analysis of the parareal time parallel time integration method the parareal algorithm is a method to solve time dependent problems parallel in time it approximates parts of the solution later in time simultaneously to parts of the solution earlier in time in this paper the relation of the parareal algorithm to space time multigrid and multiple shooting methods is first briefly discussed the focus of the paper is on new convergence results that show superlinear convergence of the algorithm when used on bounded time intervals and linear convergence for unbounded intervals

a time domain decomposition iterative method for the solution of distributed linear quadratic optimal control problems we study a class of time domain decomposition based methods for the numerical solution of large scale linear quadratic optimal control problems our methods are based on a multiple shooting reformulation of the linear quadratic optimal control problem as a discrete time optimal control dtoc problem the optimality conditions for this dtoc problem lead to a linear block tridiagonal system the diagonal blocks are invertible and are related to the original linear quadratic optimal control problem restricted to smaller time subintervals this motivates the application of block gauss seidel gs type methods for the solution of the block tridiagonal systems numerical experiments show that the spectral radii of the block gs iteration matrices are larger than one for typical applications but that the eigenvalues of the iteration matrices decay to zero fast hence while the gs method is not expected to convergence for typical applications it can be effective as a preconditioner for krylov subspace methods this is confirmed by our numerical tests a byproduct of this research is the insight that certain instantaneous control techniques can be viewed as the application of one step of the forward block gs method applied to the dtoc optimality system

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

fuzzy theoretic approach to signals and systems static systems abstract fuzzy theoretic approach to signals and systems assumes all system variables and parameters as uncertain i e being characterized by membership functions develops a mathematical theory for analytically determining the membership functions on system variables and parameters derives algorithms for estimating the parameters of membership functions and establishes robustness and convergence properties of the estimation algorithms the membership functions are analytically determined via solving a variational optimization problem that maximizes the over uncertainties averaged log membership of the observed data around an initial guess this paper develops the analytical fuzzy theory for the particular case of a multi input single output static system affected by noises the theory facilitates designing an adaptive filtering algorithm the robustness of the adaptive filtering algorithm is proved theoretically via a mathematical analysis numerical experiments further demonstrate the robustness of the filtering algorithm a comparison of the algorithm with the state of art methods is made using the practical biomedical applications related to the modeling and analysis of heart rate signals for assessing the physiological state of an individual

improved constructions of frameproof codes frameproof codes are used to preserve the security in the context of coalition when fingerprinting digital data let mc l q be the largest cardinality of a q ary c frameproof code of length l and rc l limq mc l q q l c it has been determined by blackburn that rc l 1 when l 1 mod c rc l 2 when c 2 and l is even and r3 5 5 3 in this paper we give a recursive construction for c frameproof codes of length l with respect to the alphabet size q as applications of this construction we establish the existence results for q ary c frameproof codes of length c 2 and size c 2 c q 1 2 1 for all odd q when c 2 and for all q 4 when c 3 furthermore we show that rc c 2 c 2 c meeting the upper bound given by blackburn for all integers c such that c 1 is a prime power

multiobjective evolutionary optimization of type 2 fuzzy rule based systems for financial data classification classification techniques are becoming essential in the financial world for reducing risks and possible disasters managers are interested in not only high accuracy but in interpretability and transparency as well it is widely accepted now that the comprehension of how inputs and outputs are related to each other is crucial for taking operative and strategic decisions furthermore inputs are often affected by contextual factors and characterized by a high level of uncertainty in addition financial data are usually highly skewed toward the majority class with the aim of achieving high accuracies preserving the interpretability and managing uncertain and unbalanced data this paper presents a novel method to deal with financial data classification by adopting type 2 fuzzy rule based classifiers frbcs generated from data by a multiobjective evolutionary algorithm moea the classifiers employ an approach denoted as scaled dominance for defining rule weights in such a way to help minority classes to be correctly classified in particular we have extended paes rcs an moea based approach to learn concurrently the rule and data bases of frbcs for managing both interval type 2 fuzzy sets and unbalanced datasets to the best of our knowledge this is the first work that generates type 2 frbcs by concurrently maximizing accuracy and minimizing the number of rules and the rule length with the objective of producing interpretable models of real world skewed and incomplete financial datasets the rule bases are generated by exploiting a rule and condition selection rcs approach which selects a reduced number of rules from a heuristically generated rule base and a reduced number of conditions for each selected rule during the evolutionary process the weight associated with each rule is scaled by the scaled dominance approach on the fuzzy frequency of the output class in order to give a higher weight to the minority class as regards the data base learning the membership function parameters of the interval type 2 fuzzy sets used in the rules are learned concurrently to the application of rcs unbalanced datasets are managed by using in addition to complexity selectivity and specificity as objectives of the moea rather than only the classification rate we tested our approach named it2 paes rcs on 11 financial datasets and compared our results with the ones obtained by the original paes rcs with three objectives and with and without scaled dominance the frbcs fuzzy association rule based classification model for high dimensional dataset farc hd and fuzzy unordered rules induction algorithm furia the classical c4 5 decision tree algorithm and its cost sensitive version using nonparametric statistical tests we will show that it2 paes rcs generates frbcs with on average accuracy statistically comparable with and complexity lower than the ones generated by the two versions of the original paes rcs further the frbcs generated by farc hd and furia and the decision trees computed by c4 5 and its cost sensitive version despite the highest complexity result to be less accurate than the frbcs generated by it2 paes rcs finally we will highlight how these frbcs are easily interpretable by showing and discussing one of them

interpretability of linguistic fuzzy rule based systems an overview of interpretability measures linguistic fuzzy modelling developed by linguistic fuzzy rule based systems allows us to deal with the modelling of systems by building a linguistic model which could become interpretable by human beings linguistic fuzzy modelling comes with two contradictory requirements interpretability and accuracy in recent years the interest of researchers in obtaining more interpretable linguistic fuzzy models has grown whereas the measures of accuracy are straightforward and well known interpretability measures are difficult to define since interpretability depends on several factors mainly the model structure the number of rules the number of features the number of linguistic terms the shape of the fuzzy sets etc moreover due to the subjectivity of the concept the choice of appropriate interpretability measures is still an open problem in this paper we present an overview of the proposed interpretability measures and techniques for obtaining more interpretable linguistic fuzzy rule based systems to this end we will propose a taxonomy based on a double axis complexity versus semantic interpretability considering the two main kinds of measures and rule base versus fuzzy partitions considering the different components of the knowledge base to which both kinds of measures can be applied the main aim is to provide a well established framework in order to facilitate a better understanding of the topic and well founded future works

genetic training instance selection in multiobjective evolutionary fuzzy systems a coevolutionary approach when dealing with datasets that are characterized by a large number of instances multiobjective evolutionary learning moel of fuzzy rule based systems frbss suffers from high computational costs mainly because of the fitness evaluation the use of a reduced set of representative instances in place of the overall training set ts would considerably lessen the computational effort even though a large number of papers have proposed instance selection approaches mainly in classification problems how this selection should be performed especially in the context of regression is still an open issue in this paper we tackle the instance selection problem in the framework of moel of frbss through a coevolutionary approach in the execution of the moel periodically a single objective genetic algorithm soga evolves a population of reduced tss the soga aims to maximize a purposely defined index which measures how much the pareto fronts computed by using respectively the reduced ts and the overall ts are close to each other the closer the fronts the more the reduced ts is representative of the overall ts during the execution of the moel the rule base and the membership function parameters of the fuzzy sets are concurrently learned by maximizing the accuracy and minimizing the complexity we tested our approach on 12 large datasets we adopted reduced tss composed of 5 10 and 20 of the overall ts using nonparametric statistical tests we verified that with 10 and 20 of the overall ts the pareto front approximations that are generated by our coevolutionary approach are comparable with the ones generated by applying the moel with the overall ts although the coevolution allows us to save up to 86 36 of the execution time in addition the analysis of the behavior of three representative solutions on the test set highlights that the use of the reduced tss does not affect the generalization capabilities of the generated frbss

asymptotic bounds on frameproof codes we study the asymptotic behavior of frameproof codes some lower bounds are derived from the theory of error correcting codes in particular the lower bound obtained directly by applying algebraic geometry codes is improved by employing the jacobian group structure of algebraic curves

stable parareal in time method for first and second order hyperbolic systems the parareal in time algorithm allows one to perform parallel simulations of time dependent problems this algorithm has been implemented on many types of time dependent problems with some success recent contributions have allowed one to extend the domain of application of the parareal in time algorithm so as to handle long time simulations of hamiltonian systems this improvement has managed to avoid the fatally large lack of accuracy of the plain parareal in time algorithm which does not conserve invariant quantities a somewhat similar difficulty occurs for problems where the solution lacks regularity either initially or during the evolution as is the case for hyperbolic systems of conservation laws in this paper we identify the reasons for instabilities of the parareal in time algorithm and propose a simple way to cure them we use the new method to solve a linear wave equation and a nonlinear burgers equation the results illustrate the stability of this variant of the parareal in time algorithm

an adaptive parareal algorithm in this paper we consider the problem of accelerating the numerical simulation of time dependent problems by time domain decomposition the available algorithms enabling such decompositions present severe efficiency limitations and are an obstacle for the solution of large scale and high dimensional problems our main contribution is the improvement of the parallel efficiency of the parareal in time method the parareal method is based on combining predictions made by a numerically inexpensive solver with coarse physics and or coarse resolution with corrections coming from an expensive solver with high fidelity physics and high resolution at convergence the algorithm provides a solution that has the fine solver s high fidelity physics and high resolution in the classical version the fine solver has a fixed high accuracy which is the major obstacle to achieve a competitive parallel efficiency in this paper we develop an adaptive variant that overcomes this obstacle by dynamically increasing the accuracy of the fine solver across the parareal iterations we theoretically show that the parallel efficiency becomes very competitive in the ideal case where the cost of the coarse solver is small thus proving that the only remaining factors impeding full scalability become the cost of the coarse solver and communication time the developed theory has also the merit of setting a general framework to understand the success of several extensions of parareal based on iteratively improving the quality of the fine solver and re using information from previous parareal steps we illustrate the actual performance of the method in stiff odes which are a challenging family of problems since the only mechanism for adaptivity is time and efficiency is affected by the cost of the coarse solver

selecting fuzzy if then rules for classification problems using genetic algorithms this paper proposes a genetic algorithm based method for selecting a small number of significant fuzzy if then rules to construct a compact fuzzy classification system with high classification power the rule selection problem is formulated as a combinatorial optimization problem with two objectives to maximize the number of correctly classified patterns and to minimize the number of fuzzy if then rules genetic algorithms are applied to this problem a set of fuzzy if then rules is coded into a string and treated as an individual in genetic algorithms the fitness of each individual is specified by the two objectives in the combinatorial optimization problem the performance of the proposed method for training data and test data is examined by computer simulations on the iris data of fisher

integration of an index to preserve the semantic interpretability in the multiobjective evolutionary rule selection and tuning of linguistic fuzzy systems in this paper we propose an index that helps preserve the semantic interpretability of linguistic fuzzy models while a tuning of the membership functions mfs is performed the proposed index is the aggregation of three metrics that preserve the original meanings of the mfs as much as possible while a tuning of their definition parameters is performed additionally rule selection mechanisms can be used to reduce the model complexity which involves another important interpretability aspect to this end we propose a postprocessing multiobjective evolutionary algorithm that performs rule selection and tuning of fuzzy rule based systems with three objectives accuracy semantic interpretability maximization and complexity minimization we tested our approach on nine real world regression datasets in order to analyze the interaction between the fuzzy rule selection approach and the tuning approach these are also individually proved in a multiobjective framework and compared with their respective single objective counterparts we compared the different approaches by applying nonparametric statistical tests for pairwise and multiple comparisons taking into consideration three representative points from the obtained pareto fronts in the case of the multiobjective based approaches results confirm the effectiveness of our approach and a wide range of solutions is obtained which are not only more interpretable but are also more accurate

constructing fuzzy models with linguistic integrity from numerical data afreli algorithm this paper presents an algorithm to extract rules re lating input output data and including prior knowledge the rules are created in the environment of fuzzy systems the fuzzy sets describing the system are constructed within a framework of lin guistic integrity to guarantee its interpretability in the linguistic context two algorithms are presented in this paper the main al gorithm is the autonomous fuzzy rule extractor with linguistic in tegrity afreli this algorithm is complemented with the use of the fuzion algorithm created to merge consecutive member ship functions while guaranteeing the distinguish ability between fuzzy sets comparisons with other proposed methods show a good tradeoff between accuracy and interpretability

a cloned linguistic decision tree controller for real time path planning in hostile environments abstract the idea of a cloned controller to approximate optimised control algorithms in a real time environment is introduced a cloned controller is demonstrated using linguistic decision trees ldts to clone a model predictive controller mpc based on mixed integer linear programming milp for unmanned aerial vehicle uav path planning through a hostile environment modifications to the ldt algorithm are proposed to account for attributes with circular domains such as bearings and discontinuous output functions the cloned controller is shown to produce near optimal paths whilst significantly reducing the decision period further investigation shows that the cloned controller generalises to the multi obstacle case although this can lead to situations far outside of the training dataset and consequently result in decisions with a high level of uncertainty a modification to the algorithm to improve the performance in regions of high uncertainty is proposed and shown to further enhance generalisation the resulting controller combines the high performance of mpc milp with the rapid response of an ldt while providing a degree of transparency interpretability of the decision making

monotonic parareal control for quantum systems following encouraging experimental results in quantum control numerical simulations have known significant improvements through the introduction of efficient optimization algorithms yet the computational cost still prevents using these procedures for high dimensional systems often present in quantum chemistry using parareal framework we present here a time parallelization of these schemes which allows us to reduce significantly their computational cost while still finding convenient controls

new lower bounds for secure codes and related hash families a hypergraph theoretical approach various kinds of secure codes and their related hash families are broadly studied combinatorial structures for protecting copyrighted materials the codewords in such a structure can be regarded as a subset of q n the set of all q ary vectors of given length n satisfying some constraints we use a hypergraph model to characterize the combinatorial structure by applying a result of duke et al on the lower bound of the independence number of hypergraphs we provide a new approach to evaluate the lower bounds for several kinds of secure codes and related hash families in particular the general method is illustrated via the examples of existence results on some perfect hash families frameproof codes and separable codes

genetic learning of the membership functions for mining fuzzy association rules from low quality data many methods have been proposed to mine fuzzy association rules from databases with crisp values in order to help decision makers make good decisions and tackle new types of problems however most real world problems present a certain degree of imprecision various studies have been proposed to mine fuzzy association rules from imprecise data but they assume that the membership functions are known in advance and it is not an easy task to know a priori the most appropriate fuzzy sets to cover the domains of the variables in this paper we propose farlat lqd a new fuzzy data mining algorithm to obtain both suitable membership functions and useful fuzzy association rules from databases with a wide range of types of uncertain data to accomplish this first we perform a genetic learning of the membership functions based on the 3 tuples linguistic representation model to reduce the search space and to learn the most adequate context for each fuzzy partition maximizing the fuzzy supports and the interpretability measure gm3m in order to preserve the semantic interpretability of the obtained membership functions moreover we propose a new algorithm based on the fuzzy frequent pattern growth algorithm called ffp growth lqd to efficiently mine the fuzzy association rules from inaccurate data considering the learned membership functions in the genetic process the results obtained over 3 databases of different sizes and kinds of imprecisions demonstrate the effectiveness of the proposed algorithm

design of fuzzy rule based classifiers with semantic cointension in computing with words cww knowledge is linguistically represented and has an explicit semantics defined through fuzzy information granules the linguistic representation in turn naturally bears an implicit semantics that belongs to users reading the knowledge base hence a necessary condition for achieving interpretability requires that implicit and explicit semantics are cointensive interpretability is definitely stringent when knowledge must be acquired from data through inductive learning therefore in this paper we propose a methodology for designing interpretable fuzzy models through semantic cointension we focus our analysis on fuzzy rule based classifiers frbcs where we observe that rules resemble logical propositions thus semantic cointension can be partially regarded as the fulfillment of the logical view i e the set of basic logical laws that are required in any logical system the proposed approach is grounded on the employment of a couple of tools dcf which extracts interpretable classification rules from data and espresso that is capable of fast minimization of boolean propositions our research demonstrates that it is possible to design models that exhibit good classification accuracy combined with high interpretability in the sense of semantic cointension also structural parameters that quantify model complexity show that the derived models are also simple enough to be read and understood

towards neuro linguistic modeling constraints for optimization of membership functions abstract in neuro fuzzy or fuzzy neural systems using unconstrained optimization schemes like backpropagation it is not possible to guarantee that the resulting membership functions represent human interpretable linguistic terms however one of the most interesting features of fuzzy systems is the insight provided on the linguistic relationship between their variables this work is devoted to the study of constraints which when used within an optimization scheme obviate the subjective task of interpreting membership functions to achieve this a comprehensive set of semantic properties that membership functions should have is postulated and discussed then a set of constraints is introduced and shown to be able to fulfil the properties implementation issues and one example illustrating the importance of the proposed constraints are included

a proposal for the genetic lateral tuning of linguistic fuzzy systems and its interaction with rule selection linguistic fuzzy modeling allows us to deal with the modeling of systems by building a linguistic model which is clearly interpretable by human beings however since the accuracy and the interpretability of the obtained model are contradictory properties the necessity of improving the accuracy of the linguistic model arises when complex systems are modeled to solve this problem one of the research lines in recent years has led to the objective of giving more accuracy to linguistic fuzzy modeling without losing the interpretability to a high level in this paper a new postprocessing approach is proposed to perform an evolutionary lateral tuning of membership functions with the main aim of obtaining linguistic models with higher levels of accuracy while maintaining good interpretability to do so we consider a new rule representation scheme base on the linguistic 2 tuples representation model which allows the lateral variation of the involved labels furthermore the cooperation of the lateral tuning together with fuzzy rule reduction mechanisms is studied in this paper presenting results on different real applications the obtained results show the good performance of the proposed approach in high dimensional problems and its ability to cooperate with methods to remove unnecessary rules

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

a learning process for fuzzy control rules using genetic algorithms abstract the purpose of this paper is to present a genetic learning process for learning fuzzy control rules from examples it is developed in three stages the first one is a fuzzy rule genetic generating process based on a rule learning iterative approach the second one combines two kinds of rules experts rules if there are and the previously generated fuzzy control rules removing the redundant fuzzy rules and the thrid one is a tuning process for adjusting the membership functions of the fuzzy rules the three components of the learning process are developed formulating suitable genetic algorithms

interweaving pfasst and parallel multigrid the parallel full approximation scheme in space and time pfasst introduced by emmett and minion in 2012 is an iterative strategy for the temporal parallelization of odes and discretized pdes as the name suggests pfasst is similar in spirit to a space time full approximation scheme multigrid method performed over multiple time steps in parallel however since the original focus of pfasst was on the performance of the method in terms of time parallelism the solution of any spatial system arising from the use of implicit or semi implicit temporal methods within pfasst have simply been assumed to be solved to some desired accuracy completely at each substep and each iteration by some unspecified procedure it hence is natural to investigate how iterative solvers in the spatial dimensions can be interwoven with the pfasst iterations and whether this strategy leads to a more efficient overall approach this paper presents an initial investigation on the relative performance of different strategies for cou

a globalized newton method for the accurate solution of a dipole quantum control problem a theoretical and computational framework is presented to obtain accurate controls for fast quantum state transitions that are needed in a host of applications such as nanoelectronic devices and quantum computing this method is based on a reduced hessian krylov newton scheme applied to a norm preserving discrete model of a dipole quantum control problem the use of second order numerical methods for solving the control problem is justified proving the existence of optimal solutions and analyzing first and second order optimality conditions criteria for the discretization of the nonconvex optimization problem and for the formulation of the hessian are given to ensure accurate gradients and a symmetric hessian robustness of the newton approach is obtained using a globalization strategy with a robust linesearch procedure results of numerical experiments demonstrate that the newton approach presented in this paper is able to provide fast and accurate controls for high energy state transitions

a fast and efficient multi objective evolutionary learning scheme for fuzzy rule based classifiers during the last years multi objective evolutionary algorithms moeas have been extensively used to generate fuzzy rule based systems characterized by different trade offs between accuracy and complexity in this paper we propose an moea based approach to learn concurrently the rule and data bases of fuzzy rule based classifiers frbcs in particular the rule bases are generated by exploiting a rule and condition selection rcs strategy which selects a reduced number of rules from a heuristically generated set of candidate rules and a reduced number of conditions for each selected rule during the evolutionary process rcs can be considered as a rule learning in a constrained search space as regards the data base learning the membership function parameters of each linguistic term used in the rules are learned concurrently to the application of rcs we tested our approach on twenty four classification benchmarks and compared our results with the ones obtained by two similar state of the art moea based approaches and by two well known non evolutionary classification algorithms namely furia and c4 5 using non parametric statistical tests we show that our approach generates frbcs with accuracy and complexity statistically comparable to and sometimes better than the ones generated by the two moea based approaches exploiting however only the 5 of the number of fitness evaluations used by these approaches further the classifiers generated by our approach result to be more interpretable than the ones generated by the furia and c4 5 algorithms while achieving the same accuracy level

stiff differential equations solved by radau methods radau iia methods are successful algorithms for the numerical solution of stiff differential equations this article describes radau a new implementation of these methods with a variable order strategy the paper starts with a survey on the historical development of the methods and the discoveries of their theoretical properties numerical experiments illustrate the behaviour of the code

comparison and design of interpretable linguistic vs scatter frbss gm3m generalization and new rule meaning index for global assessment and local pseudo linguistic representation this work is devoted to defining more general interpretability indexes to be applied to any scatter or linguistic model implemented by any type of membership functions they are based on metrics that should take into account the semantic and inference issues the semantic issue in order to preserve the meaning of the linguistic labels and the inference issue since this can influence the behavior of the rules on the other hand these metrics have been designed to be intuitive in order to support the analysis or selection of a final model and to favor a low computational cost within an optimization process in order to check their usefulness a multi objective evolutionary algorithm simultaneously performing a rule selection and an adjustment of the fuzzy partitions is guided by the proposed indexes on several benchmark data sets to obtain models with different degrees of accuracy and interpretability in addition using these metrics a local analysis can be carried out between models of a different nature this local analysis through the model components gives support to the user to make the best choice from among the models

a parareal waveform relaxation algorithm for semi linear parabolic partial differential equations we report a new parallel iterative algorithm for semi linear parabolic partial differential equations pdes by combining a kind of waveform relaxation wr techniques into the classical parareal algorithm the parallelism can be simultaneously exploited by wr and parareal in different directions we provide sharp error estimations for the new algorithm on bounded time domain and on unbounded time domain respectively the iterations of the parareal and the wr are balanced to optimize the performance of the algorithm furthermore the speedup and the parallel efficiency of the new approach are analyzed numerical experiments are carried out to verify the effectiveness of the theoretic work

convergence analysis for three parareal solvers we analyze in this paper the convergence properties of the parareal algorithm for the symmetric positive definite problem mathbf u a mathbf u g the coarse propagator mathcal g is fixed to the backward euler method and three time integrators are chosen for the mathcal f propagator the trapezoidal rule the third order diagonal implicit runge kutta rk dirk method and the fourth order gauss rk method it is well known that the parareal euler algorithm using the backward euler method for mathcal f and mathcal g converges rapidly but less is known when one uses for mathcal f the trapezoidal rule or the fourth order gauss rk method especially when the mesh ratio j delta t delta t is small we show that for a specified lambda max the maximal eigenvalue of a or its upper bound there exists some critical j rm cri such that the parareal solvers derived from these three choices of mathcal f converge as fast as parareal euler provided j geq j rm cri

a parareal method for time fractional differential equations in this paper a parareal method is proposed for the parallel in time integration of time fractional differential equations tfdes it is a generalization of the original parareal method proposed for classic differential equations to match the global feature of fractional derivatives the new method has in the correction step embraced the history part of the solution we provide a convergence analysis under the assumption of lipschitz stability conditions we use a multi domain spectral integrator to build the serial solvers and numerical results demonstrate the feasibility of the new approach and confirm the convergence analysis studies also show that both the coarse resolution and the nature of the differential operators can affect the performance

analysis of two parareal algorithms for time periodic problems the parareal algorithm which permits us to solve evolution problems in a time parallel fashion has created a lot of attention over the past decade the algorithm has its roots in the multiple shooting method for boundary value problems which in the parareal algorithm is applied to initial value problems with a particular coarse approximation of the jacobian matrix it is therefore of interest to formulate parareal type algorithms for time periodic problems which also couple the end of the time interval with the beginning and to analyze their performance in this context we present and analyze two parareal algorithms for time periodic problems one with a periodic coarse problem and one with a nonperiodic coarse problem an interesting advantage of the algorithm with the nonperiodic coarse problem is that no time periodic problems need to be solved during the iteration since on the time subdomains the problems are not time periodic either we prove for both linear and nonlinear problems convergence

a review of the application of multiobjective evolutionary fuzzy systems current status and further directions over the past few decades fuzzy systems have been widely used in several application fields thanks to their ability to model complex systems the design of fuzzy systems has been successfully performed by applying evolutionary and in particular genetic algorithms and recently this approach has been extended by using multiobjective evolutionary algorithms which can consider multiple conflicting objectives instead of a single one the hybridization between multiobjective evolutionary algorithms and fuzzy systems is currently known as multiobjective evolutionary fuzzy systems this paper presents an overview of multiobjective evolutionary fuzzy systems describing the main contributions on this field and providing a two level taxonomy of the existing proposals in order to outline a well established framework that could help researchers who work on significant further developments finally some considerations of recent trends and potential research directions are presented

multigrid methods for parabolic distributed optimal control problems multigrid schemes that solve parabolic distributed optimality systems discretized by finite differences are investigated accuracy properties of finite difference approximation are discussed and validated two multigrid methods are considered which are based on a robust relaxation technique and use two different coarsening strategies semicoarsening and standard coarsening the resulting multigrid algorithms show robustness with respect to changes of the value of v the weight of the cost of the control is sufficiently small fourier mode analysis is used to investigate the dependence of the linear twogrid convergence factor on v and on the discretization parameters results of numerical experiments are reported that demonstrate sharpness of fourier analysis estimates a multigrid algorithm that solves optimal control problems with box constraints on the control is considered

comments on distinguishability quantification of fuzzy sets this paper presents some comments on the above paper regarding the manipulation of gaussian fuzzy sets and provides two useful generalized formulas for both of similarity and similarity possibility relationships of gaussian fuzzy sets with different widths

a monotonic method for nonlinear optimal control problems with concave dependence on the state initially introduced in the framework of quantum control the so called monotonic algorithms have demonstrated very good numerical performance when dealing with bilinear optimal control problems this article presents a unified formulation that can be applied to more general nonlinear settings compatible with the hypothesis detailed below in this framework we show that the well posedness of the general algorithm is related to a nonlinear evolution equation we prove the existence of the solution to the evolution equation and give important properties of the optimal control functional finally we show how the algorithm works for selected models from the literature we also compare the algorithm with the gradient algorithm

towards essential improvement for the parareal tr and parareal gauss4 algorithms parareal is an iterative algorithm and is characterized by two propagators g and f which are respectively associated with large step size t and small step size t where t j t and j 2 is an integer for symmetric positive definite spd system u t a u t g t arising from semi discretizing time dependent pdes if we fix the g propagator to the backward euler method and choose for f some l stable time integrator it can be proven that the convergence factors of the corresponding parareal algorithms satisfy 1 3 j 2 and a 0 where a is the spectrum of the matrix a however this result does not hold when time integrators that lack l stability such as the trapezoidal rule and the 4th order gauss rk method are chosen as the f propagator the parareal algorithms using these two methods for the f propagator are denoted by parareal tr and parareal gauss4 in this paper we propose a strategy to let these two parareal algorithms possess such a uniform convergence property the idea is to choose an l stable propagator f and on each coarse time interval t n t n 1 we perform first two steps of f then followed by j 2 steps of f precisely for the trapezoidal rule we select the 2nd order sdirk method as the f propagator and for the 4th order gauss rk method we select the 4th order lobatto iii c method as the f propagator numerical results are given to support our theoretical conclusions

genetic tuning of fuzzy rule deep structures preserving interpretability and its interaction with fuzzy rule set reduction tuning fuzzy rule based systems for linguistic fuzzy modeling is an interesting and widely developed task it involves adjusting some of the components of the knowledge base without completely redefining it this contribution introduces a genetic tuning process for jointly fitting the fuzzy rule symbolic representations and the meaning of the involved membership functions to adjust the former component we propose the use of linguistic hedges to perform slight modifications keeping a good interpretability to alter the latter component two different approaches changing their basic parameters and using nonlinear scaling factors are proposed as the accomplished experimental study shows the good performance of our proposal mainly lies in the consideration of this tuning approach performed at two different levels of significance the paper also analyzes the interaction of the proposed tuning method with a fuzzy rule set reduction process a good interpretability accuracy tradeoff is obtained combining both processes with a sequential scheme first reducing the rule set and subsequently tuning the model

data based fuzzy rule test for fuzzy modelling in the field of fuzzy modelling the exclusive consideration of the modelling error leads to problems concerning the handling of high dimensional applications and the interpretability of the resulting rule base to solve those problems a statistically motivated fuzzy rule test is proposed it decides if a fuzzy if then statement is a relevant rule or not in this way the problem of finding a good rule base can be reduced to the problem of finding good relevant rules

newton methods for the optimal control of closed quantum spin systems an efficient and robust computational framework for solving closed quantum spin optimal control and exact controllability problems with control constraints is presented closed spin systems are of

combinatorial properties of frameproof and traceability codes in order to protect copyrighted material codes may be embedded in the content or codes may be associated with the keys used to recover the content codes can offer protection by providing some form of traceability ta for pirated data several researchers have studied different notions of ta and related concepts in previous years strong versions of ta allow at least one member of a coalition that constructs a pirate decoder to be traced weaker versions of this concept ensure that no coalition can frame a disjoint user or group of users all these concepts can be formulated as codes having certain combinatorial properties we study the relationships between the various notions and we discuss equivalent formulations using structures such as perfect hash families we use methods from combinatorics and coding theory to provide bounds necessary conditions and constructions sufficient conditions for the objects of interest

distinguishability quantification of fuzzy sets distinguishability is a semantic property of fuzzy sets that has a great relevance in the design of interpretable fuzzy models distinguishability has been mathematically defined through different measures which are addressed in this paper special emphasis is given to similarity which exhibits sound theoretical properties but its calculation is usually computationally intensive and possibility whose calculation can be very efficient but it does not exhibit the same properties of similarity it is shown that under mild conditions usually met in interpretable fuzzy modeling possibility can be used as a valid measure for assessing distinguishability thus overcoming the computational inefficiencies of similarity measures moreover procedures that minimize possibility also minimize similarity and consequently improve distinguishability in this sense the use of possibility is fully justified in interpretable fuzzy modeling

new bounds for frameproof codes frameproof codes are used to fingerprint digital data they can prevent copyrighted materials from unauthorized use in this paper we study upper and lower bounds for w frameproof codes of length n over an alphabet of size q the upper bound is based on a combinatorial approach and the lower bound is based on a probabilistic construction both bounds can improve one of the previous results when q is small compared with w say cq leq w for some constant c leq q furthermore we pay special attention to binary frameproof codes we show a binary w frameproof code of length n cannot have more than n codewords if n

editorial special issue on interpretable fuzzy systems interpretability is acknowledged as one of the most appreciated advantages of fuzzy systems in many applications especially in those with high human interaction where it actually becomes a strong requirement however it is important to remark that there is a somehow misleading but widely extended belief even in part of the fuzzy community regarding fuzzy systems as interpretable no matter how they were designed of course we are aware the use of fuzzy logic favors the interpretability of designed models thanks to their semantic expressivity close to natural language fuzzy variables and rules can be used to formalize linguistic propositions which are likely to be easily understandood by human beings obviously this fact makes easier the knowledge extraction and representation tasks carried out when modeling real world complex systems notwithstanding fuzzy logic is not enough by itself to guarantee the interpretability of the final model as it is thoroughly illustrated in this special issue achieving interpretable fuzzy systems is a matter of careful design because fuzzy systems cannot be deemed as interpretable per se thus several constraints have to be imposed along the whole design process with the aim of producing really interpretable fuzzy systems in the sense that every element of the whole system may be checked and understood by a human being otherwise fuzzy systems may even become black boxes

low level interpretability and high level interpretability a unified view of data driven interpretable fuzzy system modelling this paper aims at providing an in depth overview of designing interpretable fuzzy inference models from data within a unified framework the objective of complex system modelling is to develop reliable and understandable models for human being to get insights into complex real world systems whose first principle models are unknown because system behaviour can be described naturally as a series of linguistic rules data driven fuzzy modelling becomes an attractive and widely used paradigm for this purpose however fuzzy models constructed from data by adaptive learning algorithms usually suffer from the loss of model interpretability model accuracy and interpretability are two conflicting objectives so interpretation preservation during adaptation in data driven fuzzy system modelling is a challenging task which has received much attention in fuzzy system modelling community in order to clearly discriminate the different roles of fuzzy sets input variables and other components in achieving an interpretable fuzzy model a taxonomy of fuzzy model interpretability is first proposed in terms of low level interpretability and high level interpretability in this paper the low level interpretability of fuzzy models refers to fuzzy model interpretability achieved by optimizing the membership functions in terms of semantic criteria on fuzzy set level while the high level interpretability refers to fuzzy model interpretability obtained by dealing with the coverage completeness and consistency of the rules in terms of the criteria on fuzzy rule level some criteria for low level interpretability and high level interpretability are identified respectively different data driven fuzzy modelling techniques in the literature focusing on the interpretability issues are reviewed and discussed from the perspective of low level interpretability and high level interpretability furthermore some open problems about interpretable fuzzy models are identified and some potential new research directions on fuzzy model interpretability are also suggested

an extension on statistical comparisons of classifiers over multiple data sets for all pairwise comparisons in a recently published paper in jmlr dem sar 2006 recommends a set of non parametric statistical tests and procedures which can be safely used for comparing the performance of classifiers over multiple data sets after studying the paper we realize that the paper correctly introduces the basic procedures and some of the most advanced ones when comparing a control method however it does not deal with some advanced topics in depth regarding these topics we focus on more powerful proposals of statistical procedures for comparing n n classifiers moreover we illustrate an easy way of obtaining adjusted and comparable p values in multiple comparison procedures

fingrams visual representations of fuzzy rule based inference for expert analysis of comprehensibility since zadeh s proposal and mamdani s seminal ideas interpretability is acknowledged as one of the most appreciated and valuable characteristics of fuzzy system identification methodologies it represents the ability of fuzzy systems to formalize the behavior of a real system in a human understandable way by means of a set of linguistic variables and rules with a high semantic expressivity close to natural language interpretability analysis involves two main points of view readability of the knowledge base description regarding complexity of fuzzy partitions and rules and comprehensibility of the fuzzy system regarding implicit and explicit semantics embedded in fuzzy partitions and rules as well as the fuzzy reasoning method readability has been thoroughly treated by many authors who have proposed several criteria and metrics unfortunately comprehensibility has usually been neglected because it involves some cognitive aspects related to human reasoning which are very hard to formalize and to deal with this paper proposes the creation of a new paradigm for fuzzy system comprehensibility analysis based on fuzzy systems inference maps so called fuzzy inference grams fingrams by analogy with scientograms used for visualizing the structure of science fingrams show graphically the interaction between rules at the inference level in terms of co fired rules i e rules fired at the same time by a given input the analysis of fingrams offers many possibilities measuring the comprehensibility of fuzzy systems detecting redundancies and or inconsistencies among fuzzy rules identifying the most significant rules etc some of these capabilities are explored in this study for the case of fuzzy models and classifiers

monotonic time discretized schemes in quantum control most of the numerical simulations in quantum bilinear control have used monotonically convergent algorithms of krotov introduced by tannor et al 15 of zhu rabitz 16 or their unified formulation in 17 however the properties of the discrete version of these procedures have not been yet tackled with we present in this paper a stable time and space discretization which preserves the monotonic properties of the monotonic algorithms numerical results show that the newly derived algorithms are stable and enable various experimentations

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

optimal quantum control by an adapted coordinate ascent algorithm this paper explores an adaption of the coordinate ascent approach to quantum control problems it was motivated by the observation that several of the existing monotone converging schemes for quantum control may be viewed as approximations of the well known coordinate ascent method our implementation employs line searches in coordinate directions in control space using only evaluations of the performance functional without invoking its derivatives it is based on recasting the performance functional as a local tracking function which gives the future quality of a control at each moment in time back propagation of a basis of the target space enables these performance functional i e tracking function evaluations to be done efficiently the performance functional may include or not include regularization terms as needed convergence of the resulting algorithm and its relation to previous schemes are discussed and numerical examples are provided in our tests the coordinate ascent algorithm exhib

time domain decomposition of optimal control problems for the wave equation we consider the problem of boundary optimal control of a wave equation with boundary dissipation by the way of time domain decomposition of the corresponding optimality system we develop an iterative algorithm which shows that the decomposed optimality system corresponds to local in time optimal control problems which can be treated in parallel we show convergence of the algorithm finally we provide a time discretization which is reminiscent of an instantaneous control scheme we thereby also contribute to the problem of convergence of such schemes