numerical solution to a sideways parabolic equation the inverse heat conduction problem can be considered to be a sideways parabolic equation in the quarter plane this is a model of a problem where one wants to determine the temperature on both sides of a thick wall but one side is inaccessible to measurements a numerical procedure for this severely ill posed problem is suggested which consists of two steps namely a mollification of the data and a marching difference scheme the numerical method is proved to be stable several computational results are presented and discussed copyright 2001 john wiley sons ltd

sobolev bounds on functions with scattered zeros with applications to radial basis function surface fitting in this paper we discuss sobolev bounds on functions that vanish at scattered points in a bounded lipschitz domain that satisfies a uniform interior cone condition the sobolev spaces involved may have fractional as well as integer order we then apply these results to obtain estimates for continuous and discrete least squares surface fits via radial basis functions rbfs these estimates include situations in which the target function does not belong to the native space of the rbf

determining surface temperatures from interior observations we consider the inverse heat transfer problem for the case of the one dimensional heat equation in a quarter plane and present a new solution algorithm together with error bounds of logarithmic convexity type our method is based on regularizing the initial value problem for the heat equation run sideways we describe a numerical experiment with rapidly varying temperature profiles and explain how the method can be extended to problems with temperature dependent thermal properties

universal approximation using radial basis function networks there have been several recent studies concerning feedforward networks and the problem of approximating arbitrary functionals of a finite number of real variables some of these studies deal with cases in which the hidden layer nonlinearity is not a sigmoid this was motivated by successful applications of feedforward networks with nonsigmoidal hidden layer units this paper reports on a related study of radial basis function rbf networks and it is proved that rbf networks having one hidden layer are capable of universal approximation here the emphasis is on the case of typical rbf networks and the results show that a certain class of rbf networks with the same smoothing factor in each kernel node is broad enough for universal approximation

a new regularization method for solving a time fractional inverse diffusion problem in this paper we consider an inverse problem for a time fractional diffusion equation in a one dimensional semi infinite domain the temperature and heat flux are sought from a measured temperature history at a fixed location inside the body we show that such problem is severely ill posed and further apply a new regularization method to solve it based on the solution given by the fourier method convergence estimates are presented under the a priori bound assumptions for the exact solution finally numerical examples are given to show that the proposed numerical method is effective

note on the complexity of deciding the rainbow vertex connectedness for bipartite graphs a path in an edge colored graph is said to be a rainbow path if no two edges on the path share the same color an edge colored graph is strongly rainbow connected if there exists a rainbow geodesic path between every pair of vertices the strong rainbow connection number of g denoted by scr g respectively rc g is the smallest number of colors that are needed in order to make g strongly rainbow connected a vertex colored graph g is rainbow vertex connected if any pair of vertices in g are connected by a path whose internal vertices have distinct colors the rainbow vertex connection number of a connected graph g denoted by rvc g is the smallest number of colors that are needed in order to make g rainbow vertex connected though for a general graph g it is np complete to decide whether rc g 2 or rvc g 2 in this paper we show that the problem becomes easy when g is a bipartite graph whereas deciding whether rc g 3 or rvc g 3 is still np complete even when g is a bipartite graph moreover it is known that deciding whether a given edge vertex colored with an unbound number of colors graph is rainbow vertex connected is np complete we will prove that it is still np complete even when the edge vertex colored graph is bipartite we also show that a few np hard problems on rainbow connection are indeed np complete

on the double roman domination of graphs abstract a double roman dominating function of a graph g is a labeling f v g 0 1 2 3 such that if f v 0 then the vertex v must have at least two neighbors labeled 2 under f or one neighbor with f w 3 and if f v 1 then v must have at least one neighbor with f w 2 the double roman domination number dr g of g is the minimum value of v v g f v over such functions in this paper we firstly give some bounds of the double roman domination numbers of graphs with given minimum degree and graphs of diameter 2 and further we get that the double roman domination numbers of almost all graphs are at most n then we obtain sharp upper and lower bounds for d r g d r g moreover a linear time algorithm for the double roman domination number of a cograph is given and a characterization of the double roman cographs is provided those results partially answer two open problems posed by beeler et al 2016

note on the perfect eic graphs three edges e1 e2 and e3 in a graph g are consecutive if they form a path in this order or a cycle of length 3 the injective edge coloring number i g is the minimum number of colors permitted in a coloring of the edges of g such that if e1 e2 and e3 are consecutive edges in g then e1 and e3 receive the different colors let denote the number of edges in a maximum clique of g a graph g is called an edge injective colorable or perfect eic graph if i g in this paper we give a sharp bound of the injective coloring number of a 2 connected graph with some forbidden conditions and then we also characterize some perfect eic graph classes which extends the results of perfect eic graph of cardoso et al in injective edge chromatic index of a graph http arxiv org abs 1510 02626

an art based construction of rbf networks radial basis function rbf networks are widely used for modeling a function from given input output patterns however two difficulties are involved with traditional rbf trbf networks the initial configuration of an rbf network needs to be determined by a trial and error method and the performance suffers when the desired output has abrupt changes or constant values in certain intervals we propose a novel approach to over come these difficulties new kernel functions are used for hidden nodes and the number of nodes is determined automatically by an adaptive resonance theory art like algorithm parameters and weights are initialized appropriately and then tuned and adjusted by the gradient descent method to improve the performance of the network experimental results have shown that the rbf networks constructed by our method have a smaller number of nodes a faster learning speed and a smaller approximation error than the networks produced by other methods

an iterative lagrange method for the regularization of discrete ill posed inverse problems in this paper an iterative method is presented for the computation of regularized solutions of discrete ill posed problems in the proposed method the regularization problem is formulated as an equality constrained minimization problem and an iterative lagrange method is used for its solution the lagrange iteration is terminated according to the discrepancy principle the relationship between the proposed approach and classical tikhonov regularization is discussed results of numerical experiments are presented to illustrate the effectiveness and usefulness of the proposed method

direction dependent learning approach for radial basis function networks direction dependent scaling shaping and rotation of gaussian basis functions are introduced for maximal trend sensing with minimal parameter representations for input output approximation it is shown that shaping and rotation of the radial basis functions helps in reducing the total number of function units required to approximate any given input output data while improving accuracy several alternate formulations that enforce minimal parameterization of the most general radial basis functions are presented a novel directed graph based algorithm is introduced to facilitate intelligent direction based learning and adaptation of the parameters appearing in the radial basis function network further a parameter estimation algorithm is incorporated to establish starting estimates for the model parameters using multiple windows of the input output data the efficacy of direction dependent shaping and rotation in function approximation is evaluated by modifying the minimal resource allocating network and considering different test examples the examples are drawn from recent literature to benchmark the new algorithm versus existing methods

a modified tikhonov regularization for stable analytic continuation this paper is devoted to a new regularization method for solving the numerical analytic continuation of an analytic function f z f x iy on a strip domain omega z x iy in mathbb c mid x in mathbb r 0 y y 0 where the data is given approximately only on the line y 0 this problem is severely ill posed and has important practical applications the theoretical optimal error bound for the problem is proved which is independent of the selected regularization methods a modified tikhonov regularization method with asymptotic order optimal error estimates is proposed this method can be numerically implemented easily by the fast fourier transform some numerical examples are provided and a comparison with a fourier regularization method is given which show the modified tikhonov method works very well

two regularization methods and the order optimal error estimates for a sideways parabolic equation in this paper we consider an inverse heat conduction problem which appears in some applied subjects this problem is ill posed in the sense that the solution if it exists does not depend continuously on the data a tikhonov type s regularization method and a fourier regularization method are applied to formulate regularized solutions which are stably convergent to the exact ones with order optimal error estimates a numerical example shows that the computational effect of these methods are all satisfactory

a mollification method for a noncharacteristic cauchy problem for a parabolic equation abstract in this paper the noncharacteristic cauchy problem ncp formula is considered with lp r p 1 it is proved that a solution of ncp exists if and only if is infinitely differentiable and n lp r c 2n s2n n n for certain constantscands ncp is well known to be severely ill posed a small perturbation in the cauchy data may cause a dramatically large error in the solution the following mollification method is suggested for solving ncp in a stable way if lp r is given inexactly by lp r then we mollify by convolutions with the dirichlet kernel and the de la vallee poussin kernel the exact solution of ncp is approximated by the solution of the mollified problem with a reasonable choice of mollification parameters which yields error estimates of the holder type by the method we can work with the data inlp r p 1 and obtain several sharp stability estimates inlp andl norms of the holder type for the solution of the problem the method can easily be implemented numerically using the fast fourier transform a stable marching difference scheme based on this method is suggested several numerical examples are given which show that the method is effective

a regularization framework for mildly ill posed problems connected with pseudo differential operator abstract recently filter based regularization methods have been well investigated for ill posed problems when the forward operators are compact there are many ill posed problems connected with pseudo differential operators but there is no uniform method for this kind of problems the work on generalization of filter based regularization methods to pseudo differential operator is necessary in this paper we present a regularization framework for solving the mildly ill posed problems involved pseudo differential operators a general regularization method for this kind of problems is given the order optimal error estimates are derived under the usual source conditions as an example a new fractional tikhonov regularization method could be cast into the general framework numerical experiments are conducted for showing the validity of the new fractional tikhonov method

wavelet and fourier methods for solving the sideways heat equation we consider an inverse heat conduction problem the sideways heat equation which is a model of a problem where one wants to determine the temperature on both sides of a thick wall but where one side is inaccessible to measurements mathematically it is formulated as a cauchy problem for the heat equation in a quarter plane with data given along the line x 1 where the solution is wanted for 0 leq x 1 the problem is ill posed in the sense that the solution if it exists does not depend continuously on the data we consider stabilizations based on replacing the time derivative in the heat equation by wavelet based approximations or a fourier based approximation the resulting problem is an initial value problem for an ordinary differential equation which can be solved by standard numerical methods e g a runge kutta method we discuss the numerical implementation of fourier and wavelet methods for solving the sideways heat equation theory predicts that the fourier method and a method based on meyer wavelets will give equally good results our numerical experiments indicate that also a method based on daubechies wavelets gives comparable accuracy as test problems we take model equations with constant and variable coefficients we also solve a problem from an industrial application with actual measured data

cooperative competitive genetic evolution of radial basis function centers and widths for time series prediction in a radial basis function rbf network the rbf centers and widths can be evolved by a cooperative competitive genetic algorithm the set of genetic strings in one generation of the algorithm represents one rep network not a population of competing networks this leads to moderate computation times for the algorithm as a whole selection operates on individual rbfs rather than on whole networks selection therefore requires a genetic fitness function that promotes competition among rbfs which are doing nearly the same job while at the same time promoting cooperation among rbfs which cover different parts of the domain of the function to be approximated niche creation resulting from a fitness function of the form w sub i sup spl beta e w sub i sup spl beta 1 spl beta 2 can facilitate the desired cooperative competitive behavior the feasibility of the resulting algorithm to evolve networks of gaussian inverse multiquadric and thin plate spline rbfs is demonstrated by predicting the mackey glass time series for each type of rbf and for networks of 25 50 75 100 125 and 150 rbf units prediction errors for the evolved gaussian rbf networks are 50 70 lower than rbf networks obtained by k means clustering

orthogonal least squares learning algorithm for radial basis function networks the radial basis function network offers a viable alternative to the two layer neural network in many applications of signal processing a common learning algorithm for radial basis function networks is based on first choosing randomly some data points as radial basis function centers and then using singular value decomposition to solve for the weights of the network such a procedure has several drawbacks and in particular an arbitrary selection of centers is clearly unsatisfactory the authors propose an alternative learning procedure based on the orthogonal least squares method the procedure chooses radial basis function centers one by one in a rational way until an adequate network has been constructed in the algorithm each selected center maximizes the increment to the explained variance or energy of the desired output and does not suffer numerical ill conditioning problems the orthogonal least squares learning strategy provides a simple and efficient means for fitting radial basis function networks this is illustrated using examples taken from two different signal processing applications

multiobjective evolutionary optimization of the size shape and position parameters of radial basis function networks for function approximation this paper presents a multiobjective evolutionary algorithm to optimize radial basis function neural networks rbfnns in order to approach target functions from a set of input output pairs the procedure allows the application of heuristics to improve the solution of the problem at hand by including some new genetic operators in the evolutionary process these new operators are based on two well known matrix transformations singular value decomposition svd and orthogonal least squares ols which have been used to define new mutation operators that produce local or global modifications in the radial basis functions rbfs of the networks the individuals in the population in the evolutionary procedure after analyzing the efficiency of the different operators we have shown that the global mutation operators yield an improved procedure to adjust the parameters of the rbfnns

note on the perfect roman domination number of graphs abstract in this note we list some propositions of the perfect roman domination number of graphs and give the characterization of graphs g with special value of the perfect roman domination furthermore r p f r p f is given at the last a linear time algorithm is shown to compute the perfect roman domination number of a cograph

central difference schemes in time and error estimate on a non standard inverse heat conduction problem inverse heat conduction problems ihcp are severely ill posed in the sense that the solution if it exists does not depend continuously on the data but now the results available in the literature on ihcp are mainly devoted to the standard sideways heat equation this paper remedies this by a central difference schemes in time which itself has a regularizing effect for a non standard ihcp which appears in some applied subjects an error estimate is obtained and the error estimate also gives information about how to choose the step length in the time discretization a numerical example shows that the computational effect of this method is satisfactory

on the regularizing properties of the gmres method the gmres method is a popular iterative method for the solution of large linear systems of equations with a nonsymmetric nonsingular matrix however little is known about the behavior of this method when it is applied to the solution of nonsymmetric linear ill posed problems with a right hand side that is contaminated by errors we show that when the associated error free right hand side lies in a finite dimensional krylov subspace the gmres method is a regularization method the iterations are terminated by a stopping rule based on the discrepancy principle

central difference method of a non standard inverse heat conduction problem for determining surface heat flux from interior observations in this paper we consider a non standard inverse heat conduction problem in a quarter plane we want to know the surface heat flux from a measured temperature history at a fixed location inside the body this is a severely ill posed problem in that arbitrarily small differences in the input temperature can result in arbitrarily large differences in the surface flux a central difference schemes in time can restore the stability of heat flux solution and error estimate can be obtained a numerical experiment shows that the method works effectively

solving an ill posed cauchy problem for a two dimensional parabolic pde with variable coefficients using a preconditioned gmres method the sideways parabolic equation spe is a model of the problem of determining the temperature on the surface of a body from the interior measurements mathematically it can be formulated as a noncharacteristic cauchy problem for a parabolic partial differential equation this problem is severely ill posed in an l 2 setting we use a preconditioned generalized minimum residual method gmres to solve a two dimensional spe with variable coefficients the preconditioner is singular and chosen in a way that allows efficient implementation using the fft the preconditioner is a stabilized solver for a nearby problem with constant coefficients and it reduces the number of iterations in the gmres algorithm significantly numerical experiments are performed that demonstrate the performance of the proposed method