second order approximation of the pseudoinverse for operator deconvolutions and families of ill posed problems we identify a more accurate means of deconvolving compact nondegenerate operators than that provided by independent application of standard regularization techniques to individual members of the implied continuum of first kind integral equations for the problem defined by h x t int y f x y g y t dy where it is required to find g y t given noise corrupted versions of square integrable f x y and h x t it is shown that greater accuracy results from optimizing solutions for each member of the sequence of integral equations derived from individual components of the singular value expansion of h x t than from optimizing solutions to each of the equations determined by different fixed values of t this result has particular implications for treatment of the bioelectric biomagnetic source imaging problems where the characteristic need to perform such a deconvolution has been previously handled by producing an individually regularized solution to the above equation for each t in a collection of

gmres implementations and residual smoothing techniques for solving ill posed linear systems there are verities of useful krylov subspace methods to solve nonsymmetric linear system of equations gmres is one of the best krylov solvers with several different variants to solve large sparse linear systems any gmres implementation has some advantages as the solution of ill posed problems are important in this paper some gmres variants are discussed and applied to solve these kinds of problems residual smoothing techniques are efficient ways to accelerate the convergence speed of some iterative methods like cg variants at the end of this paper some residual smoothing techniques are applied for different gmres methods to test the influence of these techniques on gmres implementations

inverse problem in avalanche dynamics models 1 avalanche dynamics models are increasingly used to estimate the features of extremely rare events for avalanche zoning they employ a frictional coefficient which reflects something close to snow viscosity as this coefficient is more conceptual than physical it cannot be measured and must be fitted by matching avalanche dynamics model results and field data however most of the time the historical record is not long enough to fit this coefficient for extremely rare events here we propose a deterministic inversion method to obtain the probability density function of this coefficient the method has been applied to two avalanche paths in the french alps each with a sustained avalanche activity over the last century for applications the voellmy avalanche dynamics model has been used with no loss of generality it is shown that the friction coefficient is a random variable whose marginal probability distribution varies rapidly and exhibits two or more peaks

large scale tikhonov regularization via reduction by orthogonal projection abstract this paper presents a new approach to computing an approximate solution of tikhonov regularized large scale ill posed least squares problems with a general regularization matrix the iterative method applies a sequence of projections onto generalized krylov subspaces a suitable value of the regularization parameter is determined by the discrepancy principle

a regularizing lanczos iteration method for underdetermined linear systems this paper is concerned with the solution of underdetermined linear systems of equations with a very ill conditioned matrix a whose dimensions are so large to make solution by direct methods impractical or infeasible image reconstruction from projections often gives rise to such systems in order to facilitate the computation of a meaningful approximate solution we regularize the linear system i e we replace it by a nearby system that is better conditioned the amount of regularization is determined by a regularization parameter its optimal value is in most applications not known a priori we present a new iterative method based on the lanczos algorithm for determining a suitable value of the regularization parameter by the discrepancy principle and an approximate solution of the regularized system of equations

a new twist two step iterative shrinkage thresholding algorithms for image restoration iterative shrinkage thresholding 1st algorithms have been recently proposed to handle a class of convex unconstrained optimization problems arising in image restoration and other linear inverse problems this class of problems results from combining a linear observation model with a nonquadratic regularizer e g total variation or wavelet based regularization it happens that the convergence rate of these 1st algorithms depends heavily on the linear observation operator becoming very slow when this operator is ill conditioned or ill posed in this paper we introduce two step 1st twist algorithms exhibiting much faster convergence rate than 1st for ill conditioned problems for a vast class of nonquadratic convex regularizers lscrp norms some besov norms and total variation we show that twist converges to a minimizer of the objective function for a given range of values of its parameters for noninvertible observation operators we introduce a monotonic version of twist mtwist although the convergence proof does not apply to this scenario we give experimental evidence that mtwist exhibits similar speed gains over ist the effectiveness of the new methods are experimentally confirmed on problems of image deconvolution and of restoration with missing samples

poincar normal forms for partial differential equations an extension of poincare normal form theory for systems of partial differential equations which have linear ordinary equations as a main part is obtained poincare normal forms for the shallow water equation on a beta plane and in large scale with respect to rossby radius for the variable coriolis parameter are found the normal forms of the shallow water equations contain the charney equation and higher approximation equations for rossby waves and describe the nonlinear interaction of rossby and inertia gravity waves

two soft thresholding based iterative algorithms for image deblurring iterative regularization algorithms such as the conjugate gradient algorithm for least squares problems cgls and the modified residual norm steepest descent mrnsd algorithm are popular tools for solving large scale linear systems arising from image deblurring problems these algorithms however are hindered by a semi convergence behavior in that the quality of the computed solution first increases and then decreases in this paper in order to overcome the semi convergence behavior we propose two iterative algorithms based on soft thresholding for image deblurring problems one of them combines cgls with a denoising technique like soft thresholding at each iteration and another combines mrnsd with soft thresholding in a similar way we prove the convergence of mrnsd and soft thresholding based algorithm numerical results show that the proposed algorithms overcome the semi convergence behavior and the restoration results are slightly better than those of cgls and mrnsd with their optimal stopping iterations

normal forms and quadratic nonlinear klein gordon equations on construit une transformation qui change l equation a non linearite cubique que l on sait resoudre a 3 1 dimensions on demontre que la solution du probleme original a le meme comportement asymptotique que l equation de klein gordon non lineaire

model adaptivity for goal oriented inference using adjoints abstract an inverse problem seeks to infer unknown model parameters using observed data we consider a goal oriented inverse problem where the goal of inferring parameters is to use them in predicting a quantity of interest qoi recognizing that multiple models of varying fidelity and computational cost may be available to describe the physical system we formulate a goal oriented model adaptivity approach that leverages multiple models while controlling the error in the qoi prediction in particular we adaptively form a mixed fidelity model by using models of different levels of fidelity in different subregions of the domain taking the solution of the inverse problem with the highest fidelity model as our reference qoi prediction we derive an adjoint based third order estimate for the qoi error from using a lower fidelity model localization of this error then guides the formation of mixed fidelity models we demonstrate the method for example problems described by convection diffusion reaction models for these examples our mixed fidelity models use the high fidelity model over only a small portion of the domain but result in qoi estimates with small relative errors we also demonstrate that the mixed fidelity inverse problems can be cheaper to solve and less sensitive to the initial guess than the high fidelity inverse problems

tikhonov regularization and the l curve for large discrete ill posed problems discretization of linear inverse problems generally gives rise to very ill conditioned linear systems of algebraic equations typically the linear systems obtained have to be regularized to make the computation of a meaningful approximate solution possible tikhonov regularization is one of the most popular regularization methods a regularization parameter specifies the amount of regularization and in general an appropriate value of this parameter is not known a priori we review available iterative methods and present new ones for the determination of a suitable value of the regularization parameter by the l curve criterion and the solution of regularized systems of algebraic equations

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

krylov subspace iterative methods for boundary element method based near field acoustic holography the reconstruction of the acoustic field for general surfaces is obtained from the solution of a matrix system that results from a boundary integral equation discretized using boundary element methods the solution to the resultant matrix system is obtained using iterative regularization methods that counteract the effect of noise on the measurements these methods will not require the calculation of the singular value decomposition which can be expensive when the matrix system is considerably large krylov subspace methods are iterative methods that have the phenomena known as semi convergence i e the optimal regularization solution is obtained after a few iterations if the iteration is not stopped the method converges to a solution that generally is totally corrupted by errors on the measurements for these methods the number of iterations play the role of the regularization parameter we will focus our attention to the study of the regularizing properties from the krylov subspace methods like conjugate gradients least squares qr and the recently proposed hybrid method a discussion and comparison of the available stopping rules will be included a vibrating plate is considered as an example to validate our results

generalized arnoldi tikhonov method for sparse reconstruction this paper introduces two new algorithms belonging to the class of arnoldi tikhonov regularization methods which are particularly appropriate for sparse reconstruction the main idea is to consider suitable adaptively defined regularization matrices that allow the usual 2 norm regularization term to approximate a more general regularization term expressed in the p norm p geq 1 the regularization matrix can be updated both at each step and after some iterations have been performed leading to two different approaches the first one is based on the idea of the iteratively reweighted least squares method and can be obtained considering flexible krylov subspaces the second one is based on restarting the arnoldi algorithm numerical examples are given in order to show the effectiveness of these new methods and comparisons with some other already existing algorithms are made

a regularization parameter in discrete ill posed problems the tikhonov regularization method for discrete ill posed problems is considered for the practical choice of the regularization parameter alpha some authors use a plot of the norm of the regularized solution versus the norm of the residual vector for all alpha considered this paper contains an analysis of the shape of this plot and gives a theoretical justification for choosing the regularization parameter so it is related to the l corner of the plot considered in the logarithmic scale moreover a new criterion for choosing alpha is introduced independent of the shape of the plot which gives a new interpretation of the corner criterion mentioned above the existence of l corner is discussed

a novel approach for studies of multispectral bioluminescence tomography bioluminescence tomography blt is a promising new area in biomedical imaging the goal of blt is to provide quantitative reconstruction of bioluminescent source distribution within a small animal from optical signals on the animal s body surface the multispectral version of blt takes advantage of the measurement information in different spectrum bands in this paper we propose a novel approach for the multispectral blt the new feature of the mathematical framework is to use numerical prediction results based on two related but distinct boundary value problems this mathematical framework includes the conventional framework in the study of multispectral blt for the new framework introduced here we establish the solution existence uniqueness and continuous dependence on data and characterize the limiting behaviors when the regularization parameter approaches zero or when the penalty parameter approaches infinity we study two kinds of numerical schemes for multispectral blt and derive error estimates for the numerical solutions we also present numerical examples to show the performance of the numerical methods

k0 of a semilocal ring let r be a semilocal ring that is r modulo its jacobson radical j r is artinian then k0 r j r is a partially ordered abelian group with order unit isomorphic to zn u where denotes the componentwise order on zn and u is an order unit in zn moreover the canonical projection r r j r induces an embedding of partially ordered abelian groups with order unit k0 k0 r k0 r j r in this paper we prove that every embedding of partially ordered abelian groups with order unit g zn can be realized as the mapping k0 k0 r k0 r j r for a suitable hereditary semilocal ring r

on tridiagonal matrices unitarily equivalent to normal matrices abstract in this article the unitary equivalence transformation of normal matrices to tridiagonal form is studied it is well known that any matrix is unitarily equivalent to a tridiagonal matrix in case of a normal matrix the resulting tridiagonal inherits a strong relation between its super and subdiagonal elements the corresponding elements of the super and subdiagonal will have the same absolute value in this article some basic facts about a unitary equivalence transformation of an arbitrary matrix to tridiagonal form are firstly studied both an iterative reduction based on krylov sequences as a direct tridiagonalization procedure via householder transformations are reconsidered this equivalence transformation is then applied to the normal case and equality of the absolute value between the super and subdiagonals is proved self adjointness of the resulting tridiagonal matrix with regard to a specific scalar product is proved properties when applying the reduction on symmetric skew symmetric hermitian skew hermitian and unitary matrices and their relations with e g complex symmetric and pseudo symmetric matrices are presented it is shown that the reduction can then be used to compute the singular value decomposition of normal matrices making use of the takagi factorization finally some extra properties of the reduction as well as an efficient method for computing a unitary complex symmetric decomposition of a normal matrix are given

iterative regularization method in generalized inverse beamforming abstract beamforming based on microphone array is a method to identify sound sources it can visualize the sound field of the source plane and reveal interesting acoustic information generalized inverse beamforming gib is one important branch of beamforming techniques due to its high identification accuracy and computational efficiency however in real testing situation errors caused by measurement noise and configuration problems may seriously reduce the beamforming accuracy as an inverse problem the stability of gib can be improved with regularization methods we proposed a new iterative regularization method for gib by iteratively redefining the form of regularization matrix and calculating the corresponding solution moreover the new method is applied to functional beamforming and double layer antenna beamforming respectively numerical simulations and experiments are implemented the results show that the proposed regularization method leads to more robust beamforming output and higher accuracy in both the two applications

extrapolation techniques for ill conditioned linear systems in this paper the regularized solutions of an ill conditioned system of linear equations are computed for several values of the regularization parameter lambda then these solutions are extrapolated at lambda 0 by various vector rational extrapolations techniques built for that purpose these techniques are justified by an analysis of the regularized solutions based on the singular value decomposition and the generalized singular value decomposition numerical results illustrate the effectiveness of the procedures

preconditioned iterative methods for linear discrete ill posed problems from a bayesian inversion perspective in this paper we revisit the solution of ill posed problems by preconditioned iterative methods from a bayesian statistical inversion perspective after a brief review of the most popular krylov subspace iterative methods for the solution of linear discrete ill posed problems and some basic statistics results we analyze the statistical meaning of left and right preconditioners as well as projected restarted strategies computed examples illustrating the interplay between statistics and preconditioning are also presented

approximation accuracy of the krylov subspaces for linear discrete ill posed problems abstract for the large scale linear discrete ill posed problem min a x b or a x b with b contaminated by gaussian white noise the lanczos bidiagonalization based krylov solver lsqr and its mathematically equivalent cgls the conjugate gradient cg method implicitly applied to a t a x a t b are most commonly used and cgme the cg method applied to min a a t y b or a a t y b with x a t y and lsmr which is equivalent to the minimal residual minres method applied to a t a x a t b have also been choices these methods exhibit typical semi convergence feature and the iteration number k plays the role of the regularization parameter however there has been no definitive answer to the long standing fundamental question can lsqr and cgls find 2 norm filtering best possible regularized solutions the same question is for cgme and lsmr too at iteration k lsqr cgme and lsmr compute different iterates from the same k dimensional krylov subspace a first and fundamental step towards answering the above question is to accurately estimate the accuracy of the underlying k dimensional krylov subspace approximating the k dimensional dominant right singular subspace of a assuming that the singular values of a are simple we present a general sin theorem for the 2 norm distances between these two subspaces and derive accurate estimates on them for severely moderately and mildly ill posed problems we also establish some relationships between the smallest ritz values and these distances numerical experiments justify the sharpness of our results

a pseudo heuristic parameter selection rule for l1 regularized minimization problems abstract this paper considers the regularization parameter determination of l 1 regularized minimization problem we solve the l 1 regularized problem using iterative reweighted least squares irls which involves solving a linear system whose coefficient matrix has the form m 1 n 0 1 the aim of this paper is to find an efficient and computationally inexpensive algorithm to both choose the regularization parameter and solve the l 1 regularized problem in order to achieve this we propose an irls algorithm with adaptive regularization parameter selection based on a heuristic parameter determination rule de boor s parameter selection criterion compared with some of the state of the art algorithms and parameter selection rules the numerical experiments show the efficiency and robustness of the proposed method

some properties of the arnoldi based methods for linear ill posed problems in this paper we study some properties of the classical arnoldi based methods for solving infinite dimensional linear equations involving compact operators these problems are intrinsically ill posed since a compact operator does not admit a bounded inverse we study the convergence properties and the ability of these algorithms to estimate the dominant singular values of the operator

reconstruction of distributed dynamic loads on an euler beam via mode selection and consistent spatial expression this paper presents a convenient and efficient approach to reconstruct two dimensional distributed dynamic loads on a damped thin elastic plate from its steady state dynamic response the approach is based on a mode selection method which comes from an idea that an optimal range of frequency and mode exists for dynamic load reconstruction as well as on the consistent spatial expression of the distributed dynamic loads the criterion for mode selection is derived theoretically a comparison between the method of truncated singular value decomposition tsvd and the method of mode selection demonstrates that the former is unstable while the latter is not only stable but robust with the proposed criterion the method of mode selection enables one to find an optimal range for load reconstruction provided that the error bounds of both parameters and objective are given or estimated and hence to make a balance between the reconstruction range and the error control

a flexible inner outer preconditioned gmres algorithm a variant of the gmres algorithm is presented that allows changes in the preconditioning at every step there are many possible applications of the new algorithm some of which are briefly discussed in particular a result of the flexibility of the new variant is that any iterative method can be used as a preconditioner for example the standard gmres algorithm itself can be used as a preconditioner as can cgnr or cgne the conjugate gradient method applied to the normal equations however the more appealing utilization of the method is in conjunction with relaxation techniques possibly multilevel techniques the possibility of changing preconditioners may be exploited to develop efficient iterative methods and to enhance robustness a few numerical experiments are reported to illustrate this fact

the sub cauchy stokes problem solvability issues and lagrange multiplier methods with artificial boundary conditions in contrast to the conventional cauchy stokes problems in which the velocity and the stress force data are given on the accessible boundary in the present paper we reduce the accessible boundary data information and we consider a problem which deals only with shear stress data we refer to this problem as a sub cauchy stokes problem this problem is ill posed because of severe instability and even uniqueness is unknown we first address the uniqueness issues associated with this problem resorting to the domain decomposition techniques together with the duplication process of vogelius 1 we propose new lagrange multiplier methods to solve the sub cauchy stokes problem these methods consist in recasting the problem in terms of interfacial equations by equalizing two solutions of the sub cauchy stokes problem using matching conditions defined on the inaccessible boundary the matching is based on second order conditions and depends on the equations used to match the values of the unknowns on the inaccessible boundary the underlying interfacial problems are then solved by iterative procedures in which coefficients can be optimized to improve convergence rates a complete analysis of the methods is presented and various numerical results illustrate the effectiveness and the performance of the proposed approaches

some properties of range restricted gmres methods the gmres method is one of the most popular iterative schemes for the solution of large linear systems of equations with a square nonsingular matrix gmres type methods also have been applied to the solution of linear discrete ill posed problems computational experience indicates that for the latter problems variants of the standard gmres method that require the solution to live in the range of a positive power of the matrix of the linear system of equations to be solved generally yield more accurate approximations of the desired solution than standard gmres this paper investigates properties of these variants of gmres

constraint preconditioners for solving singular saddle point problems we study constraint preconditioners for solving singular saddle point problems we analyze properties of the preconditioned matrices in particular their eigenvalue distributions and prove that for solving singular saddle point problems by preconditioned gmres methods with constraint preconditioners gmres will determine the least squares solutions at breakdown in addition we present some numerical examples to demonstrate the convergence behavior of preconditioned gmres for solving singular saddle point problems

arnoldi tikhonov regularization methods tikhonov regularization for large scale linear ill posed problems is commonly implemented by determining a partial lanczos bidiagonalization of the matrix of the given system of equations this paper explores the possibility of instead computing a partial arnoldi decomposition of the given matrix computed examples illustrate that this approach may require fewer matrix vector product evaluations and therefore less arithmetic work moreover the proposed range restricted arnoldi tikhonov regularization method does not require the adjoint matrix and hence is convenient to use for problems for which the adjoint is difficult to evaluate

a projection based approach to general form tikhonov regularization we present a projection based iterative algorithm for computing general form tikhonov regularized solutions to the problem min x ax b 2 2 lambda 2 lx 2 2 where the regularization matrix l is not the identity our algorithm is designed for the common case where lambda is not known a priori it is based on a joint bidiagonalization algorithm and is appropriate for large scale problems when it is computationally infeasible to transform the regularized problem to standard form by considering the projected problem we show how estimates of the corresponding optimal regularization parameter can be efficiently obtained numerical results illustrate the promise of our projection based approach

an efficient algorithm for regularization of laplace transform inversion in real case we address design of a numerical algorithm for solving the linear system arising in numerical inversion of laplace transforms in real case l d amore a murli regularization of a fourier series method for the laplace transform inversion with real data inverse problems 18 2002 1185 1205 the matrix has a condition number that grows almost exponentially and the singular values decay gradually towards zero in such a case because of this intrinsic strong instability the main difficulty of any numerical computation is the ability of discovering at run time only using data what is the maximum attainable accuracy on the solution in this paper we use gmres with the aim of relating the current residuals to the maximum attainable accuracy of the approximate solution by using a suitable stopping rule we prove that gmres stops after at most as many iterations as the number of the largest eigenvalues compared to the machine epsilon we use a split preconditioner that symmetrically precondition the initial system by this way the largest eigenvalue dynamically provides the estimate of the condition number of the matrix

normal forms and nonlinear symmetries we give some general theorems and extensions of previous results concerning the problem of transforming an algebra of vector fields into poincare normal form by means of a unifying algebraic language we show the possibility of obtaining either a parallel or joint normal form of the vector fields in a definite way which simplifies the construction of normal forms providing a precise restriction on their structure the application to finite dimensional dynamical systems and their lie point symmetries is also discussed

on a partially ordered set associated to ring morphisms abstract we associate to any ring r with identity a partially ordered set hom r whose elements are all pairs a m where a ker and m 1 u s for some ring morphism of r into an arbitrary ring s here u s denotes the group of units of s the assignment r hom r turns out to be a contravariant functor of the category ring of associative rings with identity to the category parord of partially ordered sets the maximal elements of hom r constitute a subset max r which for commutative rings r can be identified with the zariski spectrum spec r of r every pair a m in hom r has a canonical representative that is there is a universal ring morphism r s r a m a corresponding to the pair a m where the ring s r a m a is constructed as a universal inverting r a ring in the sense of cohn several properties of the sets hom r and max r are studied

discrete ill posed least squares problems with a solution norm constraint straightforward solution of discrete ill posed least squares problems with error contaminated data does not in general give meaningful results because propagated error destroys the computed solution error propagation can be reduced by imposing constraints on the computed solution a commonly used constraint is the discrepancy principle which bounds the norm of the computed solution when applied in conjunction with tikhonov regularization another approach which recently has received considerable attention is to explicitly impose a constraint on the norm of the computed solution for instance the computed solution may be required to have the same euclidean norm as the unknown solution of the error free least squares problem we compare these approaches and discuss numerical methods for their implementation among them a new implementation of the arnoldi tikhonov method also solution methods which use both the discrepancy principle and a solution norm constraint are considered

convex non convex image segmentation a convex non convex variational model is proposed for multiphase image segmentation we consider a specially designed non convex regularization term which adapts spatially to the image structures for a better control of the segmentation boundary and an easy handling of the intensity inhomogeneities the nonlinear optimization problem is efficiently solved by an alternating directions methods of multipliers procedure we provide a convergence analysis and perform numerical experiments on several images showing the effectiveness of this procedure

automatic parameter setting for arnoldi tikhonov methods in the framework of iterative regularization techniques for large scale linear ill posed problems this paper introduces a novel algorithm for the choice of the regularization parameter when performing the arnoldi tikhonov method assuming that we can apply the discrepancy principle this new strategy can work without restrictions on the choice of the regularization matrix moreover this method is also employed as a procedure to detect the noise level whenever it is just overestimated numerical experiments arising from the discretization of integral equations and image restoration are presented

fast linearized alternating direction minimization algorithm with adaptive parameter selection for multiplicative noise removal owing to the edge preserving ability and low computational cost of the total variation tv variational models with the tv regularizer have been widely investigated in the field of multiplicative noise removal the key points of the successful application of these models lie in the optimal selection of the regularization parameter which balances the data fidelity term with the tv regularizer the efficient algorithm to compute the solution in this paper we propose two fast algorithms based on the linearized technique which are able to estimate the regularization parameter and recover the image simultaneously in the iteration step of the proposed algorithms the regularization parameter is adjusted by a special discrepancy function defined for multiplicative noise the convergence properties of the proposed algorithms are proved under certain conditions and numerical experiments demonstrate that the proposed algorithms overall outperform some state of the art methods in the psnr values and computational time

analysis of discrete ill posed problems by means of the l curve when discrete ill posed problems are analyzed and solved by various numerical regularization techniques a very convenient way to display information about the regularized solution is to plot the norm or seminorm of the solution versus the norm of the residual vector in particular the graph associated with tikhonov regularization plays a central role the main purpose of this paper is to advocate the use of this graph in the numerical treatment of discrete ill posed problems the graph is characterized quantitatively and several important relations between regularized solutions and the graph are derived it is also demonstrated that several methods for choosing the regularization parameter are related to locating a characteristic l shaped corner of the graph

flexible inner outer krylov subspace methods flexible krylov methods refers to a class of methods which accept preconditioning that can change from one step to the next given a krylov subspace method such as cg gmres qmr etc for the solution of a linear system ax b instead of having a fixed preconditioner m and the right preconditioned equation am 1 y b mx y one may have a different matrix say mk at each step in this paper the case where the preconditioner itself is a krylov subspace method is studied there are several papers in the literature where such a situation is presented and numerical examples given a general theory is provided encompassing many of these cases including truncated methods the overall space where the solution is approximated is no longer a krylov subspace but a subspace of a larger krylov space we show how this subspace keeps growing as the outer iteration progresses thus providing a convergence theory for these inner outer methods numerical tests illustrate some important implementation aspects that make the discussed inner outer methods very appealing in practical circumstances

restoring images degraded by spatially variant blur restoration of images that have been blurred by the effects of a gaussian blurring function is an ill posed but well studied problem any blur that is spatially invariant can be expressed as a convolution kernel in an integral equation fast and effective algorithms then exist for determining the original image by preconditioned iterative methods if the blurring function is spatially variant however then the problem is more difficult in this work we develop fast algorithms for forming the convolution and for recovering the original image when the convolution functions are spatially variant but have a small domain of support this assumption leads to a discrete problem involving a banded matrix we devise an effective preconditioner and prove that the preconditioned matrix differs from the identity by a matrix of small rank plus a matrix of small norm numerical examples are given related to the hubble space telescope hst wide field planetary camera the algorithms that we develop are applicable to other ill posed integral equations as well

on the computation of gauss quadrature rules for measures with a monomial denominator let d be a nonnegative measure with support on the real axis and let r be outside the convex hull of the support this paper describes a new approach to determining recursion coefficients for gauss quadrature rules associated with measures of the form d x d x x 2 the proposed method is based on determining recursion coefficients for a suitable family of orthonormal laurent polynomials numerical examples show this approach to yield higher accuracy than available methods

gmres type methods for inconsistent systems abstract the behavior of iterative methods of gmres type when applied to singular possibly inconsistent linear systems is discussed and conditions under which these methods converge to the least squares solution of minimal norm are presented error bounds for the computed iterates are shown this paper complements previous work by brown and walker p n brown h f walker siam j matrix anal appl 18 1997 37 51

conditional gradient tikhonov method for a convex optimization problem in image restoration in this paper we consider the problem of image restoration with tikhonov regularization as a convex constrained minimization problem using a kronecker decomposition of the blurring matrix and the tikhonov regularization matrix we reduce the size of the image restoration problem therefore we apply the conditional gradient method combined with the tikhonov regularization technique and derive a new method we demonstrate the convergence of this method and perform some numerical examples to illustrate the effectiveness of the proposed method as compared to other existing methods

boundary conditions and multiple image re blurring the lbt case in this paper we are concerned with deblurring problems in the case of multiple images coming from the large binocular telescope an important example of telescope of interferometric type for this problem we are interested in checking the role of the boundary conditions in the quality of the reconstructed image in particular we will consider reflective and anti reflective boundary conditions and the re blurring idea the results of the proposed combinations are quite satisfactory when compared with classical dirichlet or periodic boundary conditions especially when increasing the number of images acquired by the lbt this behavior is confirmed by a wide numerical experimentation

a note on krylov subspace methods for singular systems abstract recently calvetti et al have published an interesting paper linear algebra appl 316 2000 157 169 concerning the least squares solution of a singular system by using the so called range restricted gmres rrgmres method however one of the main results cf loc cit theorem 3 3 seems to be incomplete as a complement of paper loc cit in this note we first make an example to show the incompleteness of that theorem then we give a modified result

finite element approach to clustering of multidimensional time series we present a new approach to clustering of time series based on a minimization of the averaged clustering functional the proposed functional describes the mean distance between observation data and its representation in terms of mathbf k abstract models of a certain predefined class not necessarily given by some probability distribution for a fixed time series x t this functional depends on mathbf k sets of model parameters theta theta 1 dots theta mathbf k and mathbf k functions of cluster affiliations gamma gamma 1 t dots gamma mathbf k t characterizing the affiliation of any element x t of the analyzed time series to one of the mathbf k clusters defined by the considered model parameters we demonstrate that for a fixed set of model parameters theta the appropriate tykhonov type regularization of this functional with some regularization factor epsilon 2 results in a minimization problem similar to a variational problem usually associated with one dimensional nonhomogeneous partial differential equations this analogy allows us to apply the finite element framework to the problem of time series analysis and to propose a numerical scheme for time series clustering we investigate the conditions under which the proposed scheme allows a monotone improvement of the initial parameter guess with respect to the minimization of the discretized version of the regularized functional we also discuss the interpretation of the regularization factor in the markovian case and show its connection to metastability and exit times the computational performance of the resulting method is investigated numerically on multidimensional test data and is applied to the analysis of multidimensional historical stock market data

regula falsi based automatic regularization method for pde constrained optimization abstract many inverse problems can be described by a pde model with unknown parameters that need to be calibrated based on measurements related to its solution this can be seen as a constrained minimization problem where one wishes to minimize the mismatch between the observed data and the model predictions including an extra regularization term and use the pde as a constraint often a suitable regularization parameter is determined by solving the problem for a whole range of parameters e g using the l curve which is computationally very expensive in this paper we derive two methods that simultaneously solve the inverse problem and determine a suitable value for the regularization parameter the first one is a direct generalization of the generalized arnoldi tikhonov method for linear inverse problems the second method is a novel method based on similar ideas but with a number of advantages for nonlinear problems

perpendicular categories with applications to representations and sheaves this paper is concerned with the omnipresence of the formation of the subcategories right left perpendicular to a subcategory of objects in an abelian category we encounter these subcategories in various contexts the formation of quotient categories with respect to localizing subcategories cf section 2 the deletion of vertices and shrinking of arrows see 37 in the representation theory of finite dimensional algebras cf section 5 the comparison of the representation theories of different extended dynkin quivers cf section 10 the theory of tilting cf sections 4 and 6 the study of homological epimorphisms of rings cf section 4 the passage from graded modules to coherent sheaves on a possibly weighted projective variety or scheme cf section 7 and 21 the study of maximal cohen macaulay modules over surface singularities cf sections the comparison of weighted projective lines for different weight sequences cf section 9 the formation of affine and local algebras attached to path algebras of extended dynkin quivers canonical algebras and weighted projective lines cf section 11 and 21 and the concept of universal localization in 40 formation of the perpendicular category has many aspects in common with localization and allows one to dispose of localization techniques in situations not accessible to any of the classical concepts of localization this applies in particular to applications in the domain of finite dimensional algebras and their representations several applications of the methods presented in this paper are already in existence partly published or appearing in print in the near future see for instance 40 39 4 45 26 49 46 and have show the versatility of the notion of a perpendicular category it seems that right perpendicular categories first appeared as the subcategories of so called closed objects in the process of the formation of the quotient cateogory of an abelian category with respect to a localizing serre subcategory see 18 47 34 another natural occurrence is encountered in commutative algebra forming the possibly infinitely generated modules of depth 2 cf section 7 the concept and some of the central applications were first presented in a talk given by the first author at the honnef meeting in january 1985 we also note that the perfectly matching nomination perpendicular category was coined by a schofield who discovered independently the usefulness of this concept in dealing with hereditary algebras see 39 cf also section 7 the authors further acknowledge the support of the deutsche forschungsgemeinschaft spp darstellungstheorie von endlichen gruppen und endlichdimensionalen algebren throughout this paper rings are associative with unit and modules are unitary right modules mod r respectively mod r denotes the category of all respectively all finitely presented right r modules

sylvester tikhonov regularization methods in image restoration in this paper we consider large scale linear discrete ill posed problems where the right hand side contains noise regularization techniques such as tikhonov regularization are needed to control the effect of the noise on the solution in many applications such as in image restoration the coefficient matrix is given as a kronecker product of two matrices and then tikhonov regularization problem leads to the generalized sylvester matrix equation for large scale problems we use the global gmres method which is an orthogonal projection method onto a matrix krylov subspace we present some theoretical results and give numerical tests in image restoration

electromagnetic field synthesis by hierarchical plane wave based field transformation an optimization based field synthesis approach is presented that employs a hierarchical plane wave based field transformation concept for efficient evaluation of the inverse problem forward operator irregular transmit antenna array configurations can be used to synthesize a target field distribution in an arbitrarily shaped test volume with full consideration of the antenna element radiation characteristics for this purpose a triangular mesh forms the test zone boundary surface on which rao wilton glisson rwg vector basis functions perform electric and magnetic field testing an iterative krylov subspace solver is employed to solve the ill posed inverse problem for the transmit antenna coefficients the effectiveness accuracy and efficiency of the proposed field synthesis algorithm are demonstrated for configurations with test zone extents of up to one hundred wavelengths moreover the approach is utilized for antenna measurements where antenna near fields are transformed into far fields

square smoothing regularization matrices with accurate boundary conditions this paper is concerned with the solution of large scale linear discrete ill posed problems the determination of a meaningful approximate solution of these problems requires regularization we discuss regularization by the tikhonov method and by truncated iteration the choice of regularization matrix in tikhonov regularization may significantly affect the quality of the computed approximate solution the present paper describes the construction of square regularization matrices from finite difference equations with a focus on the boundary conditions the regularization matrices considered have a structure that makes them easy to apply in iterative methods including methods based on the arnoldi process numerical examples illustrate the properties and effectiveness of the regularization matrices described

on the discrepancy principle for iterative and parametric methods to solve linear ill posed equations for the numerical solution of non necessarily well posed linear equations in banach spaces we consider a class of iterative methods which contains well known methods like the richardson iteration if the associated resolvent operator fulfils a condition with respect to a sector it is the purpose of this paper to show that for given noisy right hand side the discrepancy principle being a stopping rule for the iteration methods belonging to the mentioned class defines a regularization method and convergence rates are proved under additional smoothness conditions on the initial error this extends similar results obtained for positive semidefinite problems in hilbert spaces then we consider a class of parametric methods which under the same resolvent condition contains the method of the abstract cauchy problem and under a weaker resolvent condition the iterated method of lavrentiev a modified discrepancy principle is formulated for them and finally numerical illustrations are presented

a tikhonov regularized penalty function approach for solving polylinear programming problems abstract this paper suggests a new regularized penalty method for poly linear functions until our knowledge it is the first time that a regularization approach solution for poly linear programming is reported in the literature we propose a penalty function depending on two parameters and for ensuring the strong convexity and the existence of a unique solution involving equality and inequality constraints we prove that if the penalty parameter tends to zero then the solution of the original problem converges to a unique solution with the minimal weighted norm we introduce a recurrent procedure based on the projection gradient method for finding the extremal points and we also prove the convergence of the method we develop an example for game theory and additional example for portfolio optimization employing the proposed regularization method for markov chains involving the definition of a poly linear function

estimating the non homogeneous elastic modulus distribution from surface deformations abstract we present a novel approach to solve the inverse problem in finite elasticity for the non homogeneous shear modulus distribution solely from known surface deformation fields the inverse problem is posed as a constrained optimization problem under regularization and solved utilizing the adjoint equations hypothetical measured surface displacement fields are created by inducing indentations on the exterior of the specimen these surface displacement fields are used to test the inverse strategy on a problem domain consisting of a stiff circular inclusion in a softer homogeneous background we observe that the shear modulus reconstruction as well as the shape of the circular inclusion improves with an increasing number of surface displacement fields furthermore with increasing noise level in the surface displacement field the contrast of the reconstructions decreases

error analysis of krylov methods in a nutshell error and residual bounds for the matrix iteration methods bicg qmr fom and gmres are derived in a simple and unified way

regularization matrices determined by matrix nearness problems abstract this paper is concerned with the solution of large scale linear discrete ill posed problems with error contaminated data tikhonov regularization is a popular approach to determine meaningful approximate solutions of such problems the choice of regularization matrix in tikhonov regularization may significantly affect the quality of the computed approximate solution this matrix should be chosen to promote the recovery of known important features of the desired solution such as smoothness and monotonicity we describe a novel approach to determine regularization matrices with desired properties by solving a matrix nearness problem the constructed regularization matrix is the closest matrix in the frobenius norm with a prescribed null space to a given matrix numerical examples illustrate the performance of the regularization matrices so obtained

superoptimal approximation for unbounded symbols abstract the superoptimal frobenius approximation of toeplitz matrices is considered in connection with the case of unbounded symbols in particular we use the superoptimal approximation as preconditioner for the cg method when a fisher hartwig singularity is present in the symbol with special regard to systems coming from times series and financial applications a theoretical discussion concerning classical circulant preconditioners and a numerical comparison with the strang and with the optimal approximations are presented particularly with reference to the presence of noise

decomposition methods for large linear discrete ill posed problems the solution of large linear discrete ill posed problems by iterative methods continues to receive considerable attention this paper presents decomposition methods that split the solution space into a krylov subspace that is determined by the iterative method and an auxiliary subspace that can be chosen to help represent pertinent features of the solution decomposition is well suited for use with the gmres rrgmres and lsqr iterative schemes

generalized hybrid iterative methods for large scale bayesian inverse problems we develop a generalized hybrid iterative approach for computing solutions to large scale bayesian inverse problems we consider a hybrid algorithm based on the generalized golub kahan bidiagonalization for computing tikhonov regularized solutions to problems where explicit computation of the square root and inverse of the covariance kernel for the prior covariance matrix is not feasible this is useful for large scale problems where covariance kernels are defined on irregular grids or are available only via matrix vector multiplication e g those from the matern class we show that iterates are equivalent to lsqr iterates applied to a directly regularized tikhonov problem after a transformation of variables and we provide connections to a generalized singular value decomposition filtered solution our approach shares many benefits of standard hybrid methods such as avoiding semiconvergence and automatically estimating the regularization parameter numerical examples from image processing demonstrate

a predictor corrector iterated tikhonov regularization for linear ill posed inverse problems many questions in science and engineering often give rise to linear ill posed inverse problems to enable meaningful approximate solutions of the inverse problems various regularization methods have subsequently been proposed to make these problems less sensitive to perturbations one of the most popular regularization techniques is iterated tikhonov regularization which has attracted considerable attention due to its interesting applications in practical however this regularization often suffers from inaccurate results and low computational efficiency in some situations in this paper we proposed an accelerated predictor corrector iterated tikhonov regularization this method combined the classical iterated tikhonov regularization with modified euler method which could improve computational efficiency without sacrificing numerical accuracy the convergence rate of the accelerated regularization was investigated theoretically when the right hand side was corrupted by noise in particular we derived an error estimate of optimal order for the convergence of the accelerated regularization both one and two dimensional numerical experiments were implemented to illustrate the accuracy and efficiency of the accelerated version of iterated tikhonov regularization

robust identification of elastic properties using the modified constitutive relation error abstract the present study concerns the identification of linear elastic material properties based on inhomogeneous tests and the use of full field measurements often based upon inverse approaches this study presents the formulation of the so called modified constitutive relation error mcre method in the context of elastostatics when dealing with uncertain data such an approach addresses the concept of the reliability of information and mainly consists in the partitioning of all the available mechanical quantities into a reliable set and a less reliable one so as to take into account the measurement uncertainties and the error made on the constitutive equation into the formulation and then allows to identify the sought material properties the method is split in two steps the first one consists in defining admissible mechanical fields from all the theoretical and experimental data for a fixed set of mechanical properties this is made by the minimization of a criterion allowing a compromise between the constitutive equation and the measurements adequacy then the second step consists in the identification of the sought material properties and takes the form of minimizing a cost function defined by using the above admissible mechanical fields a comparison with the finite element model updating femu method was performed on some numerical examples where realistic perturbations were added this comparison showed that the mcre method is more robust towards perturbations for similar input data moreover the proposed method only deals with the available information and does not need additional hypotheses to calculate the mechanical quantities eventually the method was applied to the identification of the shear modulus of an organic matrix composite from experimental data

arnoldi methods for image deblurring with anti reflective boundary conditions image deblurring with anti reflective boundary conditions and a non symmetric point spread function is considered several iterative methods based on krylov subspace projections as well as arnoldi tikhonov regularization methods with reblurring right or left preconditioners are compared the aim of the preconditioner is not to accelerate the convergence but to improve the quality of the computed solution and to increase the robustness of the regularization method right preconditioning in conjunction with methods based on the arnoldi process are found to be robust and give high quality restorations in particular when the observed image is contaminated by motion blur our new method is much more effective than other approaches described in the literature such as range restricted arnoldi methods and the conjugate gradient method applied to the normal equations implemented with the reblurring approach

implementations of range restricted iterative methods for linear discrete ill posed problems abstract this paper is concerned with iterative solution methods for large linear systems of equations with a matrix of ill determined rank and an error contaminated right hand side the numerical solution is delicate because the matrix is very ill conditioned and may be singular it is natural to require that the computed iterates live in the range of the matrix when the latter is symmetric because then the iterates are orthogonal to the null space computational experience indicates that it can be beneficial to require that the iterates live in the range of the matrix also when the latter is nonsymmetric we discuss the design and implementation of iterative methods that determine iterates with this property new implementations that are particularly well suited for use with the discrepancy principle are described

the generalized finite difference method for the inverse cauchy problem in two dimensional isotropic linear elasticity abstract in this study the generalized finite difference method gfdm was used to stably and accurately solve two dimensional 2d inverse cauchy problems in linear elasticity by using the navier equations in cauchy problems overdetermined boundary conditions are imposed on parts of the boundary whereas there are missing boundary conditions on some parts of the boundary in cauchy problems conventional numerical methods generally generate highly ill conditioned matrices and thus provide unstable numerical solutions moreover even if a slight noise is added in the boundary conditions numerical errors are evidently magnified the gfdm one of the most promising meshless methods and an extension of the classical finite difference method can avoid time consuming tasks of mesh generation and numerical quadrature the gfdm was applied in this study to stably solve the 2d cauchy problems in linear elasticity and four numerical examples are provided to illustrate the consistency and accuracy of the presented meshless numerical scheme moreover the stability of the presented scheme for inverse cauchy problems was proved by adding noise into the boundary conditions

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

a modified tikhonov regularization method tikhonov regularization and truncated singular value decomposition tsvd are two elementary techniques for solving a least squares problem from a linear discrete ill posed problem based on these two techniques a modified regularization method is proposed which is applied to an arnoldi based hybrid method theoretical analysis and numerical examples are presented to illustrate the effectiveness of the method

the discrete cosine transform each discrete cosine transform dct uses n real basis vectors whose components are cosines in the dct 4 for example the j th component of boldv k is cos j frac 1 2 k frac 1 2 frac pi n these basis vectors are orthogonal and the transform is extremely useful in image processing if the vector boldx gives the intensities along a row of pixels its cosine series sum c k boldv k has the coefficients c k boldx boldv k n they are quickly computed from a fast fourier transform but a direct proof of orthogonality by calculating inner products does not reveal how natural these cosine vectors are we prove orthogonality in a different way each dct basis contains the eigenvectors of a symmetric second difference matrix by varying the boundary conditions we get the established transforms dct 1 through dct 4 other combinations lead to four additional cosine transforms the type of boundary condition dirichlet or neumann centered at a meshpoint or a midpoint determines the applications that are appropriate for each transform the centering also determines the period n 1 or n in the established transforms n frac 1 2 or n frac 1 2 in the other four the key point is that all these eigenvectors of cosines come from simple and familiar matrices

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

improvement of space invariant image deblurring by preconditioned landweber iterations the landweber method is a simple and flexible iterative regularization algorithm whose projected variant provides nonnegative image reconstructions since the method is usually very slow we apply circulant preconditioners exploiting the shift invariance of many deblurring problems in order to accelerate the convergence this way reasonable reconstructions can be obtained within a few iterations the method becomes competitive and more robust than other approaches that although faster sometimes lead to lower accuracy some theoretical analysis of convergence is given together with numerical validations