adaptive fuzzy rule based classification systems this paper proposes an adaptive method to construct a fuzzy rule based classification system with high performance for pattern classification problems the proposed method consists of two procedures an error correction based learning procedure and an additional learning procedure the error correction based learning procedure adjusts the grade of certainty of each fuzzy rule by its classification performance that is when a pattern is misclassified by a particular fuzzy rule the grade of certainty of that rule is decreased on the contrary when a pattern is correctly classified the grade of certainty is increased because the error correction based learning procedure is not meaningful after all the given patterns are correctly classified we cannot adjust a classification boundary in such a case to acquire a more intuitively acceptable boundary we propose an additional learning procedure we also propose a method for selecting significant fuzzy rules by pruning unnecessary fuzzy rules which consists of the error correction based learning procedure and the concept of forgetting we can construct a compact fuzzy rule based classification system with high performance

constructing spectral schemes of the immersed interface method via a global description of discontinuous functions a global description of discontinuous functions is introduced in this paper by expressing a discontinuous function as the sum of a smooth function and a correction term determined by jump conditions we turn the unknown function from a discontinuous one into a sufficiently smooth one when solving a differential equation spectral schemes are developed based on this concept with the intention of eliminating or reducing the gibbs oscillation finite difference schemes are also constructed as an alternative of the current immersed interface methods both spectral and finite difference schemes are tested on one and two dimensional cases

a system of matrix equations and a linear matrix equation over arbitrary regular rings with identity abstract in this paper we consider the classical system of matrix equations a 1 xb 1 c 1 a 2 xb 2 c 2 over r an arbitrary regular ring with identity necessary and sufficient conditions for the existence and the expression of the general solution to the system are derived as an application the linear matrix equation axb cyd e over r is considered

modules and matrices abstract noncommutative ring theory was described in terms of matrices in its earliest days we give some examples showing how matrix theory can now in turn profit from the theory of rings and modules

fuzzy rule selection by multi objective genetic local search algorithms and rule evaluation measures in data mining this paper shows how a small number of simple fuzzy if then rules can be selected for pattern classification problems with many continuous attributes our approach consists of two phases candidate rule generation by rule evaluation measures in data mining and rule selection by multi objective evolutionary algorithms in our approach first candidate fuzzy if then rules are generated from numerical data and prescreened using two rule evaluation measures i e confidence and support in data mining then a small number of fuzzy if then rules are selected from the prescreened candidate rules using multi objective evolutionary algorithms in rule selection we use three objectives maximization of the classification accuracy minimization of the number of selected rules and minimization of the total rule length thus the task of multi objective evolutionary algorithms is to find a number of non dominated rule sets with respect to these three objectives the main contribution of this paper is to propose an idea of utilizing the two rule evaluation measures as prescreening criteria of candidate rules for fuzzy rule selection an arbitrarily specified number of candidate rules can be generated from numerical data for high dimensional pattern classification problems through computer simulations we demonstrate that such a prescreening procedure improves the efficiency of our approach to fuzzy rule selection we also extend a multi objective genetic algorithm moga in our former studies to a multi objective genetic local search mogls algorithm where a local search procedure adjusts the selection i e inclusion or exclusion of each candidate rule furthermore a learning algorithm of rule weights i e certainty factors is combined with our mogls algorithm such extensions to our moga for fuzzy rule selection are another contribution of this paper

mib galerkin method for elliptic interface problems material interfaces are omnipresent in the real world structures and devices mathematical modeling of material interfaces often leads to elliptic partial differential equations pdes with discontinuous coefficients and singular sources which are commonly called elliptic interface problems the development of high order numerical schemes for elliptic interface problems has become a well defined field in applied and computational mathematics and attracted much attention in the past decades despite of significant advances challenges remain in the construction of high order schemes for nonsmooth interfaces i e interfaces with geometric singularities such as tips cusps and sharp edges the challenge of geometric singularities is amplified when they are associated with low solution regularities e g tip geometry effects in many fields the present work introduces a matched interface and boundary mib galerkin method for solving two dimensional 2d elliptic pdes with complex interfaces geometric singularities and low solution regularities the cartesian grid based triangular elements are employed to avoid the time consuming mesh generation procedure consequently the interface cuts through elements to ensure the continuity of classic basis functions across the interface two sets of overlapping elements called mib elements are defined near the interface as a result differentiation can be computed near the interface as if there is no interface interpolation functions are constructed on mib element spaces to smoothly extend function values across the interface a set of lowest order interface jump conditions is enforced on the interface which in turn determines the interpolation functions the performance of the proposed mib galerkin finite element method is validated by numerical experiments with a wide range of interface geometries geometric singularities low regularity solutions and grid resolutions extensive numerical studies confirm the designed second order convergence of the mib galerkin method in the l and l 2 errors some of the best results are obtained in the present work when the interface is c 1 or lipschitz continuous and the solution is c 2 continuous

consistency for bi skew symmetric solutions to systems of generalized sylvester equations over a finite central algebra let be a finite dimensional central algebra with an involutorial antiautomorphism and char 2 n n be the set of all n n matrices over a aij n n is called bisymmetric if aij an i 1 n j 1 aji and biskewsymmetric if aij an i 1 n j 1 aji the following systems of generalized sylvester equations over a1x yb1 c1 i asx ybs cs a1xb1 c1xd1 e1 ii asxbs csxds es are considered necessary and sufficient conditions are given for the existence of constant solutions with bi skew symmetric constrains to i and ii as a particular case auxiliary results dealing with the system of sylvester equations are also presented

a galerkin formulation of the mib method for three dimensional elliptic interface problems we develop a three dimensional 3d galerkin formulation of the matched interface and boundary mib method for solving elliptic partial differential equations pdes with discontinuous coefficients i e the elliptic interface problem the present approach builds up two sets of elements respectively on two extended subdomains which both include the interface as a result two sets of elements overlap each other near the interface fictitious solutions are defined on the overlapping part of the elements so that the differentiation operations of the original pdes can be discretized as if there was no interface the extra coefficients of polynomial basis functions which furnish the overlapping elements and solve the fictitious solutions are determined by interface jump conditions consequently the interface jump conditions are rigorously enforced on the interface the present method utilizes cartesian meshes to avoid the mesh generation in conventional finite element methods fems we implement the proposed mib galerkin method with three different elements namely rectangular prism element five tetrahedron element and six tetrahedron element which tile the cartesian mesh without introducing any new node the accuracy stability and robustness of the proposed 3d mib galerkin are extensively validated over three types of elliptic interface problems in the first type interfaces are analytically defined by level set functions these interfaces are relatively simple but admit geometric singularities in the second type interfaces are defined by protein surfaces which are truly arbitrarily complex the last type of interfaces originates from multiprotein complexes such as molecular motors near second order accuracy has been confirmed for all of these problems to our knowledge it is the first time for an fem to show a near second order convergence in solving the poisson equation with realistic protein surfaces additionally the present work offers the first known near second order accurate method for c 1 continuous or h 2 continuous solutions associated with a lipschitz continuous interface in a 3d setting

voting in fuzzy rule based systems for pattern classification problems in this paper we examine two kinds of voting schemes in fuzzy rule based systems for pattern classification problems one is the voting by multiple fuzzy if then rules in a single fuzzy rule based classification system the other is the voting by multiple fuzzy rule based classification systems first we discuss the voting by multiple fuzzy if then rules which is used as a fuzzy reasoning method for classifying input patterns in a single fuzzy rule based classification system the performance of the voting by multiple fuzzy if then rules is examined by computer simulations on the iris data next we discuss the voting by multiple fuzzy rule based classification systems three voting methods i e a perfect unison rule a majority rule and a weighted voting rule are used for combining classification results by multiple fuzzy rule based classification systems finally we compare the performance of fuzzy rule based classification systems with that of other classification methods such as neural networks and statistical techniques by computer simulations on some well known test problems

design of interpretable fuzzy rule based classifiers using spectral analysis with structure and parameters optimization this paper presents a design method for fuzzy rule based systems that performs data modeling consistently according to the symbolic relations expressed by the rules the focus of the model is the interpretability of the rules and the model s accuracy such that it can be used as tool for data understanding the number of rules is defined by the eigenstructure analysis of the similarity matrix which is computed from data the rule induction algorithm runs a clustering algorithm on the dataset and associates one rule to each cluster each rule is selected among all possible combinations of one dimensional fuzzy sets as the one nearest to a cluster s center the rules are weighted in order to improve the classifier performance and the weights are computed by a bounded quadratic optimization problem the model complexity is minimized in a structure selection search performed by a genetic algorithm that selects simultaneously the most representative subset of variables and also the number of fuzzy sets in the fuzzy partition of the selected variables the resulting model is evaluated on a set of benchmark datasets for classification problems the results show that the proposed approach produces accurate and yet compact fuzzy classifiers the resulting model is also evaluated from an interpretability point of view showing how the rule weights provide additional information to help data understanding and model exploitation

mib method for elliptic equations with multi material interfaces elliptic partial differential equations pdes are widely used to model real world problems due to the heterogeneous characteristics of many naturally occurring materials and man made structures devices and equipments one frequently needs to solve elliptic pdes with discontinuous coefficients and singular sources the development of high order elliptic interface schemes has been an active research field for decades however challenges remain in the construction of high order schemes and particularly for nonsmooth interfaces i e interfaces with geometric singularities the challenge of geometric singularities is amplified when they are originated from two or more material interfaces joining together or crossing each other high order methods for elliptic equations with multi material interfaces have not been reported in the literature to our knowledge the present work develops matched interface and boundary mib method based schemes for solving two dimensional 2d elliptic pdes with geometric singularities of multi material interfaces a number of new mib schemes are constructed to account for all possible topological variations due to two material interfaces the geometric singularities of three material interfaces are significantly more difficult to handle three new mib schemes are designed to handle a variety of geometric situations and topological variations although not all of them the performance of the proposed new mib schemes is validated by numerical experiments with a wide range of coefficient contrasts geometric singularities and solution types extensive numerical studies confirm the designed second order accuracy of the mib method for multi material interfaces including a case where the derivative of the solution diverges

roth s theorems for sets of matrices abstract it is shown that roth s theorems on the equivalence and similarity of block diagonal matrices hold for finite sets of matrices over a commutative ring

a decomposed immersed interface method for variable coefficient elliptic equations with non smooth and discontinuous solutions a second order accurate finite difference method is presented for solving two dimensional variable coefficient elliptic equations on cartesian grids in which the coefficients the source term the solution and its derivatives may be nonsmooth or discontinuous across an interface a correction term is introduced to the standard central difference stencil so that the numerical discretization is well defined across the interface we also propose a new method to approximate the correction term as part of the iterative procedure the method is easy to implement since the correction term only needs to be added to the right hand side of the system therefore the coefficient matrix remains symmetric and diagonally dominant allowing for most standard solvers to be used numerical examples show good agreements with exact solutions and the order of accuracy is comparable with other immersed interface methods

a level set approach for computing solutions to incompressible two phase flow a level set approach for computing solutions to incompressible two phase flow is presented the interface between the two fluids is considered to be sharp and is described as the zero level set of a smooth function a new treatment of the level set method allows us to efficiently maintain the level set function as the signed distance from the interface we never have to explicitly reconstruct or find the zero level set consequently we are able to handle arbitrarily complex topologies large density and viscosity ratios and surface tension on relatively coarse grids we use a second order projection method along with a second order upwinded procedure for advecting the momentum and level set equations we consider the motion of air bubbles and water drops we also compute flows with multiple fluids such as air oil and water

a weighting function for improving fuzzy classification systems performance this paper considers the automatic design of fuzzy rule based classification systems from labeled data the classification accuracy and interpretability of generated rules are of major importance in fuzzy classification systems we propose a weighting function for compatibility grade of patterns that improves the performance of fuzzy classification system without degrading the interpretability of fuzzy rules our approach divides the covering subspace of each fuzzy rule into two subdivisions based on a threshold any pattern with compatibility grade above this threshold should be classified truly so the weighting function enhances their association degree for patterns below threshold their compatibility grades remain unchanged the splitting threshold for each rule i e the compatibility grade of a specific pattern is found using distribution of patterns in the covering subspace of that rule we also show that how the proposed approach is applicable when fuzzy rules have certainty grades experiments on some well known data sets are used to evaluate the performance of our approach

interpolating stabilized moving least squares mls approximation for 2d elliptic interface problems abstract the main aim of the current paper is to propose a new truly meshless numerical technique to solve the one and two dimensional elliptic interface problems the employed meshless approach is based on a new class of mls approximation the numerical procedure is based on the interpolating stabilized mls ismls approximation the new shape functions that have been made by the ismls technique have the kronecker property thus the dirichlet boundary conditions can be applied directly in the current investigation we propose a meshless collocation method for solving elliptic interface problems based on the shape functions of ismls technique several test problems with sufficient complexity have been studied to check the accuracy and efficiency of the new numerical procedure moreover test problems show the acceptable accuracy and efficiency of the proposed scheme

on the fictitious domain and interpolation formulations of the matched interface and boundary mib method this work overcomes the difficulty of dealing with large curvatures in a high order matched interface and boundary mib method proposed for solving elliptic interface problems the mib method smoothly extends the solution across the interface so that standard high order central finite difference schemes can be used without the loss of accuracy one feature of the mib is that it disassociates the discretization of the elliptic equation from the enforcement of interface jump conditions the other is to make iterative use of only the lowest order jump conditions to determine the fictitious values on extended domains it is of arbitrarily high order in convergence in principle however its applicability was hindered by the lack of sufficiently many grid points to determine all the fictitious values required for high order schemes at the location where the curvature of the interface is relatively large we remove this obstacle by introducing a new concept the disassociation between the discretization and the domain extension we show that the improved mib method is robust for handling general irregular interfaces by extensive numerical experiments on the poisson equation and the helmholtz equation to better understand the mib method and other potential high order interface schemes we propose an alternative interpolation formulation of the mib method and show that the new formulation is essentially equivalent to the improved one

how to obtain an accurate gradient for interface problems abstract it is well known that the immersed interface method iim is second order accurate for interface problems but the accuracy of the first order derivatives or gradients for short is not so clear and is often assumed to be first order accurate in this paper new strategies based on iim are proposed for elliptic interface problems to compute the gradient at grid points both regular and irregular and at the interface from each side of the interface second order in 1d or nearly second order except a factor of log h convergence in 2d of the computed gradient is obtained with almost no extra cost and has been explained in intuition and verified by non trivial numerical tests numerical examples in one two dimensions radial and axis symmetric cases in polar and spherical coordinates are presented to validate the numerical methods and analysis

the immersed interface multigrid methods for interface problems new multigrid methods are developed for the maximum principle preserving immersed interface method applied to second order linear elliptic and parabolic pdes that involve interfaces and discontinuities for elliptic interface problems the multigrid solver developed in this paper works while some other multigrid solvers do not for linear parabolic equations we have developed the second order maximum principle preserving finite difference scheme in this paper we use the crank nicolson scheme to deal with the diffusion part and an explicit scheme for the first order derivatives numerical examples are also presented

membership margin based feature selection for mixed type and high dimensional data definition of membership margin based objective function weighting the antecedents of fuzzy if then rules to improve classification performance evaluate the feature importance within membership margin framework feature selection algorithm for mixed type and high dimensional data the effectiveness of the approach is proved through an extensive experimental study the present paper describes a new feature weighting method based on a membership margin distinctive properties of the proposed method include its capability to process problems characterized by mixed type data quantitative qualitative and interval as well as a huge number of features the key idea is to map simultaneously all the features of different types into a common space the membership space once all features are represented in a homogeneous space a feature weighting task can be performed in unified way this weighting approach is integrated here within a fuzzy classifier through a fuzzy rule weighted concept in order to improve its performance each antecedent fuzzy set in the fuzzy if then rule is weighted to characterize the importance of each proposition and therefore its corresponding feature weight estimation process is based on membership margin maximization to estimate a fuzzy weight of each feature in the membership space experiments on low and high dimensional real world datasets demonstrate that the proposed approach can improve significantly the performance of the fuzzy rule based as well as other state of the art classifiers and can even outperform classical feature weighting approaches in particular we show that this approach can yield meaningful results on two real world applications for cancer prognosis and industrial process diagnosis

adaptability interpretability and rule weights in fuzzy rule based systems this paper discusses interpretability in two main categories of fuzzy systems fuzzy rule based classifiers and interpolative fuzzy systems our goal is to show that the aspect of high level interpretability is more relevant to fuzzy classifiers whereas fuzzy systems employed in modeling and control benefit more from low level interpretability we also discuss the interpretability accuracy tradeoff and observe why various rule weighting schemes that have been brought into play to increase adaptability of fuzzy systems rather just increase computational overhead and seriously compromise interpretability of fuzzy systems

distributed representation of fuzzy rules and its application to pattern classification abstract this paper introduces the concept of distributed representation of fuzzy rules and applies it to classification problems distributed representation is implemented by superimposing many fuzzy rules corresponding to different fuzzy partitions of a pattern space this means that we simulatenously employ many fuzzy rule tables corresponding to different fuzzy partitions in fuzzy inference in order to apply distributed representation of fuzzy rules to pattern classification problems we first propose an algorithm to generate fuzzy rules from numerical data next we propose a fuzzy inference method using the generated fuzzy rules the classification power of distributed representation is compared with that of ordinary fuzzy rules which can be viewed as local representation

rule weight specification in fuzzy rule based classification systems this paper shows how the rule weight of each fuzzy rule can be specified in fuzzy rule based classification systems first we propose two heuristic methods for rule weight specification next the proposed methods are compared with existing ones through computer simulations on artificial numerical examples and real world pattern classification problems simulation results show that the proposed methods outperform the existing ones in the case of multiclass pattern classification problems with many classes

a coupling interface method for elliptic interface problems we propose a coupling interface method cim under cartesian grid for solving elliptic complex interface problems in arbitrary dimensions where the coefficients the source terms and the solutions may be discontinuous or singular across the interfaces it consists of a first order version cim1 and a second order version cim2 in one dimension the cim1 is derived from a linear approximation on both sides of the interface the method is extended to high dimensions through a dimension by dimension approach to connect information from each dimension a coupled equation for the first order derivatives is derived through the jump conditions in each coordinate direction the resulting stencil uses the standard 5 grid points in two dimensions and 7 grid points in three dimensions similarly the cim2 is derived from a quadratic approximation in each dimension in high dimensions a coupled equation for the principal second order derivatives u x k x k is derived through the jump conditions in each coordinate direction the cross derivatives are approximated by one side interpolation this approach reduces the number of grid points needed for one side interpolation the resulting stencil involves 8 grid points in two dimensions and 12 14 grid points in three dimensions a numerical study for the condition number of the resulting linear system of the cim2 in one dimension has been performed it is shown that the condition number has the same behavior as that of the discrete laplacian independent of the relative location of the interface in a grid cell further we also give a proof of the solvability of the coupling equations provided the curvature k of the interface satisfies kh

roth s removal rule revisited abstract the theory of companion matrices is used to give explicit representations for the matrices needed in roth s removal rule these are then used to give simple proofs for the cyclic decomposition theorem as well as for roth s similarity theorem for matrices over a field

a method of learning weighted similarity function to improve the performance of nearest neighbor the performance of nearest neighbor nn classifier is known to be sensitive to the distance or similarity function used in classifying a test instance another major disadvantage of nn is that it uses all training instances in the generalization phase this can cause slow execution speed and high storage requirement when dealing with large datasets in the past research many solutions have been proposed to handle one or both of the above problems in the scheme proposed in this paper we tackle both of these problems by assigning a weight to each training instance the weight of a training instance is used in the generalization phase to calculate the distance or similarity of a query pattern to that instance the basic nn classifier can be viewed as a special case of this scheme that treats all instances equally by assigning equal weight to all training instances using this form of weighted similarity measure we propose a learning algorithm that attempts to maximize the leave one out lv1 classification rate of the nn rule by adjusting the weights of the training instances at the same time the algorithm reduces the size of the training set and can be viewed as a powerful instance reduction technique an instance having zero weight is not used in the generalization phase and can be virtually removed from the training set we show that our scheme has comparable or better performance than some recent methods proposed in the literature for the task of learning the distance function and or prototype reduction

weighting fuzzy classification rules using receiver operating characteristics roc analysis in fuzzy rule based classification systems rule weight has often been used to improve the classification accuracy in past research a number of heuristic methods for rule weight specification have been proposed in this paper a method of fuzzy rule weight specification using receiver operating characteristic roc analysis is proposed in order to specify the weight of a fuzzy rule using 2 class roc analysis the threshold that the rule achieves its maximum accuracy is found this threshold is used as the weight of the rule the proposed method is compared with existing ones through computer simulations on some well known classification problems with continuous attributes simulation results show that the proposed method performs better than existing methods of fuzzy rule weight specification

elicitation of fuzzy association rules from positive and negative examples the aim of this paper is to provide a crystal clear insight into the true semantics of the measures of support and confidence that are used to assess rule quality in fuzzy association rule mining to achieve this we rely on two important pillars the identification of transactions in a database as positive or negative examples of a given association between attributes and the correspondence between measures of support and confidence on one hand and measures of compatibility and inclusion on the other hand in this way we remove the mystery from recently suggested quality measures for fuzzy association rules

on decomposing any matrix as a linear combination of three idempotents abstract in a recent article we gave a full characterization of matrices that can be decomposed as linear combinations of two idempotents with prescribed coefficients in this one we use those results to improve on a recent theorem of rabanovich we establish that every square matrix is a linear combination of three idempotents for an arbitrary coefficient field rather than just one of characteristic 0

an interface fitted mesh generator and virtual element methods for elliptic interface problems abstract a simple and efficient interface fitted mesh generation algorithm which can produce a semi structured interface fitted mesh in two and three dimensions quickly is developed in this paper elements in such interface fitted meshes are not restricted to simplices but can be polygons or polyhedra especially in 3d the polyhedra instead of tetrahedra can avoid slivers virtual element methods are applied to solve elliptic interface problems with solutions and flux jump conditions algebraic multigrid solvers are used to solve the resulting linear algebraic system numerical results are presented to illustrate the effectiveness of our method

a second order virtual node method for elliptic problems with interfaces and irregular domains in three dimensions we present a numerical method for the variable coefficient poisson equation in three dimensional irregular domains and with interfacial discontinuities the discretization embeds the domain and interface into a uniform cartesian grid augmented with virtual degrees of freedom to provide accurate treatment of jump and boundary conditions the matrix associated with the discretization is symmetric positive definite and equal to the standard 7 point finite difference poisson stencil away from embedded interfaces and boundaries numerical evidence suggests second order accuracy in the l norm our approach improves the treatment of dirichlet and jump constraints in the recent work of bedrossian et al 1 and introduces innovations necessary in three dimensions specifically we construct new constraint based lagrange multiplier spaces that significantly improve the conditioning of the associated linear system of equations we provide a method for cell local polyhedral approximation to the zero isocontour surface of a level set needed for three dimensional embedding and we show that the new lagrange multiplier spaces naturally lead to a class of easy to implement multigrid methods that achieve near optimal efficiency as shown by numerical examples for the specific case of a continuous poisson coefficient in interface problems we provide an expansive treatment of the construction of a particular solution that satisfies the value jump and flux jump constraints as in 1 this is used in a discontinuity removal technique that yields the standard 7 point stencil across the interface and only requires a modification to the right hand side of the linear system

matrix equivalence and isomorphism of modules abstract it is well known that if a and b are n m matrices over a ring r then coker a coker b does not imply a and b are equivalent an elementary proof is given that the implication does hold if 1 is in the stable range of r furthermore for certain r including commutative rings if a is block diagonal and b is block upper triangular with the same diagonal blocks as a then coker a coker b implies a and b are equivalent under a special equivalence this extends results of roth and gustafson as a corollary a theorem on decomposition of modules is obtained

accurate gradient approximation for complex interface problems in 3d by an improved coupling interface method most elliptic interface solvers become complicated for complex interface problems at those exceptional points where there are not enough neighboring interior points for high order interpolation such complication increases especially in three dimensions usually the solvers are thus reduced to low order accuracy in this paper we classify these exceptional points and propose two recipes to maintain order of accuracy there aiming at improving the previous coupling interface method 26 yet the idea is also applicable to other interface solvers the main idea is to have at least first order approximations for second order derivatives at those exceptional points recipe 1 is to use the finite difference approximation for the second order derivatives at a nearby interior grid point whenever this is possible recipe 2 is to flip domain signatures and introduce a ghost state so that a second order method can be applied this ghost state is a smooth extension of the solution at the exceptional point from the other side of the interface the original state is recovered by a post processing using nearby states and jump conditions the choice of recipes is determined by a classification scheme of the exceptional points the method renders the solution and its gradient uniformly second order accurate in the entire computed domain numerical examples are provided to illustrate the second order accuracy of the presently proposed method in approximating the gradients of the original states for some complex interfaces which we had tested previous in two and three dimensions and a real molecule 1d63 which is double helix shape and composed of hundreds of atoms

maximum principle preserving schemes for interface problems with discontinuous coefficients new finite difference methods using cartesian grids are developed for elliptic interface problems with variable discontinuous coefficients singular sources and nonsmooth or even discontinuous solutions the new finite difference schemes are constructed to satisfy the sign property of the discrete maximum principle using quadratic optimization techniques the methods are shown to converge under certain conditions using comparison functions the coefficient matrix of the resulting linear system of equations is an m matrix and is coupled with a multigrid solver numerical examples are also provided to show the efficiency of the proposed methods

multipolar aggregation operators in reasoning methods for fuzzy rule based classification systems the reasoning methods in fuzzy rule based classification systems are studied and their relation to multipolar aggregation operators is discussed we describe the change in the output of the classification system based on the change of the fuzzy reasoning method we propose a new fuzzy reasoning method based on multipolar fusion owa operator which can control the transition between the maximum rule method and the maximum vote method by a weighting vector this new method uses principles of owa operators and assigns the rule weight according to the strength of activation of the rule we compare different fuzzy reasoning methods in case studies

a second order cartesian method for the simulation of electropermeabilization cell models in this paper we present a new finite differences method to simulate electropermeabilization models like the model of neu and krassowska or the recent model of kavian et al these models are based on the evolution of the electric potential in a cell embedded in a conducting medium the main feature lies in the transmission of the voltage potential across the cell membrane the jump of the potential is proportional to the normal flux thanks to the well known kirchoff law an adapted scheme is thus necessary to accurately simulate the voltage potential in the whole cell notably at the membrane separating the cell from the outer medium we present a second order finite differences scheme in the spirit of the method introduced by cisternino and weynans for elliptic problems with immersed interfaces this is a cartesian grid method based on the accurate discretization of the fluxes at the interface through the use of additional interface unknowns the main novelty of our present work lies in the fact that the jump of the potential is proportional to the flux and therefore is not explicitly known the original use of interface unknowns makes it possible to discretize the transmission conditions with enough accuracy to obtain a second order spatial convergence we prove the second order spatial convergence in the stationary linear one dimensional case and the first order temporal convergence for the dynamical non linear model in one dimension we then perform numerical experiments in two dimensions that corroborate these results

high order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular sources this paper introduces a novel high order interface scheme the matched interface and boundary mib method for solving elliptic equations with discontinuous coefficients and singular sources on cartesian grids by appropriate use of auxiliary line and or fictitious points physical jump conditions are enforced at the interface unlike other existing interface schemes the proposed method disassociates the enforcement of physical jump conditions from the discretization of the differential equation under study to construct higher order interface schemes the proposed mib method bypasses the major challenge of implementing high order jump conditions by repeatedly enforcing the lowest order jump conditions the proposed mib method is of arbitrarily high order in principle in treating straight regular interfaces we construct mib schemes up to 16th order for more general elliptic problems with curved irregular interfaces and boundary up to 6th order mib schemes have been demonstrated by employing the standard high order finite difference schemes to discretize the laplacian the present mib method automatically reduces to the standard central difference scheme when the interface is absent the immersed interface method iim is regenerated for a comparison study of the proposed method the robustness of the mib method is verified against the large magnitude of the jump discontinuity across the interface the nature of high efficiency and low memory requirement of the mib method is extensively validated via solving various elliptic immersed interface problems in two and three dimensions ee dimensions

a kernel free boundary integral method for implicitly defined surfaces the kernel free boundary integral kfbi method is a structured grid method for general elliptic partial differential equations unlike the standard boundary integral method it avoids direct evaluation of volume and boundary integrals which needs to know analytical expressions for the integral kernels to evaluate a boundary or volume integral the kfbi method first solves a corrected interface problem on a structured grid and then the numerical solution on the structured grid is interpolated to get approximate values of the integral at points on the boundary selection of control points of the boundary plays a key role in the kfbi method since both the correction for the interface equations and the interpolation with the structured grid based solution involve calculation of tangential derivatives of boundary data while stability and efficiency of the numerical differentiation critically depend on the distribution of control points this work proposes a new point selection method based on an overlapping surface decomposition of the boundary which is implicitly defined by a level set function the points selected are intersection points of the boundary with the grid lines of an underlying cartesian grid by the method the interpolation stencils can be easily chosen to be locally uniform along a coordinate axis in two space dimensions and locally uniform on a coordinate plane in three space dimensions which allows efficient numerical differentiation and boundary reconstruction representation an additional equilibrating process of boundary data further guarantees stable numerical differentiation numerical results demonstrating the method with examples in both two and three space dimensions are presented

extrinsic meshfree approximation using asymptotic expansion for interfacial discontinuity of derivative a sharp meshfree approximation for derivative discontinuities across arbitrary interfaces is proposed the interface can be arbitrarily located in a domain in which nodes are distributed uniformly or irregularly the proposed meshfree approximations consist of two parts singular and regular the moving least square meshfree approximation is used together with the local wedge function as basis functions the approximations for discontinuities are applied in a meshfree point collocation method to obtain solutions of the poisson problem with a layer delta source on the interface and second order elliptic problems with discontinuous coefficients and or the singular layer sources along the interface the numerical calculations show that this method has good performance even on irregular node models

a note on immersed interface method for three dimensional elliptic equations abstract the immersed interface method proposed by leveque and li 1 is extended to three dimensional elliptic equations of the form x u x x u x f x we study the situation in which there is an irregular interface surface s contained in the solution domain across which and f may be discontinuous or even singular as a result the solution u will usually be nonsmooth or even discontinuous a finite difference approach with a uniform cartesian grid is used in the discretization local truncation error analysis is performed to estimate the accuracy of the numerical solution

the immersed interface method for elliptic equations with discontinuous coefficients and singular sources the authors develop finite difference methods for elliptic equations of the form nabla cdot beta x nabla u x kappa x u x f x in a region omega in one or two space dimension

roth s solvability criteria for the matrix equations ax xˆb c and x axˆb c over the skew field of quaternions with an involutive automorphism q qˆ abstract the matrix equation a x x b c has a solution if and only if the matrices a c 0 b and a 0 0 b are similar this criterion was proved over a field by w e roth 1952 and over the skew field of quaternions by huang liping 1996 h k wimmer 1988 proved that the matrix equation x a x b c over a field has a solution if and only if the matrices a c 0 i and i 0 0 b are simultaneously equivalent to a 0 0 i and i 0 0 b we extend these criteria to the matrix equations a x x b c and x a x b c over the skew field of quaternions with a fixed involutive automorphism q q

a level set formulation of eulerian interface capturing methods for incompressible fluid flows a level set formulation is derived for incompressible immiscible navier stokes equations separated by a free surface the interface is identified as the zero level set of a smooth function eulerian finite difference methods based on this level set formulation are proposed these methods are robust and efficient and are capable of computing interface singularities such as merging and reconnection numerical experiments are presented to demonstrate the effectiveness of the methods

combining gp operators with sa search to evolve fuzzy rule based classifiers abstract the genotype phenotype encoding of fuzzy rule bases in ga along with their corresponding crossover and mutation operators can be used by other search schemes improving the behavior of these last ones as a practical consequence of this a simulated annealing based method for inducting both parameters and structure of a fuzzy classifier has been developed the adjacency operator in sa has been replaced with a macromutation taken from tree shaped genotype gas we will show that results of sa search are similar to those of gp in both the efficiency of the learned classifiers and in its linguistic interpretability while the memory consumption of the learning process is lower

comparison of taylor finite difference and window finite difference and their application in fdtd the finite difference time domain fdtd method is an important tool in numerical electromagnetic simulation there are many ways to construct a finite difference approximation such as the taylor series expansion theorem the filtering theory etc this paper aims to provide the comparison between the taylor finite difference tfd scheme based on the taylor series expansion theorem and the window finite difference wfd scheme based on the filtering theory their properties have been examined in detail separately in addition the formula of the generalized finite difference gfd scheme is presented which can include both the tfd scheme and the wfd scheme furthermore their application in the numerical solution of maxwell s equations is presented the formulas for the stability criterion and the numerical dispersion relation are derived and analyzed in order to evaluate their performance more accurately a new definition of error is presented upon it the effect of several factors including the grid resolution the courant number and the aspect ratio of the cell on the performance of the numerical dispersion is examined

spectral element method for three dimensional elliptic problems with smooth interfaces abstract in this paper we propose a least squares spectral element method for three dimensional elliptic interface problems the differentiability estimates and the main stability theorem using non conforming spectral element functions are proven the proposed method is free from any kind of first order reformulation a suitable preconditioner is constructed with help of the regularity estimate and proposed stability estimates which is used to control the condition number we show that these preconditioners are spectrally equivalent to the quadratic forms by which we approximate them we obtain the error estimates which show the exponential accuracy of the method numerical results are obtained for both straight and curved interfaces to show the efficiency of the proposed method

a dimension by dimension splitting immersed interface method for heat conduction equation with interfaces the numerical method proposed in this paper is an improvement of the adi method by li and mayo 1994 the proposed method is unconditionally stable for both two and three dimensional heat conduction interface problems while li s adi method is only stable for two dimensional problems the method is a modification of a locally one dimensional lod difference scheme with correction term added to the right hand side of the standard lod difference scheme at irregular points the correction term is determined so that the local truncation error is of order o h at irregular points then the method is two order convergent in both time and space directions numerical examples show good agreement with exact solutions and confirm the order of convergence and stability

a fast iterative algorithm for elliptic interface problems a fast second order accurate iterative method is proposed for the elliptic equation grad cdot beta x y grad u f x y in a rectangular region omega in two space dimensions we assume that there is an irregular interface across which the coefficient beta the solution u and its derivatives and or the source term f may have jumps we are especially interested in the cases where the coefficients beta are piecewise constant and the jump in beta is large the interface may or may not align with an underlying cartesian grid the idea in our approach is to precondition the differential equation before applying the immersed interface method proposed by leveque and li siam j numer anal 4 1994 pp 1019 1044 in order to take advantage of fast poisson solvers on a rectangular region an intermediate unknown function the jump in the normal derivative across the interface is introduced our discretization is equivalent to using a second order difference scheme for a corresponding poisson equation in the region and a second order discretization for a neumann like interface condition thus second order accuracy is guaranteed a gmres iteration is employed to solve the schur complement system derived from the discretization a new weighted least squares method is also proposed to approximate interface quantities from a grid function numerical experiments are provided and analyzed the number of iterations in solving the schur complement system appears to be independent of both the jump in the coefficient and the mesh size

improving classification performance using fuzzy mlp and two level selective partitioning of the feature space a fuzzy mlp model developed by one of the authors is used for obtaining selective two level partitioning of the feature space in order to improve its classification performance the model can handle uncertainty and or impreciseness in the input as well as the output the input to the network is modelled in terms of linguistic pi sets whose centres and radii along the feature axes in each partition are generated automatically from the distribution of the training data the performance of the model at the end of the first stage is used as a criterion for guiding the selection of the appropriate partition to be subdivided at the second stage in order to improve the effectiveness of the model a comparative study of the performance of the two level technique with other methods viz the conventional mlp linear discriminant analysis and the k nearest neighbours algorithms is also provided to demonstrate its superiority

a proposed method for learning rule weights in fuzzy rule based classification systems in fuzzy rule based classification systems frbcss rule weighting has often been used as a simple mechanism to tune the classifier in past research a number of heuristic rule weight specification methods have been proposed for this purpose a learning algorithm based on reward and punishment has also been proposed to adjust the weights of each fuzzy rule in the rule base in this paper a new method of learning rule weight in frbcss is proposed the method can be used when single winner or weighted vote methods of reasoning is used compared with reward and punishment scheme the proposed method is much faster and more effective another advantage of the proposed method is that during the learning process redundant rules are removed i e by setting their weights to zero the final rule base usually contains much fewer rules than the initial one this feature is very useful since a compact rule base is usually desired from the point of efficiency and interpretability a number of uci data sets are used to assess the performance of the proposed method in comparison with reward and punishment scheme

roth s theorems and decomposition of modules it is shown that there is a connection between roth s theorems on similarity and equivalence of block triangular matrices and decomposition of modules the module property is that if m n mn then n is a summand of m this holds for any commutative ring if m is finitely presented new proofs of roth s theorems are given for commutative rings some results are established in the noncommutative case

the immersed interface method for the navier stokes equations with singular forces abstract peskin s immersed boundary method has been widely used for simulating many fluid mechanics and biology problems one of the essential components of the method is the usage of certain discrete delta functions to deal with singular forces along one or several interfaces in the fluid domain however the immersed boundary method is known to be first order accurate and usually smears out the solutions in this paper we propose an immersed interface method for the incompressible navier stokes equations with singular forces along one or several interfaces in the solution domain the new method is based on a second order projection method with modifications only at grid points near or on the interface from the derivation of the new method we expect fully second order accuracy for the velocity and nearly second order accuracy for the pressure in the maximum norm including those grid points near or on the interface this has been confirmed in our numerical experiments furthermore the computed solutions are sharp across the interface nontrivial numerical results are provided and compared with the immersed boundary method meanwhile a new version of the immersed boundary method using the level set representation of the interface is also proposed in this paper

a low numerical dissipation immersed interface method for the compressible navier stokes equations a numerical method to solve the compressible navier stokes equations around objects of arbitrary shape using cartesian grids is described the approach considered here uses an embedded geometry representation of the objects and approximate the governing equations with a low numerical dissipation centered finite difference discretization the method is suitable for compressible flows without shocks and can be classified as an immersed interface method the objects are sharply captured by the cartesian mesh by appropriately adapting the discretization stencils around the irregular grid nodes located around the boundary in contrast with available methods no jump conditions are used or explicitly derived from the boundary conditions although a number of elements are adopted from previous immersed interface approaches a new element in the present approach is the use of the summation by parts formalism to develop stable non stiff first order derivative approximations at the irregular grid points second order derivative approximations as those appearing in the transport terms can be stiff when irregular grid points are located too close to the boundary this is addressed using a semi implicit time integration method moreover it is shown that the resulting implicit equations can be solved explicitly in the case of constant transport properties convergence studies are performed for a rotating cylinder and vortex shedding behind objects of varying shapes at different mach and reynolds numbers

a boundary condition capturing method for poisson s equation on irregular domains interfaces have a variety of boundary conditions or jump conditions that need to be enforced the ghost fluid method gfm was developed to capture the boundary conditions at a contact discontinuity in the inviscid euler equations and has been extended to treat more general discontinuities such as shocks detonations and deflagrations and compressible viscous flows in this paper a similar boundary condition capturing approach is used to develop a new numerical method for the variable coefficient poisson equation in the presence of interfaces where both the variable coefficients and the solution itself may be discontinuous this new method is robust and easy to implement even in three spatial dimensions furthermore the coefficient matrix of the associated linear system is the standard symmetric matrix for the variable coefficient poisson equation in the absence of interfaces allowing for straightforward application of standard black box solvers

immersed interface method for elliptic equations based on a piecewise second order polynomial in this paper we develop finite difference methods for elliptic equations in a domain w r d d 1 2 within the region w we suppose there is an irregular surface c of codimension 1 hereafter called an interface across which the function u or some of its derivatives are known to be discontinuous we use uniform grid and a piecewise second order polynomial to approximate u then we get a second order method for these problems at last we give several examples to show the correctness and efficiency of the scheme

genetic feature selection in a fuzzy rule based classification system learning process for high dimensional problems abstract the inductive learning of a fuzzy rule based classification system frbcs is made difficult by the presence of a large number of features that increases the dimensionality of the problem being solved the difficulty comes from the exponential growth of the fuzzy rule search space with the increase in the number of features considered in the learning process in this work we present a genetic feature selection process that can be integrated in a multistage genetic learning method to obtain in a more efficient way frbcss composed of a set of comprehensible fuzzy rules with high classification ability the proposed process fixes a priori the number of selected features and therefore the size of the search space of candidate fuzzy rules the experimentation carried out using sonar example base shows a significant improvement on simplicity precision and efficiency achieved by adding the proposed feature selection processes to the multistage genetic learning method or to other learning methods

a continuum method for modeling surface tension abstract a new method for modeling surface tension effects on fluid motion has been developed interfaces between fluids of different properties or colors are represented as transition regions of finite thickness across which the color variable varies continuously at each point in the transition region a force density is defined which is proportional to the curvature of the surface of constant color at that point it is normalized so that the conventional description of surface tension on an interface is recovered when the ratio of local transition region thickness to local radius of curvature approaches zero the continuum method eliminates the need for interface reconstruction simplifies the calculation of surface tension enables accurate modeling of two and three dimensional fluid flows driven by surface forces and does not impose any modeling restrictions on the number complexity or dynamic evolution of fluid interfaces having surface tension computational results for two dimensional flows are given to illustrate the properties of the method

analysis of a one dimensional model for the immersed boundary method numerical methods are studied for the one dimensional heat equation with a singular forcing term u t u xx c t delta x alpha t the delta function delta x is replaced by a discrete approximation d h x and the resulting equation is solved by a crank nicolson method on a uniform grid the accuracy of this method is analyzed for various choices of d h the case where c t is specified and also the case where c is determined implicitly by a constraint on the solution at the point a are studied these problems serve as a model for the immersed boundary method of peskin for incompressible flow problems in irregular regions some insight is gained into the accuracy that can be achieved and the importance of choosing appropriate discrete delta functions

solving elliptic interface problems with jump conditions on cartesian grids abstract we present a simple numerical algorithm for solving elliptic equations where the diffusion coefficient the source term the solution and its flux are discontinuous across an irregular interface the algorithm produces second order accurate solutions and first order accurate gradients in the l norm on cartesian grids the condition number is bounded regardless of the ratio of the diffusion constant and scales like that of the standard 5 point stencil approximation on a rectangular grid with no interface numerical examples are given in two and three spatial dimensions

a front tracking method for viscous incompressible multi fluid flows abstract a method to simulate unsteady multi fluid flows in which a sharp interface or a front separates incompressible fluids of different density and viscosity is described the flow field is discretized by a conservative finite difference approximation on a stationary grid and the interface is explicitly represented by a separate unstructured grid that moves through the stationary grid since the interface deforms continuously it is necessary to restructure its grid as the calculations proceed in addition to keeping the density and viscosity stratification sharp the tracked interface provides a natural way to include surface tension effects both two and three dimensional full numerical simulations of bubble motion are presented

three objective genetics based machine learning for linguisitc rule extraction abstract this paper shows how a small number of linguistically interpretable fuzzy rules can be extracted from numerical data for high dimensional pattern classification problems one difficulty in the handling of high dimensional problems by fuzzy rule based systems is the exponential increase in the number of fuzzy rules with the number of input variables another difficulty is the deterioration in the comprehensibility of fuzzy rules when they involve many antecedent conditions our task is to design comprehensible fuzzy rule based systems with high classification ability this task is formulated as a combinatorial optimization problem with three objectives to maximize the number of correctly classified training patterns to minimize the number of fuzzy rules and to minimize the total number of antecedent conditions we show two genetic algorithm based approaches one is rule selection where a small number of linguistically interpretable fuzzy rules are selected from a large number of prespecified candidate rules the other is fuzzy genetics based machine learning where rule sets are evolved by genetic operations these two approaches search for non dominated rule sets with respect to the three objectives

discontinuous bubble scheme for elliptic problems with jumps in the solution abstract we propose a new numerical method to solve an elliptic problem with jumps both in the solution and derivative along an interface by considering a suitable function which has the same jumps as the solution we transform the problem into one without jumps then we apply the immersed finite element method in which we allow uniform meshes so that the interface may cut through elements to discretize the problem as introduced in 1 2 3 some convenient way of approximating the jumps of the solution by piecewise linear functions is suggested our method can also handle the case when the interface passes through grid points we believe this paper presents the first resolution of such cases numerical experiments for various problems show second order convergence in l2 and first order in h1 norms moreover the convergence order is very robust for all problems tested

a neuro fuzzy method to learn fuzzy classification rules from data abstract neuro fuzzy systems have recently gained a lot of interest in research and application neuro fuzzy models as we understand them are fuzzy systems that use local learning strategies to learn fuzzy sets and fuzzy rules neuro fuzzy techniques have been developed to support the development of e g fuzzy controllers and fuzzy classifiers in this paper we discuss a learning method for fuzzy classification rules the learning algorithm is a simple heuristics that is able to derive fuzzy rules from a set of training data very quickly and tunes them by modifying parameters of membership functions our approach is based on nefclass a neuro fuzzy model for pattern classification we also discuss some results obtained by our software implementation of nefclass which is freely available on the internet

the explicit jump immersed interface method finite difference methods for pdes with piecewise smooth solutions many boundary value problems bvps or initial bvps have nonsmooth solutions with jumps along lower dimensional interfaces the explicit jump immersed interface method ejiim was developed following li s fast iterative immersed interface method fiiim recognizing that the foundation for the efficient solution of many such problems is a good solver for elliptic bvps ejiim generalizes the class of problems for which fiiim is applicable it handles interfaces between constant and variable coefficients and extends the immersed interface method iim to bvps on irregular domains with neumann and dirichlet boundary conditions proofs of second order convergence for a one dimensional 1d problem with piecewise constant coefficients and for two dimensional 2d problems with singular sources are given other problems are reduced to the singular sources case with additional equations determining the source strengths the advantages of ejiim are high quality of solutions even on coarse grids and easy adaptation to many problems with complicated geometries while still maintaining the efficiency of the fiiim

three dimensional matched interface and boundary mib method for treating geometric singularities this paper reports the three dimensional 3d generalization of our previous 2d higher order matched interface and boundary mib method for solving elliptic equations with discontinuous coefficients and non smooth interfaces new mib algorithms that make use of two sets of interface jump conditions are proposed to remove the critical acute angle constraint of our earlier mib scheme for treating interfaces with sharp geometric singularities such as sharp edges sharp wedges and sharp tips the resulting 3d mib schemes are of second order accuracy for arbitrarily complex interfaces with sharp geometric singularities of fourth order accuracy for complex interfaces with moderate geometric singularities and of sixth order accuracy for curved smooth interfaces a systematical procedure is introduced to make the mib matrix optimally symmetric and banded by appropriately choosing auxiliary grid points consequently the new mib linear algebraic equations can be solved with fewer number of iterations the proposed mib method makes use of cartesian grids standard finite difference schemes lowest order interface jump conditions and fictitious values the interface jump conditions are enforced at each intersecting point of the interface and mesh lines to overcome the staircase phenomena in finite difference approximation while a pair of fictitious values are determined along a mesh at a time an iterative procedure is proposed to determine all the required fictitious values for higher order schemes by repeatedly using the lowest order jump conditions a variety of mib techniques are developed to overcome geometric constraints the essential strategy of the mib method is to locally reduce a 2d or a 3d interface problem into 1d like ones the proposed mib method is extensively validated in terms of the order of accuracy the speed of convergence the number of iterations and cpu time numerical experiments are carried out to complex interfaces including the molecular surfaces of a protein a missile interface and van der waals surfaces of intersecting spheres

an augmented matched interface and boundary mib method for solving elliptic interface problem abstract in this paper a second order accurate augmented matched interface and boundary mib is introduced for solving two dimensional 2d elliptic interface problems with piecewise constant coefficients the augmented mib seamlessly combines several key ingredients of the standard mib augmented immersed interface method iim and explicit jump iim to produce a new fast interface algorithm based on the mib zeroth and first order jump conditions are enforced across an arbitrarily curved interface which yields fictitious values on cartesian nodes near the interface by using such fictitious values a simple procedure is proposed to reconstruct cartesian derivative jumps as auxiliary variables and couple them with the jump corrected taylor series expansions which allow us to restore the order of the central difference across the interface to two moreover by using the schur complement to disassociate the algebraic computation of auxiliary variables and function values the discrete laplacian can be efficiently inverted by using the fast fourier transform fft it is found in our numerical experiments that the iteration number in solving the auxiliary system weakly depends on the mesh size as a consequence the total computational cost of the augmented mib is about o n 2 log n for a cartesian grid with dimension n n in 2d therefore the augmented mib outperforms the classical mib in all cases by significantly reducing the cpu time while keeping the same second order of accuracy in dealing with complicated interfaces

approximative fuzzy rules approaches for classification with hybrid ga techniques abstract in this paper the use of different methods from the fuzzy modeling field for classification tasks is evaluated and the potential of their integration in producing better classification results is investigated the methods considered approximative in their nature consider different integrations of techniques with an initial rule generation step and a following rule tuning approach using different evolutionary algorithms we analyse the adaptation of existing techniques in the fuzzy modeling context for the classification problem and the integration of these techniques in order to improve the classifiers performance finally a genetic algorithm ga for translation from approximative rules to similar descriptive ones trying to preserve the accuracy of the approximative classifier is presented the classical iris and cancer data set are used throughout the evaluation process to form a common ground for comparison and performance analysis