residual bounds of approximate solutions of the discrete time algebraic riccati equation let tilde x be a hermitian matrix which approximates the unique hermitian positive semi definite solution x to the discrete time algebraic riccati equation dare where f in cal c n times n c 2 in cal c m times m is hermitian positive definite g 1 in cal c n times m c in cal c r times n the pair f g 1 is stabilizable and the pair c f is detectable assume that i g tilde x is nonsingular and i g tilde x 1 f is stable let g g 1g 2 1 g 1 rm h h c rm h c and let be the residual of the dare with respect to tilde x define the linear operator vec l by the main result of this paper is if where denotes any unitarily invariant norm and then

accurate solutions of m matrix algebraic riccati equations this paper is concerned with the relative perturbation theory and its entrywise relatively accurate numerical solutions of an m matrix algebraic riccati equations mare xdx ax xb c 0 by which we mean the following conformally partitioned matrix left begin array ll b d c a end array right is a nonsingular or an irreducible singular m matrix it is known that such an mare has a unique minimal nonnegative solution phi it is proved that small relative perturbations to the entries of a b c and d introduce small relative changes to the entries of the nonnegative solution phi thus the smaller entries phi do not suffer bigger relative errors than its larger entries unlike the existing perturbation theory for general algebraic riccati equations we then discuss some minor but crucial implementation changes to three existing numerical methods so that they can be used to compute phi as accurately as the input data deserve current study is based on a previous paper of the authors on m matrix sylvester equation for which d 0

on the augmented system approach to sparse least squares problems we study the augmented system approach for the solution of sparse linear least squares problems it is well known that this method has better numerical properties than the method based on the normal equations we use recent work by arioli et al 1988 to introduce error bounds and estimates for the components of the solution of the augmented system in particular we find that using iterative refinement we obtain a very robust algorithm and our estimates of the error are accurate and cheap to compute the final error and all our error estimates are much better than the classical or skeel s error analysis 1979 indicates moreover we prove that our error estimates are independent of the row scaling of the augmented system and we analyze the influence of the bjorck scaling 1967 on these estimates we illustrate this with runs both on large scale practical problems and contrived examples comparing the numerical behaviour of the augmented systems approach with a code using the normal equations these experiments show that while the augmented system approach with iterative refinement can sometimes be less efficient than the normal equations approach it is comparable or better when the least squares matrix has a full row and is in any case much more stable and robust

a new class of nonsymmetric algebraic riccati equations abstract motivated by the study of linear quadratic differential games we introduce a new class of nonsymmetric algebraic riccati equations it is shown that every equation in this class has a unique stabilizing solution which is the solution required to find the open loop nash equilibrium for the differential game we show that the doubling algorithm can be used to find this solution efficiently the solution may also be found by the schur method and under further assumptions by newton s method and a basic fixed point iteration

mixed and componentwise condition numbers for a linear function of the solution of the total least squares problem abstract in this paper we consider the mixed and componentwise condition numbers for a linear function lx of the solution to the total least squares tls problem we derive the explicit expressions of the mixed and componentwise condition numbers through the dual techniques under both unstructured and structured componentwise perturbations the sharp upper bounds for condition numbers are obtained an efficient condition estimation algorithm is proposed which can be integrated into the iterative method for solving large scale tls problems moreover the new derived condition number expressions can recover the previous results on the condition analysis for the tls problem when l i n numerical experiments show the effectiveness of the introduced condition numbers and condition estimation algorithm

on level 2 condition number for the weighted moore penrose inverse in this paper we present characterizations for the level 2 condition number of the weighted moore penrose inverse i e cond m n a cond m n 2 a cond m n a 1 where cond m n a is the condition number of the weighted moore penrose inverse of a rectangular matrix and cond m n 2 a is the level 2 condition number of this problem this paper extends the result by cucker diao and wei f cucker h diao y wei on the level 2 condition number for moore penrose inversion 2005 unpublished report and improves the results by wei and wang y wei d wang condition numbers and perturbation of weighted moore penrose inverse and weighted linear least squares problem appl math comput 145 2003 45 58

stochastic analysis of well type flows in randomly heterogeneous porous formations 1 well type flow takes place in a heterogeneous porous formation where the transmissivity is modeled as a stationary random space function rsf general expressions for the covariances of the head and flux are obtained and analyzed the second order approximation of the mean radial flux is represented as the product between the solution valid in a homogeneous domain and a distortion term which adjusts according to the medium heterogeneity the spatial dependence of the function is studied in view of the formation identification problem the equivalent t eq and apparent t ap transmissivity are computed the important result is the relationship may be either eq or ap where th and tg represent the harmonic and the geometric means of the transmissivity respectively the position dependent weight is explicitly calculated indeed close to the well it yields which is understandable in view of the fact that the limit is equivalent to i which is the heterogeneity structure of a stratified formation nevertheless the effective transmissivity of a stratified formation is precisely th in contrast far from the well one has with the flow being slowly varying in the mean there it is shown that grows with increasing in the case of t eq the rate of growing is found similar to dagan and lessoff 2007 to be strongly dependent upon the position in the flow domain whereas t ap is a more robust property finally it is shown how the general results can be used for practical applications

a system of generalized sylvester quaternion matrix equations and its applications let h m n be the set of all m n matrices over the real quaternion algebra we call that a h n n is hermitian if a a i j k where i j k are the quaternion units denote a a in this paper we derive some necessary and sufficient conditions for the solvability to the system of generalized sylvester real quaternion matrix equations a i x i y i b i c i z d i e i i 1 2 and give an expression of the general solution to the above mentioned system as applications we give some solvability conditions and general solution for the generalized sylvester real quaternion matrix equation a 1 x y b 1 c 1 z d 1 e 1 where z is required to be hermitian we also present some solvability conditions and general solution for the system of real quaternion matrix equations involving hermicity a i x i a i x i b i y b i c i i 1 2 where y is required to be hermitian our results include some well known results as special cases

computational methods for linear matrix equations given the square matrices a b d e and the matrix c of conforming dimensions we consider the linear matrix equation a mathbf x e d mathbf x b c in the unknown matrix mathbf x our aim is to provide an overview of the major algorithmic developments that have taken place over the past few decades in the numerical solution of this and related problems which are producing reliable numerical tools in the formulation and solution of advanced mathematical models in engineering and scientific computing

two dimensional characterization of hydraulic heterogeneity by multiple pumping tests 1 the conventional analysis of pumping tests by type curve methods is based on the assumption of a homogeneous aquifer applying these techniques to pumping test data from real heterogeneous aquifers leads to estimates of the hydraulic parameters that depend on the choice of the pumping and observation well positions in this paper we test whether these values may be viewed as pseudo local values of transmissivity and storativity which can be interpolated by kriging we compare such estimates to those obtained by geostatistical inverse modeling where heterogeneity is assumed in all stages of estimation we use drawdown data from multiple pumping tests conducted at the test site in krauthausen germany the geometric mean values of transmissivity and storativity determined by type curve analysis are very close to those obtained by geostatistical inversion but the conventional approach failed to resolve the spatial variability of transmissivity in contrast the estimate from geostatistical inversion reveals more structure this indicates that the estimates of the type curve approaches can not be treated as pseudo local values concerning storativity both analysis methods show strong fluctuations because the variability of all terms making up the storativity is small we believe that the estimated variability of storativity is biased we examine the influence of measurement error on estimating structural parameters of covariance functions in the inversion we obtain larger correlation lengths and smaller prior variances if we trust the measured data less

correlation structure of flow variables for steady flow toward a well with application to highly anisotropic heterogeneous formations the study a continuation of that of indelman et al 1996 aims at deriving the second order moments of flow variables such as hydraulic head its gradient and the specific discharge for steady flow toward a fully penetrating well in a confined heterogeneous aquifer the log conductivity y ln k is modeled as a three dimensional stationary function of gaussian correlation of anisotropy ratio e by using first order approximations in 2y and e we derive the variance and the vertical integral scale of the piezometric head h of its radial gradient er and of the radial component of the specific discharge qr owing to the nonuniformity of the average flow these quantities are functions of the distance from the well it is shown that the variances of the head 2h and of its gradient 2er as well as the crossvariance e y between er and y vanish at the well whereas the discharge variance 2qr tends to the product between the log conductivity variance 2y and the squared mean discharge qr 2 this behavior pertains to a stratified formation surrounding the well far from the well 75 horizontal y integral scales i the head variance approaches a constant value for r 10i the moments 2er 2qr and ery tend to the corresponding values for uniform flow but with the local mean head gradient replacing the constant one the head vertical integral scale grows indefinitely with r whereas the vertical integral scale of the flux is larger by one log conductivity vertical scale than the one prevailing in uniform flow this latter property is explained by the presence of the source line which increases the correlations in the vertical direction the present results may be used in identifying the log conductivity statistical parameters from flowmeter velocity measurements in piezometers surrounding pumping or injecting wells

a class of solutions to the quantum colored yang baxter equation abstract in this paper we give a new method to solve the quantum colored yang baxter matrix equation qcybe and a class of solutions to the qcybe is given

a contribution to the theory of condition in this paper the use of the condition number of a problem as defined by rice in 1966 is discussed for the eigenvalue eigenvector and linear least squares problems either condition numbers according to various norms are determined or lower and upper bounds for them are derived

equivalent transmissivity of heterogeneous leaky aquifers for steady state radial flow 1 the purpose of this study is to estimate the equivalent transmissivity for two dimensional steady state radial flow toward a fully penetrating well in a heterogeneous leaky aquifer the log transmissivity of the pumped aquifer is modeled as a multivariate random spatial function with stationary mean and exponential semivariogram while the aquitard conductance is assumed to be spatially uniform an approximate expression for the upscaled or block transmissivity defined as the transmissivity of an equivalent homogeneous leaky aquifer system with the same pumping is proposed the proposed relation expresses the upscaled transmissivity as a weighted average of the log transforms of the point transmissivity values monte carlo simulations are used to numerically validate the proposed upscaling expression for wide ranges of aquifer parameters the relation of the weighing function to the statistical parameters of the transmissivity field is discussed the first and second moments of the upscaled transmissivity are also estimated as a function of the problem parameters the expected value of the equivalent transmissivity is shown to increase from the geometric mean to the arithmetic mean as the aquitard conductance increases from zero to infinity

iterative least squares solutions of coupled sylvester matrix equations abstract in this paper we present a general family of iterative methods to solve linear equations which includes the well known jacobi and gauss seidel iterations as its special cases the methods are extended to solve coupled sylvester matrix equations in our approach we regard the unknown matrices to be solved as the system parameters to be identified and propose a least squares iterative algorithm by applying a hierarchical identification principle and by introducing the block matrix inner product the star product for short we prove that the iterative solution consistently converges to the exact solution for any initial value the algorithms proposed require less storage capacity than the existing numerical ones finally the algorithms are tested on computer and the results verify the theoretical findings

a survey of nonsymmetric riccati equations abstract we survey recent and also older results on nonsymmetric matrix riccati differential equations and in the time invariant case on the corresponding algebraic riccati equations in particular we cite various applications connected with matrix riccati equations

exact sensitivity matrix and influence of the number of pilot points in the geostatistical inversion of moment equations of groundwater flow received 3 august 2009 revised 12 july 2010 accepted 21 july 2010 published 4 november 2010 1 we present novel equations for the exact sensitivity matrix of the ensemble mean hydraulic head under steady state groundwater flow conditions these equations are embedded in a geostatistical inverse procedure to condition approximations of stochastic moment equations of flow on measured hydraulic conductivities and heads our formulation allows considerable improvement of the methodology proposed by hernandez et al 2003 2006 and renders the inversion of moment equations feasible for a large number of unknown hydraulic parameters the spatial distribution of the natural logarithm y of conductivity is parameterized within the pilot points framework whereas prior values of y at pilot points are obtained by a variant of kriging posterior estimates at pilot points are obtained through a maximum likelihood fit of computed to measured heads the maximum likelihood function also includes a regularization term by means of a synthetic example and upon adopting formal model information criteria we explore the influence of 1 the number of pilot points and 2 the order of approximation of the governing mean flow equation on our ability to properly estimate the log conductivity and head fields and identify the relative weight of the regularization term and the parameters of the underlying y variogram we find that none of the adopted information criteria can identify the optimum number of pilot points and the plausibility weight and variogram parameters values can be determined by the kashyap s bayesian measure

spatial averaging of transmissivity in heterogeneous fields with flow toward a well steady state nonuniform flow in a heterogeneous medium is investigated using a combined numerical empirical approach the objective of the study is to determine the effective transmissivity of a square field with a constant head external boundary and a constant rate well at its center transmissivity at the point scale is modeled as an isotropic and multivariate lognormal spatial random function block transmissivities are defined using an empirical scaling up of point support values within the field the scaling process is a weighted spatial geometric averaging where log transmissivities are weighted by the inverse square of their distance from the well block transmissivities thus obtained and true effective values calculated using a numerical flow model are compared for realizations a finely discretized transmissivity field the principal finding of this study is that effective transmissivities obtained from the numerical flow model and block values obtained by spatial averaging are in excellent agreement for low to moderate variances of log transmissivity and moderately eccentric field geometries the geostatistical model for point scale transmissivity and the deterministic spatial averaging law are combined in order to create a geostatistical model for transmissivity at the block scale exact expressions are derived for the ensemble moments of block averaged transmissivities and well drawdowns in both the nonconditional case and the conditional case when transmissivity at the well bore is known the ensemble mean of block averaged transmissivity is found to decrease from the ensemble arithmetic mean toward the ensemble geometric mean as the field size becomes large compared to the integral range of spatial correlation this result contradicts earlier published work numerical results presented in this study also disagree with the analytical results of a recent stochastic analysis of specific discharge in radial systems

the yang baxter equation and invariants of links on considere le fait que l on peut construire des invariants d isotopie p et f pour des liens en utilisant certaines solutions des equations de yang baxter

on condition numbers for least squares with quadric inequality constraint abstract in this paper we will study normwise mixed and componentwise condition numbers for the linear mapping of the solution for general least squares with quadric inequality constraint glsqi and its standard form lsqi we will introduce the mappings from the data space to the interested data space and the frechet derivative of the introduced mapping can deduced through matrix differential techniques based on condition number theory we derive the explicit expressions of normwise mixed and componentwise condition numbers for the linear function of the solution for glsqi and lsqi also easier computable upper bounds for mixed and componentwise condition numbers are given numerical example shows that the mixed and componentwise condition numbers can tell us the true conditioning of the problem when its data is sparse or badly scaled compared with normwise condition numbers the mixed and componentwise condition number can give sharp perturbation bounds

componentwise perturbation theory for linear systems with multiple right hand sides existing definitions of componentwise backward error and componentwise condition number for linear systems are extended to systems with multiple right hand sides and to a general class of componentwise measure of perturbations involving holder p norms it is shown that for a system of order n with r right hand sides the componentwise backward error can be computed by finding the minimum p norm solutions to n underdetermined linear systems and an explicit expression is obtained in the case r 1 a perturbation bound is derived and from this the componentwise condition number is obtained to within a multiplicative constant applications of the results are discussed to invariant subspace computations quasi newton methods based on multiple secant equations and an inverse ode problem

three dimensional steady state flow to a well in a randomly heterogeneous bounded aquifer 1 we consider flow in a confined aquifer of uniform thickness due to a well of zero radius that fully penetrates the aquifer and discharges at a constant rate if the lateral extent of the aquifer is infinite a steady state flow regime never develops it is however well known that if the aquifer is additionally uniform a quasi steady state region extends from the well out to a cylindrical surface whose radius expands as the square root of time on the expanding surface head is uniform and time invariant inside this surface head at any time is described by a steady state solution a rigorous analysis of the analogous situation in a randomly heterogeneous aquifer would require the solution of a three dimensional transient stochastic flow problem in an aquifer of infinite lateral extent here we take a different approach by developing a three dimensional steady solution for mean flow to a well in a randomly heterogeneous aquifer with a cylindrical prescribed head boundary in analogy to the uniform case we expect our solution to approximate a quasi steady state region whose radius is initially small in comparison to the horizontal correlation scale of log conductivity but grows with time to become eventually much larger we treat log conductivity as a statistically homogeneous random field characterized by a gaussian spatial covariance function that may have different horizontal and vertical correlation scales our solution consists of analytical expressions for the ensemble mean and variance of head in the aquifer to second order in the standard deviation of log conductivity it is based on recursive approximations of exact nonlocal moment equations that are free of distributional assumptions and so apply to both gaussian and non gaussian log conductivity fields the analytical solution is supported by numerical monte carlo simulations it clarifies the manner in which relationships between the horizontal and vertical scales of the quasi steady state region and those of statistical anisotropy impact the statistical moments of drawdown and the equivalent and apparent hydraulic conductivities of the aquifer both conductivities are shown to exhibit a scale effect by growing with distance from the well within a radius of one to two horizontal integral scales from it

the extended thiem s solution including the impact of heterogeneity 1 in this study we present a formula for the hydraulic head describing the mean drawdown of a three dimensional steady state pumping test in heterogeneous anisotropic porous media effectively by modeling the hydraulic conductivity k x as spatial random function and using the upscaling method coarse graining we succeed in deriving a closed form solution hefw r which we understand as an extension of thiem s formula to heterogeneous media the solution hefw r does not only depend on the radial distance r but accounts also for the statistics of k x namely geometric mean kg variance 2 horizontal correlation length l and anisotropy ratio e we perform a sensitivity analysis on the parameters of hefw r and implement an inverse estimation strategy using numerical pumping tests we show the applicability of hefw r on the interpretation of drawdown data this will be done for both an ensemble of as well as for single pumping tests making use of the inverse estimation method we find excellent agreement of estimated parameters with initial values in particular for the horizontal correlation length

condition numbers of algebraic riccati equations in the frobenius norm consider the continuous time algebraic riccati equation care and the discrete time algebraic riccati equation dare which arise in linear control and system theory it is known that appropriate assumptions on the coefficient matrices guarantee the existence and uniqueness of hermitian positive semidefinite stabilizing solutions in this note we apply the theory of condition developed by rice to define condition numbers of the care and dare in the frobenius norm and derive explicit expressions of the condition numbers in a uniform manner both the complex case and real case are considered and connections to certain existing condition numbers of the care and dare are discussed

mixed and componentwise condition numbers of nonsymmetric algebraic riccati equation abstract in this paper the mixed and componentwise condition numbers of nonsymmetric algebraic riccati equation are considered the explicit expressions of two kinds of condition numbers are derived in the uniform manner and the upper bounds are proposed some simple numerical experiments show that the mixed and componentwise condition numbers can be sharp

classification of eight vertex solutions of the coloured yang baxter equation in this paper all eight vertex type solutions of the coloured yang baxter equation dependent on spectral as well as colour parameters are given it is proved that they are composed of three groups of basic solutions three groups of their degenerate forms and two groups of trivial solutions up to five solution transformations moreover all non trivial solutions can be classified into two types called baxter type and free fermion type

scaling of the discrete time algebraic riccati equation to enhance stability of the schur solution method a simple scaling procedure for discrete time riccati equations is introduced this procedure eliminates instabilities which can be associated with the linear equation solution step of the generalized schur method without changing the condition of the underlying problem a computable bound for the relative error of the solution of the riccati equation is also derived

condition numbers for a linear function of the solution of the linear least squares problem with equality constraints abstract in this paper we consider the normwise mixed and componentwise condition numbers for a linear function l x of the solution x to the linear least squares problem with equality constraints lse the explicit expressions of the normwise mixed and componentwise condition numbers are derived also we revisit some previous results on the condition numbers of linear least squares problem ls and lse it is shown that some previous explicit condition number expressions on ls and lse can be recovered from our new derived condition numbers formulas the sharp upper bounds for the derived normwise mixed and componentwise condition numbers are obtained which can be estimated efficiently by means of the classical hager higham algorithm for estimating matrix one norm moreover the proposed condition estimation methods can be incorporated into the generalized qr factorization method for solving lse the numerical examples show that when the coefficient matrices of lse are sparse and badly scaled the mixed and componentwise condition numbers can give sharp perturbation bounds on the other hand normwise condition numbers can severely overestimate the exact relative errors because normwise condition numbers ignore the data sparsity and scaling however from the numerical experiments for random lse problems if the data is not either sparse or badly scaled it is more suitable to adopt the normwise condition number to measure the conditioning of lse since the explicit formula of the normwise condition number is more compact

radially convergent flow in heterogeneous porous media we present an analytical solution for apparent effective transmissivity under radially convergent steady state flow conditions produced by constant pumping from a single well of finite radius rw apparent effective transmissivity te is defined as the value that relates the expected values of flow and head gradient at a certain location the domain is two dimensional of annular shape and the size of the pumping well is explicitly taken into account the solution for the steady state heads is obtained by solving the perturbed flow equation and substituting it into darcy s law to obtain a consistent second order expansion for te we show that apparent effective transmissivity is a scalar for any choice of isotropic covariance model with an expression given in integral form our main result is that te in a heterogeneous statistically isotropic random field under radial steady state flow conditions is a monotonie increasing function of r distance from the well that rises from the harmonic mean of the point transmissivity values close to the well and tends asymptotically towards the geometric mean far from the well the asymptotic value is reached at a distance of a few integral scales 1 5 2 for the gaussian model and 3 5 for the exponential one the apparent effective transmissivity versus normalized r curves are in excellent agreement with previously published numerical work carried out using monte carlo method

nonlocal and localized analyses of conditional mean steady state flow in bounded randomly nonuniform domains 2 computational examples in paper 1 guadagnini and neuman this issue of this two part series we presented exact nonlocal equations for first and second conditional ensemble moments of hydraulic head and flux under steady state groundwater flow in bounded randomly nonuniform domains in the presence of uncertain source and boundary terms we derived recursive approximations for these equations based on expansion in powers of a small parameter y which represents the standard estimation error of log hydraulic conductivity developed a two dimensional finite element scheme for their solution under the action of deterministic forcing terms to first order in y2 showed how to localize the conditional mean flow equations so as to yield familiar looking darcian differential equations in which however all quantities are conditional on data and therefore nonunique and proposed that traditional deterministic flow models be reinterpreted in light of these localized equations we stressed that all quantities which enter into our nonlocal and localized conditional moment equations are defined on a consistent support scale thereby obviating the need for upscaling from support to grid scale or the a priori definition of a computational grid that such a grid can therefore be defined a posteriori based on the degree of smoothness one expects the conditional moments to exhibit and that conditioning points are often spaced widely enough to render these moments relatively smooth and hence resolvable on a coarser grid than is necessary for monte carlo simulations this notwithstanding we use in this paper a high resolution grid to explore superimposed mean uniform and convergent flows in unconditional and conditional log hydraulic conductivity domains and to compare the relative accuracies of corresponding nonlocal localized and monte carlo finite element solutions whereas our nonlocal solution is nominally restricted to mildly nonuniform media with y2 1 we find that it actually yields remarkably accurate results for strongly nonuniform media with y2 at least as large as 4 localized solutions deteriorate relative to nonlocal results as one approaches some points of conditioning and singularity or as y2 increases though they require much less computer time than do nonlocal solutions localized solutions have the added disadvantage that they yield no information about predictive uncertainty

two kinds of condition numbers for the quadratic matrix equation in the situation that the quadratic matrix equation has the dominant or minimal solvents the mixed and componentwise condition numbers of the quadratic matrix equation are considered the explicit approximate expressions of two kinds of condition numbers are derived in the uniform manner and the upper bounds are proposed the theoretical results are illustrated by using some simple numerical examples

general solution of a kind of quantum coloured yang baxter equation i abstract in this paper we give a new method to solve the quantum coloured yang baxter matrix equation qcybe and the general solution for a kind of qcybe is given

a structured condition number for self adjoint polynomial matrix equations with applications in linear control abstract based on the classic definition of condition number a structured condition number is proposed for a class of self adjoint polynomial matrix equations the explicit formula of the structured condition number is derived with applying newly defined linear operators the structured condition number can be applied to some important polynomial matrix equations including the continuous time algebraic riccati equation care the discrete time algebraic lyapunov equation dale etc compared with the state of the art condition numbers for care and dale the newly proposed structured condition number can measure the sensitivity of the solution better which is validated by numerical examples

mean travel time of conservative solutes in randomly heterogeneous unbounded domains under mean uniform flow 1 we derive a closed form expression for mean travel time of a conservative solute migrating under uniform in the mean flow conditions within an infinite stationary field with simple exponential correlation of the natural logarithm of hydraulic conductivity our expression is developed from a consistent second order expansion in y standard deviation of the log hydraulic conductivity of the equations for moments of travel time and trajectories of conservative solutes in two dimensional randomly nonuniform flows of guadagnini et al 2001 as such it is nominally valid for moderately heterogeneous fields with y2 1 its validity for larger heterogeneity degrees is tested against numerical monte carlo simulations our results clarify the nonlinear effect in the mean travel time with respect to distance that has been observed numerically and modeled empirically in the literature

the condition numbers for weighted moore penrose inverse and weighted linear least squares problem condition numbers play an important role in numerical analysis classical normise condition numbers are used to measure the size of both input perturbations and output errors in this paper we study the weighted normwise relative condition numbers for the weighted moore penrose inverse and the weighted linear least squares wls problems in the case of the full column rank matrix the bounds or formulas for the weighted condition numbers are presented the obtained results can be viewed as extensions of the earlier works studied by others

condition number and backward errors of nonsymmetric algebraic riccati equation abstract the normwise condition number and two kinds of backward errors are considered when nonsymmetric algebraic riccati equation nare has the minimal nonnegative solution based on the techniques for the symmetric case we apply the condition number theory developed by rice to define condition number for nare the explicit expression is derived in a uniform manner meanwhile two kinds of backward errors are defined and evaluated by the explicit formulas numerical experiments are listed to illustrate and compare the practical performance

sensitivity analysis of the discrete time algebraic riccati equation abstract consider the discrete time algebraic riccati equation dare a t xa x a t xb s r b t xb 1 b t xa s t q 0 where a r n n b s r n m r rt r m m q qt r n n the available perturbation theory for the dare can only be applied to the case r 0 however in some control problems the matrix r can be singular in this paper we study perturbation properties of the dare without the restriction r 0 perturbation bounds and a relative condition number for the stabilizing solution of the dare are derived computable residual bounds for an approximate solution are also derived the theoretical results are illustrated by numerical examples

on mixed and componentwise condition numbers for indefinite least squares problem in this paper we mainly consider the mixed and componentwise condition numbers for the indefinite least squares problem without constraints or with least squares constraints the explicit expressions and the easily computable upper bounds for these condition numbers are presented two numerical examples are given to test these results and illustrate their superiority compared with the corresponding normwise condition numbers furthermore we also provide the mixed and componentwise condition numbers for the moore penrose inverse in indefinite inner product spaces

the solution of the matrix equations axb cxd e and ya dz yc bz e f the conditions for the existence of a unique solution of the matrix equation axb cxd e are proved to be that i the pencils a c and d b are regular and ii the spectra of the pencils have an empty intersection a numerical algorithm for solving the equation is proposed the possibility of a least squares type solution is briefly discussed the set of equations ya dz yc bz e f is proved to be equivalent to the aforementioned equation and its solution is also investigated a numerical algorithm is proposed

low rank approximation to the solution of a nonsymmetric algebraic riccati equation from transport theory abstract we consider the solution of the large scale nonsymmetric algebraic riccati equation xcx xd ax b 0 from transport theory juang 1995 with m d c b a r 2 n 2 n being a nonsingular m matrix in addition a d are rank 1 updates of diagonal matrices with the products a 1 u a u d 1 v and d v computable in o n complexity for some vectors u and v and b c are rank 1 the structure preserving doubling algorithm by guo et al 2006 is adapted with the appropriate applications of the sherman morrison woodbury formula and the sparse plus low rank representations of various iterates the resulting large scale doubling algorithm has an o n computational complexity and memory requirement per iteration and converges essentially quadratically as illustrated by the numerical examples

accurate solutions of m matrix sylvester equations this paper is concerned with a relative perturbation theory and its entrywise relatively accurate numerical solutions of an m matrix sylvester equation ax xb c by which we mean both a and b have positive diagonal entries and nonpositive off diagonal entries and p i m otimes a b rm t otimes i n is a nonsingular m matrix and c is entrywise nonnegative it is proved that small relative perturbations to the entries of a b and c introduce small relative errors to the entries of the solution x thus the smaller entries of x do not suffer bigger relative errors than its larger entries unlikely the existing perturbation theory for general sylvester equations we then discuss some minor but crucial implementation changes to three existing numerical methods so that they can be used to compute x as accurately as the input data deserve

a structure preserving doubling algorithm for nonsymmetric algebraic riccati equation in this paper we propose a structure preserving doubling algorithm sda for the computation of the minimal nonnegative solution to the nonsymmetric algebraic riccati equation nare based on the techniques developed for the symmetric cases this method allows the simultaneous approximation to the minimal nonnegative solutions of the nare and its dual equation requiring only the solutions to two linear systems and several matrix multiplications per iteration similar to newton s method and the fixed point iteration methods for solving nares we also establish global convergence for sda under suitable conditions using only elementary matrix theory we show that sequences of matrices generated by sda are monotonically increasing and quadratically convergent to the minimal nonnegative solutions of the nare and its dual equation numerical experiments show that the sda algorithm is feasible and effective and outperforms newton s iteration and the fixed point iteration methods

ergodicity of pumping tests 1 standard interpretations of pumping tests in heterogeneous formations rely on effective representations of porous media which replace spatially varying hydraulic properties with their constant counterparts averaged over the support volume of a test rigorous approaches for deriving representative effective apparent upscaled etc parameters employ either ensemble or spatial averaging we derive a set of conditions under which these two paradigms yield identical results we refer to them as conditions for the ergodicity of pumping tests this allows one to use stochastic approaches to estimate the statistics of the spatial variability of hydraulic parameters on scales smaller than the support volume of a pumping test

structured mixed and componentwise condition numbers of some structured matrices structured matrices such as cauchy vandermonde toeplitz hankel and circulant matrices are considered in this paper we apply a kronecker product based technique to deduce the structured mixed and componentwise condition numbers for the matrix inversion and for the corresponding linear systems

flow to partially penetrating wells in unconfined heterogeneous aquifers mean head and interpretation of pumping tests 1 a partially penetrating well of length lw and radius rw starts to pump at constant discharge qw at t 0 from an unconfined aquifer of thickness d the aquifer is of random and stationary conductivity characterized by kg geometric mean y2 log conductivity variance and i and iv the horizontal and vertical integral scales the flow problem is solved under a few simplifying assumptions commonly adopted in the literature for homogeneous media rw lw 1 linearization of the free surface condition and constant drainable porosity n additionally it is assumed that rw i 1 and lw iv 1 to simplify the well boundary conditions and that a first order approximation in y2 extended to finite y2 on a conjectural basis is adopted the solution is obtained for the mean head field h r z t and the associated water table equation the main result of the analysis is that the flow domain can be divided into three zones for h 1 the neighborhood of the well r i where h qw lwka h0 r z tkefuv nd with h0 being the zero order solution pertaining to a homogeneous and isotropic aquifer ka being the conductivity arithmetic mean and kefuv being the effective vertical conductivity in mean uniform flow 2 an exterior zone r i in which h qw lwkefuh h0 r z tkefuv nd with kefuh being the horizontal effective conductivity and 3 an intermediate zone in which the solution requires a few numerical quadratures not carried out here the application to pumping tests reveals that identification of the aquifer parameters for homogeneous and anisotropic aquifers by commonly used methods can be applied for the drawdown measured in an observation well of length lowiv to ensure exchange of space and ensemble head averages in the second zone in order to identify kefuh kefuv and n in contrast the use of the drawdown in the well first zone leads to an overestimation of kefuh by the factor ka kefuh

mixed and componentwise condition numbers for matrix decompositions abstract we present normwise and componentwise perturbation bounds for the lu the cholesky the l d l t and the qr decompositions by using a new approach the explicit expressions of mixed and componentwise condition numbers for these matrix decompositions are derived the condition numbers improve known results of the normwise and componentwise cases and reveal the characterizations of the structured perturbations the exact explicit perturbation expressions are derived for the factors l and u of the lu decomposition and the rigorous normwise and componentwise perturbation bounds are presented for the lu decomposition

norm estimations for the moore penrose inverse of multiplicative perturbations of matrices abstract let b be a multiplicative perturbation of a c m n given by b d 1 a d 2 where d 1 c m m and d 2 c n n are both nonsingular new norm upper bounds for b a f and b a 2 are derived where a and b are the moore penrose inverses of a and b respectively the main results of meng and zheng 2015 10 are improved in the cases of the frobenius norm and the spectral norm

uncertainty quantification in scale dependent models of flow in porous media equations governing flow and transport in randomly heterogeneous porous media are stochastic and scale dependent in the moment equations me method exact deterministic equations for the leading moments of state variables are obtained at the same support scale as the governing equations computable approximations of the mes can be derived via perturbation expansion in orders of the standard deviation of the random model parameters as such their convergence is guaranteed only for standard deviation smaller than one here we consider steady state saturated flow in a porous medium with random second order stationary conductivity field we show it is possible to identify a support scale where the typically employed approximate formulations of me yield accurate statistical moments of a target state variable therefore at support scale and larger me presents an attractive alternative to slowly convergent monte carlo mc methods whenever lead order statistical moments of a target state variable are needed we also demonstrate that a surrogate model for statistical moments could be constructed from mc simulations at larger support scales and be used to accurately estimate moments at smaller scales where mc simulations are expensive and the me method is not applicable

condition numbers of the generalized sylvester equation this note is devoted to the perturbation analysis for the generalized sylvester equations ax yb dx ye c f which have some important applications in control and system theory expressions and upper bounds for normwise mixed and componentwise condition numbers are presented the results are illustrated by numerical examples

representations and computations of 2 3 we investigate full rank representations of 2 3 and 2 4 inverses with prescribed rank as well as with prescribed range and null space in an indefinite inner product space in certain particular cases we obtain new results or generalizations related with 2 3 and 2 4 inverses in a hilbert unitary space direct and iterative methods for computing 2 3 and 2 4 inverses in an indefinite inner product space are developed numerical examples with different metric matrices are presented

a note on well boundary condition for flow through heterogeneous formations 1 in modeling flow toward wells it is mathematically convenient to replace the well by a singularity line along its axis in the case of homogeneous aquifers confined flow and fully penetrating wells the strength of the sources is constant and the common condition of constant head on the well is satisfied exactly it also leads to constant flux on the well envelope in the case of aquifers of spatially variable conductivity the constant head boundary condition can be satisfied accurately if the well radius is much smaller than the log conductivity horizontal integral scale by selecting a variable source strength that is proportional to the local conductivity along the well indelman et al 1996 fiori et al 1998 indelman 2003a 2003b in this case the equivalent conductivity tends to the arithmetic mean near the well it changes with distance approaching from above the effective conductivity in uniform horizontal mean flow far from the well selecting inadvertently a constant source strength violates the condition of constant well head the equivalent conductivity tends to the harmonic mean near the well and approaches from below the effective one in uniform mean flow the equivalent conductivity is derived for both types of sources for stratified formations as well as for aquifers of a three dimensional heterogeneous structure the implications on interpretation of head measurements by the two models is illustrated by a few examples

brief paper perturbation analysis and condition numbers of symmetric algebraic riccati equations this paper is devoted to the perturbation analysis of symmetric algebraic riccati equations based on our perturbation analysis the upper bounds for the normwise mixed and componentwise condition numbers are presented the results are demonstrated by our preliminary numerical experiments

effective groundwater model parameter values influence of spatial variability of hydraulic conductivity leakance and recharge a stochastic simulation approach was used to inspect the influence of spatial variability in aquifer and recharge properties on effective averaged groundwater model parameters a two dimensional unconfined aquifer model was set up for an area similar to that near livermore california monte carlo simulations were generated for different sets of spatial correlation structures assuming stationarity and an exponential semivariogram for each monte carlo realization groundwater flow was simulated and estimates of the means and standard deviations of the hydraulic head were obtained nine different cases were studied involving different correlation lengths of hydraulic conductivity conditional simulations parameter zonation and spatial recharge distributions results indicate that there is no single uniform value of the hydraulic conductivity that reproduces the expected head values spatial averaging to obtain effective values was based on the p norm power norm which ranges from 1 for the arithmetic mean to 1 for the harmonic mean with 0 representing the geometric mean for the unconditional case the 0 4 p norm seems to best reproduce the expected values the 0 2 p norm gave the best result for the conditional simulation case if the geometric mean were used instead the heads would deviate from the expected heads by about 2 m near the wells some value between the arithmetic and the geometric means of riverbed leakance will give results close to the expected head values the arithmetic mean of aerially distributed recharge produced results with small deviations from the expected head values these particular results may only apply to this modeled system as the pumping pattern and magnitudes exhibited strong influences on zones of maximum head deviation the implication of this work is that if an effective hydraulic conductivity is used in a simulation model it must be selected for a particular set of wells and pumping rates in an aquifer with significant pumping centers the best effective value for a two dimensional unconfined flow model is most likely between the geometric and harmonic mean if the wells are turned off the effective hydraulic conductivity for that flow model reverts to the geometric mean

analytical solutions to statistical moments for transient flow in two dimensional bounded randomly heterogeneous media 1 in this paper we derive analytical solutions to statistical moments for transient saturated flow in two dimensional bounded randomly heterogeneous porous media by perturbation expansions we first derive partial differential equations governing the zeroth order head h 0 and the first order head term h 1 where orders are in terms of the standard deviation of the log transmissivity we then solve h 0 and h 1 analytically both of which are expressed as infinite series the head perturbation h 1 is then used to derive expressions for autocovariance of the hydraulic head and the cross covariance between the log transmissivity and head the expressions for the mean flux and flux covariance tensor are formulated from the head moments based on darcy s law using numerical examples we demonstrate the convergence of these solutions we also examine the accuracy of these first order solutions by comparing them with solutions from both monte carlo simulations and the numerical moment equation method

structured perturbations of group inverse and singular linear system with index one in this paper we study the normwise perturbation theory of singular linear structured system with index one the structures under investigation are toeplitz circulant hankel matrices we show that the structured condition number is smaller than unstructured condition number for specific right hand side we present the condition number for structured system

the sensitivity of the algebraic and differential riccati equations in this paper it is shown that the ideas developed by byers in proc summer research conference ams vol 47 contemporary math american mathematical society providence ri 1984 pp 35 49 on the sensitivity of the algebraic riccati equation can be sharpened and extended to norms other than the frobenius norm this extension is crucial from an interpretive point of view because use of the spectral norm allows an identification between the condition number of the algebraic riccati equation and the damping properties of the closed loop dynamical system moreover this approach has the pleasant feature that it carries over to a completely parallel theory for the sensitivity of the differential riccati equation an area that has not been considered previously

condition number related to the outer inverse of a complex matrix in this paper we obtain the formula for computing the condition number of a complex matrix which is related to the outer generalized inverse of a given matrix we use the schur decomposition of a matrix we characterize the spectral norm and the frobenius norm of the relative condition number of the generalized inverse and the level 2 condition number of the generalized inverse the sensitivity for the generalized drazin inverse solution of linear systems is presented we also present the structured perturbation of the generalized inverse

generalized schur methods with condition estimators for solving the generalized sylvester equation stable algorithms are presented for solving the generalized sylvester equation they are based on orthogonal equivalence transformations of the original problem perturbation theory and rounding error analysis are included condition estimators dif sup 1 estimators are developed which when substituted into derived error bounds give accuracy estimates of a computed solution results from numerical experiments on well conditioned and ill conditioned problems are reported

three dimensional numerical analysis of steady state pumping tests in heterogeneous confined aquifers 1 we study three dimensional steady state flow field generated by a fully penetrating well extracting a constant water discharge from a confined aquifer of constant thickness the hydraulic conductivity k is modeled as an axisymmetric stationary random space function and the flow equation is solved by using the seven node finite volume numerical scheme in view of application in aquifer characterization we analyze how the equivalent conductivity keq varies as a function of the distance from the extraction well the numerical simulations showed that the boundary condition at the well exerts a large impact on keq and that the first order solution in y2 obtained by indelman et al 1996 is valid up to y2 0 5 for higher values of y2 the first order solution overestimates keq with differences that for a given y2 reduce progressively as the formation becomes more anisotropic a simple inverse procedure based on keq obtained through cross well interference is finally proposed and tested with the objective of estimating the parameters of the geostatistical model of variability of the hydraulic conductivity a few parametric examples showed that a suitable interpretation of pumping tests can be used to safely estimate parameters such as the geometric mean of the hydraulic conductivity its variance and although with larger uncertainty the horizontal integral scale while the inference of the anisotropy ratio is highly uncertain and error prone

some results on condition numbers of the scaled total least squares problem under the golub van loan condition for the existence and uniqueness of the scaled total least squares stls solution a first order perturbation estimate for the stls solution and upper bounds for condition numbers of a stls problem have been derived by zhou et al recently in this paper a different perturbation analysis approach for the stls solution is presented the analyticity of the solution to the perturbed stls problem is explored and a new expression for the first order perturbation estimate is derived based on this perturbation estimate for some stls problems with linear structure we further study the structured condition numbers and derive estimates for them numerical experiments show that the structured condition numbers can be markedly less than their unstructured counterparts

an investigation of the validity of first order stochastic dispersion theories in isotropie porous media monte carlo simulations are used to 1 investigate the accuracy of approximations that are implicit in first order stochastic dispersion theories and 2 identify the accuracy limits of first order dispersion theories in isotropic porous media the fickian theory of gelhar and axness 1983 as well as the fickian and non fickian theories of dagan 1984 and neuman and zhang 1990 are investigated all monte carlo simulations are in three dimensions confidence limits of ensemble averaged monte carlo results in isotropic porous media are established for 0 1 y 1 5 these results showed that first order theoretical estimates of the eulerian velocity covariance function are quite accurate for y 1 theoretical estimates of the non fickian longitudinal dispersivity do not deviate significantly from theory for at least y 1 5 theoretical estimation of the transverse dispersivity is limited to y 1 and the fickian longitudinal dispersivity is overestimated by the theory of gelhar and axness 1983 of all first order dispersion theories the theory of dagan 1984 is most robust in estimating the dispersivity tensor

bayesian identification of the local transmissivity using time drawdown data from pumping tests 1 conventional pumping test analysis techniques focus mostly on estimating the effective transmissivity of the perturbed subsurface volume surrounding the extraction and observation wells ignoring the existing local scale heterogeneity the purpose of this paper is to explore whether transient drawdown data derived from pumping tests can also be used to identify the statistical spatial structure of the local transmissivity field the log transmissivity is modeled as a multivariate random spatial function with stationary mean and exponential semivariogram monte carlo simulations are used to compute the transient drawdown for different values of the log transmissivity variance and integral scale on the basis of these simulations a bayesian procedure is developed to estimate the probability density functions of these two statistical parameters the main benefit of using a bayesian approach is that it readily accounts for the uncertainty of the estimated parameters application of the proposed procedure to simulated drawdown data shows that it is possible to characterize the local transmissivity from pumping test data and that the reliability of the estimates increases with an increase in the number of available tests

the structured sensitivity of vandermonde like systems we consider a general class of structured matrices that includes possibly confluent vandermonde and vandermonde like matrices here the entries in the matrix depend nonlinearly upon a vector of parameters we define condition numbers that measure the componentwise sensitivity of the associated primal and dual solutions to small componentwise perturbations in the parameters and in the right hand side convenient expressions are derived for the infinity norm based condition numbers and order of magnitude estimates are given for condition numbers defined in terms of a general vector norm we then discuss the computation of the corresponding backward errors after linearising the constraints we derive an exact expression for the infinity norm dual backward error and show that the corresponding primal backward error is given by the minimum infinity norm solution of an underdetermined linear system exact componentwise condition numbers are also derived for matrix inversion and the least squares problem and the linearised least squares backward error is characterised

structured condition numbers and small sample condition estimation of symmetric algebraic riccati equations abstract this paper is devoted to a structured perturbation analysis of the symmetric algebraic riccati equations by exploiting the symmetry structure based on the analysis the upper bounds for the structured normwise mixed and componentwise condition numbers are derived due to the exploitation of the symmetry structure our results are improvements of the previous work on the perturbation analysis and condition numbers of the symmetric algebraic riccati equations our preliminary numerical experiments demonstrate that our condition numbers provide accurate estimates for the change in the solution caused by the perturbations on the data moreover by applying the small sample condition estimation method we propose a statistical algorithm for practically estimating the condition numbers of the symmetric algebraic riccati equations

the ubiquitous kronecker product abstract the kronecker product has a rich and very pleasing algebra that supports a wide range of fast elegant and practical algorithms several trends in scientific computing suggest that this important matrix operation will have an increasingly greater role to play in the future first the application areas where kronecker products abound are all thriving these include signal processing image processing semidefinite programming and quantum computing second sparse factorizations and kronecker products are proving to be a very effective way to look at fast linear transforms researchers have taken the kronecker methodology as developed for the fast fourier transform and used it to build exciting alternatives third as computers get more powerful researchers are more willing to entertain problems of high dimension and this leads to kronecker products whenever low dimension techniques are tensored together

displacement rank of the drazin inverse in this paper we study the displacement rank of the drazin inverse both sylvester displacement and the generalized displacement are discussed we present upper bounds for the ranks of the displacements of the drazin inverse the general results are applied to the group inverse of a structured matrix such as close to toeplitz generalized cauchy toeplitz plus hankel and bezoutians

on mixed and componentwise condition numbers for moore penrose inverse and linear least squares problems classical condition numbers are normwise they measure the size of both input perturbations and output errors using some norms to take into account the relative of each data component and in particular a possible data sparseness componentwise condition numbers have been increasingly considered these are mostly of two kinds mixed and componentwise in this paper we give explicit expressions computable from the data for the mixed and componentwise condition numbers for the computation of the moore penrose inverse as well as for the computation of solutions and residues of linear least squares problems in both cases the data matrices have full column row rank

a note on the componentwise perturbation bounds of matrix inverse and linear systems in this note we present sharp bounds for the componentwise perturbation of matrix inverse and linear systems especially for nonsingular m matrix we also use matrix derivatives to deduce the matrix componentwise condition number and present some numerical tests to show our results

linear estimation in krein spaces i theory the authors develop a self contained theory for linear estimation in krein spaces the derivation is based on simple concepts such as projections and matrix factorizations and leads to an interesting connection between krein space projection and the recursive computation of the stationary points of certain second order or quadratic forms the authors use the innovations process to obtain a general recursive linear estimation algorithm when specialized to a state space structure the algorithm yields a krein space generalization of the celebrated kalman filter with applications in several areas such as h sup spl infin filtering and control game problems risk sensitive control and adaptive filtering

nonlocal and localized analyses of conditional mean steady state flow in bounded randomly nonuniform domains 1 theory and computational approach we consider the effect of measuring randomly varying hydraulic conductivitiesk x on one s ability to predict numerically without resorting to either monte carlo simulation or upscaling steady state flow in bounded domains driven by random source and boundary terms our aim is to allow optimum unbiased prediction of hydraulic heads h x and fluxes q x by means of their ensemble moments h x c and q x c respectively conditioned on measurements of k x these predictors have been shown by neuman and orr 1993a to satisfy exactly an integrodifferential conditional mean flow equation in which q x c is nonlocal and non darcian here we develop complementary integrodifferential equations for second conditional moments of head and flux which serve as measures of predictive uncertainty obtain recursive closure approximations for both the first and second conditional moment equations through expansion in powers of a small parameter y which represents the standard estimation error of ln k x and show how to solve these equations to first order in y2 by finite elements on a rectangular grid in two dimensions in the special case where one treats k x as if it was locally homogeneous and mean flow as if it was locally uniform one obtains a localized darcian approximation q x c kc x h x c in which kc x is a space dependent conditional hydraulic conductivity tensor this leads to the traditional deterministic darcian steady state flow equation which however acquires a nontraditional meaning in that its parameters and state variables are data dependent and therefore inherently nonunique it further explains why parameter estimates obtained by traditional inverse methods tend to vary as one modifies the database localized equations yield no information about predictive uncertainty our stochastic derivation of these otherwise standard deterministic flow equations makes clear that uncertainty measures associated with estimates of head and flux obtained by traditional inverse methods are generally smaller often considerably so than measures of corresponding predictive uncertainty which can be assessed only by means of stochastic models such as ours we present a detailed comparison between finite element solutions of nonlocal and localized moment equations and monte carlo simulations under superimposed mean uniform and convergent flow regimes in two dimensions paper 1 presents the theory and computational approach and paper 2 guadagnini and neuman this issue describes unconditional and conditional computational results

fuzzy static output feedback may be possible in lmi framework takagi sugeno t s fuzzy model based static output feedback controller design is a challenging issue due to the nonconvexity in the lyapunov stability condition recently a convexified linear matrix inequality lmi condition was proposed for the problem by additionally importing as many equality constraints as the number of the fuzzy rules with a variable in the paper from the study of fang et al however numerical experiences seem to imply the infeasibility of the proposed lmi this letter shows that their lmi problem may be feasible

residual bounds of approximate solutions of the algebraic riccati equation let tilde x geq 0 approximate the unique hermitian positive semi definite solution x to the algebraic riccati equation are where f g geq 0 a f is stabilizable and a g is detectable let be the residual of the are with respect to tilde x and define the linear operator vec l by by applying a new forward perturbation bound to the optimal backward perturbation corresponding to the approximate solution tilde x we obtained the following result if a f tilde x is stable and if 4 vec l 1 vec l 1 hat r f 1 for any unitarily invariant norm then

perturbation analysis of a quadratic matrix equation associated with an m matrix this paper is devoted to the perturbation analysis for the quadratic matrix equation x 2 e x f 0 where e is a diagonal matrix and f is an m matrix the quadratic matrix equation of this type arises in noisy wiener hopf problems for markov chains the solution of practical interest is a particular m matrix solution in this paper the perturbation bound of the m matrix solution is presented meanwhile the residual bound for an approximate solution to the m matrix solution is obtained the theoretical results are illustrated by using simple numerical examples

steady flow toward wells in heterogeneous formations mean head and equivalent conductivity we consider steady flow of water in a confined aquifer toward a fully penetrating well of radius rw figure 1 the hydraulic conductivity k is modeled as a three dimensional stationary random space function the two point covariance of y in k kg is of axisymmetric anisotropy with i and i the horizontal and vertical integral scales respectively and kg the geometric mean of k unlike previous studies which assumed constant flux the well boundary condition is of given constant head figure 1 the aim of the study is to derive the mean head h and the mean specific discharge q as functions of the radial coordinate r and of the parameters y2 e i i and rw i an approximate solution is obtained at first order in y2 by replacing the well by a line source of strength proportional to k and by assuming ergodicity i e equivalence between space averages over the vertical and h q ensemble means an equivalent conductivity keq is defined as the fictitious one of a homogeneous aquifer which conveys the same discharge q as the actual one for the given head hw in the well and a given head in a piezometer at distance r from the well this definition corresponds to the transmissivity determined in a pumping test by an observer that measures hw andq the main result of the study is the relationship 19 keq ka 1 kefu where ka is the conductivity arithmetic mean and kefu is the effective conductivity for mean uniform flow in the horizontal direction in the same aquifer the weight coefficient 10 has the simple approximate expression ln r i in r rw near the well 0 and keq ka which is easily understood since for rw i 1 the formation behaves locally like a stratified one in contrast far from the well 1 and keq kefu the flow being slowly varying there since ka kefu our result indicates that the transmissivity is overestimated in a pumping test in a steady state and it decreases with the distance from the well however the difference between ka and kefu is small for highly anisotropic formations for which e 1 a nonlocal effective conductivity which depends only on the heterogeneous structure is derived in appendix a along the lines of indelman and abramovich 1994

structured condition numbers of structured tikhonov regularization problem and their estimations both structured componentwise and structured normwise perturbation analysis of the tikhonov regularization are presented the structured matrices under consideration include toeplitz hankel vandermonde and cauchy matrices structured normwise mixed and componentwise condition numbers for the tikhonov regularization are introduced and their explicit expressions are derived for the general linear structure based on the derived expressions we prove structured condition numbers are smaller than their corresponding unstructured counterparts by means of the power method and small sample statistical condition estimation sce fast condition estimation algorithms are proposed our estimation methods can be integrated into tikhonov regularization algorithms that use the generalized singular value decomposition gsvd for large scale linear structured tikhonov regularization problems we show how to incorporate the sce into the preconditioned conjugate gradient pcg method to get the posterior error estimations the structured condition numbers and perturbation bounds are tested on some numerical examples and compared with their unstructured counterparts our numerical examples demonstrate that the structured mixed condition numbers give sharper perturbation bounds than existing ones and the proposed condition estimation algorithms are reliable also an image restoration example is tested to show the effectiveness of the sce for large scale linear structured tikhonov regularization problems

condition numbers for the nonlinear matrix equation and their statistical estimation abstract in this paper we consider four different types of condition numbers i e two normwise ones mixed and componentwise ones for the nonlinear matrix equation x a f x a q and derive their explicit expressions using the probabilistic spectral norm estimator and the statistical condition estimation method these condition numbers can be estimated with high reliability two numerical examples are provided to illustrate the difference between the normwise condition numbers and the mixed and componentwise condition numbers and the efficiency of the estimators respectively

prediction of steady state flow in nonuniform geologic media by conditional moments exact nonlocal formalism effective conductivities and weak approximation we consider the effect of measuring randomly varying local hydraulic conductivities k x on one s ability to predict steady state flow within a bounded domain driven by random source and boundary functions more precisely we consider the prediction of local hydraulic head h x and darcy flux q x by means of their unbiased ensemble moments h x and q x conditioned on measurements of k x these predictors satisfy a deterministic flow equation in which q x x h x r x where x is a relatively smooth unbiased estimate of k x and r x is a residual flux we derive a compact integral expression for r x which is rigorously valid for a broad class of k x fields including fractals it demonstrates that q x is nonlocal and non darcian so that an effective hydraulic conductivity does not generally exist we show analytically that under uniform mean flow the effective conductivity may be a scalar a symmetric or a nonsymmetric tensor or a set of directional scalars which do not form a tensor we demonstrate numerically that in two dimensional mean radial flow it may increase from the harmonic mean of k x near interior and boundary sources to the geometric mean far from such sources for cases where r x can neither be expressed nor approximated by a local expression we propose a weak integral approximation closure which appears to work well in media with pronounced heterogeneity and improves as the quantity and quality of k x measurements increase the nonlocal deterministic flow equation can be solved numerically by standard methods our theory shows clearly how the scale of grid discretization should relate to the scale quantity and quality of available data after providing explicit approximations for the second moments of head and flux prediction errors we conclude by discussing practical methods to compute x from noisy measurements of k x and to calculate required second moments of the associated estimation errors when k x is lognormal

a stochastic analysis of pumping tests in laterally nonuniform media conventional pumping test analysis methodology assumes that aquifer transmissivity is invariant in space the ramifications of this assumption are examined for hypothetical units whose variations in transmissivity are considered to be reasonable representations of variations possible in natural systems the dependence of pumping test transmissivity on spatial angular and radial and temporal location of observations and the method of drawdown analysis is assessed the dependence on the angular position of an observation well appears of little significance dependence on radial position however can be strong the dependence on the interval of time used in the pumping test analysis is only important in highly variable systems methods of drawdown analysis that yield identical estimates in uniform units yield differing estimates in nonuniform ones as a result of a difference in data fitting procedures a reasonable estimate of transmissivity at a regional scale can be obtained from a cooper jacob analysis applied at a large duration of pumpage in general conventional approaches for pumping test analysis should be viable in the nonuniform aquifers of the type considered here

stochastic modeling of groundwater flow by unconditional and conditional probabilities 1 conditional simulation and the direct problem mathematical modeling of groundwater flow in heterogeneous porous formations of large extent is investigated the formation properties hydraulic conductivity transmissivity as well as flow variables head specific discharge solute concentration are regarded as random variables subjected to uncertainty the main aim of the study is to analyze the influence of conditional probability of input variables upon the statistical structure of the dependent variables the unconditional probability density functions are supposed to be stationary multivariate normal while conditioning accounts for the measured values at a few points of the formation two problems of groundwater flow are discussed in part 1 conditional simulation and the direct problem for steady flow analytical results are obtained by using perturbation approximations average uniform head gradient flows as well as recharge and flow to wells are discussed in unconditional modeling the input variable the conductivity or transmissivity log is regarded as stationary and is represented by its constant mean and variogram the ensemble of formations on which statistical calculations are carried out represents all aquifers with same probability density distributions in conditional modeling the measured values at a few points are kept fixed and uncertainty prevails only at other points consequently statistical computations are performed for the subensemble of aquifers which preserve the measured values and as a result both input and output variables are nonstationary the main effect of conditioning is to reduce the variance i e the uncertainty of these variables this effect is particularly important in the case of flow toward wells application of the modeling method to field problems is outlined