the leja method revisited backward error analysis for the matrix exponential the leja method is a polynomial interpolation procedure that can be used to compute matrix functions in particular computing the action of the matrix exponential on a given vector is a typical application this quantity is required e g in exponential integrators the leja method essentially depends on three parameters the scaling parameter the location of the interpolation points and the degree of interpolation we present here a backward error analysis that allows us to determine these three parameters as a function of the prescribed accuracy additional aspects that are required for an efficient and reliable implementation are discussed numerical examples illustrating the performance of our matlab code are included

numerical low rank approximation of matrix differential equations abstract the efficient numerical integration of large scale matrix differential equations is a topical problem in numerical analysis and of great importance in many applications standard numerical methods applied to such problems require an unduly amount of computing time and memory in general based on a dynamical low rank approximation of the solution a new splitting integrator is proposed for a quite general class of stiff matrix differential equations this class comprises differential lyapunov and differential riccati equations that arise from spatial discretizations of partial differential equations the proposed integrator handles stiffness in an efficient way and it preserves the symmetry and positive semidefiniteness of solutions of differential lyapunov equations numerical examples that illustrate the benefits of this new method are given in particular numerical results for the efficient simulation of the weather phenomenon el nino are presented

overcoming order reduction in diffusion reaction splitting part 2 oblique boundary conditions splitting methods constitute a well established class of numerical schemes for the time integration of partial differential equations their main advantages over more traditional schemes are computational efficiency and superior geometric properties in the presence of nontrivial boundary conditions however splitting methods usually suffer from order reduction and some additional loss of accuracy for diffusion reaction equations with inhomogeneous oblique boundary conditions a modification of the classic second order strang splitting is proposed that successfully solves the problem of order reduction the same correction also improves the accuracy of the classic first order lie splitting the proposed modification only depends on the available boundary data and in the case of non dirichlet boundary conditions on the computed numerical solution consequently this modification can be implemented in an efficient way which makes the modified splitting schemes superior to their classic versions the fr

a labeling algorithm for the fuzzy assignment problem abstract this paper concentrates on the assignment problem where costs are not deterministic numbers but imprecise ones here the elements of the cost matrix of the assignment problem are subnormal fuzzy intervals with increasing linear membership functions whereas the membership function of the total cost is a fuzzy interval with decreasing linear membership function by the max min criterion suggested by bellman and zadeh the fuzzy assignment problem can be treated as a mixed integer nonlinear programming problem we show that this problem can usually be simplified into either a linear fractional programming problem or a bottleneck assignment problem here we propose an efficient algorithm based on the labeling method for solving the linear fractional programming case the algorithm begins with primal feasibility and proceeds to obtain dual feasibility while maintaining complementary slackness until the primal optimal solution is found the computational results show that the proposed labeling algorithm offers an effective and efficient way for handling the fuzzy assignment problem

study on multiobjective transportation problem with fuzzy numbers abstract in this paper we investigate the efficient solutions and stability of multiobjective transportation problem fmtp with fuzzy coefficient c ij r c r and or fuzzy supply quantities a i and or fuzzy demands quantities b j the concept of fuzzy efficient was introduced in which the ordinary efficient solution is extended based on the level of fuzzy numbers a necessary and sufficient condition for such a solution is established the basic notions as the solvability set and the stability set of the first kind are defined and characterized the parametric analysis is used to characterize the parametric optimal solutions to the auxiliary problems an algorithm for the determination of the stability set is proposed finally a numerical example is given

efficient matrix valued algorithms for solving stiff riccati differential equations in the time varying case a classical approach which has been widely used to compute the solution of the riccati matrix equation of for example size n n is to unroll the matrices into vectors and integrate the resulting system of n sup 2 vector differential equations directly if the system of vectorized differential equations is stiff the cost computation time and storage requirements of applying the popular backward differentiation formulas bdfs to the stiff equations will be very high for large n because a linear system of algebraic equations of size n sup 2 n sup 2 must be solved at each time step new matrix valued algorithms based on a matrix generalization of the bdfs are proposed for solving stiff riccatti differential equations the amount of work required to compute the solution per time step is only o n sup 3 flops by using the matrix valued algorithms whereas the classical approach requires o n sup 6 flops per time step

low rank second order splitting of large scale differential riccati equations we apply first and second order splitting schemes to the differential riccati equation such equations are very important in e g linear quadratic regulator lqr problems where they provide a link between the state of the system and the optimal input the methods can also be extended to generalized riccati equations e g arising from lqr problems given in implicit form in contrast to previously proposed schemes such as bdf or rosenbrock methods the splitting schemes exploit the fact that the nonlinear and affine parts of the problem when considered in isolation have closed form solutions we show that if the solution possesses low rank structure which is frequently the case then this is preserved by the method this feature is used to implement the methods efficiently for large scale problems the proposed methods are expected to be competitive as they at most require the solution of a small number of linear equation systems per time step finally we apply our low rank implementations to the riccati equations arising from two lqr problems the results show that the rank of the solutions stay low and the expected orders of convergence are observed

a new iterative method for solving large scale lyapunov matrix equations in this paper we propose a new projection method to solve large scale continuous time lyapunov matrix equations the new approach projects the problem onto a much smaller approximation space generated as a combination of krylov subspaces in a and a 1 the reduced problem is then solved by means of a direct lyapunov scheme based on matrix factorizations the reported numerical results show the competitiveness of the new method compared to a state of the art approach based on the factorized alternating direction implicit iteration

estimating intensity variance due to noise in registered images applications to diffusion tensor mri image registration techniques which require image interpolation are widely used in neuroimaging research we show that signal variance in interpolated images differs significantly from the signal variance of the original images in native space we describe a simple approach to compute the signal variance in registered images based on the signal variance and covariance of the original images the spatial transformations computed by the registration procedure and the interpolation or approximation kernel chosen the method is general and could handle various sources of signal variability such as thermal noise and physiological noise provided that their effects can be assessed in the original images our approach is applied to diffusion tensor dt mri data assuming only thermal noise as the source of variability in the data we show that incorrect noise variance estimates in registered diffusion weighted images can affect dt parameters as well as indices of goodness of fit such as chi square maps in addition to dt mri we believe that this methodology would be useful any time parameter extraction methods are applied to registered or interpolated data such as in relaxometry and functional mri studies

numerical integration of the differential riccati equation and some related issues abstract in this paper the problem of direct numerical integration of differential riccati equations dres and some related issues are considered the dre is an expression of a particular change of variables for a linear system of ordinary differential equations the error that an approximate solution of the dre induces on the original variables of the system is considered and it is related to geometrical properties of the system itself sharp bounds on the global error for the computed solution are also given in terms of local errors and geometrical properties of the original system nonstiff and stiff dres of unsymmetric and symmetric type are considered a useful matrix interpretation is given for many integration schemes such as the backward differentiation formulas bdf when applied to the dre this allows the matrix structure of the problem to be exploited in particular for stiff dres the resulting strategy allows for a saving of three orders of magnitude with respect to the standard reform

computing a nearest symmetric positive semidefinite matrix the nearest symmetric positive semidefinite matrix in the frobenius norm to an arbitrary real matrix a is shown to be b h 2 where h is the symmetric polar factor of b a at 2 in the 2 norm a nearest symmetric positive semidefinite matrix and its distance 2 a from a are given by a computationally challenging formula due to halmos we show how the bisection method can be applied to this formula to compute upper and lower bounds for 2 a differing by no more than a given amount a key ingredient is a stable and efficient test for positive definiteness based on an attempted choleski decomposition for accurate computation of 2 a we formulate the problem as one of zero finding and apply a hybrid newton bisection algorithm some numerical difficulties are discussed and illustrated by example

matrix oriented discretization methods for reaction diffusion pdes comparisons and applications abstract systems of reaction diffusion partial differential equations rd pdes are widely applied for modeling life science and physico chemical phenomena in particular the coupling between diffusion and nonlinear kinetics can lead to the so called turing instability giving rise to a variety of spatial patterns like labyrinths spots stripes etc attained as steady state solutions for large time intervals to capture the morphological peculiarities of the pattern itself a very fine space discretization may be required limiting the use of standard vector based ode solvers in time because of excessive computational costs by exploiting the structure of the diffusion matrix we show that matrix based versions of time integrators such as implicit explicit imex and exponential schemes allow for much finer problem discretizations we illustrate our findings by numerically solving the schnakenberg model prototype of rd pde systems with turing pattern solutions and the dib morphochemical model describing metal growth during battery charging processes

a fuzzy transportation algorithm this paper proposes an algorithm for solving transportation problems with fuzzy constraints and investigates the relationship between the algebraic structure of the optimum solution of the deterministic problem and its fuzzy equivalent

fuzzy integer transportation problem an algorithm has been proposed which solves the transportation problem with fuzzy supply and demand values and the integrality condition imposed on the solution this algorithm is exact and computationally effective although the problem is formulated in the general way i e its fuzzy supply and demand values can differ from each other and be fuzzy numbers of any type

rosenbrock methods for solving riccati differential equations the riccati differential equation rde arises in several fields like optimal control optimal filtering h control of linear time varying systems differential games etc in the literature there is a large variety of approaches to compute its solution particularly for stiff rdes matrix valued versions of the standard multi step methods for solving ordinary differential equations have given good results in this technical note we discuss a particular class of one step methods these are the linear implicit runge kutta methods or rosenbrock methods we show that they offer a practical alternative for solving stiff rdes they can be implemented with good stability properties and allow for a cheap step size control the matrix valued version of the rosenbrock methods for rdes requires the solution of one sylvester equation in each stage of the method for the case in which the coefficient matrices of the sylvester equation are dense the bartels stewart method can be efficiently applied for solving the equations the computational cost computing time and memory requirements is smaller than for multi step methods

a model based method for retrospective correction of geometric distortions in diffusion weighted epi the self diffusion tensor may be calculated from several echo planar image acquisitions preceded by different diffusion gradients unfortunately these diffusion gradients cause geometric distortion that must be corrected before estimation of the tensor in the present paper we suggest and implement a method for retrospective correction of these distortions firmly based on a physical model for the diffusion weighted images this method simultaneously estimates subject movement and distortion parameters by finding the set of parameters that minimizes residual error when fitting data to the diffusion tensor model we show how this notion can be formalized as a quadratic form thereby facilitating the implementation of a rapid algorithm in addition we suggest models for how distortions vary with slice position and gradient direction that allow us to substantially reduce the dimensionality of the parameter space our results indicate that we are able to estimate both eddy current induced distortion and subject movement directly from the data without need of any additional measurements

an efficient spde approach for el nino abstract we consider the numerical approximation of stochastic partial differential equations spdes based models for a quasi periodic climate pattern in the tropical pacific ocean known as el nino phenomenon we show that for these models the mean and the covariance are given by a deterministic partial differential equation and by an operator differential equation respectively in this context we provide a numerical framework to approximate these parameters directly we compare this method to stochastic differential equations and spdes based models from the literature solved by taylor methods and stochastic galerkin methods respectively numerical results for different scenarios taking as a reference measured data of the years 2014 and 2015 last nino event validate the efficiency of our approach

a fuzzy approach to the transportation problem abstract the transportation problem with fuzzy supply values of the deliverers and with fuzzy demand values of the receivers is analysed for the solution of the problem the technique of parametric programming is used this makes it possible to obtain not only the maximizing solution according to the bellman zadeh criterion but also other alternatives close to the optimal solution

analysis of some krylov subspace approximations to the matrix exponential operator in this note a theoretical analysis of some krylov subspace approximations to the matrix exponential operation exp a v is presented and a priori and a posteriors error estimates are established several such approximations are considered the main idea of these techniquesis to approximately project the exponential operator onto a small krylov subspace and to carry out the resulting small exponential matrix computation accurately this general approach which has been used with success in several applications provides a systematic way of defining high order explicit type schemes for solving systems of ordinary differential equations or time dependent partial differential equations

process planning in a fuzzy environment abstract mixed integer optimization models for chemical process planning typically assume that model parameters can be accurately predicted as precise forecasts are difficult to obtain process planning usually involves uncertainty and ambiguity in the data this paper presents an application of fuzzy programming to process planning the forecast parameters are assumed to be fuzzy with a linear or triangular membership function the process planning problem is then formulated in terms of decision making in a fuzzy environment with fuzzy constraints and fuzzy net present value goals the model is transformed to a deterministic mixed integer linear program or mixed integer nonlinear program depending on the type of uncertainty involved in the problem for the nonlinear case a global optimization algorithm is developed for its solution this algorithm is applicable to general possibilistic programs and can be used as an alternative to the commonly used bisection method illustrative examples and computational results for a petrochemical complex with 38 processes and 24 products illustrate the applicability of the developed models and algorithms

b spline signal processing i theory the use of continuous b spline representations for signal processing applications such as interpolation differentiation filtering noise reduction and data compressions is considered the b spline coefficients are obtained through a linear transformation which unlike other commonly used transforms is space invariant and can be implemented efficiently by linear filtering the same property also applies for the indirect b spline transform as well as for the evaluation of approximating representations using smoothing or least squares splines the filters associated with these operations are fully characterized by explicitly evaluating their transfer functions for b splines of any order applications to differentiation filtering smoothing and least squares approximation are examined the extension of such operators for higher dimensional signals such as digital images is considered

new method for solving fuzzy transportation problems with lr flat fuzzy numbers transportation problems have various applications in logistics and supply chains for reducing costs in real life situations the parameters of transportation problems may not be known precisely because of uncontrollable factors herein we propose a new method for solving fuzzy transportation problems ftps in which the transportation costs and supply and demand are represented by non negative lr flat fuzzy numbers the ftp is converted into four transportation problems which are solved using standard transportation simplex algorithms the advantages of the proposed method over existing methods are discussed in the context of two application examples the results show that the proposed method is simpler and computationally more efficient than existing methods in the literature

the use of parametric programming in fuzzy linear programming in this paper the possibility of the identification of a complete fuzzy decision not only the maximizing alternative in fuzzy linear programming by use of the parametric programming technique is presented also it is shown that this fact can be useful in the zimmermann approach to multiple objective linear programming the presented remarks are illustrated by some numerical examples

solving fuzzy transportation problems based on extension principle abstract transportation models play an important role in logistics and supply chain management for reducing cost and improving service this paper develops a procedure to derive the fuzzy objective value of the fuzzy transportation problem in that the cost coefficients and the supply and demand quantities are fuzzy numbers the idea is based on the extension principle a pair of mathematical programs is formulated to calculate the lower and upper bounds of the fuzzy total transportation cost at possibility level from different values of the membership function of the objective value is constructed two different types of the fuzzy transportation problem are discussed one with inequality constraints and the other with equality constraints it is found that the membership function of the objective value of the equality problem is contained in that of the inequality problem since the objective value is expressed by a membership function rather than by a crisp value more information is provided for making decisions

positive definiteness in the numerical solution of riccati differential equations in this work we address the issue of integrating symmetric riccati and lyapunov matrix differential equations in many cases typical in applications the solutions are positive definite matrices our goal is to study when and how this property is maintained for a numerically computed solution there are two classes of solution methods direct and indirect algorithms the first class consists of the schemes resulting from direct discretization of the equations the second class consists of algorithms which recover the solution by exploiting some special formulae that these solutions are known to satisfy we show first that using a direct algorithm a one step scheme or a strictly stable multistep scheme explicit or implicit limits the order of the numerical method to one if we want to guarantee that the computed solution stays positive definite then we show two ways to obtain positive definite higher order approximations by using indirect algorithms the first is to apply a symplectic integrator to an associated hamiltonian system the other uses stepwise linearization

on the decay rate of hankel singular values and related issues this paper investigates the decay rate of the hankel singular values of linear dynamical systems this issue is of considerable interest in model reduction by means of balanced truncation for instance since the sum of the neglected singular values provides an upper bound for an appropriate norm of the approximation error the decay rate involves a new set of invariants associated with a linear system which are obtained by evaluating a modified transfer function at the poles of the system these considerations are equivalent to studying the decay rate of the eigenvalues of the product of the solutions of two lyapunov equations the related problem of determining the decay rate of the eigenvalues of the solution to one lyapunov equation will also be addressed very often these eigenvalues like the hankel singular values are rapidly decaying this fact has motivated the development of several algorithms for computing low rank approximate solutions to lyapunov equations however until now conditions assuring rapid decay have not been well understood such conditions are derived here by relating the solution to a numerically low rank cauchy matrix determined by the poles of the system bounds explaining rapid decay rates are obtained under some mild conditions

distribution planning decisions using interactive fuzzy multi objective linear programming this work develops an interactive fuzzy multi objective linear programming i fmolp method for solving the fuzzy multi objective transportation problems with piecewise linear membership function the proposed i fmolp method aims to simultaneously minimize the total distribution costs and the total delivery time with reference to fuzzy available supply and total budget at each source and fuzzy forecast demand and maximum warehouse space at each destination additionally the proposed method provides a systematic framework that facilitates the fuzzy decision making process enabling a decision maker dm to interactively modify the fuzzy data and related parameters until a set of satisfactory solutions is obtained an industrial case is presented to demonstrate the feasibility of applying the proposed method to real transportation problems consequently the proposed method yields an efficient solution and overall degree of dm satisfaction with the determined objective values especially several significant management implications regarding the practical application of the proposed i fmolp method are presented

the fuzzy shortest path problem and its most vital arcs abstract the shortest path problem is to find the shortest distance between two specified nodes in a network an arc is called a single most vital arc in the network if its removal from the network results in the greatest increase in the shortest distance the availability of arcs plays a key role in network problems the most vital arcs problems provide a means by which the importance of arc s availability can be measured in the traditional shortest path problem the arc lengths are assumed to be crisp numbers in this paper we consider the case that the arc lengths are fuzzy numbers in particular we derive the membership function of the shortest distance by using a fuzzy linear programming approach based on this result we then propose an algorithm for finding the single most vital arc in a network

computing the action of the matrix exponential with an application to exponential integrators a new algorithm is developed for computing e ta b where a is an n times n matrix and b is n times n 0 with n 0 ll n the algorithm works for any a its computational cost is dominated by the formation of products of a with n times n 0 matrices and the only input parameter is a backward error tolerance the algorithm can return a single matrix e ta b or a sequence e t ka b on an equally spaced grid of points t k it uses the scaling part of the scaling and squaring method together with a truncated taylor series approximation to the exponential it determines the amount of scaling and the taylor degree using the recent analysis of al mohy and higham siam j matrix anal appl 31 2009 pp 970 989 which provides sharp truncation error bounds expressed in terms of the quantities a k 1 k for a few values of k where the norms are estimated using a matrix norm estimator shifting and balancing are used as preprocessing steps to reduce the cost of the algorithm numerical experiments show that the algorithm performs in a numerically stable fashion across a wide range of problems and analysis of rounding errors and of the conditioning of the problem provides theoretical support experimental comparisons with matlab codes based on krylov subspace chebyshev polynomial and laguerre polynomial methods show the new algorithm to be sometimes much superior in terms of computational cost and accuracy an important application of the algorithm is to exponential integrators for ordinary differential equations it is shown that the sums of the form sum k 0 p varphi k a u k that arise in exponential integrators where the varphi k are related to the exponential function can be expressed in terms of a single exponential of a matrix of dimension n p built by augmenting a with additional rows and columns and the algorithm of this paper can therefore be employed

interpolation revisited medical images application based on the theory of approximation this paper presents a unified analysis of interpolation and resampling techniques an important issue is the choice of adequate basis functions the authors show that contrary to the common belief those that perform best are not interpolating by opposition to traditional interpolation the authors call their use generalized interpolation they involve a prefiltering step when correctly applied the authors explain why the approximation order inherent in any basis function is important to limit interpolation artifacts the decomposition theorem states that any basis function endowed with approximation order ran be expressed as the convolution of a b spline of the same order with another function that has none this motivates the use of splines and spline based functions as a tunable way to keep artifacts in check without any significant cost penalty the authors discuss implementation and performance issues and they provide experimental evidence to support their claims

a concept of the optimal solution of the transportation problem with fuzzy cost coefficients abstract in the paper a definition of the optimal solution of the transportation problem with fuzzy cost coefficients as well as an algorithm determining this solution are proposed

discretized dynamical low rank approximation in the presence of small singular values low rank approximations to large time dependent matrices and tensors are the subject of this paper these matrices and tensors either are given explicitly or are the unknown solutions of matrix and tensor differential equations based on splitting the orthogonal projection onto the tangent space of the low rank manifold novel time integrators for obtaining approximations by low rank matrices and low rank tensor trains were recently proposed by standard theory the lie trotter and strang projector splitting methods are first and second order accurate respectively but the usual error bounds break down when the low rank approximation has small singular values this happens when the singular values of the solution decay without a distinct gap or when the effective rank of the solution is overestimated on the other hand the integrators are exact when given time dependent matrices or tensors are already of the prescribed rank we provide an error analysis which unifies these properties we show that in c