error estimates of stochastic optimal neumann boundary control problems we study mathematically and computationally optimal control problems for stochastic partial differential equations with neumann boundary conditions the control objective is to minimize the expectation of a cost functional and the control is of the deterministic boundary value type mathematically we prove the existence of an optimal solution and of a lagrange multiplier we represent the input data in terms of their karhunen loeve expansions and deduce the deterministic optimality system of equations computationally we approximate the finite element solution of the optimality system and estimate its error through the discretizations with respect to both spatial and random parameter spaces

stability robustness of linear state space models with structured perturbations abstract a new approach is proposed to analyze the stability robustness of linear state space models under structured uncertainties the presented method is computationally simple to use and can be calculated easily by computer the merit of the proposed method is demonstrated by two examples where the results achieved are much better than those already obtained by the lyapunov stability method and the transition matrix approach

proximal thresholding algorithm for minimization over orthonormal bases the notion of soft thresholding plays a central role in problems from various areas of applied mathematics in which the ideal solution is known to possess a sparse decomposition in some orthonormal basis using convex analytical tools we extend this notion to that of proximal thresholding and investigate its properties providing in particular several characterizations of such thresholders we then propose a versatile convex variational formulation for optimization over orthonormal bases that covers a wide range of problems and we establish the strong convergence of a proximal thresholding algorithm to solve it numerical applications to signal recovery are demonstrated

control of the stochastic burgers model of turbulence we consider a control problem for a stochastic burgers equation this problem is motivated by a model from the control of turbulence see choi et al j fluid mech 253 1993 pp 509 543 we study a sequence of approximated hamilton jacobi equations by using dynamic programming

global regular solutions of second order hamilton jacobi equations in hilbert spaces with locally lipschitz nonlinearities abstract this paper is devoted to the study of a second order hamilton jacobi equation in infinite dimensions with a locally lipschitz continuous hamiltonian h related with a stochastic optimal control problem driven by white noise in an earlier paper the author studied the case of globally lipschitz continuous hamiltonian h this paper is concerned with the case of locally lipschitz continuous h that include e g the quadratic case which appears very frequently in the applications existence uniqueness and c 2 regularity of a local solution follow by methods given previously the heart of the work is the proof of a priori estimates of the c 1 norm of the local solution this has been done by studying properties of nonlinear transition semigroups and by a careful application of them to our problem the results have been applied to a stochastic optimal control problem proving existence of feedback optimal controls

quadratic stability and stabilization of uncertain linear discrete time systems with state delay abstract this paper deals with the problem of quadratic stability analysis and quadratic stabilization for uncertain linear discrete time systems with state delay the system under consideration involves state time delay and time varying norm bounded parameter uncertainties appearing in all the matrices of the state space model necessary and sufficient conditions for quadratic stability and quadratic stabilization are presented in terms of certain matrix inequalities respectively a robustly stabilizing state feedback controller can be constructed by using the corresponding feasible solution of the matrix inequalities two examples are presented to demonstrate the effectiveness of the proposed approach

robust stability and stabilization of discrete singular systems an equivalent characterization this note deals with the problems of robust stability and stabilization for uncertain discrete time singular systems the parameter uncertainties are assumed to be time invariant and norm bounded appearing in both the state and input matrices a new necessary and sufficient condition for a discrete time singular system to be regular causal and stable is proposed in terms of a strict linear matrix inequality lmi based on this the concepts of generalized quadratic stability and generalized quadratic stabilization for uncertain discrete time singular systems are introduced necessary and sufficient conditions for generalized quadratic stability and generalized quadratic stabilization are obtained in terms of a strict lmi and a set of matrix inequalities respectively with these conditions the problems of robust stability and robust stabilization are solved an explicit expression of a desired state feedback controller is also given which involves no matrix decomposition finally an illustrative example is provided to demonstrate the applicability of the proposed approach

comparison of approaches for random pde optimization problems based on different matching functionals in this article we consider an optimal control problem for an elliptic partial differential equation with random inputs to determine an applicable deterministic control f x we consider the four cases which we compare for efficiency and feasibility we prove the existence of optimal states adjoint states and optimality conditions for each cases we also derive the optimality systems for the four cases the optimality system is then discretized by a standard finite element method and sparse grid collocation method for physical space and probability space respectively the numerical experiments are performed for their efficiency and feasibility

blind motion deblurring using multiple images recovery of degraded images due to motion blurring is a challenging problem in digital imaging most existing techniques on blind deblurring are not capable of removing complex motion blurring from the blurred images of complex structures one promising approach is to recover the clear image using multiple images captured for the scene however in practice it is observed that such a multi frame approach can recover a high quality clear image of the scene only after multiple blurred image frames are accurately aligned during pre processing which is a very challenging task even with user interactions in this paper by exploring the sparsity of the motion blur kernel and the clear image under certain domains we propose an alternative iteration approach to simultaneously identify the blur kernels of given blurred images and restore a clear image our proposed approach is not only robust to image formation noises but is also robust to the alignment errors among multiple images a modified version of linearized bregman iteration is then developed to efficiently solve the resulting minimization problem experiments show that our proposed algorithm is capable of accurately estimating the blur kernels of complex camera motions with minimal requirements on the accuracy of image alignment as a result our method is capable of automatically recovering a high quality clear image from multiple blurred images

some results on non linear optimal control problems and hamilton jacobi equations in infinite dimensions abstract the optimal control of a distributed parameter system is connected to the solution of the corresponding hamilton jacobi equation this is a first order equation in infinite dimensions with discontinuous coefficients we study the hamilton jacobi equation of a system governed either by a semilinear or by a monotone dynamics replacing the unbounded terms of this equation by their yosida approximations we prove convergence of the approximate solutions to the value function of the original problem by using the uniqueness result for viscosity solutions examples and applications are included

linearized bregman iterations for compressed sensing finding a solution of a linear equation au f with various minimization properties arises from many applications one such application is compressed sensing where an efficient and robust to noise algorithm to find a minimal l 1 norm solution is needed this means that the algorithm should be tailored for large scale and completely dense matrices a while au and a t u can be computed by fast transforms and the solution we seek is sparse recently a simple and fast algorithm based on linearized bregman iteration was proposed in 28 32 for this purpose this paper is to analyze the convergence of linearized bregman iterations and the minimization properties of their limit based on our analysis here we derive also a new algorithm that is proven to be convergent with a rate furthermore the new algorithm is simple and fast in approximating a minimal l 1 norm solution of au f as shown by numerical simulations hence it can be used as another choice of an efficient tool in compressed sensing

robust control for linear systems with structured state space uncertainty abstract a recent sufficient condition for the stability robustness of linear systems with time varying norm bounded state space uncertainty is extended to include the structure of the uncertainty our new result requires that the nominal eigenvalues lie to the left of a vertical line in the complex plane which is determined by a norm involving the structure of the uncertainty and the nominal closed loop eigenvector matrix therefore this robustness result is especially well suited to the design of control systems using eigenstructure assignment when the uncertainty is time invariant our norm is also an upper bound on the incremental eigenvalue perturbations we also consider the use of perron weightings to reduce conservatism and the extension of the results to discrete time systems

split bregman iteration algorithm for total bounded variation regularization based image deblurring many existing algorithms taking the seminorm in bv for regularization have achieved great success in image processing however this paper considers the total bounded variation regularization based approach to perform image deblurring based on this novel model we introduce an extended split bregman iteration to obtain the optimum solution quickly we also provide the rigorous convergence analysis of the iterative algorithm here compared with the results of the rof method numerical simulations illustrate the more excellent reconstruction performance of the proposed algorithm

improved measures of stability robustness for linear state space models in this paper the aspect of stability robustness of linear systems is analyzed in the time domain a bound on the structured perturbation of an asymptotically stable linear system is obtained to maintain stability using a lyapunov matrix equation solution the resulting bound is shown to be an improved bound over the ones recently reported in the literature also special cases of the nominal system matrix are considered for which the bound is given in terms of the nominal matrix thereby avoiding the solution of the lyapunov matrix equation examples given include comparison of the proposed approach with the recently reported results

a chaotic iterative algorithm based on linearized bregman iteration for image deblurring in this paper we propose a new chaotic iterative algorithm for image deblurring the algorithm is based on generalized inverse iteration and linearized bregman iteration for the basis pursuit problem min u r n u 1 au f numerical experiments show that the chaotic algorithm for image restoration is effective and efficient

local strong homogeneity of a regularized estimator this paper deals with regularized pointwise estimation of discrete signals which contain large strongly homogeneous zones where typically they are constant or linear or more generally satisfy a linear equation the estimate is defined as the minimizer of an objective function combining a quadratic data fidelity term and a regularization prior term the latter term is the sum of the values obtained by applying a potential function pf to each component called a difference of a linear transform of the signal minimizers of functions of this form arise in various settings in statistics and optimization the features exhibited by such an estimate are closely related to the shape of the pf our goal is to determine estimators providing solutions which involve large strongly homogeneous zones where more precisely the differences are null in spite of the noise corrupting the data to this end we require that the strongly homogeneous zones recovered by the estimator be insensitive to any variation of th

the stochastic burgers equation we study burgers equation perturbed by a white noise in space and time we prove the existence of solutions by showing that the cole hopf transformation is meaningful also in the stochastic case the problem is thus reduced to the anaylsis of a linear equation with multiplicative half white noise an explicit solution of the latter is constructed through a generalized feynman kac formula typical properties of the trajectories are then discussed a technical result concerning the regularizing effect of the convolution with the heat kernel is proved for stochastic integrals

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

a trust region algorithm with adaptive stochastic collocation for pde optimization under uncertainty the numerical solution of optimization problems governed by partial differential equations pdes with random coefficients is computationally challenging because of the large number of deterministic pde solves required at each optimization iteration this paper introduces an efficient algorithm for solving such problems based on a combination of adaptive sparse grid collocation for the discretization of the pde in the stochastic space and a trust region framework for optimization and fidelity management of the stochastic discretization the overall algorithm adapts the collocation points based on the progress of the optimization algorithm and the impact of the random variables on the solution of the optimization problem it frequently uses few collocation points initially and increases the number of collocation points only as necessary thereby keeping the number of deterministic pde solves low while guaranteeing convergence currently an error indicator is used to estimate gradient errors due to adaptive

analysis of stability robustness for generalized state space systems with structured perturbations abstract a new approach is proposed to analyze the stability robustness of generalized state space systems with structured perturbations the presented method is computationally simple to use and can easily be calculated by computer as far as we are aware this paper seems to be the first one to solve the robust stability problems for generalized state space systems with structured uncertainties the robust stability problem of generalized state space systems is more complicated than that of regular state space systems because it needs consideration of not only stability robustness but also system regularity and impulse elimination the latter two ones need not be considered in regular state space systems

viscosity solutions of fully nonlinear second order equations and optimal stochastic control in infinite dimensions part i the case of bounded stochastic evolutions on etudie des equations elliptiques degenerees d ordre 2 non lineaires de la forme f d 2 u du u x 0 dans h ou h est un espace de hilbert separable x un point generique dans h u une fonction de h r

sparse canonical correlation analysis new formulation and algorithm in this paper we study canonical correlation analysis cca which is a powerful tool in multivariate data analysis for finding the correlation between two sets of multidimensional variables the main contributions of the paper are 1 to reveal the equivalent relationship between a recursive formula and a trace formula for the multiple cca problem 2 to obtain the explicit characterization for all solutions of the multiple cca problem even when the corresponding covariance matrices are singular 3 to develop a new sparse cca algorithm and 4 to establish the equivalent relationship between the uncorrelated linear discriminant analysis and the cca problem we test several simulated and real world datasets in gene classification and cross language document retrieval to demonstrate the effectiveness of the proposed algorithm the performance of the proposed method is competitive with the state of the art sparse cca algorithms

an adaptive algorithm for tv based model of three norms lq q 12 1 2 in image restoration abstract in this paper we present an adaptive method for the tv based model of three norms l q q 1 2 1 2 for the image restoration problem the algorithm with the l2 norm is used in the smooth regions where the value of u is small the algorithm with the l 1 2 norm is applied for the jumps where the value of u is large when the value of u is moderate the algorithm with the l1 norm is employed thus the three algorithms are applied for different regions of a given image such that the advantages of each algorithm are adopted the numerical experiments demonstrate that our adaptive algorithm can not only keep the original edge and original detailed information but also weaken the staircase phenomenon in the restored images specifically in contrast to the l1 norm as in the rudin osher fatemi model the l2 norm yields better results in the smooth and flat regions and the l 1 2 norm is more suitable in regions with strong discontinuities therefore our adaptive algorithm is efficient and robust even for images with large noises

an iterative algorithm for signal reconstruction from bispectrum the author presents a procedure to reconstruct a minimum or nonminimum phase discrete time signal from its bispectrum the algorithm is iterative and based on using the method of projections onto convex sets pocs in an inner product space in which the vector space consists of real sequences and vector addition is defined in terms of the convolution operation prior information such as an energy bound on the signal can be incorporated into this algorithm an algorithm that employs energy information was found to produce better results in simulation than one that ignores energy constraints

optimal control problems for stochastic reaction diffusion systems with non lipschitz coefficients by using the dynamic programming approach we study a control problem for a class of stochastic reaction diffusion systems with coefficients having polynomial growth in the cost functional a non lipschitz term appears and this allows us to treat the quadratic case which is of interest in the applications the corresponding hamilton jacobi bellman equation is first resolved by a fixed point argument in a small time interval and then is extended to arbitrary time intervals by suitable a priori estimates the main ingredient in the proof is the smoothing effect of the transition semigroup associated with the uncontrolled system

an improved algorithm for basis pursuit problem and its applications abstract we propose an algorithm for solving the basis pursuit problem min u c n u 1 a u f our starting motivation is the algorithm for compressed sensing proposed by qiao li and wu which is based on linearized bregman iteration with generalized inverse qiao li and wu defined new algorithm for solving the basis pursuit problem in compressive sensing using a linearized bregman iteration and the iterative formula of linear convergence for computing the matrix generalized inverse in our proposed approach we combine a partial application of the newton s second order iterative scheme for computing the generalized inverse with the bregman iteration our scheme takes lesser computational time and gives more accurate results in most cases the effectiveness of the proposed scheme is illustrated in two applications signal recovery from noisy data and image deblurring

regular solutions of second order stationary hamilton jacobi equations abstract we study a second order stationary hamilton jacobi equation in infinite dimension this equation is nonlinear and convex with respect to the first order term we use properties of the transition semigroup associated to the linear equation to write the hamilton jacobi equation in integral form and we prove existence uniqueness and regularity of a solution by the theory of maximal monotone operators we also prove that this solution is the pointwise limit of a uniformly bounded sequence of classical solutions of approximating problems finally the solution is the value function of the associated optimal stochastic control problem some examples are given

restoration of chopped and nodded images by framelets in infrared astronomy an observed image from a chop and nod process can be considered as the result of passing the original image through a high pass filter here we propose a restoration algorithm which builds up a tight framelet system that has the high pass filter as one of the framelet filters our approach reduces the solution of restoration problem to that of recovering the missing coefficients of the original image in the tight framelet decomposition the framelet approach provides a natural setting to apply various sophisticated framelet denoising schemes to remove the noise without reducing the intensity of major stars in the image a proof of the convergence of the algorithm based on convex analysis is also provided simulated and real images are tested to illustrate the efficiency of our method over the projected landweber method

technical communique robust stabilization for uncertain discrete singular systems this paper considers the problem of robust stabilization for uncertain discrete singular systems subject to time invariant norm bounded unstructured uncertainty in the state matrix the problem we address is the design of state feedback controllers such that the resulting closed loop system is regular causal as well as stable for all admissible uncertainties in this connection the concept of generalized quadratic stabilizability is defined in terms of matrix inequalities a necessary and sufficient condition for generalized quadratic stabilizability is derived under certain assumptions when this condition holds the explicit formula of a state feedback controller solving the problem is also given the results can be viewed as extensions of existing results on robust stabilization of uncertain discrete standard state space systems

deconvolution a wavelet frame approach this paper devotes to analyzing deconvolution algorithms based on wavelet frame approaches which has already appeared in chan et al siam j sci comput 24 4 1408 1432 2003 appl comput hormon anal 17 91 115 2004a int j imaging syst technol 14 91 104 2004b as wavelet frame based high resolution image reconstruction methods we first give a complete formulation of deconvolution in terms of multiresolution analysis and its approximation which completes the formulation given in chan et al siam j sci comput 24 4 1408 1432 2003 appl comput hormon anal 17 91 115 2004a int j imaging syst technol 14 91 104 2004b this formulation converts deconvolution to a problem of filling the missing coefficients of wavelet frames which satisfy certain minimization properties these missing coefficients are recovered iteratively together with a built in denoising scheme that removes noise in the data set such that noise in the data will not blow up while iterating this approach has already been proven to be efficient in solving various problems in high resolution image reconstructions as shown by the simulation results given in chan et al siam j sci comput 24 4 1408 1432 2003 appl comput hormon anal 17 91 115 2004a int j imaging syst technol 14 91 104 2004b however an analysis of convergence as well as the stability of algorithms and the minimization properties of solutions were absent in those papers this paper is to establish the theoretical foundation of this wavelet frame approach in particular a proof of convergence an analysis of the stability of algorithms and a study of the minimization property of solutions are given

the augmented lagrangian method for equality and inequality constraints in hilbert spaces in this paper we consider an augmented lagrangian method for the minimization of a nonlinear functional in the presence of an equality constraint whose image space is in a hilbert space an inequality constraint whose image space is finite dimensional and an affine inequality constraint whose image space is in an infinite dimensional hilbert space we obtain local convergence of this method without imposing strict complementarity conditions when the equality as well as the inequality constraint with finite dimensional image space are augmented to the author s knowledge this result even generalizes the convergence results which are known when all spaces are finite dimensional

convergence of the linearized bregman iteration for l1 norm minimization one of the key steps in compressed sensing is to solve the basis pursuit problem min u n u au f bregman iteration was very successfully used to solve this problem in 40 also a simple and fast iterative algorithm based on linearized bregman iteration was proposed in 40 which is described in detail with numerical simulations in 35 a convergence analysis of the smoothed version of this algorithm was given in 11 the purpose of this paper is to prove that the linearized bregman iteration proposed in 40 for the basis pursuit problem indeed converges

projected landweber iteration for matrix completion recovering an unknown low rank or approximately low rank matrix from a sampling set of its entries is known as the matrix completion problem in this paper a nonlinear constrained quadratic program problem concerning the matrix completion is obtained a new algorithm named the projected landweber iteration plw is proposed and the convergence is proved strictly numerical results show that the proposed algorithm can be fast and efficient under suitable prior conditions of the unknown low rank matrix

the burgers equation with a random force and a general model for directed polymers in random environments the study of the burgers equation with a random force leads via a hopf cole type transformation to a stochastic heat equation having a white noise with spatial parameters type potential the latter can be studied by means of a general model of directed polymers in random environments with two point random potentials these models exhibit a gaussian behavior at large times and have certain stationary distributions which yield the corresponding results for the above stochastic heat and burgers equations

wavelet algorithms for high resolution image reconstruction high resolution image reconstruction refers to the reconstruction of high resolution images from multiple low resolution shifted degraded samples of a true image in this paper we analyze this problem from the wavelet point of view by expressing the true image as a function in cal l bbb r 2 we derive iterative algorithms which recover the function completely in the cal l sense from the given low resolution functions these algorithms decompose the function obtained from the previous iteration into different frequency components in the wavelet transform domain and add them into the new iterate to improve the approximation we apply wavelet packet thresholding methods to denoise the function obtained in the previous step before adding it into the new iterate our numerical results show that the reconstructed images from our wavelet algorithms are better than that from the tikhonov least squares approach extension to super resolution image reconstruction where some of the low resolution images are missing is also considered

multiscale methods for polyhedral regularizations in this paper we present the extension and generalization of the adaptive inverse scale space aiss method proposed for ell 1 regularization in math comp 82 2013 pp 269 299 to arbitrary polyhedral functions we will see that the representation of a convex polyhedral function as a finitely generated function allows us to conclude that the inverse scale space flow the time continuous formulation of the augmented lagrangian method can be solved exactly without discretization we present an algorithm for the computation of such a solution for arbitrary polyhedral functions analyze its convergence and interpret the well known forward scale space flow as the inverse scale space flow on the convex conjugate functional thus including this class of flows in our analysis a surprising result is the equivalence of the scale space or gradient flow with a standard variational problem finally we give examples of the applications for the aiss algorithm for polyhedral functions which illustrate th

stationary hamilton jacobi equations in hilbert spaces and applications to a stochastic optimal control problem we study an infinite horizon stochastic control problem associated with a class of stochastic reaction diffusion systems with coefficients having polynomial growth the hamiltonian is assumed to be only locally lipschitz continuous so that the quadratic case can be covered we prove that the value function v corresponding to the control problem is given by the solution of the stationary hamilton jacobi equation associated with the state system to this purpose we write the hamilton jacobi equation in integral form and by using the smoothing properties of the transition semigroup relative to the state system and the theory of m dissipative operators we show that it admits a unique solution moreover the value function v is obtained as the limit of minima for some approximating control problems which admit unique optimal controls and states

second order hamilton jacobi equations in hilbert spaces and stochastic boundary control the paper is concerned with fully nonlinear second order hamilton jacobi bellman isaacs equations of elliptic type in separable hilbert spaces which have unbounded first and second order terms the viscosity solution approach is adapted to the equations under consideration and the existence and uniqueness of viscosity solutions are proved a stochastic optimal control problem driven by a parabolic stochastic pde with control of dirichlet type on the boundary is considered it is proved that the value function of this problem is the unique viscosity solution of the associated hamilton jacobi bellman equation

a dual split bregman method for fast l minimization in this paper we propose a new algorithm for fast l1 minimization as frequently arising in compressed sensing our method is based on a split bregman algorithm applied to the dual of the problem of minimizing u 1 1 2 u 2 such that u solves the under determined linear system au f which was recently investigated in the context of linearized bregman methods furthermore we provide a convergence analysis for split bregman methods in general and show with our compressed sensing example that a split bregman approach to the primal energy can lead to a different type of convergence than split bregman applied to the dual thus making the analysis of different ways to minimize the same energy interesting for a wide variety of optimization problems

regularization preconditioners for frame based image deblurring with reduced boundary artifacts thresholding iterative methods were recently successfully applied to image deblurring problems in this paper we investigate the modified linearized bregman algorithm mlba used in image deblurring problems with a proper treatment of the boundary artifacts we consider two standard approaches the imposition of boundary conditions and the use of the rectangular blurring matrix the fast convergence of the mlba depends on a regularizing preconditioner that could be computationally expensive and hence it is usually chosen as a block circulant circulant block bccb matrix diagonalized by the discrete fourier transform we show that the standard approach based on the bccb preconditioner may provide low quality restored images and we propose different preconditioning strategies that improve the quality of the restoration and save some computational cost at the same time motivated by a recent nonstationary preconditioned iteration we propose a new algorithm that combines such a method with the mlba we p

robust control analysis and design for discrete time singular systems abstract in this paper we propose a simple approach to analyse stability robustness of discrete time singular systems under structured perturbations the developed robustness criteria are then applied to solve robust regional pole assignment problems of singular systems a robust control design algorithm via state feedback is also given the robust stability problem of singular systems is more complicated than that of regular systems not only stability robustness but system regularity and impulse elimination should be considered simultaneously since the results of robust control and analysis for singular systems is not available in the literature as much as other fields the paper may be viewed as a complementary result in this field although only discrete time case is discussed several results can be directly applied to continuous time systems as well

adaptive wavelet galerkin methods for linear inverse problems we introduce and analyze numerical methods for the treatment of inverse problems based on an adaptive wavelet galerkin discretization these methods combine the theoretical advantages of the wavelet vaguelette decomposition wvd in terms of optimally adapting to the unknown smoothness of the solution together with the numerical simplicity of galerkin methods in a first step we simply combine a thresholding algorithm on the data with a galerkin inversion on a fixed linear space in a second step a more elaborate method performs the inversion by an adaptive procedure in which a smaller space adapted to the solution is iteratively constructed this leads to a significant reduction of the computational cost

reduced conservatism in stability robustness bounds by state transformation this note addresses the issue of conservatism in the time domain stability robustness bounds obtained by the lyapunov approach a state transformation is employed to improve the upper bounds on the linear time varying perturbation of an asymptotically stable linear time invariant system for robust stability this improvement is due to the variance of the conservatism of the lyapunov approach with respect to the basis of the vector space in which the lyapunov function is constructed improved bounds are obtained using a transformation on elemental and vector norms of perturbations i e structured perturbations as well as on a matrix norm of perturbations i e unstructured perturbations for the case of a diagonal transformation an algorithm is proposed to find the optimal transformation several examples are presented to illustrate the proposed analysis