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

constant depth reducibility the purpose of this paper is to study reducibilities that can be computed by combinational logic networks of polynomial size and constant depth containing and s or s and not s with no bound placed on the fan in of and gates and or gates two such reducibilities are defined and reductions and equivalences among several common problems such as parity sorting integer multiplication graph connectivity bipartite matching and network flow are given certain problems are shown to be complete with respect to these reducibilities in the complexity classes deterministic logarithmic space nondeterministic logarithmic space and deterministic polynomial time new upper bounds on the size depth unbounded fan in circuit complexity of symmetric boolean functions are established

analysis of a continuous finite element method for hyperbolic equations a finite element method for hyperbolic equations is analyzed in the context of a first order linear problem in r 2 the method is applicable over a triangulation of the domain and produces a continuous piecewise polynomial approximation which can be developed in an explicit fashion from triangle to triangle in a sense it extends the basic upwind difference scheme to higher order the method is shown to be stable and error estimates are obtained for nth degree approximation the errors in the approximate solution and its gradient are shown to be at least of order h n 1 4 and h n 1 2 respectively assuming sufficient regularity in the solution

level set and total variation regularization for elliptic inverse problems with discontinuous coefficients we propose a level set approach for elliptic inverse problems with piecewise constant coefficients the geometry of the discontinuity of the coefficient is represented implicitly by level set functions the inverse problem is solved using a variational augmented lagrangian formulation with total variation regularization of the coefficient the corresponding euler lagrange equation gives the evolution equation for the level set functions and the constant values of the coefficients we use a multiple level set representation which allows the coefficient to have multiple constant regions knowledge of the exact number of regions is not required only an upper bound is needed numerical experiments show that the method can recover coefficients with rather complicated geometries of discontinuities under moderate amount of noise in the observation data the method is also robust with respect to the initial guess for the geometry of the coefficient discontinuities

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

a review of some recent results on the output least squares formulation of parameter estimation problems abstract in this paper recent advances in the analysis of the output least squares formulation of parameter estimation problems for distributed parameter systems are summarized we address problems of identifiability discuss the notion of output least squares stability involving tikhonov regularization in a nonlinear environment give results on the approximation of these infinite dimensional problems by finite dimensional ones and review numerical techniques

residual based stabilized formulation for the solution of inverse elliptic partial differential equations abstract we consider an inverse problem where the forward problem is that of linear plane stress elasticity or equivalently that of linear heat hydraulic conduction we demonstrate that the linearized version of the saddle point problem obtained from the minimization problem inherits some stability from the forward elliptic problem in particular it is stable for the response variable and the lagrange multiplier but not for the material property field this lack of stability implies that we are unable to prove optimal convergence with mesh refinement for the overall problem we overcome this difficulty by adding to the saddle point problem a residual based term that provides sufficient stability and prove optimal convergence in an energy like norm we verify these estimates through simple numerical examples we note that while we have considered a specific model for an inverse elliptic problem in this manuscript similar ideas could be developed for a broad class of inverse elliptic problems

the estimation of l diffusion coefficients in the parabolic equations abstract the estimation of diffusion coefficients having only l bounds in parabolic equations is considered existence is obtained within a more general class of parabolic operators by means of g convergence the corresponding semi discrete problems are related to the original problem

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

a note on some languages in uniform acc 0 acc is the class of languages recognized by circuit families with polynomial size constant depth and unbounded fan in where gates may calculate the and or or mod c function for constant c robust uniformity definitions for acc and related classes were given by barrington immerman and straubing 3 here we show that uniform acc contains all semi linear or rational sets of integer vectors using binary notation sharpening a result of ibarra jiang ravikumar and chang 10

numerical methods for nonlinear inverse problems inverse problems of distributed parameter systems with applications to optimal control and identification are considered numerical methods and their numerical analysis for solving this kind of inverse problems are presented main emphasis being on the estimates of the rate of convergence for various schemes finally based on the given error estimates a two grid method and related algorithms are introduced which can be used to solve nonlinear inverse problems effectively

a c0 finite element method for an inverse problem a c 0 finite element method is presented for an inverse problem in which the coefficient in the differential operator is to be determined from the measurement of the solution of a boundary value problem the unknown in the inverse problem is approximated by a minimizer of a cost function that includes both the output error and equation error error estimates in a weighted h 1 norm and l 2 are given numerical examples are presented to show features of the method

an equation error approach for the elasticity imaging inverse problem for predicting tumor location the primary objective of this work is to study the elasticity imaging inverse problem of identifying cancerous tumors in the human body this nonlinear inverse problem not only represents an important and interesting application it also brings forth noteworthy mathematical challenges since the underlying model is a system of elasticity equations involving incompressibility due to the locking effect classical finite element methods are not effective for incompressible elasticity equations therefore special treatment is necessary for both the direct and inverse problems to study the inverse problem in an optimization framework we propose an extension of the equation error approach we focus on two cases namely when the material parameter is sufficiently smooth and when it is may be discontinuous for the latter case we extend the total variation regularization method to the elasticity imaging inverse problem we give the existence results for the proposed equation error approach and give the convergence analysis for the discretized problem we give sufficient details on the discrete formulas as well as on the implementation issues numerical examples for smooth and discontinuous coefficients are given

strange attractors across the boundary of hyperbolic systems we study a bifurcation of axiom a hyperbolic vector fields in dimension three leading to robust strange attractors with singularities the axiom a vector fields involved in the bifurcation exhibit a basic set equivalent to the suspension of a three symbol subshift the attractors arising from this kind of bifurcation are not equivalent to the geometric lorenz attractors

numerical identification of a spatially varying diffusion coefficient we consider the inverse problem of identifying a spatially varying diffusion coefficient on the basis of an observed solution to the forward problem under appropriate conditions this inverse problem can be solved as a first order hyperbolic problem in the unknown coefficient we provide a modified upwind difference scheme for this hyperbolic problem and prove that its convergence rate is 0 h when certain conditions are met

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

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

a priori error estimates for elliptic optimal control problems with a bilinear state equation in this paper a priori error analysis for the finite element discretization of an optimal control problem governed by an elliptic state equation is considered the control variable enters the state equation as a coefficient and is subject to pointwise inequality constraints we derive a priori error estimates for the discretization error in the control variable and confirm our theoretical results by numerical examples

descent gradient methods for nonsmooth minimization problems in ill posed problems descent gradient methods are the most frequently used algorithms for computing regularizers of inverse problems they are either directly applied to the discrepancy term which measures the difference between operator evaluation and data or to a regularized version incorporating suitable penalty terms in its basic form gradient descent methods converge slowly we aim at extending different optimization schemes which have been recently introduced for accelerating these approaches by addressing more general penalty terms in particular we use a general setting in infinite hilbert spaces and examine accelerated algorithms for regularization methods using total variation or sparsity constraints to illustrate the efficiency of these algorithms we apply them to a parameter identification problem in an elliptic partial differential equation using total variation regularization

a priori error estimates for the finite element discretization of elliptic parameter identification problems with pointwise measurements we develop an a priori error analysis for the finite element galerkin discretization of parameter identification problems the state equation is given by an elliptic partial differential equation of second order with a finite number of unknown parameters which are estimated using pointwise measurements of the state variable