analysis of multilevel methods for eddy current problems in papers by arnold falk and winther and by hiptmair novel multigrid methods for discrete h curl elliptic boundary value problems have been proposed such problems frequently occur in computational electromagnetism particularly in the context of eddy current simulation this paper focuses on the analysis of those nodal multilevel decompositions of the spaces of edge finite elements that form the foundation of the multigrid methods it provides a significant extension of the existing theory to the case of locally vanishing coefficients and nonconvex domains in particular asymptotically uniform convergence of the multigrid method with respect to the number of refinement levels can be established under assumptions that are satisfied in realistic settings for eddy current problems the principal idea is to use approximate helmholtz decompositions of the function space h curl into an h1 regular subspace and gradients the main results of standard multilevel theory for h1 elliptic problems can then be applied to both subspaces this yields preliminary decompositions still outside the edge element spaces judicious alterations can cure this

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

long time asymptotics of non degenerate non linear diffusion equations we study the long time asymptotics of prototypical non linear diffusion equations specifically we consider the case of a non degenerate diffusivity function that is a non negative polynomial of the dependent variable of the problem we motivate these types of equations using einstein s random walk paradigm leading to a partial differential equation in non divergence form on the other hand using conservation principles leads to a partial differential equation in divergence form a transformation is derived to handle both cases then a maximum principle on both an unbounded and a bounded domain is proved in order to obtain bounds above and below for the time evolution of the solutions to the non linear diffusion problem specifically these bounds are based on the fundamental solution of the linear problem the so called aronson s green function having thus sandwiched the long time asymptotics of solutions to the non linear problems between two fundamental solutions of the linear problem we prove that unlike the case of degenerate diffusion a non degenerate diffusion equation s solution converges onto the linear diffusion solution at long times select numerical examples support the mathematical theorems and illustrate the convergence process our results have implications on how to interpret asymptotic scalings of potentially anomalous diffusion processes such as in the flow of particulate materials that have been discussed in the applied physics literature

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

on a mathematical model of the productivity index of a well from reservoir engineering motivated by the reservoir engineering concept of the productivity index of a produc ing oil well in an isolated reservoir we analyze a time dependent functional diffusive capacity on the solutions to initial boundary value problems for a parabolic equation sufficient conditions providing for time independent diffusive capacity are given for different boundary conditions the dependence of the constant diffusive capacity on the type of the boundary condition dirichlet neumann or third boundary condition is investigated using a known variational principle and confirmed numer ically for various geometrical settings an important comparison between two principal constant values of a diffusive capacity is made leading to the establishment of criteria when the so called pseudo steady state and boundary dominated productivity indices of a well significantly differ from each other the third boundary condition is shown to model the thin skin effect for the constant wellbore pressure production regime for a damaged well the questions of stabilization and unique ness of the time independent values of the diffusive capacity are addressed the derived formulas are used in numerical study of evaluating the productivity index of a well in a general three dimensional reservoir for a variety of well configurations

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

interpreting idr as a petrov galerkin method the induced dimension reduction idr method of sonneveld and van gijzen siam j sci comput 31 2008 pp 1035 1062 is shown to be a petrov galerkin projection method with a particular choice of left krylov subspaces these left subspaces are rational krylov spaces consequently other methods such as bicgstab and ml s bicgstab which are mathematically equivalent to some versions of idr can also be interpreted as petrov galerkin methods the connection with rational krylov spaces inspired a new version of idr called ritz idr where the poles of the rational function are chosen as certain ritz values experiments are presented illustrating the effectiveness of this new version

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

analysis of generalized forchheimer flows of compressible fluids in porous media this work is focused on the analysis of nonlinear flows of slightly compressible fluids in porous media not adequately described by darcy s law we study a class of generalized nonlinear momentum equations which covers all three well known forchheimer equations the so called two term power and three term laws the generalized forchheimer equation is inverted to a nonlinear darcy equation with implicit permeability tensor depending on the pressure gradient this results in a degenerate parabolic equation for the pressure two classes of boundary conditions are considered given pressure and given total flux in both cases they are allowed to be unbounded in time the uniqueness lyapunov and asymptotic stabilities and other long time dynamical features of the corresponding initial boundary value problems are analyzed the results obtained in this paper have clear hydrodynamic interpretations and can be used for quantitative evaluation of engineering parameters some numerical simulations are also included

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

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

finding structure with randomness probabilistic algorithms for constructing approximate matrix decompositions low rank matrix approximations such as the truncated singular value decomposition and the rank revealing qr decomposition play a central role in data analysis and scientific computing this work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low rank matrix approximation these techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets this paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions these methods use random sampling to identify a subspace that captures most of the action of a matrix the input matrix is then compressed either explicitly or implicitly to this subspace and the reduced matrix is manipulated deterministically to obtain the desired low rank factorization in many cases this approach beats its classical competitors in terms of accuracy robustness and or speed these claims are supported by extensive numerical experiments and a detailed error analysis the specific benefits of randomized techniques depend on the computational environment consider the model problem of finding the k dominant components of the singular value decomposition of an m times n matrix i for a dense input matrix randomized algorithms require bigo mn log k floating point operations flops in contrast to bigo mnk for classical algorithms ii for a sparse input matrix the flop count matches classical krylov subspace methods but the randomized approach is more robust and can easily be reorganized to exploit multiprocessor architectures iii for a matrix that is too large to fit in fast memory the randomized techniques require only a constant number of passes over the data as opposed to bigo k passes for classical algorithms in fact it is sometimes possible to perform matrix approximation with a single pass over the data

fluid flows of mixed regimes in porous media in porous media there are three known regimes of fluid flows namely pre darcy darcy and post darcy because of their different natures these are usually treated separately in the literature to study complex flows when all three regimes may be present in different portions of a same domain we use a single equation of motion to unify them several scenarios and models are then considered for slightly compressible fluids a nonlinear parabolic equation for the pressure is derived which is degenerate when the pressure gradient is either small or large we estimate the pressure and its gradient for all time in terms of initial and boundary data we also obtain their particular bounds for large time which depend on the asymptotic behavior of the boundary data but not on the initial one moreover the continuous dependence of the solutions on initial and boundary data and the structural stability for the equation are established in porous media there are three known regimes of fluid flows namely pre darcy darcy and post darcy because of their different natures these are usually treated separately in the literature to study complex flows when all three regimes may be present in different portions of a same domain we use a single equation of motion to unify them several scenarios and models are then considered for slightly compressible fluids a nonlinear parabolic equation for the pressure is derived which is degenerate when the pressure gradient is either small or large we estimate the pressure and its gradient for all time in terms of initial and boundary data we also obtain their particular bounds for large time which depend on the asymptotic behavior of the boundary data but not on the initial one moreover the continuous dependence of the solutions on initial and boundary data and the structural stability for the equation are established

generalized integrating factor methods for stiff pdes the integrating factor if method for numerical integration of stiff nonlinear pdes has the disadvantage of producing large error coefficients when the linear term has large norm we propose a generalization of the if method and in particular construct multistep type methods with several orders of magnitude improved accuracy we also consider exponential time differencing etd methods and point out connections with a particular application of the commutator free lie group methods we present a new fourth order etdrk method with improved accuracy the methods considered are compared in several numerical examples

a class of explicit multistep exponential integrators for semilinear problems a class of explicit multistep exponential methods for abstract semilinear equations is introduced and analyzed it is shown that the k step method achieves order k for appropriate starting values which can be computed by auxiliary routines or by one strategy proposed in the paper together with some implementation issues numerical illustrations are also provided

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

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

exponential integrators for large systems of differential equations we study the numerical integration of large stiff systems of differential equations by methods that use matrix vector products with the exponential or a related function of the jacobian for large problems these can be approximated by krylov subspace methods which typically converge faster than those for the solution of the linear systems arising in standard stiff integrators the exponential methods also offer favorable properties in the integration of differential equations whose jacobian has large imaginary eigenvalues we derive methods up to order 4 which are exact for linear constant coefficient equations the implementation of the methods is discussed numerical experiments with reaction diffusion problems and a time dependent schrodinger equation are included

ml n bicgstabt a ml n bicgstab variant with a transpose the 1980 idr method wesseling and sonneveld 1980 12 plays an important role in the history of krylov subspace methods it started the research of transpose free krylov subspace methods the ml n bicgstab method yeung 2012 is one of such methods in this paper we present a new ml n bicgstab variant that involves a transpose in its implementation comparison of this new algorithm with the existing ml n bicgstab algorithms and some other krylov subspace algorithms will be presented

ml k bicgstab a bicgstab variant based on multiple lanczos starting vectors we present a variant of the popular bicgstab method for solving nonsymmetric linear systems the method which we denote by ml k bicgstab is derived from a variant of the bicg method based on a lanczos process using multiple k 1 starting left lanczos vectors compared with the original bicgstab method our new method produces a residual polynomial which is of lower degree after the same number of steps but which also requires fewer matrix vector products to generate on average requiring only 1 1 k matvecs per step empirically it also seems to be more stable and more quickly convergent the new method can be implemented as a k term recurrence and can be viewed as a bridge connecting the arnoldi based fom gmres methods and the lanczos based bicgstab methods

idr a new generation of krylov subspace methods the induced dimension reduction idr technique developed by sonneveld and van gijzen is a powerful concept resulting in a variety of transpose free krylov subspace methods based on short term recurrences we present the main di erences between and similarities of idr methods and classical krylov subspace methods our tool of trade is the so called generalized hessenberg decomposition the concept of transfer of techniques from the setting of classical krylov subspace methods to the idr based methods is introduced for simplicity we only sketch some recent results in the fields of eigenvalue computations and of solution of linear systems

maximum estimates for generalized forchheimer flows in heterogeneous porous media this article continues the study in 4 of generalized forchheimer flows in heterogeneous porous media such flows are used to account for deviations from darcy s law in heterogeneous media the derived nonlinear partial differential equation for the pressure can be singular and degenerate in the spatial variables in addition to being degenerate for large pressure gradient here we obtain the estimates for the l l norms of the pressure and its time derivative in terms of the initial and the time dependent boundary data they are established by implementing de giorgi moser s iteration in the context of weighted norms with the weights specifically defined by the forchheimer equation s coefficient functions with these weights we prove suitable weighted parabolic poincare sobolev inequalities and use them to facilitate the iteration moreover local in time l l bounds are combined with uniform gronwall type energy inequalities to obtain long time l l estimates

on krylov subspace approximations to the matrix exponential operator krylov subspace methods for approximating the action of matrix exponentials are analyzed in this paper we derive error bounds via a functional calculus of arnoldi and lanczos methods that reduces the study of krylov subspace approximations of functions of matrices to that of linear systems of equations as a side result we obtain error bounds for galerkin type krylov methods for linear equations namely the biconjugate gradient method and the full orthogonalization method for krylov approximations to matrix exponentials we show superlinear error decay from relatively small iteration numbers onwards depending on the geometry of the numerical range the spectrum or the pseudospectrum the convergence to exp tau a v is faster than that of corresponding krylov methods for the solution of linear equations i tau a x v which usually arise in the numerical solution of stiff ordinary differential equations odes we therefore propose a new class of time integration methods for large systems of nonlinear differential equations which use krylov approximations to the exponential function of the jacobian instead of solving linear or nonlinear systems of equations in every time step

variants of bicgstab for matrices with complex spectrum recently van der vorst siam j sci statist comput 13 1992 pp 631 644 proposed for solving nonsymmetric linear systems az b a biconjugate gradient bicg based krylov space method called bicgstab that like the biconjugate gradient squared bicgs method of sonneveld does not require matrix vector multiplications with the transposed matrix a t and that has typically a much smoother convergence behavior than bicg and bicgs its nth residual polynomial is the product of the one of bicg i e the nth lanczos polynomial with a polynomial of the same degree with real zeros therefore nonreal eigenvalues of a are not approximated well by the second polynomial factor here the author presents for real nonsymmetric matrices a method bicgstab2 in which the second factor may have complex conjugate zeros moreover versions suitable for complex matrices are given for both methods

one dimensional two phase generalized forchheimer flows of incompressible fluids abstract we derive a nonlinear system of parabolic equations to describe the one dimensional two phase generalized forchheimer flows of incompressible immiscible fluids in porous media with the presence of capillary forces under relevant constraints on relative permeabilities and capillary pressure non constant steady state solutions are found and classified into sixteen types according to their monotonicity and asymptotic behavior for a steady state whose saturation can never attain either value 0 or 1 we prove that it is stable with respect to a certain weight this weight is a function comprised of the steady state relative permeabilities and capillary pressure the proof is based on specific properties of the steady state weighted maximum principle and bernstein s estimate

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

exploiting bicgstab ell strategies to induce dimension reduction idr s p sonneveld and m b van gijzen siam j sci comput 31 2008 pp 1035 1062 and bicgstab ell g l g sleijpen and d r fokkema electron trans numer anal 1 1993 pp 11 32 are two of the most efficient short recurrence iterative methods for solving large nonsymmetric linear systems of equations which of the two is best depends on the specific problem class in this paper we describe idrstab a new method that combines the strengths of idr s and bicgstab ell to derive idrstab we extend the results that we reported on in g l g sleijpen p sonneveld and m b van gijzen appl numer math 2009 doi 10 1016 j apnum 2009 07 001 where we considered bi cgstab as an induced dimension reduction idr method we will analyze the relation between hybrid bi cg methods and idr and introduce the new concept of the sonneveld subspace as a common framework through numerical experiments we will show that idrstab can outperform both idr s and bicgstab ell

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

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

existence of a solution for generalized forchheimer flow in porous media with minimal regularity conditions this paper is focused on the generalized forchheimer flows for slightly compressible fluids described as a system of two nonlinear degenerating partial differential equations of first order we prove the existence and uniqueness of the dirichlet problem for the stationary case the technique of semidiscretization in time is used to prove the existence for the time dependent case this paper is focused on the generalized forchheimer flows for slightly compressible fluids described as a system of two nonlinear degenerating partial differential equations of first order we prove the existence and uniqueness of the dirichlet problem for the stationary case the technique of semidiscretization in time is used to prove the existence for the time dependent case

exponential time differencing for stiff systems we develop a class of numerical methods for stiff systems based on the method of exponential time differencing we describe schemes with second and higher order accuracy introduce new runge kutta versions of these schemes and extend the method to show how it may be applied to systems whose linear part is nondiagonal we test the method against other common schemes including integrating factor and linearly implicit methods and show how it is more accurate in a number of applications we apply the method to both dissipative and dispersive partial differential equations after illustrating its behavior using forced ordinary differential equations with stiff linear parts

spaces snu diametral dimension and property omega bar abstract we provide new developments to extend the formula of the diametral dimension of locally p convex spaces s aubry and bastin 2010 to some locally pseudoconvex ones we also use our method to prove that the spaces s verify the property of vogt and wagner in case they are locally p convex and for some cases of local pseudoconvexity

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

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

idr s a family of simple and fast algorithms for solving large nonsymmetric systems of linear equations we present idr s a new family of efficient short recurrence methods for large nonsymmetric systems of linear equations the new methods are based on the induced dimension reduction idr method proposed by sonneveld in 1980 idr s generates residuals that are forced to be in a sequence of nested subspaces although idr s behaves like an iterative method in exact arithmetic it computes the true solution using at most n n s matrix vector products with n the problem size and s the codimension of a fixed subspace we describe the algorithm and the underlying theory and present numerical experiments to illustrate the theoretical properties of the method and its performance for systems arising from different applications our experiments show that idr s is competitive with or superior to most bi cg based methods and outperforms bi cgstab when s 1

accelerating the induced dimension reduction method using spectral information the induced dimension reduction method idr s sonneveld and van gijzen 2008 is a short recurrences krylov method to solve systems of linear equations in this work we accelerate this method using spectral information we construct a hessenberg relation from the idr s residual recurrences formulas from which we approximate the eigenvalues and eigenvectors using the ritz values we propose a self contained variant of the ritz idr s method simoncini and szyld 2010 for solving a system of linear equations in addition the ritz vectors are used to speed up idr s for the solution of sequence of systems of linear equations 2018 elsevier b v

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

gbi cgstab s l idr s with higher order stabilization polynomials idr s is now recognized as one of the most effective methods often superior to other krylov subspace methods for large nonsymmetric linear systems of equations in this paper we propose an improvement upon idr s by incorporating a higher order stabilization polynomial into idr s the proposed algorithm named gbi cgstab s l shares desirable features with both idr s and bi cgstab l

advanced topology on the multiscale sequence spaces s nu abstract we pursue the study of the multiscale spaces s introduced by jaffard in the context of multifractal analysis we give the necessary and sufficient condition for s to be locally p convex and exhibit a sequence of p norms that defines its natural topology the strong topological dual of s is identified to another sequence space depending on endowed with an inductive limit topology as a particular case we describe the dual of a countable intersection of besov spaces

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

the effect of non optimal bases on the convergence of krylov subspace methods there are many examples where non orthogonality of a basis for krylov subspace methods arises naturally these methods usually require less storage or computational effort per iteration than methods using an orthonormal basis optimal methods but the convergence may be delayed truncated krylov subspace methods and other examples of non optimal methods have been shown to converge in many situations often with small delay but not in others we explore the question of what is the effect of having a non optimal basis we prove certain identities for the relative residual gap i e the relative difference between the residuals of the optimal and non optimal methods these identities and related bounds provide insight into when the delay is small and convergence is achieved further understanding is gained by using a general theory of superlinear convergence recently developed our analysis confirms the observed fact that in exact arithmetic the orthogonality of the basis is not important only the need to maintain linear independence is numerical examples illustrate our theoretical results

topological properties of the sequence spaces s nu abstract we define sequence spaces based on the distributions of the wavelet coefficients in the spirit of s jaffard beyond besov spaces part i distributions of wavelet coefficients j fourier anal appl 10 2004 221 246 we study their topology and especially show that they can be endowed with a unique complete metric for which compact sets can be explicitly described and we study properties of this metric we also give relationships with besov spaces

stability of runge kutta methods for quasilinear parabolic problems we consider a quasilinear parabolic problem u t q u t u t u t 0 u 0 d where q w d c x x w w c x is a family of sectorial operators in a banach space x with fixed domain d this problem is discretized in time by means of a strongly a stable 0 2 runge kutta method we prove that the resulting discretization is stable under some natural assumptions on the dependence of q w with respect to w our results are useful for studying in l p norms 1 p many problems arising in applications some auxiliary results for time dependent parabolic problems are also provided

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

a family of steady two phase generalized forchheimer flows and their linear stability analysis we model multi dimensional two phase flows of incompressible fluids in porous media using generalized forchheimer equations and the capillary pressure first we find a family of steady state solutions whose saturation and pressure are radially symmetric and velocities are rotation invariant their properties are investigated based on relations between the capillary pressure each phase s relative permeability and forchheimer polynomial second we analyze the linear stability of those steady states the linearized system is derived and reduced to a parabolic equation for the saturation this equation has a special structure depending on the steady states which we exploit to prove two new forms of the lemma of growth of landis type in both bounded and unbounded domains using these lemmas qualitative properties of the solution of the linearized equation are studied in details in bounded domains we show that the solution decays exponentially in time in unbounded domains in addition to their stability the solution decays to zero as the spatial variables tend to infinity the bernstein technique is also used in estimating the velocities all results have a clear physical interpretation

residual replacement strategies for krylov subspace iterative methods for the convergence of true residuals in this paper a strategy is proposed for alternative computations of the residual vectors in krylov subspace methods which improves the agreement of the computed residuals and the true residuals to the level of o u a x building on earlier ideas on residual replacement and on insights in the finite precision behavior of the krylov subspace methods computable error bounds are derived for iterations that involve occasionally replacing the computed residuals by the true residuals and they are used to monitor the deviation of the two residuals and hence to select residual replacement steps so that the recurrence relations for the computed residuals which control the convergence of the method are perturbed within safe bounds numerical examples are presented to demonstrate the effectiveness of this new residual replacement scheme

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

shifted gmres for oscillatory integrals none of the existing methods for computing the oscillatory integral int a b f x rm e rm i omega g x rm d x achieve all of the following properties high asymptotic order stability avoiding deformation into the complex plane and insensitivity to oscillations in f we present a new method that satisfies these properties based on applying the gmres algorithm to a shifted linear differential operator

on the convergence behavior of idr s and related methods an explanation is given of the convergence behavior of idr s methods the convergence mechanism of these algorithms has two components the first consists of damping properties of certain factors in the residual polynomials which becomes less important for large values of s the second component depends on the behavior of lanczos polynomials that occur in the theoretical description this part of the residual polynomials is related to lanczos methods with s left starting vectors as described in a paper by yeung and chan on their ml k bicgstab method in siam j sci comput 21 1999 pp 1263 1290 in this paper the behavior of the second component is compared with the full gmres method siam j sci statist comput 7 1986 pp 856 869 and an expected rate of convergence is given based on a random choice of the s shadow vectors

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

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 restarted induced dimension reduction method to approximate eigenpairs of large unsymmetric matrices this work presents a new algorithm to compute eigenpairs of large unsymmetric matrices using the induced dimension reduction method idr s which was originally proposed for solving systems of linear equations we obtain a hessenberg decomposition from which we approximate the eigenvalues and eigenvectors of a matrix this decomposition has two main advantages first idr s is a short recurrence method which is attractive for large scale computations second the idr s polynomial used to create this hessenberg decomposition is also used as a filter to discard the unwanted eigenvalues additionally we incorporate the implicitly restarting technique proposed by d c sorensen in order to approximate specific portions of the spectrum and improve the convergence

solution of the mixed formulation for generalized forchheimer flows of isentropic gases this paper focuses on the generalized forchheimer flows of isentropic gas in a porous medium described by a system of two nonlinear degenerate partial differential equations of first order we prove the existence and uniqueness of the dirichlet problem for the stationary problem the technique of semi discretization in time is used to prove the existence for the time dependent problem

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

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