stability analysis in magnetic resonance elastography ii abstract we consider the inverse problem of finding unknown elastic parameters from internal measurements of displacement fields for tissues in the sequel to 5 we use pseudodifferential methods for the problem of recovering the shear modulus for stokes systems from internal data we prove stability estimates in d 2 3 with reduced regularity on the estimates and show that the presence of a finite dimensional kernel can be removed this implies the convergence of the landweber numerical iteration scheme we also show that these hypotheses are natural for experimental use in constructing shear modulus distributions

iterative regularization for elliptic inverse problems elliptic inverse problems can be formulated using coefficient dependent energy least squares functionals resulting in a smooth convex objective functional a variational inequality emerges as a necessary and sufficient optimality condition the principle of iterative regularization when coupled with the auxiliary problem principle results in a strongly convergent scheme for the solution of elliptic inverse problems

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

incremental identification of transport coefficients in convection diffusion systems in this paper an incremental approach for the identification of a model for transport coefficients in convection diffusion systems on the basis of high resolution measurement data is presented the transport is represented by a convection term with known convective velocity and by a diffusion term with an unknown generally state dependent transport coefficient the identification of the transport model for this transport coefficient constitutes an ill posed nonlinear inverse problem we present a novel decomposition approach in which this inverse problem is split into a sequence of inverse subproblems in the first identification step of this incremental approach a source is estimated by solving an affine linear inverse problem by means of the conjugate gradient method in the second identification step a nonlinear inverse problem has to be solved to reconstruct a transport coefficient a newton type method using the conjugate gradient method in its inner iteration is used to solve this nonlinear inverse problem of coefficient estimation finally in the third identification step a transport model structure is proposed and identified on the basis of the model free transport coefficient reconstructed in the two previous steps the ill posedness of each inverse problem is examined by using artificially perturbed transient simulation data and appropriate regularization techniques the identification methodology is illustrated for a three dimensional convection diffusion equation which has its origin in the modeling and simulation of energy transport in a laminar wavy film flow

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

critical points for elliptic equations with prescribed boundary conditions this paper concerns the existence of critical points for solutions to second order elliptic equations of the form nabla cdot sigma x nabla u 0 posed on a bounded domain x with prescribed boundary conditions in spatial dimension n 2 it is known that the number of critical points where nabla u 0 is related to the number of oscillations of the boundary condition independently of the positive coefficient sigma we show that the situation is different in dimension n geq3 more precisely we obtain that for any fixed dirichlet or neumann boundary condition for u on partial x there exists an open set of smooth coefficients sigma x such that nabla u vanishes at least at one point in x by using estimates related to the laplacian with mixed boundary conditions the result is first obtained for a piecewise constant conductivity with infinite contrast a problem of independent interest a second step shows that the topology of the vector field nabla u on a subdomain is not modified for appropriate bounded sufficiently high contrast smooth coefficients sigma x these results find applications in the class of hybrid inverse problems where optimal stability estimates for parameter reconstruction are obtained in the absence of critical points our results show that for any finite number of prescribed boundary conditions there are coefficients sigma x for which the stability of the reconstructions will inevitably degrade

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

error analysis of an equation error method for the identification of the diffusion coefficient in a quasi linear parabolic differential equation we consider the inverse problem of reconstructing the diffusion coefficient in a quasi linear parabolic differential equation in divergence form from measurements of the solution at a finite number of points in the interior of the domain an equation error method is developed which transforms the inverse problem into a system of linear operator equations for the diffusion coefficient and which can be solved by the conjugate gradient method in a very efficient and stable manner a detailed error analysis relates the required number of measurements with their accuracy numerical results illustrate the performance of the method

eeg mec error bounds for a static dipole source with a realistic head model we derive cramer rao bounds crbs on the errors of estimating the parameters location and moment of a static current dipole source using data from electro encephalography eeg magneto encephalography meg or the combined eeg meg modality we use a realistic head model based on knowledge of surfaces separating tissues of different conductivities obtained from magnetic resonance mr or computer tomography ct imaging systems the electric potentials and magnetic field components at the respective sensors are functions of the source parameters through integral equations these potentials and field are formulated for solving them by the boundary or the finite element method bem or fem with a weighted residuals technique we present a unified framework for the measurements computed by these methods that enables the derivation of the bounds the resulting bounds may be used for instance to choose the best configuration of the sensors for a given patient and region of expected source location numerical results are used to demonstrate an application for showing expected accuracies in estimating the source parameters as a function of its position in the brain based on real eeg meg system and mr or ct images

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

identification of transmissivity coefficients by mollification techniques part ii n dimensional elliptic problems abstract we analyze the numerical identification of the transmissivity coefficient in the n dimensional model for linear elliptic equations this problem arises in a number of important practical problems in science the identification of this coefficient is an inverse problem which is highly ill posed and difficult to solve in order to address the ill posedness of this parameter estimation problem we use mollification techniques combined with finite differences to get a numerical method which is easy to implement in rectangular domains some numerical examples of interest for n 2 are presented

on finite element subspaces over quadrilateral and hexahedral meshes for incompressible viscous flow problems optimal rates of convergence for the approximate solution of the stationary stokes equations are obtained for finite element schemes which use piecewise constants to approximate the pressure and piecewise linear or piecewise bilinear or trilinear polynomials to approximate the velocity over fairly general quadrilateral or hexahedral meshes

distribution of lattice points visible from the origin let be a region in the plane which contains the origin is star shaped with respect to the origin and has a piecewise c 1 boundary for each integer q 1 we consider the integer lattice points from which are visible from the origin and prove that the 1 st consecutive spacing distribution of the angles formed with the origin exists this is a probability measure supported on an interval m with m 0 its repartition function is explicitly expressed as the convolution between the square of the distance from origin function and a certain kernel

inverse problems for heat equation and space time fractional diffusion equation with one measurement abstract given a connected compact riemannian manifold m g without boundary dim m 2 we consider a space time fractional diffusion equation with an interior source that is supported on an open subset v of the manifold the time fractional part of the equation is given by the caputo derivative of order 0 1 and the space fractional part by g where 0 1 and g is the laplace beltrami operator on the manifold the case 1 which corresponds to the standard heat equation on the manifold is an important special case we construct a specific source such that measuring the evolution of the corresponding solution on v determines the manifold up to a riemannian isometry

the determination of the leading coefficient in the monotone potential sturm liouville operator from boundary measurements this paper deals with the problem of determining the leading coefficient k k u 2 of the nonlinear monotone potential sturm liouville operator au k u 2 u x q x u x x a b as an additional condition only two measured data at the boundary x a x b are used solvability and linearization of the corresponding nonlinear direct problem are given an existence of a quasi solution of the inverse problem is obtained in a suitable compact class of admissible coefficients in the second part of the paper an approximate analytical solution for the inverse problem is derived the approach presented permits to analyze well posed as well as all ill posed situations for the inverse coefficient problem numerical examples corresponding to the all considered situations are presented

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

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

a new approach for numerical identification of conductivity this paper studies the parameter estimation problem for a steady state flow in an inhomogeneous medium for identifying the spatially varying diffusion coefficient from an observed solution to the forward problem we propose a direct method using local green s function technique this could be used when the diffusion coefficient is discontinuous the convergence order is calculated and numerical simulations are performed

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

asymptotic behavior of matrix coefficients of admissible representations introduction 869 1 generalities on reductive groups 873 2 the infinitesimal cartan decomposition 876 3 the t radial components 880 4 differential equations satisfied by sphereical functions 885 5 asymptotic behavior of spherical functions on a 888 6 asymptotic behavior of spherical functions along the walls of a 892 7 leading characters and growth estimates on the group 898 8 admissible representations and their matrix coefficients 906 appendix 917

inversion formulas for the linearized problem for an inverse boundary value problem in elastic prospection the problem of determining an unknown pair gamma lambda mu of lame parameters inside a body by its dirichlet to neumann data map lambda gamma is considered using explicit exact solutions for the case of constant gamma it is seen that the deflection h between gamma h and gamma is uniquely determined by the first order approximation of lambda gamma h at gamma

error analysis of a multisingular inverse coefficient problem for the sturm liouville operator based on boundary measurement in this paper we study the inverse problem of determining the unknown leading coefficient k k x of the linear sturm liouville operator au k x u x q x u x x 0 1 from boundary measurements when u x or and u x vanishes at several called singular points of the interval 0 1 as a result the considered inverse problem has simultaneously different types moderate or severely of ill conditioned situations in different parts of the interval 0 1 the presented inverse polynomial method permits use of a priori information about singular points either to increase the order of the polynomial approximation in each subinterval or to obtain an artificial cauchy data for the unknown coefficient error estimations for the polynomial approximations are presented for well conditioned as well as for ill conditioned situations the behaviour of the inverse problem solution with respect to both types dirichlet and neumann of noisy data is analyzed the obtained results are illustrated by various numerical examples

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

blowing up and desingularizing constant scalar curvature kahler manifolds this paper is concerned with the existence of constant scalar curvature kahler metrics on blow ups at finitely many points of compact manifolds which already carry constant scalar curvature kahler metrics we also consider the desingularization of isolated quotient singularities of compact orbifolds which carry constant scalar curvature kahler metrics

boundary control of elliptic solutions to enforce local constraints abstract we present a constructive method to devise boundary conditions for solutions of second order elliptic equations so that these solutions satisfy specific qualitative properties such as i the norm of the gradient of one solution is bounded from below by a positive constant in the vicinity of a finite number of prescribed points ii the determinant of gradients of n solutions is bounded from below in the vicinity of a finite number of prescribed points such constructions find applications in recent hybrid medical imaging modalities the methodology is based on starting from a controlled setting in which the constraints are satisfied and continuously modifying the coefficients in the second order elliptic equation the boundary condition is evolved by solving an ordinary differential equation ode defined via appropriate optimality conditions unique continuations and standard regularity results for elliptic equations are used to show that the ode admits a solution for sufficiently long times

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

compact self dual manifolds with positive scalar curvature on etudie l existence d une metrique auto duale a courbure scalaire positive sur une variete compacte simplement connexe qui n est pas conformement equivalente a ces metriques standard sur s 4 ou cp 2

the uniqueness of inverse problems for a fractional equation with a single measurement this article is concerned with an inverse problem on simultaneously determining some unknown coefficients and or an order of derivative in a multidimensional time fractional evolution equation either in a euclidean domain or on a riemannian manifold based on a special choice of the dirichlet boundary input we prove the unique recovery of at most two out of four varvec x dependent coefficients possibly with an extra unknown fractional order by a single measurement of the partial neumann boundary output especially both a vector valued velocity field of a convection term and a density can also be uniquely determined the key ingredient turns out to be the time analyticity of the decomposed solution which enables the construction of dirichlet to neumann maps in the frequency domain and thus the application of inverse spectral results

identifiablity under approximation for an elliptic boundary value problem necessary and sufficient conditions for identifiability of the diffusion coefficient in galerkin approximations to a two point boundary value problem are derived for various choices of galerkin subspaces the results are further used to investigate output least squares identifiability and output least squares stability of the diffusion coefficient

determination of leading coefficients in sturm liouville operator from boundary measurements ii unicity and an engineering approach we study the problem of determining the leading coefficient k k x of the sturm liouville operator au k x u x q x u x x a b from a measured data given in the form of dirichlet or and neumann type additional boundary conditions the contribution of the each measured data to the unicity of the inverse problem solution is analysed based on the analysis the problem is treated as a cauchy problem for the first order ordinary differential equation with respect to k x for the case when the direct problem solution u u x belongs to the class of the second order polynomials and u x 0 we present an analytical formula for an approximate solution of the inverse problem numerical examples related to accuracy of the solution and to essential features of the inverse problem are demonstrated

inherent identifiability of parameters in elliptic differential equations together with appropriate boundary conditions for u on the boundary r of q for given a and f this is a well known equation in the unknown u we shall consider the inverse problem which givenf consists in determining a from u more precisely we study the question of uniqueness and continuous dependence of a in terms of u the uniqueness problem is generally referred to as the identifiability problem well known contributions are the borg and the gelfand levitan theories which require knowledge of spectral data of 1 1 see 5 61 for recent work an identifiability of a from neumann boundary data for all g in the elliptic equation v avu o 1 f g we refer to s the problem of continuous dependence of a with respect to u is of equal importance as identifiability but has received much less attention so far in 121 the study of the inverse problem associated with 1 1 is based on the observation that 1 1 is a hyperbolic equation in a employing the method of characteristics a continuous dependence analysis is given under combinations of the assumptions vul 0 and ldul 0 in this paper we concentrate on uniqueness and continuous dependence results without a priori assumptions on the sign of vul the basic and very simple idea for which we give an analytical justification is that a in 1 1

the asymptotic distribution of lattice points in euclidean and non euclidean spaces the counting numbers for discrete subgroups of motions in euclidean and non euclidean spaces are obtained using the wave equation as the principal tool in dimensions 2 and 3 the error estimates are close to the best known

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

a full field image conversion method for the inverse conductivity problem with internal measurements this article investigates a fourier based algorithm for computing heterogeneous material parameter distributions from internal measurements of physical fields within the framework of the periodic scalar conductivity model a pair of dual lippmann schwinger integral equations is derived for the sought constitutive parameters based on full intensity or current density field measurements a numerical method based on the fast fourier transform and fixed point iterations is proposed convergence stability and approximation quality of the method are analysed for materials with small contrast a first order born like approximation is also obtained overall the proposed reconstruction approach enables a direct conversion of full field measurement images possibly noisy into maps of material conductivity a set of numerical results is presented to illustrate the performance of the method

identification of q x in u_t delta u qu from boundary observations we consider the problem of recovering the coefficient q x in the equation u t delta u qu from boundary observations uniqueness of q based on knowledge of the neumann mapsto dirichlet response operator is shown as an implication of known corresponding results concerning the inverse problem for the corresponding hyperbolic equation w tt delta w qw this is then reduced to use of the response to a single input with some consideration of computational approximation

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

recovery of multiple coefficients in a reaction diffusion equation abstract this paper considers the inverse problem of recovering both the unknown spatially dependent conductivity a x and the potential q x in a parabolic equation from overposed data consisting of the value of solution profiles taken at a later time t we show both uniqueness results and the convergence of an iteration scheme designed to recover these coefficients we also allow a more general setting in particular when the usual time derivative is replaced by one of fractional order and when the potential term is coupled with a known nonlinearity f of the form q x f u

a wavelet multiscale method for the inverse problem of a nonlinear convection diffusion equation abstract this paper is concerned with the problem of identifying the diffusion parameters in a nonlinear convection diffusion equation which arises as the saturation equation in the fractional flow formulation of the two phase porous media flow equations the forward problem is discretized using finite difference methods and the inverse problem is formulated as a minimization problem with regularization terms in order to overcome disturbance of local minimum a wavelet multiscale method is applied to solve this parameter identification inverse problem this method works by decomposing the inverse problem into multiple scales with wavelet transform so that the original inverse problem is reformulated to a set of sub inverse problems relying on scale variables and successively solving these sub inverse problems according to the size of scale from the smallest to the largest the stable and fast regularized gauss newton method is applied to each scale numerical simulations show that the proposed algorithm is widely convergent computationally efficient and has the anti noise and de noising abilities

unstable polarized del pezzo surfaces we provide new examples of k unstable polarized smooth del pezzo surfaces using a flopped version first used by cheltsov and rubinstein of the test configurations introduced by ross and thomas as an application we provide new obstructions for the existence of constant scalar curvature kahler metrics on polarized smooth del pezzo surfaces

radial rapid decay property for cocompact lattices abstract we study haagerup inequality for radial functions on uniform lattices in semisimple lie groups with respect to riemannian metrics and in some case to word metrics in particular we extend the swiatkowski valette results to any lattice acting properly and essentially transitively on classical buildings

stability analysis for magnetic resonance elastography we consider the inverse problem of finding unknown elastic parameters from internal measurements of displacement fields for tissues the measurements are made on the entirety of a smooth domain since tissues can be modeled as quasi incompressible fluids we examine the stokes system and consider only the recovery of shear modulus distributions our main result is to establish lipschitz stable estimates on the shear modulus distributions from internal measurements of displacement fields these estimates imply convergence of a numerical scheme known as the landweber iteration scheme for reconstructing the shear modulus distributions

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

on multiple frequency power density measurements ii the full maxwell s equations abstract we shall give conditions on the illuminations i such that the solutions to maxwell s equations curl e i i h i in curl h i i e i e i in e i i on satisfy certain non zero qualitative properties inside the domain provided that a finite number of frequencies are chosen in a fixed range the illuminations are explicitly constructed this theory finds applications in several hybrid imaging problems where unknown parameters have to be imaged from internal measurements some of these examples are discussed this paper naturally extends a previous work of the author 2013 7 where the helmholtz equation was studied

on the correlations of directions in the euclidean plane let r v x y q denote the repartition of the v level correlation measure of the finite set of directions p x y p where p x y is the fixed point x y 0 1 2 and p is an integer lattice point in the square q q 2 we show that the average of the pair correlation repartition r 2 x y q over x y in a fixed disc d 0 converges as q oo more precisely we prove for every a r and 0 1 10 the estimate formula math we also prove that for each individual point x y 0 1 2 the 6 level correlation r 6 x y q diverges at any point a r 5 as q and we give an explicit lower bound for the rate of divergence

the distribution of free path lengths in the periodic lorentz gas and related lattice point problems the periodic lorentz gas describes the dynamics of a point particle in a periodic array of spherical scatterers and is one of the fundamental models for chaotic diffusion in the present paper we investigate the boltzmann grad limit where the radius of each scatterer tends to zero and prove the existence of a limiting distribution for the free path length we also discuss related problems such as the statistical distribution of directions of lattice points that are visible from a fixed position

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

parameter identification by regularization for surface representation via the moving grid approach we consider the identification of a diffusion parameter in a second order elliptic equation in two dimensions by interior measurements the diffusion parameter is assumed to have discontinuities for its reconstruction we propose regularization algorithms with an adaptive grid the grid is adapted according to a measure of the smoothness of the regularized solution for the numerical computation we compare several iterative methods such as the minimal error method the steepest descent method and an inexact iteratively regularized gauss newton method the computations show that these algorithms can effectively identify the discontinuities

parameter estimation by spectral approximation in this paper chebyshev and legendre approximations are proposed for estimating parameters in differential equations which are easy to be performed the convergence and the spectral accuracy are proved even without some conditions as imposed in other papers the numerical results show the advantages of this new approach

on the inverse problem of identifying lame coefficients in linear elasticity an output least squares type functional is employed to identify the lame parameters in linear elasticity to be able to identify even the discontinuous lame parameters the regularization is performed by the bv seminorm finite element discretization is used and convergence analysis is given numerical examples are given to show the feasibility of the approach

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

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

an inverse polynomial method for the identification of the leading coefficient in the sturm liouville operator from boundary measurements an inverse polynomial method of determining the unknown leading coefficient k k x of the linear sturm liouville operator au k x u x q x u x x 0 1 is presented as an additional condition only two measured data at the boundary x 0 x 1 are used in absence of a singular point u x 0 u x 0 x 0 1 the inverse problem is classified as a well conditioned if there exists at least one singular point then the inverse problem is classified as moderately ill conditioned u x 0 0 x 0 0 1 u x 0 x x 0 u x 0 x 0 1 and severely ill conditioned u x 0 u x 0 0 x 0 0 1 u x 0 u x 0 x x 0 for each of the cases direct problem solution is approximated by corresponding polynomials and the inverse problem is reformulated as a cauchy problem for to the first order differential equation with respect the unknown function k k x an approximate analytical solution of the each cauchy problems are derived in explicit form numerical simulations all the above cases are given for noise free and noisy data an accuracy of the presented approach is demonstrated on numerical test solutions

conformal duality and compact complex surfaces on etudie les structures demi conformement plates sur une variete de riemann a 4 dimensions m g avec l hypothese supplementaire que m est complexe

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

reconstruction of coefficients in scalar second order elliptic equations from knowledge of their solutions this paper concerns the reconstruction of possibly complex valued coefficients in a second order scalar elliptic equation that is posed on a bounded domain from knowledge of several solutions of that equation we show that for a sufficiently large number of solutions and for an open set of corresponding boundary conditions all coefficients can be uniquely and stably reconstructed up to a well characterized gauge transformation we also show that in some specific situations a minimum number of such available solutions equal to is sufficient to uniquely and globally reconstruct the unknown coefficients this theory finds applications in several coupled physics medical imaging modalities including photo acoustic tomography transient elastography and magnetic resonance elastography 2013 wiley periodicals inc

reconstruction of a fully anisotropic elasticity tensor from knowledge of displacement fields we present explicit reconstruction algorithms for fully anisotropic unknown elasticity tensors from knowledge of a finite number of internal displacement fields with applications to transient elastography under certain rank maximality assumptions satisfied by the strain fields explicit algebraic reconstruction formulas are provided a discussion ensues on how to fulfill these assumptions describing the range of validity of the approach we also show how the general method can be applied to more specific cases such as the transversely isotropic one

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

sparse equidistribution problems period bounds and subconvexity we introduce a geometric method to bound periods of automorphic forms the key features of this method are the use of equidistribution results in place of mean value theorems and the systematic use of mixing and the spectral gap applications are given to equidistribution of sparse subsets of horocycles and to equidistribution of cm points to subconvexity of the triple product period in the level aspect over number fields which implies subconvexity for certain standard and rankin selberg l functions and to bounding fourier coefficients of automorphic forms

perturbation method for abstract second order inverse problems abstract we are concerned with an abstract second order inverse problem we restrict ourselves here to abstract inverse problems in which in order to solve the differential equation for u t it is necessary to know the value of some operator or functional u t g t as a function of the argument t a new approach based on the perturbation theory of linear operators is introduced among the results obtained we point out sufficient conditions of existence and uniqueness of a solution as a possible application of the abstract theorems some applications from elasticity theory are given

on the pair correlation density for hyperbolic angles we prove a conjecture of boca popa zaharescu on the density function for the pair correlation of hyperbolic angles in a growing orbit of a discrete lattice in psl2 r

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