the reconstruction of groundwater parameters from head data in an unconfined aquifer the estimation of groundwater flow parameters from head measurements and other ancillary data is fundamental to the process of modelling a groundwater system in an unconfined aquifer the problem is more complex because the governing equation for the well heads the boussinesq equation is non linear we consider here a new method that allows for the simultaneous computation of the unconfined groundwater parameters as the unique minimum of a convex functional

convergence rates for tikhonov regularization of a two coefficient identification problem in an elliptic boundary value problem we investigate the convergence rates for tikhonov regularization of the problem of simultaneously estimating the coefficients q and a in the neumann problem for the elliptic equation rm div q nabla u au f rm in omega q partial u partial n g on the boundary partial omega omega subset mathbb r d d geq 1 when u is imprecisely given by z delta in h 1 omega u z delta h 1 omega le delta delta 0 we regularize this problem by minimizing the strictly convex functional of q a begin array lll int limits omega left q nabla u q a z delta 2 a u q a z delta 2 right dx quad rho left q q 2 l 2 omega a a 2 l 2 omega right end array over the admissible set k where 0 is the regularization parameter and q a is an a priori estimate of the true pair q a which is identified and consider the unique solution of these minimization problem as the regularized one to that of the inverse problem we obtain the convergence rate mathcal o sqrt delta as 0 and for the regularized solutions under the simple and weak source condition rm there is a function w in v rm such that u prime q dagger a dagger w q dagger q a dagger a with q dagger a dagger being the q a minimum norm solution of the coefficient identification problem u the frechet derivative of u v the sobolev space on which the boundary value problem is considered our source condition is without the smallness requirement on the source function which is popularized in the theory of regularization of nonlinear ill posed problems furthermore some concrete cases of our source condition are proved to be simply the requirement that the sought coefficients belong to certain smooth function spaces

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

topological gauge theories and group cohomology we show that three dimensional chern simons gauge theories with a compact gauge groupg not necessarily connected or simply connected can be classified by the integer cohomology grouph4 bg z in a similar way possible wess zumino interactions of such a groupg are classified byh3 g z the relation between three dimensional chern simons gauge theory and two dimensional sigma models involves a certain natural map fromh4 bg z toh3 g z we generalize this correspondence to topological spin theories which are defined on three manifolds with spin structure and are related to what might be calledz2 graded chiral algebras or chiral superalgebras in two dimensions finally we discuss in some detail the formulation of these topological gauge theories for the special case of a finite group establishing links with two dimensional holomorphic orbifold models

an explicit description of the second cohomology group of a quandle we use the inflation restriction sequence and a result of etingof and grana on the rack cohomology to give a explicit description of 2 cocycles of finite indecomposable quandles with values in an abelian group several applications are given

from racks to pointed hopf algebras abstract a fundamental step in the classification of finite dimensional complex pointed hopf algebras is the determination of all finite dimensional nichols algebras of braided vector spaces arising from groups the most important class of braided vector spaces arising from groups is the class of braided vector spaces c x c q where x is a rack and q is a 2 cocycle on x with values in c racks and cohomology of racks appeared also in the work of topologists this leads us to the study of the structure of racks their cohomology groups and the corresponding nichols algebras we will show advances in these three directions we classify simple racks in group theoretical terms we describe projections of racks in terms of general cocycles we introduce a general cohomology theory of racks containing properly the existing ones we introduce a fourier transform on racks of certain type finally we compute some new examples of finite dimensional nichols algebras

quandle homotopy invariants of knotted surfaces given a finite quandle we introduce a quandle homotopy invariant of knotted surfaces in the 4 sphere modifying that of classical links this invariant is valued in the third homotopy group of the quandle space and is universal among the generalized quandle cocycle invariants we compute the second and third homotopy groups with respect to regular alexander quandles as a corollary any quandle cocycle invariant using the dihedral quandle of prime order is a scalar multiple of mochizuki 3 cocycle invariant as another result we determine the third quandle homology group of the dihedral quandle of odd order

schur multipliers and second quandle homology abstract we define a map from second quandle homology to the schur multiplier and examine its properties furthermore we express the second homology of alexander quandles in terms of exterior algebras additionally we present a self contained proof of its structure and provide some computational examples

a variational method for numerical differentiation a new method is given for effecting numerical differentiation by means of an optimization procedure the method is shown to be effective for the differentiation of noisy functions and stability and convergence results for a steepest descent implementation are proved

remarks on hierarchic control abstract this paper contains results about hierarchic control in domains of r n 1 with moving boundaries it develops the stackelberg nash strategy for this type of domains it also contains a proof of weak and strong solutions for an optimality system

homologies of algebraic structures via braidings and quantum shuffles abstract in this paper we construct structural pre braidings characterizing different algebraic structures a rack an associative algebra a leibniz algebra and their representations some of these pre braidings seem original on the other hand we propose a general homology theory for pre braided vector spaces and braided modules based on the quantum co shuffle comultiplication applied to the structural pre braidings above it gives a generalization and a unification of many known homology theories all the constructions are categorified resulting in particular in their super and co versions loday s hyper boundaries as well as certain homology operations are efficiently treated using the shuffle tools

energy minimization related to the nonlinear schrodinger equation in this the window of the sobolev gradient technique to the problem of minimizing a schrodinger functional associated with a nonlinear schrodinger equation we show that gradients act in a suitably chosen sobolev space sobolev gradients can be used in finite difference and finite element settings in a computationally efficient way to find minimum energy states of schrodinger functionals

lattice topological field theory in two dimensions the lattice definition of a two dimensional topological field theory tft is given generically and the exact solution is obtained explicitly in particular the set of all lattice topological field theories is shown to be in one to one correspondence with the set of all associative algebrasr and the physical hilbert space is identified with the centerz r of the associative algebrar perturbations of tft s are also considered in this approach showing that the form of topological perturbations is automatically determined and that all tft s are obtained from one tft by such perturbations several examples are presented including twistedn 2 minimal topological matter and the case wherer is a group ring

approximating time evolution related to ginzburg landau functionals via sobolev gradient methods in a finite element setting sobolev gradients have previously 1 been used to approximate time evolution related to a model a functional in a finite difference setting in this journal here a related approach in a finite element setting is discussed

one point recovery of a parabolic equation abstract we recover the shape of the domain and the diffusion coefficient of a parabolic equation from observations of an infinite sequence of solutions taken at an arbitrary single point inside or on the boundary of the unknown domain the method is to reconstruct the principal eigenfunction of the associated elliptic operator from the observed data

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

on the recovery of transport parameters in groundwater modelling the problem of recovering the coefficient functions in the groundwater transport equation from piezomet tic head and contaminant concentration measurements is solved by minimization of an associated convex functional

sobolev gradient approach for the time evolution related to energy minimization of ginzburg landau functionals the sobolev gradient technique has been discussed previously in this journal as an efficient method for finding energy minima of certain ginzburg landau type functionals s sial j neuberger t lookman a saxena energy minimization using sobolev gradients application to phase separation and ordering j comput phys 189 2003 88 97 in this article a sobolev gradient method for the related time evolution is discussed

exact controllability for multidimensional semilinear hyperbolic equations in this paper we obtain a global exact controllability result for a class of multidimensional semilinear hyperbolic equations with a superlinear nonlinearity and variable coefficients for this purpose we establish an observability estimate for the linear hyperbolic equation with an unbounded potential in which the crucial observability constant is estimated explicitly by a function of the norm of the potential such an estimate is obtained by a combination of a pointwise estimate and a global carleman estimate for the hyperbolic differential operators and analysis on the regularity of the optimal solution to an auxiliary optimal control problem

identification of the coefficient in elliptic equations this paper seeks to identify the space distributed diffusion coefficient a in the steady state diffusion equation operatorname div a operatorname grad u f in a bounded region of the real plane u f and the flux at the boundary are observed this problem is solved by a least square method choosing for the cost function the equation error defined in a natural way by the riesz representation theorem the method is stable if a smoothing term is added to the cost function tikhonov regularisation this paper shows that selection strategies of the smoothing coefficient exist that insure convergence under the only condition that a be sufficiently smooth the case is also considered where a is piecewise smooth and it is shown how to extend the previous formulation in both cases computational results are presented

on the observability inequalities for exact controllability of wave equations with variable coefficients this paper deals with boundary exact controllability for the dynamics governed by the wave equation with nonconstant coefficients in the principal part subject to dirichlet or neumann boundary controls the observability inequalities are established by the riemannian geometry method under some geometric condition for the dirichlet problem and for the neumann problem respectively next a number of nontrivial examples are presented to verify the observability inequality in particular a counterexample is given without boundary exact controllability where the control is exerted on the whole boundary

parameter estimation in flow through partially saturated porous materials a class of numerical simulators were developed and critically evaluated to be incorporated as the solver of a forward problem in the framework of an inverse modeling strategy the strategy couples a mass lumped galerkin linear finite element solution of the mixed form richards equation with an experimental time space series and the osborne more revised form of the levenberg marquardt algorithm to retrieve hydraulic parameters of a partially saturated porous medium the numerical simulator shows excellent agreement with a reference solution obtained on a dense grid and infinitesimal time step in terms of fluid pressure head fluid content and fluid volumetric flux density and perfectly conserves the global mass an adaptive algorithm was implemented to estimate sensitivity matrix in the inverse algorithm a multi criterion stopping rule was developed and successfully implemented to end the inverse code at the solution the result of the optimization was compared with a large scale in situ soil moisture space time series measured during the course of a drainage experiment and excellent agreements were found analysis of the parameter response surfaces and hyper space plots closeness of the gradient of the penalty function at minimum to zero and positive definiteness of the approximation for the hessian at the solution eigs h 0 indicate that the obtained solution is a strong local minimum a state of the art sensitivity analysis carried out to quantify sensitivity of the state variable with respect to uncertainty and changes in different model parameters

controllability of the linear one dimensional wave equation with inner moving forces we address the null inner controllability of the one dimensional wave equation with supports of the controls that may vary with respect to the time variable we consider theoretical and numerical aspects assuming a geometric optic condition on the support of the controls we first prove a generalized observability inequality for the homogeneous wave equation implying the controllability we then introduce and prove the well posedness of a mixed formulation that characterizes the controls of minimal square integrable norm a numerical approximation of this formulation based on a conformal space time finite element method is then presented several numerical experiments are finally reported

an inversion method for parabolic equations based on quasireversibility abstract this paper is concerned with a new method to solve a linearized inverse problem for one dimensional parabolic equations the inverse problem seeks to recover the subsurface absorption coefficient function based on the measurements obtained at the boundary the method considers a temporal interval during which time dependent measurements are provided it linearizes the working equation around the system response for a background medium it is then possible to relate the inverse problem of interest to an ill posed boundary value problem for a differential integral equation whose solution is obtained by the method of quasireversibility this approach leads to an iterative method a number of numerical results are presented which indicate that a close estimate of the unknown function can be obtained based on the boundary measurements only

existence of solution to parabolic equations with mixed boundary condition on non cylindrical domains abstract in this paper we study the existence of weak solutions to initial boundary value problems of linear and semi linear parabolic equations with mixed boundary conditions on non cylindrical domains t 0 t t t of spatial temporal space r n r in the case of the linear equation each boundary condition is given on any open subset of the boundary surface t 0 t t t under a condition that the boundary portion for dirichlet condition 0 is nonempty at any time t due to this it is difficult to reduce the problem to the one on a cylindrical domain by diffeomorphism of the spatial domains t by a transformation of the unknown function and the penalty method we connect the problem to a monotone operator equation for functions defined on the non cylindrical domain we are also concerned with a semilinear problem when the boundary portion for dirichlet condition is cylindrical

an explicit update scheme for inverse parameter and interface estimation of piecewise constant coefficients in linear elliptic pdes we introduce a general and efficient method to recover piecewise constant coefficients occurring in elliptic partial differential equations as well as the interface where these coefficients have jump discontinuities for this purpose we use an output least squares approach with level set and augmented lagrangian methods our formulation incorporates the inherent nature of the piecewise constant coefficients which eliminates the need for a complicated nonlinear solve at every iteration instead we obtain an explicit update formula and therefore vastly speed up computation we employ our approach to the example problems of poisson s equation and linear elasticity and provide the combination of simultaneously recovering coefficients and interface

quandle cohomology and state sum invariants of knotted curves and surfaces the 2 twist spun trefoil is an example of a sphere that is knotted in 4 dimensional space a proof is given in this paper that this sphere is distinct from the same sphere with its orientation reversed our proof is based on a state sum invariant for knotted surfaces developed via a cohomology theory of racks and quandles also known as distributive groupoids a quandle is a set with a binary operation the axioms of which model the reidemeister moves in classical knot theory colorings of diagrams of knotted curves and surfaces by quandle elements together with cocycles of quandles are used to define state sum invariants for knotted circles in 3 space and knotted surfaces in 4 space cohomology groups of various quandles are computed herein and applied to the study of the state sum invariants non triviality of the invariants is proved for a variety of knots and links and conversely knot invariants are used to prove non triviality of cohomology for a variety of quandles

exact controllability of wave equations with locally distributed control in non cylindrical domain abstract exact controllability of a one dimensional wave equation with locally distributed control in non cylindrical domain is considered this equation characterizes the motion of a string with a fixed endpoint and the other moving one if the adjoint system is observable this establishes exact controllability of the original system the adjoint system is observable by multiplier method therefore we obtain exact controllability of this equation when the speed of the moving endpoint is less than wave speed

exact controllability stabilization and perturbations for distributed systems exact controllability is studied for distributed systems of hyperbolic type or for petrowsky systems like plate equations the control is a boundary control or a local distributed control exact controllability consists in trying to drive the system to rest in a given finite time the solution of the problems depends on the function spaces where the initial data are taken and also depends on the function space where the control can be chosen a systematic method named hum for hilbert uniqueness method is introduced as the terminology indicates it is based on uniqueness results classical or new and on hilbert spaces constructed in infinitely many ways by using uniqueness a number of applications are indicated having a general method for exact controllability implies having a general method for stabilization this leads to new and complicated nonlinear riccati s type pdes to be compared with direct methods when available in the last part of the paper we consider how all this behaves fo

dynamics of wave equations with moving boundary abstract this paper is concerned with long time dynamics of weakly damped semilinear wave equations defined on domains with moving boundary since the boundary is a function of the time variable the problem is intrinsically non autonomous under the hypothesis that the lateral boundary is time like the solution operator of the problem generates an evolution process u t x x t where x t are time dependent sobolev spaces then by assuming the domains are expanding we establish the existence of minimal pullback attractors with respect to a universe of tempered sets defined by the forcing terms our assumptions allow nonlinear perturbations with critical growth and unbounded time dependent external forces

optimal energy control in finite time by varying the length of the string we consider a finite string where at both end points a homogeneous dirichlet boundary condition holds one boundary point is fixed and the other is moving hence the length of the string is changing in time the string is controlled through the movement of this boundary point we consider movements of the boundary that are lipschitz continuous only movements for which at the given finite terminal time the string has the same length as at the beginning are admissible moreover we impose an upper bound for the lipschitz constant of the movement that is smaller than the speed of wave propagation we consider the optimal control problem to find an admissible movement for which at the given terminal time the energy of the string is minimal we give a sufficient condition for the existence and uniqueness of an optimal movement and construct an optimal control movement

hp finite element discretisation of the electrical impedance tomography problem abstract this paper is concerned with the accurate recovery of the permittivity and conductivity from the solution of the electrical impedance tomography eit inverse problem given the knowledge of either the neumann to dirichlet or the dirichlet to neumann map the linearised eit inverse problem for the fourier regime is discretised by hp finite elements which allows the accurate calculation of the sensitivity matrix entries the variational algorithm of knowles 1998 31 is extended to allow the recovery of both material parameters by solution of the eit problem in the laplace regime the associated forward problem is again discretised by hp finite elements a series of numerical results are included to demonstrate the effectiveness of the proposed techniques

conditional well posedness for an elliptic inverse problem for the elliptic equation nabla cdot p x nabla v lambda q x v f x in omega subset mathbb r n the problem of determining when one or more of the coefficient functions p q and f depends continuously on the solution functions v v p q f lambda is considered

uniqueness for an elliptic inverse problem for the elliptic equation nabla cdot p x nabla v lambda q x v f x in omega subset bbr n the problem of determining when one or more of the coefficient functions p q and f are defined uniquely by a knowledge of one or more of the solution functions v v p q f lambda is considered

sharp sufficient conditions for the observation control and stabilization of waves from the boundary for the observation or control of solutions of second order hyperbolic equation in mathbb r t times omega ralston s construction of localized states comm pure appl math 22 1969 pp

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

geostatistical inverse modeling of transient pumping tests using temporal moments of drawdown 1 pumping tests belong to the most common techniques of hydrogeological site assessment while the steady state drawdown is determined by the distribution of transmissivity alone the transient behavior is also influenced by the storativity field in geostatistical inverse modeling the spatial distributions of both transmissivity and storativity are inferred from the drawdown curves and prior information on the spatial correlation of the parameter fields so far however transient data have hardly been analyzed by geostatistical inverse methods because the computational effort is rather high in the present study we characterize the drawdown by its temporal moments we present moment generating equations and corresponding equations to compute the sensitivity of the temporal moments of drawdown with respect to the distributions of transmissivity and storativity we utilize these equations to infer the transmissivity and storativity fields from transient pumping tests using the quasi linear geostatistical approach of inverse modeling considering temporal moments rather than full drawdown curves drastically reduces the computational effort of the estimation procedure in test cases we show that the first two temporal moments are sufficient to characterize the drawdown curves we investigate how erroneous assumptions regarding the spatial variability of storativity affect the estimate of the transmissivity field and we analyze the effect of truncating the measured drawdown curves

variational method for multiple parameter identification in elliptic pdes abstract in the present paper we investigate the inverse problem of identifying simultaneously the diffusion matrix source term and boundary condition in the neumann boundary value problem for an elliptic partial differential equation pde from a measurement data which is weaker than required of the exact state a variational method based on energy functions with tikhonov regularization is here proposed to treat the identification problem we discretize the pde with the finite element method and prove the convergence as well as analyze error bounds of this approach

exact controllability for a one dimensional wave equation in non cylindrical domains abstract this paper is addressed to a study of the controllability for a one dimensional wave equation in domains with moving boundary this equation characterizes the motion of a string with a fixed endpoint and the other moving one when the speed of the moving endpoint is less than the characteristic speed by the hilbert uniqueness method the exact controllability of this equation is established also an explicit dependence of the controllability time on the speed of the moving endpoint is given moreover when the speed of the moving endpoint is equal to the characteristic speed by a constructive method we characterize a target set for the exact controllability with smooth controllers

parameter identification for elliptic problems abstract the problem of computing a principal coefficient function p in the differential equation p x u f x r m m 1 on a bounded region from a knowledge of the solution function u and the right hand side f where u f are known only approximately and p may have mild discontinuities is solved by minimization of an associated functional

control and stabilization for the wave equation with variable coefficients in domains with moving boundary abstract the study of stabilization and control for pdes with variable coefficients involves higher level of complexity than the corresponding case of constant coefficients further compounding the complexity are the concern and effects of dynamic boundary motion the problem in its general form is extremely challenging to treat but under certain specific physical and geometric conditions such as the time likeness of the boundary and a limited speed of domain expansion energy decay estimates can be established and the exact controllability can also be obtained by control theoretic and riemannian geometric methods our approach here is based on the bochner technique of differential geometry in terms of riemannian metric and geometric multipliers by generalizing an energy identity method used earlier in bardos and chen 1981 concrete examples are also given to illustrate the geometric conditions and the theorems

multi objective optimization in variably saturated fluid flow a general purpose multi objective inverse modeling strategy was developed and implemented to quantify fluid flow parameters in variably saturated porous materials the strategy combines a robust and mass conservative numerical simulator of the flow equation with an optimization algorithm and an experimental data set to estimate the parameters the numerical simulator of the direct problem shows excellent agreement with a reference solution and conserves global mass with near perfection an adaptive method was proposed in which the sensitivity matrix was calculated by one sided finite difference approximation at the early stages of the optimization and the more accurate two sided differentiation as the search approaches the minimum a combined termination criterium was developed to stop the inverse code at the solution the results of the multi objective optimization were compared with those of single objective minimization while single objective optimization generates reasonable results for either the fluid pressure head profile or the fluid content data the proposed multi objective optimization shows excellent agreement with both profiles simultaneously

control and stabilization for the wave equation part iii domain with moving boundary we use energy invariants to study the growth and decay estimates for solutions of the wave equation in a domain with moving boundary sufficient conditions are formulated which insure the exact distributed parameter controllability of the wave equation

nichols algebras of group type with many quadratic relations abstract we classify nichols algebras of irreducible yetter drinfeld modules over nonabelian groups satisfying an inequality for the dimension of the homogeneous subspace of degree two all such nichols algebras are finite dimensional and all known finite dimensional nichols algebras of nonabelian group type appear in the result of our classification we find a new finite dimensional nichols algebra over fields of characteristic two

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

approximate controllability for a one dimensional wave equation with the fixed endpoint control abstract this paper is devoted to the study of the approximate controllability for a one dimensional wave equation in domains with moving boundary this equation models the motion of a string where an endpoint is fixed and the other one is moving when the speed of the moving endpoint is less than the characteristic speed the controllability of this equation is established we present the following results the existence and uniqueness of nash equilibrium the approximate controllability with respect to the leader control and the optimality system for the leader control

an inverse problem for the steady state diffusion equation we consider the problem of identifying a spatially varying diffusion coefficient using an observed solution to the forward problem under appropriate conditions this problem can be treated as a first order hyperbolic equation in the unknown coefficient we develop some continuous dependence results for this problem and also propose a particularly advantageous set of test conditions for observing the forward solution

convergence rates for total variation regularization of coefficient identification problems in elliptic equations ii abstract we investigate the convergence rates for total variation regularization of the problem of identifying i the coefficient q in the neumann problem for the elliptic equation div q u f in q u n g on ii the coefficient a in the neumann problem for the elliptic equation u a u f in u n g on r d d 1 when u is imprecisely given by z h 1 u z h 1 0 we regularize these problems by correspondingly minimizing the strictly convex functionals 1 2 q u q z 2 d x 1 2 q l 2 2 q and 1 2 u a z 2 d x 1 2 a u a z 2 d x 1 2 a l 2 2 a over admissible sets where u q u a is the solution of the first second neumann boundary value problem 0 is the regularization parameter taking the solutions of these optimization problems as the regularized solutions to the corresponding identification problems we obtain the convergence rates of them to the solution of the inverse problem in the sense of the bregman distance and in the l 2 norm under relatively simple source conditions without the smallness requirement on the source functions

development of objective oriented groundwater models 1 robust parameter identification 1 when the structure of an aquifer is complex and unknown determining the complexity of model structure and identifying its spatially distributed parameters become very difficult this two paper series describes an approach for objective oriented model construction that can identify a model with simplified structure and assure its reliability for predetermined model applications the theoretical basis of the methodology includes the definition of structure error the conditions of structure reducibility the calculation of the worst case parameter wcp and the solution of a generalized inverse problem in the first paper of the series we extend the basic theory established in our previous works from zonation parameterization to any kind of parameterization the extension makes the methodology more general and complete we present an effective direct search algorithm that can form a nonnested structure series with an optimized pattern for solving the generalized inverse problem we use numerical examples to show how the wcp depends on the structure complexity prior information boundary conditions and the objectives of model application generally speaking with more prior information simpler boundary conditions fewer objectives and lower accuracy requirement the wcp will have a simpler shape as a result a reliable model can be identified with less observation data in part 2 of this series we will give the conditions of structure reducibility and the method for robust experimental design a robust experimental design is independent of the true parameter and can provide sufficient information for identifying an objective oriented model