polynomial lienard equations near infinity abstract we study the behaviour near infinity of the generalized lienard equations x y y m k 0 a k x k y n k 0 b k x k providing a complete classification using poincare compactification as well as poincare lyapunov compactification we show that all the necessary information is contained in a m and b n except for the so called center focus problem occuring in case m 2 n 1 with m n odd and where the behaviour also depends on the value of the other coefficients a i and b j we also indicate how to take care about obtaining uniform information near infinity

on unifying the concepts of scale instrumentation and stochastics in the development of multiphase transport theory a generalized theory of multiphase transport is presented which combines the concepts of scale instrumentation stochastics and time series with the development of transport equations by defining the filtering process as a convolution of a measure p with a field property we are able to exploit the fourier transform to place plausible restrictions on an instrument in frequency space so as to make its measurement relevant to its physical environment an ideal instrument is defined which filters out high frequency noise corresponding to short distances and yet does not alter the structure of low frequencies using ideal instruments we successively filter out lower frequency noise in a multiscale multiphase environment formulas are developed to relate the autocorrelation of a field property on one scale of motion to that on any other scale while taking into account the types of instruments used in the measuring process an equation relating the integral scale on one scale of motion to the integral scale on any other scale of motion is developed power spectra are developed which relate spectra on different scales to the measuring instrument used by successively applying filtering theorems a hierarchy of multiscale transport equations is developed filtered properties in the transport equations are mass averages different properties are allowed to be measured by different instruments and different instruments are allowed on different scales of motion and for different phases the concept of a wide sense stationary ergodic process is introduced to develop mass average autocorrelations and spectra over scales of motion as a function of measuring devices

an efficient laplace transform solution for multiaquifer systems an efficient numerical algorithm is developed for groundwater flow in multiaquifer systems the method is based on herrera s formulation which eliminates the need of explicitly modeling aquitards laplace transform is applied to the governing integrodifferential equations which removes not only the time dependence in the solution system but also the integral expression the final system is a set of nontransient linear partial differential equations in two spatial dimensions a regular finite difference method is used for its solution the solution in the laplace transform domain is then inverted to the time domain using either a slow or a quick scheme depending on the requirement of speed and accuracy examples of a two aquifer one aquitard system involving radial and regional geometries are examimed

estimation of aquifer parameters under transient and steady state conditions 2 uniqueness stability and solution algorithms paper 2 of this three part series starts with a discussion of the question under what conditions is the aquifer inverse problem well posed after defining the terms uniqueness identifiability and stability theoretical considerations and synthetic examples are used to demonstrate that ill posedness can be mitigated by including prior information about the parameters in the estimation criterion to be minimized at the same time the inclusion of such information is shown to be insufficient to guarantee uniqueness and stability in all cases several test problems in the recent literature which have resulted in pessimistic conclusions about the solvability of the aquifer inverse problem are shown to be ill posed a question is thus raised about the validity of these conclusions in the general case various conjugate gradient algorithms coupled with the adjoint state finite element method for computing the gradient of the estimation criterion and with newton s method for determining the optimum step size downgradient are compared a marked improvement in the rate at which these algorithms converge is shown to be achieved by switching from one method to another when the former slows down or fails to converge

on measurement scale and scaling the relationships between measurement scale and scaling are discussed and analyzed in detail it is pointed out that only certain fundamental properties can be measured and that these properties do not include constitutive variables the interrelationships between scale and measurement are shown to manifest themselves in two types of heterogeneity which we label as explicit and implicit an important issue that is addressed for the first time is the necessity of scaling instruments when scaling laboratory experiments up to field scale or vice versa explicit constraints that scaled instruments must satisfy are derived for the frequency response of the instrument it is shown that linear interpolators can be viewed as filters instruments but that these interpolators don t preserve the scale constraints of the instrument used in making the field measurements the relationship between signal conditioning and the instrument s window is isolated and shown to be a multiple convolution over the spatial coordinates corresponding to the window followed by a temporal convolution for the signal conditioning

temporal moments revisited why there is no better way for physically based model reduction in time 1 many hydro geo logical problems are highly complex in space and time coupled with scale issues variability and uncertainty especially time dependent models often consume enormous computational resources but model reduction techniques can alleviate this problem temporal moments tm offer an approach to reduce the time demands of transient hydro geo logical simulations tm reduce transient governing equations to steady state and directly simulate the temporal characteristics of the system if the equations are linear and coefficients are time independent this is achieved by an integral transform projecting the dynamic system response onto monomials in time in comparison to classical approaches of model reduction that involve orthogonal base functions however the monomials for tm are nonorthogonal which might impair the quality and efficiency of model reduction thus we raise the question of whether there are more suitable temporal base functions than the monomials that lead to tm in this work we will derive theoretically that there is only a limited class of temporal base functions that can reduce hydro geo logical models by comparing those to tm we conclude that in terms of gained efficiency versus maintained accuracy tm are the best possible choice while our theoretical results hold for all systems of linear partial or ordinary differential equations pdes odes with any order of space and time derivatives we illustrate our study with an example of pumping tests in a confined aquifer for that case we demonstrate that two four tm are sufficient to represent more than 80 90 of the dynamic behavior and that the information content strictly increases with increasing tm order

computationally efficient algorithms for parameter estimation and uncertainty propagation in numerical models of groundwater flow finite difference and finite element methods are frequently used to study aquifer flow however additional analysis is required when model parameters and hence predicted heads are uncertain computational algorithms are presented for steady and transient models in which aquifer storage coefficients transmissivities distributed inputs and boundary values may all be simultaneously uncertain innovative aspects of these algorithms include a new form of generalized boundary condition a concise discrete derivation of the adjoint problem for transient models with variable time steps an efficient technique for calculating the approximate second derivative during line searches in weighted least squares estimation and a new efficient first order second moment algorithm for calculating the covariance of predicted heads due to a large number of uncertain parameter values the techniques are presented in matrix form and their efficiency depends on the structure of sparse matrices which occur repeatedly throughout the calculations details of matrix structures are provided for a two dimensional linear triangular finite element model

on the modified successive overrelaxation method new forms of the modified successive overrelaxation msor mksor mksor1 and mksor2 are introduced these forms depend on the updated successive overrelaxation method ksor and the combination of the sor and the ksor methods we present analysis of the convergence of the mksor methods with fixed parameters for solving linear system of equations ax b where a belongs to a subclass of consistently ordered matrices these matrices appear in the numerical treatment of elliptic partial differential equations corresponding to the red black ordering also functional relations for the eigenvalues of the iteration matrices of the jacobi the mksor methods and the relaxation parameters are established a general representative numerical example illustrating this treatment is considered different values of the relaxation parameters are used on each set of equations we have implemented our treatment using computer algebra software mathematica 8 0 new results on this subject are presented and compared with those of young this approach is promising and will help in the numerical treatment of boundary value problems other extensions and applications for further work are mentioned

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

parameter identification of groundwater aquifer models a generalized least squares approach this paper concerns the methods of estimating aquifer transmissivities on the basis of unsteady state hydraulic head data traditionally the criterion of minimizing the sum of the squares of errors has been used to match the observed data with the model response the data used for optimization usually contain noise that is not necessarily uncorrelated it is well understood that the results of identification methods are very sensitive to measurement errors in data in this study the ordinary least squares ols technique is carried out along with a generalized least squares gls technique specifically designed to reduce the effect of correlated errors the trace of the covariance matrix is used as a measure of overall accuracy and reliability of the estimated parameters the effectiveness of the ols and gls techniques in dealing with noisy data is demonstrated by using a hypothetical example the results of numerical experiments suggest that gls offers a promising approach in efficiently improving the reliability of the estimated parameters

on the probabilistic structure of water age the age distribution of water in hydrologic systems has received renewed interest recently especially in relation to watershed response to rainfall inputs the purpose of this contribution is first to draw attention to existing theories of age distributions in population dynamics fluid mechanics and stochastic groundwater and in particular to the mckendrick von foerster equation and its generalizations and solutions a second and more important goal is to clarify that when hydrologic fluxes are modeled by means of time varying stochastic processes the age distributions must themselves be treated as random functions once their probabilistic structure is obtained it can be used to characterize the variability of age distributions in real systems and thus help quantify the inherent uncertainty in the field determination of water age we illustrate these concepts with reference to a stochastic storage model which has been used as a minimalist model of soil moisture and streamflow dynamics

error estimates for the numerical identification of a variable coefficient error estimates are derived for the approximate identification of an unknown transmissivity coefficient in a partial differential equation describing a model problem in groundwater now the approximation scheme considered determines the coefficient by least squares fitting of the observed pressure data 1 introduction in this paper we wish to present an error analysis of a common numerical scheme used in the identification of parameters in distributed systems specifically we shall concern ourselves here with a model problem in groundwater flow the problem is to identify a spatially varying transmissivity coefficient a x from observations of the piezometric head u x in a two dimensional static aquifer s where a and u are related by the equations 1 div avw ins2

simulation of groundwater age distributions the objective of our work is to examine how to simulate the age of groundwater in such a way that it can be compared to actual measurements we start by showing that computation of kinematic age the one obtained by tracking water along streamlines is ill posed in heterogeneous aquifers this together with its inability to account for mixing processes makes it inadequate for comparison with age measurements which are the result of some averaging of the age distribution in the water sample the type of averaging depends on the measurement procedure therefore we go on to write the equations for the cumulative distribution function of residence time under transient flow conditions this allows us to derive transient equations for the mean age as well as for the higher order moments of its distribution which generalize previous results by others these moments can be used for approximating age measurements which need not be equal to the mean age of the water sample using both a synthetic and a real example we show that mean age is an acceptable estimate of radiometric age measurements in many cases including a second order correction variance of residence time distribution always improves results on the other hand higher order approximations converge slowly for old compared to half life waters to the point that third order approximations often worsen the results

on convergence and semi convergence of ssor like methods for augmented linear systems abstract in this paper we analyze the convergence and semi convergence of a class of ssor like methods with four real functions for augmented systems the class takes most existed ssor like methods as its special cases for nonsingular systems we obtain the minimum of convergence factors of all the ssor like methods in the class and study when it can be reached by the convergence factors of methods in the class by considering the equivalence of methods we show that most of the existed ssor like methods have the same minimum of convergence factors

symmetric sor method for augmented systems in this paper we develop symmetric successive overrelaxation ssor method to find solution of augmented systems these systems have appeared in many different applications of scientific computing e g the finite element approximation to solve the navier stokes equation and the constrained optimization the convergence of ssor method are studied finally we apply the new method to solve augmented systems the iteration number of this method to receive the solution is very better than sor like method

cauchy data requirement of the inverse problem of the mean age equation a classical theoretical inverse problem in groundwater hydraulics is the calibration of a heterogeneous isotropic hydraulic conductivity field using known hydraulic head data at each point in the domain it has been long known that a necessary condition for the uniqueness of the solution to this theoretical inverse problem is the cauchy data i e the hydraulic conductivity values or equivalently flux along a line crossing all streamtubes in the entire domain this result has been cited in identifying the value of boundary flux data in practical inverse problems although in general such data are generally not available in this work we show that groundwater mean age data replace this cauchy data requirement

symmetric block sor methods for rank deficient least squares problems in this article we develop symmetric block successive overrelaxation s block sor methods for finding the solution of the rank deficient least squares problems we propose an s2 block sor and an s3 block sor method for solving such problems and the convergence of these two methods is studied the comparisons between the s2 block and the s3 block methods are presented with some numerical examples

symmetric successive overrelaxation methods for rank deficient linear systems in this paper we develop symmetric successive overrelaxation symmetric sor or ssor methods for finding the least square solution of minimal norm to the linear system ax b where a is an mxn matrix of rank r the methods are obtained by first augmenting the system to a block 4x4 consistent system the augmented coefficient matrix is then split by a subproper ssor splitting we state and prove some theorems and by some numerical examples we show the number of iterations for ssor is less than sor and accelerated overrelaxation methods for finding the least square solution of minimal norm to the linear system

on semi convergence of ult iterative method for the singular saddle point problems zheng and ma recently proposed an efficient upper and lower triangular ult splitting iterative method for solving the large sparse nonsingular saddle point problems see zheng and ma 2016 in this paper we further prove the semi convergence of this method when it is applied to solve the large sparse singular saddle point problems under suitable conditions the characteristic of eigenvalues of the iteration matrix of the ult method is analyzed also the pseudo optimal iteration parameters and the corresponding pseudo optimal semi convergence factor for some special cases of the ult method are determined in addition numerical experiments are used to show the feasibility and effectiveness of the ult iterative method for solving singular saddle point problems

incorporation of prior information on parameters into nonlinear regression groundwater flow models 1 theory prior information on the parameters of a groundwater flow model can be used to improve parameter estimates obtained from nonlinear regression solution of a modeling problem two scales of prior information can be available 1 prior information having known reliability that is bias and random error structure and 2 prior information consisting of best available estimates of unknown reliability a regression method that incorporates the second scale of prior information assumes the prior information to be fixed for any particular analysis to produce improved although biased parameter estimates approximate optimization of two auxiliary parameters of the formulation is used to help minimize the bias which is almost always much smaller than that resulting from standard ridge regression it is shown that if both scales of prior information are available then a combined regression analysis may be made

on the recovery of multiple flow parameters from transient head data the problem of estimating groundwater flow parameters from head measurements and other ancillary data is fundamental to the process of modelling a groundwater system we consider here a new method that allows for the simultaneous computation of multiple parameters as the unique minimum of a convex functional

stochastic modeling of groundwater flow by unconditional and conditional probabilities the inverse problem the inverse problem is defined here as follows determine the transmissivity at varius points given the shape and boundary of the aquifer and recharge intensity and given a set of measured log transmissivity y and head h values at a few points the log transmissivity distribution is regarded as a realization of a random function of normal and stationary unconditional probability density function pdf the solution of the inverse problem is the conditional normal pdf of y conditioned on measured h and y which is expressed in terms of the unconditional joint pdf of y and h the problem is reduced to determining the unconditional head log transmissivity covariance and head variogram for a selected y covariance which depends on a few unknown parameters this is achieved by solving a first order approximation of the flow equations the method is illustrated for an exponential y covariance and the effect of head and transmissivity measurements upon the reduction of uncertainty of y is investigated systematically it is shown that measurement of h has a lesser impact than those of y but a judicious combination may lead to significant reduction of the predicted variance of y possible applications to real aquifers are outlined

on generalized symmetric sor method for augmented systems in this paper we establish the generalized symmetric sor method gssor for solving the large sparse augmented systems of linear equations which is the extension of the ssor iteration method the convergence of the gssor method for augmented systems is studied numerical resume shows that this method is effective

continuous groundwater velocity fields and path lines in linear bilinear and trilinear finite elements darcy velocity fields obtained from finite element solutions of heads in groundwater flow exhibit discontinuities at element boundaries thus giving rise to inaccuracies in path line construction several methods have been developed to resolve those inaccuracies including global postprocessing of velocity fields mixed hybrid finite element formulations and stream function formulations all of these techniques either lead to a considerable increase in the computational effort or are not general enough for all purposes the point of view taken here is that standard finite element flow models yield exact water balances which in the usual approach to path line computation are not fully used by starting out from patches with flux conserving boundaries a new postprocessing procedure is derived that allows construction of a continuous flux distribution over the whole model domain in this procedure only the head gradients of adjacent elements are used thus avoiding the solution of another global system of equations furthermore application of a stream function interpolation inside each patch allows the fast computation of path lines without any time stepping the path lines computed with the proposed procedure are compared to path lines computed by usual particle tracking and examples are presented showing that the new method is vastly superior to traditional particle tracking

configurations of limit cycles in lienard equations abstract we show that every finite configuration of disjoint simple closed curves in the plane is topologically realizable as the set of limit cycles of a polynomial lienard equation the related vector field x is morse smale moreover it has the minimum number of singularities required for realizing the configuration in a lienard equation we provide an explicit upper bound on the degree of x which is lower than the results obtained before obtained in the context of general polynomial vector fields

a modified sor like method for the augmented systems for the augmented system of linear equations by considering a new splitting of the coefficient matrix a modified sor like msor like method is presented in this paper the convergence of this method is discussed under suitable restrictions on iteration parameters the optimal parameters and the corresponding optimal convergence factor are determined numerical examples are reported to confirm the efficiency of the proposed method

the inverse problem for confined aquifer flow identification and estimation with extensions the contributions of this work are twofold first a methodology for estimating the elements of parameter matrices in the governing equation of flow in a confined aquifer is developed the estimation techniques for the distributed parameter inverse problem pertain to linear least squares and generalized least squares methods the linear relationship among the known heads and unknown parameters of the flow equation provides the background for developing criteria for determining the identifiability status of unknown parameters under conditions of exact or overidentification it is possible to develop statistically consistent parameter estimators and their asymptotic distributions the estimation techniques namely two stage least squares and three stage least squares are applied to a specific groundwater inverse problem and compared between themselves and with an ordinary least squares estimator the three stage estimator provides the closer approximation to the actual parameter values but it also shows relatively large standard errors as compared to the ordinary and two stage estimators the estimation techniques provide the parameter matrices required to simulate the unsteady groundwater flow equation second a nonlinear maximum likelihood estimation approach to the inverse problem is presented the statistical properties of maximum likelihood estimators are derived and a procedure to construct confidence intervals and do hypothesis testing is given the relative merits of the linear and maximum likelihood estimators are analyzed other topics relevant to the identification and estimation methodologies i e a continuous time solution to the flow equation coping with noise corrupted head measurements and extension of the developed theory to nonlinear cases are also discussed a simulation study is used to evaluate the methods developed in this study

an application of the geostatistical approach to the inverse problem in two dimensional groundwater modeling the geostatistical approach to the estimation of transmissivity from head and transmissivity measurements is developed for two dimensional steady flow the field of the logarithm of transmissivity log transmissivity is represented as a zero order intrinsic random field its spatial structure is described in this application through a two term covariance function that is linear in the parameters 1 and 2 linearization of the discretized flow equations allows the construction of the joint covariance matrix of the head and log transmissivity measurements as a linear function of 1 and 2 in this particular application the coefficient matrices are calculated numerically in a noniterative fashion maximum likelihood estimation is employed to estimate 1 and 2 as well as additional parameters from measurements linear estimation theory cokriging then yields point or block averaged estimates of transmissivity the approach is first applied to a test case with favorable results it is shown that the application of the methodology gives good estimates of transmissivities it is also shown that when the transmissivities are used in a numerical model they reproduce the head measurements quite well results from the application of the methodology to the jordan aquifer in iowa are also presented

estimation of aquifer parameters under transient and steady state conditions 1 maximum likelihood method incorporating prior information in this series of three papers a method is presented to estimate the parameters of groundwater flow models under steady and nonsteady state conditions the parameters include values and directions of principal hydraulic conductivities or transmissivities in anisotropic media specific storage or storativity interior and boundary recharge or leakage rates coefficients of head dependent interior and boundary sources and boundary heads in transient situations the initial head distribution can also be estimated if the system is originally at a steady state paper 1 of the series discusses some of the advantage in treating the inverse problem statistically and in regularizing its solution by means of penalty criteria based on prior estimates of the parameters the inverse problem is posed in the framework of maximum likelihood theory cast in a manner that accounts for prior information about the parameters since not all the factors which contribute to the prior errors can be quantified statistically at the outset the covariance matrices of these errors are expressed in terms of several parameters which if unknown can be estimated jointly with the hydraulic parameters by a stagewise optimization process when transient head data are separated by a fixed time interval the temporal structure of these data is approximated by a lag one autoregressive model with a correlation coefficient that can be treated as another unknown parameter estimation errors are analyzed by examining the lower bound of their covariance matrix in the eigenspace paper 1 concludes by suggesting that certain model identification criteria developed in the time series literature all of which are based on the maximum likelihood concept might be useful for selecting the best groundwater model or the best method of parameterizing a particular model among a number of given alternatives

an eigenvalue numerical technique for solving unsteady linear groundwater models continuously in time a procedure for solving the differential groundwater flow equation is presented herein using a finite difference or finite element discretization scheme a set of simultaneous linear equations is obtained the eigenvalues and eigenvectors of a matrix which is a function of the coefficients of the set are the key to the solution a vector l is obtained straightforwardly by combining the eigenvector matrix a the eigenvalue vector the pumping vector p and the initial head vector h vector l which depends on time can be expressed simply and explicitly as a function of the eigenvalues piezometric heads can be obtained by combining a and l l is the only vector that needs to be computed as p changes with time in this way influence functions of a piezometric head flow velocity and flow depletion of a stream connected with the aquifer under a unit stress can be obtained explicitly and continuously in time the method can be applied to confined as well as to leaky aquifers and to one two or three dimensional linear models its main advantage lies in the fact that it is unnecessary to repeatedly solve a matrix for every time increment the method is particularly useful for groundwater management problems in which a large number of alternatives have to be evaluated

phase portraits of piecewise linear continuous differential systems with two zones separated by a straight line abstract this paper provides the classification of the phase portraits in the poincare disc of all piecewise linear continuous differential systems with two zones separated by a straight line having a unique finite singular point which is a node or a focus the sufficient and necessary conditions for existence and uniqueness of limit cycles are also given

kinematics of age mixing in advection dispersion models this paper investigates age mixing processes arising in advection dispersion models where large scale travel and residence time distributions can be explicitly calculated based on the underlying velocity field in particular we analyze spatially integrated age mixing dynamics by comparing the age distributions of the storage and of the outflow s the relevance of the work lies in the impact of age mixing dynamics on the shape of travel time distributions ttds which ultimately control the long term memory of catchment transport processes we set up a theoretical framework that bridges previous lagrangian and eulerian water age theories in heterogeneous media the framework allows for the analysis of the dynamical connection between water age distributions in large scale volume and flux samples theoretical advances are then illustrated through the application to a finite one dimensional domain with constant advection and dispersion coefficient therein we analyze the type of mixing emerging for different peclet numbers and diverse spatiotemporal patterns of solute input we find that in spite of the enhanced nonstationarity of ttds the type of mixing is markedly invariant moreover for relatively low peclet numbers the different ages available in the control volume are systematically removed from the domain at a rate nearly proportional to their relative abundance random sampling emerging large scale patterns of age mixing yield theoretical and practical implications for watershed hydrology where ttds can be used to infer general patterns of catchment response across scales

temporal moments for transport with mass transfer described by an arbitrary memory function in heterogeneous media 1 temporal moment equations are generalized for transport under linear mass transfer which has been used to model a broad range of small scale processes kinetic sorption diffusion into immobile regions and transport through heterogeneous aquifers solving the moment equations which are formally identical to steady state transport equations is computationally more efficient than evaluating the temporal moments by integrating the transient flux concentrations we derive recursive relations for the moments of the flux concentration which involve the moments of the memory function but do not dependent on its shape it turns out that two mass transfer models have the same kth temporal moment if the moments of order lower than k are equal particularly the mean retention time i e the first moment of the retention probability density function pdf in the immobile domain decides the second temporal moment of concentration a mass transfer model with two first order rate coefficients can match up to the fourth temporal moment described by a multirate model with a predescribed pdf of the mass transfer rate coefficient the kth temporal moment is finite when the k 1 th moment of the memory function exists

a geostatistical approach to the inverse problem in groundwater modeling steady state and one dimensional simulations the problem of estimating hydrogeologic parameters in particular permeability from input output measurements is reexamined in a geostatistical framework the field of the unknown parameters is represented as a random field and the estimation procedure consists of two main steps first the structure of the parameter field is identified i e mathematical representations of the variogram and the trend are selected and their parameters are established by using all available information including measurements of hydraulic head and permeability second linear estimation theory is applied to provide minimum variance and unbiased point estimates of hydrogeologic parameters kriging structure identification is achieved iteratively in three substeps structure selection maximum likelihood estimation and model validation and diagnostic checking the methodology was extensively tested through simulations on a simple one dimensional case the results are remarkably stable and well behaved the estimated field is smooth while small scale variability is statistically described as the quality of measurements improves the procedure reproduces more features of the original field the results are also shown to be rather insensitive to deviations from assumptions about the geostatistical structure of the field

numerical methods for generalized least squares problems usually generalized least squares problems are solved by transforming them into regular least squares problems which can then be solved by well known numerical methods however this approach is not very effective in some cases and besides is very expensive for large scale problems in 1979 paige suggested another approach which consists of solving an equivalent equality constrained least squares problem by the orthogonal decomposition the bnp algorithm or the james implicit nullspace iterative methods in this paper we present some new developments of the numerical methods for example 2 cycle sor method and preconditioned conjugate gradient method for generalized least squares problems some numerical comparisons are included as well

aquifer parameter identification with kriging and optimum parameterization a new parameter identification method is developed for a two dimensional unsteady state groundwater flow model the method is a direct approach as it requires no iterative searching of parameters the unknown functional coefficients parameters are optimally determined by a generalized matrix inversion scheme kriging an unbiased minimum variance estimator is used as a presampling filter to reconstruct the head distribution for the entire flow domain the ill posedness of the inverse problem as characterized by instability and nonuniqueness is controlled by reparameterization utilizing the technique of finite elements a set of statistics are developed to quantify explicitly the tradeoff between the system modeling error and the error associated with the parameter uncertainty these two types of errors are shown to be a function of the parameter dimension residual mean square and trace of the covariance matrix of the estimated parameters serve as good indicators for the determination of the optimum parameter dimension a set of numerical experiments are conducted to illustrate the methodology

bayesian inversion for facies detection an extensible level set framework 1 in many cases it has been assumed that the variability in hydrologic parameters can be adequately described through a simple geostatistical model with a given variogram in other cases variability may be best described as a series of jumps in parameter behavior e g those that occur at geologic facies contacts when using indirect measurements such as pump tests to try to map such heterogeneity during inverse modeling the resulting images of the subsurface are always affected by the assumptions invoked in this paper we discuss inversion for parameter fields where prior information has suggested that major variability can be described by boundaries between geologic units or facies in order to identify such parameter fields we propose a bayesian level set inversion protocol framework which allows for flexible zones of any shape size and number we review formulas for defining facies locations using the level set method and for moving the boundaries between zones using a gradient based technique that improves fit through iterative deformation of the boundaries we describe the optimization algorithm employed when multiple level set functions are used to represent a field with more than two facies we extend these formulas to the investigation of the inverse problem in a bayesian context in which prior information is taken into account and through which measures of uncertainty can be derived we also demonstrate that the level set method is well suited for joint inversion problems and present a strategy for integrating different data types such as hydrologic and geophysical without assuming strict petrophysical relations our framework for joint inversion also contrasts with many previous methods in that all data sources e g both hydrologic and geophysical contribute to boundary delineation at once

on generalized successive overrelaxation methods for augmented linear systems for the augmented system of linear equations golub wu and yuan recently studied an sor like method bit 41 2001 71 85 by further accelerating it with another parameter in this paper we present a generalized sor gsor method for the augmented linear system we prove its convergence under suitable restrictions on the iteration parameters and determine its optimal iteration parameters and the corresponding optimal convergence factor theoretical analyses show that the gsor method has faster asymptotic convergence rate than the sor like method also numerical results show that the gsor method is more effective than the sor like method when they are applied to solve the augmented linear system this gsor method is further generalized to obtain a framework of the relaxed splitting iterative methods for solving both symmetric and nonsymmetric augmented linear systems by using the techniques of vector extrapolation matrix relaxation and inexact iteration besides we also demonstrate a complete version about the convergence theory of the sor like method

the direct inverse problem in aquifers solution of the inverse problem for aquifers requires identification of a distributed parameter system in addition to ill posedness the inverse problem may be underdetermined if recharge is treated as a spatially and temporally variable distribution parameter galerkin s method is applied to a weak formulation of the direct inverse problem leading to a possibly nonlinear weighted least squares problem consideration of generic errors suggests that if the estimate for some derivative of head contains a significant error on one element for all time then significant estimation errors are possible for all nodes and for all reservoir parameters in contrast if some derivative of head is small in some spatial region for all time causing the matrix to be inverted in the least squares algorithm to be nearly singular then significant estimation errors can occur in the corresponding reservoir parameters in that spatial region although it is still possible that other reservoir parameters in that region as well as all reservoir parameters elsewhere may be satisfactorily estimated

effects of kriging and inverse modeling on conditional simulation of the avra valley aquifer in southern arizona the avra valley aquifer in southern arizona is modeled stochastically at three levels of uncertainty the highest level of uncertainty occurs when log transmissivity estimates are based on measured values of this parameter but without regard to the geographic location of each measurement point the resulting steady state hydraulic heads in the aquifer computed by unconditional simulation with the aid of a multivariate normal random number generator coupled with a finite element model have a relatively large variance this variance can be reduced by conditioning the log transmissivity estimates on the spatial arrangement of the data by means of kriging when conditional simulation of the aquifer is performed at this intermediate level of uncertainty with the aid of the same technique as in the unconditional case the variance of the predicted head values is reduced by a factor of 3 2 the lowest level of uncertainty is achieved when the log transmissivity estimates are further conditioned on data relating to the flow regime such as flow rates and water levels in observation wells by means of a statistical inverse procedure conditional simulation at this level of uncertainty is a novel concept it results in a hydraulic head prediction variance that is 14 3 times lower than the corresponding variance based on kriged log transmissivities the net effect of conditioning by means of kriging and inverse modeling is to reduce the prediction variance by a factor of 46 0 a similar study performed by binsarti 1980 on the cortaro aquifer in southern arizona showed insignificant variance reduction due to kriging but a factor of four reduction after inverse modeling these results indicate that the conditioning effect of the inverse method may be much greater than that of kriging the fact that conditioning can reduce the hydraulic head prediction variance by as much as a factor of 46 implies that one should be cautious when dealing with the results of stochastic aquifer models in which conditioning effects are disregarded

improved methodology for parameter inference in nonlinear hydrologic regression models a new method is developed for the construction of reliable marginal confidence intervals and joint confidence regions for the parameters of nonlinear hydrologic regression models a parameter power transformation is combined with measures of the asymptotic bias and asymptotic skewness of maximum likelihood estimators to determine the transformation constants which cause the bias or skewness to vanish these optimized constants are used to construct confidence intervals and regions for the transformed model parameters using linear regression theory the resulting confidence intervals and regions can be easily mapped into the original parameter space to give close approximations to likelihood method confidence intervals and regions for the model parameters unlike many other approaches to parameter transformation the procedure does not use a grid search to find the optimal transformation constants an example involving the fitting of the michaelis menten model to velocity discharge data from an australian gauging station is used to illustrate the usefulness of the methodology

a global multilevel coordinate search procedure for estimating the unsaturated soil hydraulic properties we present a new inverse modeling procedure to characterize the hydraulic properties of partially saturated soils from soil moisture measurements during a natural transient flow experiment the inversion of the governing one dimensional richards equation is carried out using the global multilevel coordinate search optimization algorithm in sequential combination with the local nelder mead simplex algorithm gmcs nms we introduce this optimization method in the area of unsaturated zone hydrology since it is adapted for solving accurately and efficiently complex nonlinear problems several numerical experiments have been conducted to evaluate the proposed inversion method using synthetic error free and error contaminated data for different textured soils inversion of the simulated error free data and examination of the related response surfaces demonstrated the uniqueness of the inverse solution and the suitability of the gmcs nms strategy when identifying four key parameters of the hydraulic functions described by the mualem van genuchten model inversion of the error contaminated data proved further the good stability of the inverse solution that is consistent with the needs required by real experiments

aquifer parameter identification with optimum dimension in parameterization this paper presents a systematic procedure whereby the functional coefficients imbedded in a two dimensional partial differential equation which governs unsteady groundwater flow are optimally identified the coefficients to be identified are transmissivities which vary spatially finite elements are used to represent the unknown transmissivity function parametrically in terms of nodal values over a suitable discretization of a flow region a modified gauss newton algorithm is used for parameter optimization covariance analysis is used to estimate the reliability of the estimated parameters as the dimension of the unknown parameter increases the modeling error represented by a least squares criterion will generally decrease but errors in data would be propagated to a greater degree into the estimated parameters thus reducing the reliability of estimation the reliability of the estimated parameters is characterized by a norm of the covariance matrix this information is used for the determination of the optimum dimension in parameterization

a class of vectorfields on s2 that are topologically equivalent to polynomial vectorfields abstract let x be a c1 vectorfield on s2 x y z r x2 y2 z2 1 such that no open subset of s2 is the union of closed orbits of x if x has only a finite number of singular orbits and satisfies one additional condition then it is shown that x is topologically equivalent to a polynomial vectorfield a corollary is that a foliation f of the plane is topologically equivalent to a foliation by orbits of a polynomial vectorfield if and only if f has only a finite number of inseparable leaves

semi convergence analysis of gmssor methods for singular saddle point problems recently some authors darvishi and hessari 2011 zhang et al 2012 discussed the generalized modified symmetric successive over relaxation gmssor method for saddle point problems in this paper we further study the gmssor method for solving singular saddle point problems we prove the semi convergence of the gmssor method without the assumption that the 1 1 block sub matrix a should be symmetric and analyze the spectral property of the corresponding preconditioned matrix numerical experiments are given to illustrate the efficiency of gmssor with appropriate parameters

the problem of complex eigensystems in the semianalytical solution for advancement of time in solute transport simulations a new method using real arithmetic in the numerical modeling of groundwater solute transport explicit solutions may be obtained for the concentration field at any future time without computing concentrations at intermediate times the spatial variables are discretized and time is left continuous in the governing differential equation these semianalytical solutions have been presented in the literature and involve the eigensystem of a coefficient matrix this eigensystem may be complex i e have imaginary components due to the asymmetry created by the advection term in the governing advection dispersion equation previous investigators have either used complex arithmetic to represent a complex eigensystem or chosen large dispersivity values for which the imaginary components of the complex eigenvalues may be ignored without significant error it is shown here that the error due to ignoring the imaginary components of complex eigenvalues is large for small dispersivity values a new algorithm that represents the complex eigensystem by converting it to a real eigensystem is presented the method requires only real arithmetic

non fickian dispersion of groundwater age 1 we expand the governing equation of groundwater age to account for non fickian dispersive fluxes using continuous random walks groundwater age is included as an additional fifth dimension on which the volumetric mass density of water is distributed and we follow the classical random walk derivation now in five dimensions the general solution of the random walk recovers the previous conventional model of age when the low order moments of the transition density functions remain finite at their limits and describes non fickian age distributions when the transition densities diverge previously published transition densities are then used to show how the added dimension in age affects the governing differential equations depending on which transition densities diverge the resulting models may be nonlocal in time space or age and can describe asymptotic or preasymptotic dispersion a joint distribution function of time and age transitions is developed as a conditional probability and a natural result of this is that time and age must always have identical transition densities this implies that a transition density defined for age can substitute for a density in time and this has implications for transport model parameter estimation we present examples of simulated age distributions from a geologically based heterogeneous domain that exhibit non fickian behavior and show that the non fickian model provides better descriptions of the distributions than the fickian model

multimodel bayesian analysis of groundwater data worth we explore the way in which uncertain descriptions of aquifer heterogeneity and groundwater flow impact one s ability to assess the worth of collecting additional data we do so on the basis of maximum likelihood bayesian model averaging mlbma by accounting jointly for uncertainties in geostatistical and flow model structures and parameter hydraulic conductivity as well as system state hydraulic head estimates given uncertain measurements of one or both variables previous description of our approach was limited to geostatistical models based solely on hydraulic conductivity data here we implement the approach on a synthetic example of steady state flow in a two dimensional random log hydraulic conductivity field with and without recharge by embedding an inverse stochastic moment solution of groundwater flow in mlbma a moment equations based geostatistical inversion method is utilized to circumvent the need for computationally expensive numerical monte carlo simulations the approach is compatible with either deterministic or stochastic flow models and consistent with modern statistical methods of parameter estimation admitting but not requiring prior information about the parameters it allows but does not require approximating lead predictive statistical moments of system states by linearization while updating model posterior probabilities and parameter estimates on the basis of potential new data both before and after such data are actually collected

optimal parameters of the generalized symmetric sor method for augmented systems for the augmented system of linear equations zhang and lu 2008 recently studied the generalized symmetric sor method gssor with two parameters in this note the optimal parameters of the gssor method are obtained and numerical examples are given to illustrate the corresponding results

a methodology to compute mixing ratios with uncertain end members 1 mixing calculations involve computing the ratios in which two or more end members are mixed in a sample mixing calculations are useful for a number of tasks in hydrology such as hydrograph separation water or solute mass balances and identification of groundwater recharge sources most methods available for computing mixing ratios are based on assuming that end member concentrations are perfectly known which is rarely the case often end members cannot be sampled and their concentrations vary in time and space still much information about them is contained in the mixtures to take advantage of this information we present here a maximum likelihood method to estimate mixing ratios while acknowledging uncertainty in end member concentrations maximizing the likelihood of concentration measurements with respect to both mixing ratios and end member concentrations leads to a general constrained optimization problem an algorithm for solving this problem is presented and applied to two synthetic examples of water mixing problems results allow us to conclude that the method outperforms traditional approaches such as least squares or linear mixing in the computation of mixing ratios the method also yields improved estimates of end member concentrations thus enlarging the potential of mixing calculations the method requires defining the reliability of measurements but results are quite robust with respect to the assumed standard deviations a nice feature of the method is that it allows for improving the quality of computations by increasing the number of samples and or analyzed species

coupled estimation of flow and solute transport parameters a methodology for solving the coupled linear flow and transport inverse problem is presented it allows the estimation of aquifer parameters transmissivity storativity recharge boundary heads and flows leakage dispersivity molecular diffusion porosity retardation linear decay boundary concentrations using their prior estimates as well as head and concentration measurements as basic information transient flow and transport equations are solved using the finite element method in space and a weighted finite difference scheme in time flow and transport domains can be one two or quasi three dimensional applying linear one dimensional elements linear two dimensional triangles or bilinear two dimensional rectangles mixed at will time regimes can be transient or steady state in any equation flow or transport parameters are estimated on the basis of maximum likelihood theory which allows us to obtain information about parameters uncertainty minimization of the negative log likelihood is performed using marquardt s method the applicability of the method is shown by means of two real examples they illustrate that the proposed approach can successfully address model selection parameter uncertainty and nontrivial identifiability problems estimation of time dependent sources and the role of prior estimates are also addressed in the examples while the formulation of the problem leaves room for both stochastic and deterministic treatment of hydraulic parameters the latter has been adopted in both examples limitations of the approach are also apparent

review of parameter identification procedures in groundwater hydrology the inverse problem the purpose of this survey is to review parameter identification procedures in groundwater hydrology and to examine computational techniques which have been developed to solve the inverse problem parameter identification methods are classified under the error criterion used in the formulation of the inverse problem the problem of ill posedness in connection with the inverse problem is addressed typical inverse solution techniques are highlighted the review also includes the evaluation of methods used for computing the sensitivity matrix statistics which can be used to estimate the parameter uncertainty are outlined attempts have been made to compare and contrast representative inverse procedures and direction for future research is suggested

strategic placement of localization devices such as pilot points and anchors in inverse modeling schemes 1 localization schemes using devices such as pilot points or anchors localization devices are used in inversion schemes as a tool for gleaning information from measurements that can be used to constrain inversion schemes with the added benefit in the form of a potential for reduction in the number of parameters employed in the inversion scheme this paper proposes and demonstrates a method for strategic placement of the localization devices the method combines stochastic singular value decomposition with parameter rejection methods to produce a map of intensity scores which allows the inverse modeler to place the localization devices at locations defined by their probability for producing informative probabilistic constraints on the statistical distributions of the target variables we also show that the method can be used for selecting measurement locations when designing data acquisition campaigns the method is demonstrated for the case of steady state flow in a heterogeneous conductivity field using mad method of anchored distributions in summary the proposed localization scheme intends to increase the information gleaned from measurements while reducing the associated computational costs associated with stochastic inversion

a characterization of theω limit sets of planar continuous dynamical systems abstract if the family of curves g i i and the set of points s are given we find necessary and sufficient conditions for the existence of a c k map f from the plane into itself 0 k if k 0 we also assume that f is locally lipschitz such that the curves from g and the points from s are respectively nondegenerate trajectories and singular points of the system x f x and additionally s i i is the limit set of some trajectory of the system in such cases we also provide a detailed geometric description of s i i after contraction of the connected components of s

block sor methods for rank deficient least squares problems many papers have discussed preconditioned block iterative methods for solving full rank least squares problems however very few papers studied iterative methods for solving rank deficient least squares problems miller and neumann 1987 proposed the 4 block sor method for solving the rank deficient problem here a 2 block sor method and a 3 block sor method are proposed to solve such problem the convergence of the block sor methods is studied the optimal parameters are determined comparison between the 2 block sor method and the 3 block sor method is given also

efficient posterior exploration of a high dimensional groundwater model from two stage markov chain monte carlo simulation and polynomial chaos expansion 1 this study reports on two strategies for accelerating posterior inference of a highly parameterized and cpu demanding groundwater flow model our method builds on previous stochastic collocation approaches e g marzouk and xiu 2009 and marzouk and najm 2009 and uses generalized polynomial chaos gpc theory and dimensionality reduction to emulate the output of a large scale groundwater flow model the resulting surrogate model is cpu efficient and serves to explore the posterior distribution at a much lower computational cost using two stage mcmc simulation the case study reported in this paper demonstrates a two to five times speed up in sampling efficiency

modified accelerated parameterized inexact uzawa method for singular and nonsingular saddle point problems abstract recently bai and wang 2008 9 presented an efficient accelerated parameterized inexact uzawa apiu method for solving both symmetric and nonsymmetric nonsingular saddle point problems in this work we propose a modified apiu mapiu method for solving saddle point problems which is an extension of the apiu iteration method and covers a series of existing iterative methods the mapiu method can be applied not only to the nonsingular case but also to the singular case the convergence properties for the nonsingular saddle point problems and the semi convergence properties for the singular one are carefully discussed under suitable restrictions we prove that the mapiu method has a wider convergence or semi convergence region than the apiu method

realization problems for limit cycles of planar polynomial vector fields abstract we show that for any finite configuration of closed curves r 2 one can construct an explicit planar polynomial vector field that realizes up to homeomorphism as the set of its limit cycles with prescribed periods multiplicities and stabilities the only obstruction given on this data is the obvious compatibility relation between the stabilities and the parity of the multiplicities the constructed vector fields are darboux integrable and admit a polynomial inverse integrating factor

a continuous time inverse operator for groundwater and contaminant transport modeling model identifiability general identifiability conditions are derived for the model coefficients of the spatially discretized continuous time diffusion equation these conditions in turn determine some sufficient and necessary conditions for the existence and uniqueness of the continuous time inverse operator described in a previous article the identifiability conditions are in terms of the initial conditions and the flux vector containing boundary condition information and the eigenspace of the spatially discretized system the derivation of these conditions gives some insight into the location of the solution of the inverse problem in terms of the direct operator under investigation in the space of the laplace transform and limited insight into the stability of the solution near that location

preconditioned conjugate gradient method for generalized least squares problems abstract a variant of the preconditioned conjugate gradient method to solve generalized least squares problems is presented if the problem is min ax b t w 1 ax b with a r m n and w r m m symmetric and positive definite the method needs only a preconditioner a 1 r n n but not the inverse of matrix w or of any of its submatrices freund s comparison result for regular least squares problems is extended to generalized least squares problems an error bound is also given

determination of the optimal value of relaxation parameter in symmetric sor method for rectangular coefficient matrices in this paper we obtain the optimal value of relaxation parameter in symmetric sor ssor method by ssor method we find the least square solution of minimal norm to the linear system ax b where a is an m by n matrix

incorporation of prior information on parameters into nonlinear regression groundwater flow models 2 applications this paper investigates factors influencing the degree of improvement in estimates of parameters of a nonlinear regression groundwater flow model by incorporating prior information of unknown reliability consideration of expected behavior of the regression solutions and results of a hypothetical modeling problem lead to several general conclusions first if the parameters are properly scaled linearized expressions for the mean square error mse in parameter estimates of a nonlinear model will often behave very nearly as if the model were linear second by using prior information the mse in properly scaled parameters can be reduced greatly over the mse of ordinary least squares estimates of parameters third plots of estimated mse and the estimated standard deviation of mse versus an auxiliary parameter the ridge parameter specifying the degree of influence of the prior information on regression results can help determine the potential for improvement of parameter estimates fourth proposed criteria can be used to make appropriate choices for the ridge parameter and another parameter expressing degree of overall bias in the prior information results of a case study of truckee meadows reno sparks area washoe county nevada conform closely to the results of the hypothetical problem in the truckee meadows case incorporation of prior information did not greatly change the parameter estimates from those obtained by ordinary least squares however the analysis showed that both sets of estimates are more reliable than suggested by the standard errors from ordinary least squares

configurations of limit cycles and planar polynomial vector fields abstract we show that every finite configuration of disjoint simple closed curves of the plane is topologically realizable as the set of limit cycles of a polynomial vector field moreover the realization can be made by algebraic limit cycles and we provide an explicit polynomial vector field exhibiting any given finite configuration of limit cycles

multiple outflows spatial components and nonlinearities in age theory water age has become an important variable for the characterization of hydrologic systems the goal of this paper is to analyze the role of multiple outflows spatial components and nonlinearities in age theory we first extend the theory to linear systems with multiple outflows including the relationship between age distribution at death and survival time distribution at birth we further show that for each outflow there is a survival time distribution at birth which normalized corresponds to the impulse response function for the specific outflow we also analyze how the impulse response function affects both the amplitude gain and time delay of the outflow and the long term average partitioning with regard to linear spatially extended systems we link the impulse response function to the green s function this allows us to easily compute the loss function and the age distribution for the system finally we focus on nonlinear systems to analyze the effects of storage dependent and age distribution dependent loss functions by considering the burgers equation we show how the relationships between spatial dynamics and the age distribution are complicated by nonlinearities this article is protected by copyright all rights reserved

inverse problems for dynamical systems in this paper we consider the problem of reconstructing a structurally stable vector field from its global phase portrait or from certain local pieces of the phase portrait we show that for arbitrary finite sets of points and of simple closed curves we can construct a structurally stable system having the elements of those sets as its critical points and limit cycles and a prescribed topological type in a neighborhood of each of them given a global structurally stable phase portrait we can construct a structurally stable polynomial system which is topologically equivalent to it some of our results were originally announced in 28 1 the general theory of dynamical systems orginated in the study of solution curves for systems of ordinary differential equations in a phase space this qualitative theory had its first major development in the work of poincare on the topology of integral curves a typical problem is to give a geometric description of the solutions of a particular system or class of systems with emphasis on the location of special features such as critical points and periodic orbits an inverse problem then is the construction of a system of equations given geometric properties of the solutions which may be in the form of a picture of the flow a schematic diagram of phase space or a list of critical orbits defined analytically a method for solving such problems is potentially useful in building models in the various fields in which the theory of dynamical systems is now being applied it could provide a basic model to which statistical identification techniques 141 could then be applied there is also an intrinsic interest in methods for constructing polynomial systems of prescribed type which is related to the lack of any progress toward a solution of hilbert s sixteenth problem the recent appearance of a counterexample 27 to the petrovsky landis conjecture for quadratic systems leaves the problem completely open one of the earliest results of this kind was obtained by forster in 1938

a modified symmetric successive overrelaxation method for augmented systems in this paper we establish a modified symmetric successive overrelaxation mssor method to solve augmented systems of linear equations which uses two relaxation parameters this method is an extension of the symmetric sor ssor iterative method the convergence of the mssor method for augmented systems is studied numerical examples show that the new method is an efficient method

a continuous time inverse operator for groundwater and contaminant transport modeling deterministic case the value of time dependent data in the calibration of numerical models of groundwater flow has been recently highlighted in the literature in this paper the background for a new development in the inverse problem is placed and a new operator for inversion of the spatially discretized diffusion equation is introduced the motivation for the development of an inverse problem solution procedure which uses information in transient data is illustrated with a focused review of recent literature such an approach which implicitly incorporates the continuous time dependency of the flow model is then presented this approach is based upon the new concept of inverting the model after the for example finite element spatial discretization but before the typically finite difference temporal discretization of the model not afterward as has been done in the past the laplace transform is used to replace discretization in time of the system of ordinary differential equations to discretization in the laplace variable of the algebraic system in laplace space the existence of the deterministic solution in the laplace transform space does not require knowledge of the eigenvalues of the diffusion matrix but only the specification of the laplace variables near zero a simple numerical example from the literature is used to generate a set of three simplified test cases upon which the proposed method is tested the implications of the theoretical investigation and the results of the numerical experiments are discussed with a view toward prospects of practical implementation

extended shift splitting preconditioners for saddle point problems abstract in this paper we consider to solve the linear systems of the saddle point problems by preconditioned krylov subspace methods the preconditioners are based on a special splitting of the saddle point matrix the convergence theory of this class of the extended shift splitting preconditioned iteration methods is established the spectral properties of the preconditioned matrices are analyzed numerical implementations show that the resulting preconditioners lead to fast convergence when they are used to precondition krylov subspace iteration methods such as gmres

recovery of a variable coefficient in a coastal evolution equation a method is presented for reconstructing an unknown coefficient in a linear diffusion equation from measured data this equation arises in the description of coastline evolution and preliminary results are presented here the unknown term may vary with both space and time although time variation is assumed to be slow inversion is carried out by first expressing the solution of the direct problem formally in terms of the governing operators and making explicit approximations to these expressions using data at two time steps this then allows equations to be derived and solved to give explicit expressions for the required function