontinuous galerkin method by probing the fine scale data on sampling domains within a macroscopic partition of the computational domain macroscopic numerical fluxes an essential ingredient of discontinuous galerkin finite elements can be recovered from the computation on the sampling domains with negligible computation overhead fully discrete a priori error bounds are derived in the l 2 and h 1 norms

a reduced basis method for coercive equations with an exact solution certificate and spatio parameter adaptivity energy norm and output error bounds we develop a reduced basis method for linear coercive parametrized partial differential equations pdes with two objectives providing an energy norm or functional output a posteriori error bound

robust a posteriori error control and adaptivity for multiscale multinumerics and mortar coupling we consider discretizations of a model elliptic problem by means of different numerical methods applied separately in different subdomains termed multinumerics coupled using the mortar technique the grids need not match along the interfaces we are also interested in the multiscale setting where the subdomains are partitioned by a mesh of size h whereas the interfaces are partitioned by a mesh of much coarser size h and where lower order polynomials are used in the subdomains and higher order polynomials are used on the mortar interface mesh we derive several fully computable a posteriori error estimates which deliver a guaranteed upper bound on the error measured in the energy norm our estimates are also locally efficient and one of them is robust with respect to the ratio h h under an assumption of sufficient regularity of the weak solution the present approach allows bounding separately and comparing mutually the subdomain and interface errors a subdomain interface adaptive refinement s

a new certification framework for the port reduced static condensation reduced basis element method abstract in this paper we introduce a new certification framework for the port reduced static condensation reduced basis element pr scrbe method which has been developed for the simulation of large component based applications such as bridges or acoustic waveguides in an offline computational stage we construct a library of interoperable parametrized reference components in the subsequent online stage we instantiate and connect the components at the interfaces ports to form a system of components to compute a truth finite element approximation of the say coercive elliptic partial differential equation on the component based system we use a domain decomposition approach for an efficient simulation we employ two different types of model reduction a reduced basis rb approximation within the interior of the component huynh et al 2013 and empirical port reduction eftang and patera 2013 on the ports where the components connect we demonstrate the well posedness of the pr scrbe approximation and introduce a new certification framework to assess the quality of the port reduction we use conservative fluxes we adapt the standard estimators from rb methods to the scrbe setting to derive an a posteriori error estimator for the rb error contribution in order to combine the a posteriori estimators for both error contributions and derive a rigorous a posteriori error estimator for pr scrbe we adapt techniques from multi scale methods and component mode synthesis finally we prove that the effectivity of the derived estimator can be bounded we provide numerical experiments for heat conduction and linear elasticity to show that the derived a posteriori error estimator provides an effective estimator moreover we demonstrate the applicability of the introduced certification framework by analyzing the computational online costs

guaranteed and robust discontinuous galerkin a posteriori error estimates for convection diffusion reaction problems we propose and study a posteriori error estimates for convection diffusion reaction problems with inhomogeneous and anisotropic diffusion approximated by weighted interior penalty discontinuous galerkin methods our twofold objective is to derive estimates without undetermined constants and to analyze carefully the robustness of the estimates in singularly perturbed regimes due to dominant convection or reaction we first derive locally computable estimates for the error measured in the energy semi norm these estimates are evaluated using h div w conforming diffusive and convective flux reconstructions thereby extending the previous work on pure diffusion problems the resulting estimates are semi robust in the sense that local lower error bounds can be derived using suitable cutoff functions of the local peclet and damkohler numbers fully robust estimates are obtained for the error measured in an augmented norm consisting of the energy semi norm a dual norm of the skew symmetric part of the differential operator and a suitable contribution of the interelement jumps of the discrete solution numerical experiments are presented to illustrate the theoretical results

a reduced basis localized orthogonal decomposition in this work we combine the framework of the reduced basis method rb with the framework of the localized orthogonal decomposition lod in order to solve parametrized elliptic multiscale problems the idea of the lod is to split a high dimensional finite element space into a low dimensional space with comparably good approximation properties and a remainder space with negligible information the low dimensional space is spanned by locally supported basis functions associated with the node of a coarse mesh obtained by solving decoupled local problems however for parameter dependent multiscale problems the local basis has to be computed repeatedly for each choice of the parameter to overcome this issue we propose an rb approach to compute in an offline stage lod for suitable representative parameters the online solution of the multiscale problems can then be obtained in a coarse space thanks to the lod decomposition and for an arbitrary value of the parameters thanks to a suitable interpolation of the selected rb the online rb lod has a basis with local support and leads to sparse systems applications of the strategy to both linear and nonlinear problems are given

discontinuous galerkin finite element heterogeneous multiscale method for advection diffusion problems with multiple scales a discontinuous galerkin finite element heterogeneous multiscale method is proposed for advection diffusion problems with highly oscillatory coefficients the method is based on a coupling of a discontinuous galerkin discretization for an effective advection diffusion problem on a macroscopic mesh whose a priori unknown data are recovered from micro finite element calculations on sampling domains within each macro element the computational work involved is independent of the high oscillations in the problem at the smallest scale the stability of our method depending on both macro and micro mesh sizes is established for both diffusion dominated and advection dominated regimes without any assumptions about the type of heterogeneities in the data fully discrete a priori error bounds are derived for locally periodic data numerical experiments confirm the theoretical error estimates

on the liouville type theorems for self similar solutions to the navier stokes equations we prove liouville type theorems for the self similar solutions to the navier stokes equations one of our results generalizes the previous ones by necas ru icka sverak and tsai using a liouville type theorem we also remove a scenario of asymptotically self similar blow up for the navier stokes equations with the profile belonging to l p infty mathbb r 3 with p frac 3 2

on cluster validity index for estimation of the optimal number of fuzzy clusters a new cluster validity index is proposed that determines the optimal partition and optimal number of clusters for fuzzy partitions obtained from the fuzzy c means algorithm the proposed validity index exploits an overlap measure and a separation measure between clusters the overlap measure which indicates the degree of overlap between fuzzy clusters is obtained by computing an inter cluster overlap the separation measure which indicates the isolation distance between fuzzy clusters is obtained by computing a distance between fuzzy clusters a good fuzzy partition is expected to have a low degree of overlap and a larger separation distance testing of the proposed index and nine previously formulated indexes on well known data sets showed the superior effectiveness and reliability of the proposed index in comparison to other indexes

a regularity criterion at one scale without pressure for suitable weak solutions to the navier stokes equations abstract in this paper we continue our work in 15 to derive e regularity criteria at one scale without pressure for suitable weak solutions to the navier stokes equations we establish an e regularity criterion below of suitable weak solutions for any 0 q 1 u 5 2 d x d t e as an application we extend the previous corresponding results concerning the improvement of the classical caffarelli kohn nirenberg theorem by a logarithmic factor

lower bounds on blow up solutions of the three dimensional navier stokes equations in homogeneous sobolev spaces suppose that u t is a solution of the three dimensional navier stokes equations either on the whole space or with periodic boundary conditions that has a singularity at time t in this paper we show that the norm of u t t in the homogeneous sobolev space hs must be bounded below by cst 2s 1 4 for 1 2 5 2 the result for 1 2 s 3 2 follows from well known lower bounds on blowup in lp spaces we show in particular that the local existence time in hs r3 depends only on the hs norm for 1 2 s 5 2 s 3 2

a validity measure for fuzzy clustering the authors present a fuzzy validity criterion based on a validity function which identifies compact and separate fuzzy c partitions without assumptions as to the number of substructures inherent in the data this function depends on the data set geometric distance measure distance between cluster centroids and more importantly on the fuzzy partition generated by any fuzzy algorithm used the function is mathematically justified via its relationship to a well defined hard clustering validity function the separation index for which the condition of uniqueness has already been established the performance of this validity function compares favorably to that of several others the application of this validity function to color image segmentation in a computer color vision system for recognition of ic wafer defects which are otherwise impossible to detect using gray scale image processing is discussed

unsupervised optimal fuzzy clustering this study reports on a method for carrying out fuzzy classification without a priori assumptions on the number of clusters in the data set assessment of cluster validity is based on performance measures using hypervolume and density criteria an algorithm is derived from a combination of the fuzzy k means algorithm and fuzzy maximum likelihood estimation the unsupervised fuzzy partition optimal number of classes algorithm performs well in situations of large variability of cluster shapes densities and number of data points in each cluster the algorithm was tested on different classes of simulated data and on a real data set derived from sleep eeg signal

adaptive iterative multiscale finite volume method the multiscale finite volume msfv method is a computationally efficient numerical method for the solution of elliptic and parabolic problems with heterogeneous coefficients it has been shown for a wide range of test cases that the msfv results are in close agreement with those obtained with a classical computationally expensive technique the method however fails to give accurate results for highly anisotropic heterogeneous problems due to weak localization assumptions recently a convergent iterative msfv i msfv method was developed to enhance the quality of the multiscale results by improving the localization conditions although the i msfv method proved to be efficient for most practical problems it is still favorable to improve the localization condition adaptively i e only for a sub domain where the original msfv localization conditions are not acceptable e g near shale layers and long coherent structures with high permeability contrasts in this paper a space time adaptive i msfv ai msfv method is introduced it is shown how to improve the msfv results adaptively in space and simulation time the fine scale smoother which is necessary for convergence of the i msfv method is also applied locally finally for multiphase flow problems two criteria are investigated for adaptively updating the msfv interpolation functions 1 a criterion based on the total mobility change for the transient coefficients and 2 a criterion based on the pressure equation residual for the accuracy of the results for various challenging test cases it is demonstrated that iterations in order to obtain accurate results even for highly anisotropic heterogeneous problems are required only in small sub domains and not everywhere the findings show that the error introduced in the msfv framework can be controlled and improved very efficiently with very little additional computational cost compared to the original non iterative msfv method

adaptive finite element methods with convergence rates adaptive finite element methods for numerically solving elliptic equations are used often in practice only recently 12 17 have these methods been shown to converge however this convergence analysis says nothing about the rates of convergence of these methods and therefore does in principle not guarantee yet any numerical advantages of adaptive strategies versus non adaptive strategies the present paper modifies the adaptive method of morin nochetto and siebert 17 for solving the laplace equation with piecewise linear elements on domains in 2 by adding a coarsening step and proves that this new method has certain optimal convergence rates in the energy norm which is equivalent to the h1 norm namely it is shown that whenever s 0 and the solution u is such that for each n 1 it can be approximated to accuracy o n s in the energy norm by a continuous piecewise linear function on a triangulation with n cells using complete knowledge of u then the adaptive algorithm constructs an approximation of the same type with the same asymptotic accuracy while using only information gained during the computational process moreover the number of arithmetic computations in the proposed method is also of order o n for each n 1 the construction and analysis of this adaptive method relies on the theory of nonlinear approximation

reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation we present a new approach to treating nonlinear operators in reduced basis approximations of parametrized evolution equations our approach is based on empirical interpolation of nonlinear differential operators and their frechet derivatives efficient offline online decomposition is obtained for discrete operators that allow an efficient evaluation for a certain set of interpolation functionals an a posteriori error estimate for the resulting reduced basis method is derived and analyzed numerically we introduce a new algorithm the podei greedy algorithm which constructs the reduced basis spaces for the empirical interpolation and for the numerical scheme in a synchronized way the approach is applied to nonlinear parabolic and hyperbolic equations based on explicit or implicit finite volume discretizations we show that the resulting reduced scheme is able to capture the evolution of both smooth and discontinuous solutions in case of symmetries of the problem the approach realizes an automatic and intuitive space compression or even space dimensionality reduction we perform empirical investigations of the error convergence and run times in all cases we obtain a good run time acceleration

on the a priori estimates for the euler the navier stokes and the quasi geostrophic equations abstract we prove new a priori estimates for the 3d euler the 3d navier stokes and the 2d quasi geostrophic equations by the method of similarity transforms

a cluster validity index combining an overlap measure and a separation measure based on fuzzy aggregation operators since a clustering algorithm can produce as many partitions as desired one needs to assess their quality in order to select the partition that most represents the structure in the data if there is any this is the rationale for the cluster validity cv problem and indices this paper presents a cv index that helps to find the optimal number of clusters of data from partitions generated by a fuzzy clustering algorithm such as the fuzzy c means fcm or its derivatives given a fuzzy partition this new index uses a measure of multiple cluster overlap and a separation measure for each data point both based on an aggregation operation of membership degrees experimental results on artificial and benchmark datasets are given to demonstrate the performance of the proposed index as compared with traditional and recent indices

the heterogeneous multiscale finite element method for elliptic homogenization problems in perforated domains in this contribution we analyze a generalization of the heterogeneous multiscale finite element method for elliptic homogenization problems in perforated domains the method was originally introduced by e and engquist commun math sci 1 1 87 132 2003 for homogenization problems in fixed domains it is based on a standard finite element approach on the macroscale where the stiffness matrix is computed by solving local cell problems on the microscale a posteriori error estimates are derived in l 2 by reformulating the problem into a discrete two scale formulation see also ohlberger in multiscale model simul 4 1 88 114 2005 and using duality methods afterwards numerical experiments are given in order to numerically evaluate the efficiency of the error estimate

evaluating the numerical instability in fuzzy clustering validation of high dimensional data abstract fuzzy clustering validation of high dimensional datasets is only possible using a reliable cluster validity index cvi a good cvi must correctly recognize a data structure and its validations must be independently of any parameter of a clustering algorithm or data property however some classical fuzzy cvis as partition coefficient pc partition entropy pe and fukuyama sugeno fs have the monotonic tendency in function of the number of clusters although the literature presents extensive investigations about such tendency they were conducted for low dimensional data in which such data property does not affect the clustering behavior in order to investigate how such aspects affect the fuzzy clustering results of high dimensional data in this work we have clustered objects of thirteen real datasets using the fuzzy c means algorithm the fuzzy partitions were validated by pc pe fs and some proposed improvements of them to lead with the monotonic tendency totaling eight fuzzy cvis analyzed besides the analysis made about the number of clusters selected by the cvis the mann kendall test was performed to verify statistically the monotonic trend of the cvis results from the two analysis made the modified partition coefficient and scaled partition entropy indices were successful in respectively improving the pc and pe indices

adaptive reduced basis finite element heterogeneous multiscale method an adaptive reduced basis finite element heterogeneous multiscale method rb fe hmm is proposed for elliptic problems with multiple scales the multiscale method is based on the rb fe hmm introduced in a abdulle y bai reduced basis finite element heterogeneous multiscale method for high order discretizations of elliptic homogenization problems j comput phys 231 21 2012 7014 7036 it couples a macroscopic solver with effective data recovered from the solution of micro problems solved on sampling domains unlike classical numerical homogenization methods the micro problems are computed in a finite dimensional space spanned by a small number of accurately computed representative micro solutions the reduced basis obtained by a greedy algorithm in an offline stage in this paper we present a residual based a posteriori error analysis in the energy norm as well as an a posteriori error analysis in quantities of interest for both type of adaptive strategies rigorous a posteriori error estimates are derived and corresponding error estimators are proposed in contrast to the adaptive finite element heterogeneous multiscale method fe hmm there is no need to adapt the micro mesh simultaneously to the macroscopic mesh refinement up to an offline preliminary stage the rb fe hmm has the same computational complexity as a standard adaptive fem for the effective problem two and three dimensional numerical experiments confirm the efficiency of the rb fe hmm and illustrate the improvements compared to the adaptive fe hmm c 2013 elsevier b v all rights reserved

generalized multiscale finite element methods gmsfem in this paper we propose a general approach called generalized multiscale finite element method gmsfem for performing multiscale simulations for problems without scale separation over a complex input space as in multiscale finite element methods msfems the main idea of the proposed approach is to construct a small dimensional local solution space that can be used to generate an efficient and accurate approximation to the multiscale solution with a potentially high dimensional input parameter space in the proposed approach we present a general procedure to construct the offline space that is used for a systematic enrichment of the coarse solution space in the online stage the enrichment in the online stage is performed based on a spectral decomposition of the offline space in the online stage for any input parameter a multiscale space is constructed to solve the global problem on a coarse grid the online space is constructed via a spectral decomposition of the offline space and by choosing the eigenvectors corresponding to the largest eigenvalues the computational saving is due to the fact that the construction of the online multiscale space for any input parameter is fast and this space can be re used for solving the forward problem with any forcing and boundary condition compared with the other approaches where global snapshots are used the local approach that we present in this paper allows us to eliminate unnecessary degrees of freedom on a coarse grid level we present various examples in the paper and some numerical results to demonstrate the effectiveness of our method

boundary characteristic point regularity for navier stokes equations blow up scaling and petrovskii type criterion a formal approach the three dimensional 3d navier stokes equations less thanbrgreater than less thanbrgreater thanu t u del u del p delta u divu 0 in q 0 0 1 less thanbrgreater than less thanbrgr

solutions for semilinear parabolic equations in lp and regularity of weak solutions of the navier stokes system abstract we construct a unique local regular solution in l q 0 t l p for a class of semilinear parabolic equations which includes the semilinear heat equation u t u u u 0 and the navier stokes system here p and q are so chosen that the norm of l q 0 t l p is dimensionless or scaling invariant the main relation between p and q for the semilinear heat equation is 1 q 1 p 1 q n 2 provided that initial data are in l r with r n 2 1 where n is the space dimension applying our regular solutions to the navier stokes system we show that the k 2 dimensional hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in l q 0 t l p where k 2 q nq p p n 1 q we show moreover that a turbulent solution is regular if it is in c 0 t l n

validity guided re clustering with applications to image segmentation when clustering algorithms are applied to image segmentation the goal is to solve a classification problem however these algorithms do not directly optimize classification duality as a result they are susceptible to two problems 1 the criterion they optimize may not be a good estimator of true classification quality and 2 they often admit many suboptimal solutions this paper introduces an algorithm that uses cluster validity to mitigate problems 1 and 2 the validity guided re clustering vgc algorithm uses cluster validity information to guide a fuzzy re clustering process toward better solutions it starts with a partition generated by a soft or fuzzy clustering algorithm then it iteratively alters the partition by applying novel split and merge operations to the clusters partition modifications that result in improved partition validity are retained vgc is tested on both synthetic and real world data for magnetic resonance image mri segmentation evaluations by radiologists show that vgc outperforms the unsupervised fuzzy c means algorithm and vgc s performance approaches that of the supervised k nearest neighbors algorithm

numerical model reduction with error control in computational homogenization of transient heat flow numerical model reduction nmr is exploited for solving the finite element problem on a representative volume element rve that arises from the computational homogenization of a model problem of transient heat flow since the problem is linear an orthogonal basis is obtained via the classical method of spectral decomposition a symmetrized version of the space time variational format is adopted for estimating the error from the model reduction in i energy norm and in ii given quantities of interest this technique which was recently developed in the context of the non selfadjoint stationary diffusion convection problem is novel in the present context of nmr by considering the discrete unreduced model as exact we are able to obtain guaranteed bounds on the error while using only the reduced basis and with minor computational effort the performance of the error estimates is demonstrated via numerical results where the subscale is modeled in both one and three spatial dimensions

a k order fuzzy or operator for pattern classification with k order ambiguity rejection in pattern recognition the membership of an object to classes is often measured by labels this article mainly deals with the mathematical foundations of labels combination operators built on t norms that extend previous ambiguity measures of objects by dealing not only with two classes ambiguities but also with k classes k lying between 1 and the number of classes c mathematical properties of this family of combination operators are established and a weighted extension is proposed allowing to give more or less importance to a given class a classifier with reject options built on the proposed measure is presented and applied on synthetic data a critical analysis of the results led to derivate some new operators by aggregating previous measures a modified classifier is proposed and applied to synthetic data as well as to standard real data

convergence of adaptive finite element methods adaptive finite element methods fems have been widely used in applications for over 20 years now in practice they converge starting from coarse grids although no mathematical theory has been able to prove this assertion ensuring an error reduction rate based on a posteriori error estimators together with a reduction rate of data oscillation information missed by the underlying averaging process we construct a simple and efficient adaptive fem for elliptic partial differential equations we prove that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance extensions to higher order elements and applications to saddle point problems are discussed as well

on partial regularity problem for 3d boussinesq equations abstract in this paper we study the partial regularity of the solutions for the three dimensional boussinesq equations we first prove a criterion of local holder continuous of the suitable weak solutions of the boussinesq equations and show that one dimensional hausdorff measure of the singular point set is zero secondly we present a local uniform gradient estimate on the suitable weak solutions and assert that the local behavior of the solution can be dominated by some scaled quantities such as the scaled local l 3 norm of the velocity besides when the initial data v 0 and 0 decay sufficiently rapidly at the distribution of the regular point set of the suitable weak solutions is also considered based on it one can find that mhd equations are more similar to navier stokes equations than boussinesq equations finally we give a local regularity criterion of the suitable weak solutions near the boundary

liouville type theorems for the forced euler equations and the navier stokes equations in this paper we study the liouville type properties for solutions to the steady incompressible euler equations with forces in mathbb r n if we assume single signedness condition on the force then we can show that a c 1 mathbb r n solution v p with v 2 p in l frac q 2 mathbb r n q in frac 3n n 1 infty is trivial v 0 for the solution of the steady navier stokes equations satisfying v x to 0 as x to infty the condition int mathbb r 3 delta v frac 6 5 dx infty which is stronger than the important d condition int mathbb r 3 nabla v 2 dx infty but both having the same scaling property implies that v 0 in the appendix we reprove theorem 1 1 chae commun math phys 273 203 215 2007 using the self similar euler equations directly

a posteriori error estimates for lowest order mixed finite element discretizations of convection diffusion reaction equations we establish residual a posteriori error estimates for lowest order raviart thomas mixed finite element discretizations of convection diffusion reaction equations on simplicial meshes in two or three space dimensions the upwind mixed scheme is considered as well and the emphasis is put on the presence of an inhomogeneous and anisotropic diffusion dispersion tensor and on a possible convection dominance global upper bounds for the approximation error in the energy norm are derived where in particular all constants are evaluated explicitly so that the estimators are fully computable our estimators give local lower bounds for the error as well and they hold from the cases where convection or reaction are not present to convection or reaction dominated problems we prove that their local efficiency depends only on local variations in the coefficients and on the local peclet number moreover the developed general framework allows for asymptotic exactness and full robustness with respect to inhomogeneities and anisotropies the main idea of the proof is a construction of a locally postprocessed approximate solution using the mean value and the flux in each element known in the mixed finite element method and a subsequent use of the abstract framework arising from the primal weak formulation of the continuous problem numerical experiments confirm the guaranteed upper bound and excellent efficiency and robustness of the derived estimators

error control for the localized reduced basis multiscale method with adaptive on line enrichment in this contribution we consider localized robust and efficient a posteriori error estimation of the localized reduced basis multiscale lrbms method for parametric elliptic problems with possibly heterogeneous diffusion coefficient the numerical treatment of such parametric multiscale problems is characterized by a high computational complexity arising from the multiscale character of the underlying differential equation and the additional parameter dependence the lrbms method can be seen as a combination of numerical multiscale methods and model reduction using reduced basis rb methods to efficiently reduce the computational complexity with respect to the multiscale as well as the parametric aspect of the problem simultaneously in contrast to the classical residual based error estimators currently used in rb methods we are considering error estimators that are based on conservative flux reconstruction and provide an efficient and rigorous bound on the full error with respect to the weak solut

regularity criteria for suitable weak solutions of the navier stokes equations near the boundary abstract we present some new regularity criteria for suitable weak solutions of the navier stokes equations near the boundary in dimension three we prove that suitable weak solutions are holder continuous up to the boundary provided that the scaled mixed norm l x t p q with 3 p 2 q 2 2 q p q 3 2 is small near the boundary our methods yield new results in the interior case as well partial regularity of weak solutions is also analyzed under some additional integral conditions

interior regularity criteria for suitable weak solutions of the navier stokes equations we present new interior regularity criteria for suitable weak solutions of the 3 d navier stokes equations a suitable weak solution is regular near an interior point z if either the scaled l p q x t norm of the velocity with 3 p 2 q 2 1 q or the l p q x t norm of the vorticity with 3 p 2 q 3 1 q or the l p q x t norm of the gradient of the vorticity with 3 p 2 q 4 1 q 1 p is sufficiently small near z

a reduced basis hybrid method for the coupling of parametrized domains represented by fluidic networks abstract in this paper we propose a reduced basis hybrid method rbhm for the approximation of partial differential equations in domains represented by complex networks where topological features are recurrent the rbhm is applied to stokes equations in domains which are decomposable into smaller similar blocks that are properly coupled the rbhm is built upon the reduced basis element method rbem and it takes advantage from both the reduced basis methods rb and the domain decomposition method we move from the consideration that the blocks composing the computational domain are topologically similar to a few reference shapes on the latter representative solutions corresponding to the same governing partial differential equations are computed for different values of some parameters of interest representing for example the deformation of the blocks a generalized transfinite mapping is used in order to produce a global map from the reference shapes of each block to any deformed configuration the desired solution on the given original computational domain is recovered as projection of the previously precomputed solutions and then glued across subdomain interfaces by suitable coupling conditions the geometrical parametrization of the domain by transfinite mapping induces non affine parameter dependence an empirical interpolation technique is used to recover an approximate affine parameter dependence and a subsequent offline online decomposition of the reduced basis procedure this computational decomposition yields a considerable reduction of the problem complexity results computed on some combinations of 2d and 3d geometries representing cardiovascular networks show the advantage of the method in terms of reduced computational costs and the quality of the coupling to guarantee continuity of both stresses pressure and velocity at subdomain interfaces

forward self similar solutions of the fractional navier stokes equations abstract we study forward self similar solutions to the 3 d navier stokes equations with the fractional diffusion first we construct a global time forward self similar solutions to the fractional navier stokes equations with 5 6 1 for arbitrarily large self similar initial data by making use of the so called blow up argument moreover we prove that this solution is smooth in r 3 0 in particular when 1 we prove that the solution constructed by korobkov and tsai 2016 16 satisfies the decay estimate by establishing regularity of solution for the corresponding elliptic system which implies this solution has the same properties as a solution which was constructed in jia and sverak 2014 13

a posteriori error estimates for a discontinuous galerkin approximation of second order elliptic problems several a posteriori error estimators are introduced and analyzed for a discontinuous galerkin formulation of a model second order elliptic problem in addition to residual type estimators we introduce some estimators that are couched in the ideas and techniques of domain decomposition results of numerical experiments are presented

nonexistence of self similar singularities for the 3d incompressible euler equations we prove that there exists no self similar finite time blowing up solution to the 3d incompressible euler equations if the vorticity decays sufficiently fast near infinity in mathbb r 3 by a similar method we also show nonexistence of self similar blowing up solutions to the divergence free transport equation in mathbb r n this result has direct applications to the density dependent euler equations the boussinesq system and the quasi geostrophic equations for which we also show nonexistence of self similar blowing up solutions

nonexistence of self similar singularities in the viscous magnetohydrodynamics with zero resistivity abstract we are concerned on the possibility of finite time singularity in a partially viscous magnetohydrodynamic equations in r n n 2 3 namely the mhd with positive viscosity and zero resistivity in the special case of zero magnetic field the system reduces to the navier stokes equations in r n in this paper we exclude the scenario of finite time singularity in the form of self similarity under suitable integrability conditions on the velocity and the magnetic field we also prove the nonexistence of asymptotically self similar singularity this provides us information on the behavior of solutions near possible singularity of general type as described in corollary 1 1

validity index for crisp and fuzzy clusters abstract in this article a cluster validity index and its fuzzification is described which can provide a measure of goodness of clustering on different partitions of a data set the maximum value of this index called the pbm index across the hierarchy provides the best partitioning the index is defined as a product of three factors maximization of which ensures the formation of a small number of compact clusters with large separation between at least two clusters we have used both the k means and the expectation maximization algorithms as underlying crisp clustering techniques for fuzzy clustering we have utilized the well known fuzzy c means algorithm results demonstrating the superiority of the pbm index in appropriately determining the number of clusters as compared to three other well known measures the davies bouldin index dunn s index and the xie beni index are provided for several artificial and real life data sets

the reduced basis element method application to a thermal fin problem the reduced basis element method is a new approach for the approximation of partial differential equations that takes its roots in the domain decomposition method and in the reduced basis discretization the basic idea is to decompose the domain of computation into a series of subdomains the elements that are similar to a few reference domains these reference domains are actually filled with reduced basis functional spaces that are mapped to each subdomain together with the geometry the discrete approximation space is then composed of functions with the property that a function restricted to a subdomain belongs to the mapped reduced spaces finally a mortar type method is applied to glue the various local functions in this paper we focus on the definition of the reference shapes and together with theoretical and numerical justifications of the method we provide a posteriori error analysis tools that allow us to certify the computational results

on error controlled numerical model reduction in fe2 analysis of transient heat flow numerical model reduction is exploited for solving the two scale problem that arises from the computational homogenization of a model problem of transient heat flow since the problem is linear an orthogonal basis is obtained via the method of spectral decomposition a key feature is the construction of a symmetrized version of the space time variational format by which it is possible to estimate the error from the model reduction in i energy norm and in ii a given quantity of interest in previous work by the authors this strategy has been applied to the solution of an individual representative volume element problem whereas the error transport to the macroscale problem in the finite element squared fe2 approach was ignored in this paper such transport of error is included in the error estimate it is remarkable that it is still possible to obtain guaranteed bounds on the error as compared to the fully resolved discrete finite element problem while using only the reduced basis and with minor extra computational effort the performance of the error estimates is demonstrated via numerical results whereby the subscale is modeled in three dimensions whereas the macroscale problem is either one or two dimensional

on blow up space jets for the navier stokes equations in r3 with convergence to euler equations formation of blow up singularities for the navier stokes equations ut u u p u and div u 0 in r3 0 t is studied in cylindrical polar coordinates r z in r3 their restriction to the linear subspace w2 span 1 z is shown to be consistent using links with blow up theory for nonlinear reaction diffusion partial differential equations the following questions are under scrutiny ii introducing a self similar blow up swirl mechanism with the angular swirl divergences t ln t t and t t t as t t where r is a parameter ii existence of a countable family of space jets via a nonlocal semilinear parabolic equation with effective regional global blow up of the z component of the velocity iii as an intrinsic part of the construction convergence of the above rescaled patterns as t t to smooth blow up self similar solutions of the corresponding three dimensional euler equations ut u u p and div u 0 in r3 0 t which iv are also shown to admit single point blow up in

on partial regularity results for the navier stokes equations looking at the regularity results of scheffer respectively caffarelli kohn and nirenberg from a new point of view indicates that estimates for the pressure do not play an essential role in partial regularity results for the navier stokes equations

adaptive variational multiscale methods based on a posteriori error estimation energy norm estimates for elliptic problems we develop a new adaptive multiscale finite element method using the variational multiscale framework together with a systematic technique for approximation of the fine scale part of the solution the fine scale is approximated by a sum of solutions to decoupled localized problems which are solved numerically on a fine grid partition of a patch of coarse grid elements the sizes of the patches of elements may be increased to control the error caused by localization we derive an a posteriori error estimate in the energy norm which captures the dependency of the crucial discretization parameters the coarse grid mesh size the fine grid mesh size and the sizes of the patches based on the a posteriori error estimate we present an adaptive algorithm that automatically tunes these parameters finally we show how the method works in practice by presenting various numerical examples

forward discretely self similar solutions of the navier stokes equations extending the work of jia and sverak on self similar solutions of the navier stokes equations we show the existence of large forward discretely self similar solutions

on leray s self similar solutions of the navier stokes equations satisfying local energy estimates this paper proves that leray s self similar solutions of the three dimensional navier stokes equations must be trivial under very general assumptions for example if they satisfy local energy estimates

ecm an evidential version of the fuzzy c means algorithm a new clustering method for object data called ecm evidential c means is introduced in the theoretical framework of belief functions it is based on the concept of credal partition extending those of hard fuzzy and possibilistic ones to derive such a structure a suitable objective function is minimized using an fcm like algorithm a validity index allowing the determination of the proper number of clusters is also proposed experiments with synthetic and real data sets show that the proposed algorithm can be considered as a promising tool in the field of exploratory statistics

a new fuzzy clustering validity index with a median factor for centroid based clustering determining the number of clusters which is usually approved by domain experts or evaluated by clustering validity indexes is an important issue in clustering analysis this study discusses the effectiveness of clustering validity indexes for centroid based partitional clustering algorithms most general purpose clustering validity indexes take the minimum maximum distance between a pair of data objects a pair of cluster centroids or an object and a centroid as an important evaluation factor however they may present unstable results especially when two centroids are allocated closely to alleviate this problem a new clustering validity index which is termed the wu and li index wli is proposed in this paper our proposed wli partially allows to some extent the existence of closely allocated centroids in the clustering results by considering not only the minimum but the median distances between a pair of centroids as well therefore possessing better stability the performances of wli and some existing clustering validity indexes are evaluated and compared by running the fuzzy c means algorithm for clustering various types of datasets including artificial datasets uci datasets and images experimental results have shown that wli has the more accurate and satisfactory performance than other indexes

regularity for the stationary navier stokes equations in bounded domain let u p be a weak solution of the stationary navier stokes equations in a bounded domain n 5 n if u p satisfy the additional conditions begin gathered text a text left frac u text 2 text 2 p right in l text loc q omega text q in left left frac n 2 infty right right hfill hfill text b text int limits omega nabla u nabla u gamma dx leqq text int limits omega left frac u text 2 text 2 p right text u nabla gamma dx int limits omega fu gamma dx text forall gamma in c 0 infty omega text gamma geqq 0 hfill end gathered they become regular moreover it is proved that every weak solution u p satisfying a with q is regular the existence of such solutions for n 5 has been established in a former paper 3

on fuzzy cluster validity indices cluster analysis aims at identifying groups of similar objects and helps to discover distribution of patterns and interesting correlations in large data sets especially fuzzy clustering has been widely studied and applied in a variety of key areas and fuzzy cluster validation plays a very important role in fuzzy clustering this paper introduces the fundamental concepts of cluster validity and presents a review of fuzzy cluster validity indices available in the literature we conducted extensive comparisons of the mentioned indices in conjunction with the fuzzy c means clustering algorithm on a number of widely used data sets and make a simple analysis of the experimental results

a component based hybrid reduced basis finite element method for solid mechanics with local nonlinearities abstract the scrbe static condensation reduced basis element method is a component to system model order reduction approach for efficient many query and real time treatment of linear partial differential equations characterized by many spatially distributed constitutive geometry and topology parameters in this paper we incorporate the scrbe approach into a framework for analysis of problems in solid mechanics which are largely linear with the exception of local nonlinearities in particular we exploit a linear nonlinear domain decomposition to develop a hybrid formulation we consider a scrbe approximation over the assumed predominant part of the domain associated to a linear elasticity model we revert to a full finite element fe approximation over the part of the domain associated to the locally nonlinear model we adapt the scrbe port training procedures to anticipate the behavior of the field over the linear nonlinear interface and hence ensure an accurate solution over the entire domain we choose a globally conforming approximation which exploits the intrinsic port structure of the scrbe method and yields a decoupled non invasive formulation of the respective linear and nonlinear blocks of the residual vector and newton jacobian matrix we present numerical results for several local nonlinearities elastic contact with and without friction plastic contact which demonstrate substantial savings relative to standard fe approximation over the entire domain in particular in the regime in which the linear part of the domain represents the majority of the fe degrees of freedom

minimal initial data for potential navier stokes singularities abstract assuming some initial data u 0 h 1 2 r 3 lead to a singularity for the 3d navier stokes equations we show that there are also initial data with the minimal h 1 2 norm which will produce a singularity

estimation of penalty parameters for symmetric interior penalty galerkin methods this paper presents computable lower bounds of the penalty parameters for stable and convergent symmetric interior penalty galerkin methods in particular we derive the explicit dependence of the coercivity constants with respect to the polynomial degree and the angles of the mesh elements numerical examples in all dimensions and for different polynomial degrees are presented we show the numerical effects of loss of coercivity

liouville theorems for the navier stokes equations and applications we study bounded ancient solutions of the navier stokes equations these are solutions with bounded velocity defined in rn 1 0 in two space dimensions we prove that such solutions are either constant or of the form u x t b t depending on the exact definition of admissible solutions the general 3 dimensional problem seems to be out of reach of existing techniques but partial results can be obtained in the case of axisymmetric solutions we apply these results to some scenarios of potential singularity formation for axi symmetric solutions and obtain extensions of results in a recent paper by chen strain tsai and yau 4

non existence of finite time self similar solutions of the keller segel system in the scaling invariant class let us consider the problem whether there does exist a finite time self similar solution of the backward type to the semilinear keller segel system in the case of parabolic elliptic type for n 3 we show that there is no such a solution with a finite mass in the scaling invariant class on the other hand in the case of parabolic parabolic type for n 2 non existence of finite time self similar solutions is proved in a larger class of a finite mass with some local bounds

element boundary terms in reduced order models for flow problems domain decomposition and adaptive coarse mesh hyper reduction abstract in this paper we present a finite element based reduced order model and in particular we consider two aspects related to the introduction of inter element boundary terms in the formulation the first is a domain decomposition strategy in which the transmission conditions involve boundary terms to account for non matching meshes and discontinuous physical properties the second is a coarse mesh hyper reduction for which we propose an adaptive refinement driven by an a posteriori error estimator that contains element boundary terms as the finite element full order model the reduced order model is based on the variational multi scale framework with sub grid scales defined not only in the element interiors but also on the inter element boundaries we present some examples of application using the incompressible navier stokes equations and the boussinesq approximation

overlapping schwarz methods with adaptive coarse spaces for multiscale problems in 3d we propose two variants of the overlapping additive schwarz method for the finite element discretization of scalar elliptic problems in 3d with highly heterogeneous coefficients the methods are efficient and simple to construct using the abstract framework of the additive schwarz method and an idea of adaptive coarse spaces in one variant the coarse space consists of finite element functions associated with the wire basket nodes and functions based on solving some generalized eigenvalue problems on the faces in the other variant it contains functions associated with the vertex nodes with functions based on solving some generalized eigenvalue problems on subdomain faces and subdomain edges the functions that constitute the coarse spaces are chosen adaptively and they correspond to the eigenvalues that are smaller than a given threshold the convergence rate of the preconditioned conjugate gradients method in both cases is shown to be independent of the variations in the coefficients for the sufficient number of eigenfunctions in the coarse space numerical results are given to support the theory

discretely self similar solutions to the navier stokes equations with besov space data we construct self similar solutions to the three dimensional navier stokes equations for divergence free self similar initial data that can be large in the critical besov space dot b p infty 3 p 1 where 3 1 these results extend those of bradshaw and tsai ann henri poincare 2016 https doi org 10 1007 s00023 016 0519 0 which dealt with initial data in l 3 w since l 3 w subsetneq dot b p infty 3 p 1 for p 3 we also provide several concrete examples of vector fields in the relevant function spaces