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33
Convergence of Spectra of Mesoscopic Systems Collapsing Onto a Graph.
 J. Math. Anal. Appl
, 1999
"... Let M be a finite graph in the plane and M " be a domain that looks like the "fattened graph M (exact conditions on the domain are given). It is shown that the spectrum of the Neumann Laplacian on M " converges when " ! 0 to the spectrum of an ODE problem on M . Presence of an electromagnetic f ..."
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Cited by 55 (2 self)
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Let M be a finite graph in the plane and M " be a domain that looks like the "fattened graph M (exact conditions on the domain are given). It is shown that the spectrum of the Neumann Laplacian on M " converges when " ! 0 to the spectrum of an ODE problem on M . Presence of an electromagnetic field is also allowed. Considerations of this kind arise naturally in mesoscopic physics and other areas of physics and chemistry. The results of the paper extend the ones previously obtained by J. Rubinstein and M. Schatzman. 2000 MSC: 35Q40, 35P15, 35J10, 81V99 Key words and phrases: mesoscopic system, Schrodinger operator, spectrum 1 Introduction In recent years one has witnessed growing interest in spectral theory of differential (versus difference) operators on graphs. Although probably one of the first such studies was done in physical chemistry [47], the main thrust 1 in this direction came from the mesoscopic physics [29]. Recent progress in nanotechnology and microelectronics en...
A Multigrid Method For Distributed Parameter Estimation Problems
 Trans. Numer. Anal
, 2001
"... . This paper considers problems of distributed parameter estimation from data measurements on solutions of partial differential equations (PDEs). A nonlinear least squares functional is minimized to approximately recover the sought parameter function (i.e., the model). This functional consists of a ..."
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Cited by 39 (13 self)
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. This paper considers problems of distributed parameter estimation from data measurements on solutions of partial differential equations (PDEs). A nonlinear least squares functional is minimized to approximately recover the sought parameter function (i.e., the model). This functional consists of a data fitting term, involving the solution of a finite volume or finite element discretization of the forward differential equation, and a Tikhonovtype regularization term, involving the discretization of a mix of model derivatives. We develop a multigrid method for the resulting constrained optimization problem. The method directly addresses the discretized PDE system which defines a critical point of the Lagrangian. The discretization is cellbased. This system is strongly coupled when the regularization parameter is small. Moreover, the compactness of the discretization scheme does not necessarily follow from compact discretizations of the forward model and of the regularization term. We therefore employ a Marquardttype modification on coarser grids. Alternatively, fewer grids are used and a preconditioned Krylovspace method is utilized on the coarsest grid. A collective point relaxation method (weighted Jacobi or a GaussSeidel variant) is used for smoothing. We demonstrate the efficiency of our method on a classical model problem from hydrology. 1.
On effective methods for implicit piecewise smooth surface recovery
 SIAM J. Scient. Comput
"... Abstract. This paper considers the problem of reconstructing a piecewise smooth model function from given, measured data. The data are compared to a field which is given as a possibly nonlinear function of the model. A regularization functional is added which incorporates the a priori knowledge that ..."
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Cited by 32 (24 self)
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Abstract. This paper considers the problem of reconstructing a piecewise smooth model function from given, measured data. The data are compared to a field which is given as a possibly nonlinear function of the model. A regularization functional is added which incorporates the a priori knowledge that the model function is piecewise smooth and may contain jump discontinuities. Regularization operators related to total variation (TV) are therefore employed. Two popular methods are modified TV and Huber’s function. Both contain a parameter which must be selected. The Huber variant provides a more natural approach for selecting its parameter, and we use this to propose a scheme for both methods. Our selected parameter depends both on the resolution and on the model average roughness; thus, it is determined adaptively. Its variation from one iteration to the next yields additional information about the progress of the regularization process. The modified TV operator has a smoother generating function; nonetheless we obtain a Huber variant with comparable, and occasionally better, performance. For large problems (e.g., high resolution) the resulting reconstruction algorithms can be tediously slow. We propose two mechanisms to improve efficiency. The first is a multilevel continuation approach aimed mainly at obtaining a cheap yet good estimate for the regularization parameter and the solution. The second is a special multigrid preconditioner for the conjugate gradient algorithm used to solve the linearized systems encountered in the procedures for recovering the model function.
Preconditioned AllAtOnce Methods for Large, Sparse Parameter Estimation Problems
, 2000
"... The problem of recovering a parameter function based on measurements of solutions of a system of partial differential equations in several space variables leads to a number of computational challenges. Upon discretization of a regularized formulation a large, sparse constrained optimization prob ..."
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Cited by 24 (4 self)
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The problem of recovering a parameter function based on measurements of solutions of a system of partial differential equations in several space variables leads to a number of computational challenges. Upon discretization of a regularized formulation a large, sparse constrained optimization problem is obtained. Typically in the literature, the constraints are eliminated and the resulting unconstrained formulation is solved by some variant of Newton's method, usually the GaussNewton method. A preconditioned conjugate gradient algorithm is applied at each iteration for the resulting reduced Hessian system. In this paper we apply instead a preconditioned Krylov method directly to the KKT system arising from a Newtontype method for the constrained formulation (an "allatonce" approach). A variant of symmetric QMR is employed, and an effective preconditioner is obtained by solving the reduced Hessian system approximately. Since the reduced Hessian system presents significa...
A framework for the adaptive finite element solution of large inverse problems. I. Basic techniques
, 2004
"... Abstract. Since problems involving the estimation of distributed coefficients in partial differential equations are numerically very challenging, efficient methods are indispensable. In this paper, we will introduce a framework for the efficient solution of such problems. This comprises the use of a ..."
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Cited by 14 (7 self)
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Abstract. Since problems involving the estimation of distributed coefficients in partial differential equations are numerically very challenging, efficient methods are indispensable. In this paper, we will introduce a framework for the efficient solution of such problems. This comprises the use of adaptive finite element schemes, solvers for the large linear systems arising from discretization, and methods to treat additional information in the form of inequality constraints on the parameter to be recovered. The methods to be developed will be based on an allatonce approach, in which the inverse problem is solved through a Lagrangian formulation. The main feature of the paper is the use of a continuous (function space) setting to formulate algorithms, in order to allow for discretizations that are adaptively refined as nonlinear iterations proceed. This entails that steps such as the description of a Newton step or a line search are first formulated on continuous functions and only then evaluated for discrete functions. On the other hand, this approach avoids the dependence of finite dimensional norms on the mesh size, making individual steps of the algorithm comparable even if they used differently refined meshes. Numerical examples will demonstrate the applicability and efficiency of the method for problems with several million unknowns and more than 10,000 parameters. Key words. Adaptive finite elements, inverse problems, Newton method on function spaces. AMS subject classifications. 65N21,65K10,35R30,49M15,65N50 1. Introduction. Parameter
Matching Pursuit for Imaging High Contrast Conductivity
, 1999
"... We show that imaging an isotropic, high contrast conducting medium is asymptotically equivalent to the identification of a unique resistor network, given measurements of currents and voltages at the boundary. We show that a matching pursuit approach can be used effectively towards the numerical solu ..."
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Cited by 8 (4 self)
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We show that imaging an isotropic, high contrast conducting medium is asymptotically equivalent to the identification of a unique resistor network, given measurements of currents and voltages at the boundary. We show that a matching pursuit approach can be used effectively towards the numerical solution of the high contrast imaging problem, if the library of functions is constructed carefully and in accordance with the asymptotic theory. We also show how other libraries of functions that at first glance seem reasonable, in fact, do not work well. When the contrast in the conductivity is not so high, we show that wavelets can be used, especially nonorthogonal wavelet libraries. However, the library of functions that is based on the high contrast asymptotic theory is more robust, even for intermediate contrasts, and especially so in the presence of noise. Key words. Impedance tomography, high contrast, asymptotic resistor network, imaging. Contents 1 Introduction 1 2 The Neumann to Dir...
Inverse problems for nonsmooth first order perturbations of the Laplacian
, 2004
"... We consider inverse boundary value problems in Rn, n ≥ 3, for operators which may be written as first order perturbations of the Laplacian. The purpose is to obtain global uniqueness theorems for such problems when the coefficients are nonsmooth. We use complex geometrical optics solutions of Sylves ..."
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Cited by 8 (4 self)
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We consider inverse boundary value problems in Rn, n ≥ 3, for operators which may be written as first order perturbations of the Laplacian. The purpose is to obtain global uniqueness theorems for such problems when the coefficients are nonsmooth. We use complex geometrical optics solutions of SylvesterUhlmann type to achieve this. A main tool is an extension of the NakamuraUhlmann intertwining method to operators which have continuous coefficients. For the inverse conductivity problem for a C 1+ε conductivity, we construct complex geometrical optics solutions whose properties depend explicitly on ε. This implies the uniqueness result of PäivärintaPanchenkoUhlmann for C 3/2 conductivities. For the magnetic Schrödinger equation, the result is that the DirichlettoNeumann map uniquely determines the magnetic field corresponding to a Dini continuous magnetic potential in C 1,1 domains. For the steady state heat equation with a convection term, we obtain global uniqueness of Lipschitz continuous convection terms in Lipschitz
On the Parametrization of Illposed Inverse Problems Arising from Elliptic Partial Differential Equations
, 2006
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INVISIBILITY AND INVERSE PROBLEMS
"... Abstract. We describe recent theoretical and experimental progress on making objects invisible. Ideas for devices that would have once seemed fanciful may now be at least approximately realized physically, using a new class of artificially structured materials, metamaterials. The equations that gove ..."
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Cited by 6 (4 self)
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Abstract. We describe recent theoretical and experimental progress on making objects invisible. Ideas for devices that would have once seemed fanciful may now be at least approximately realized physically, using a new class of artificially structured materials, metamaterials. The equations that govern a variety of wave phenomena, including electrostatics, electromagnetism, acoustics and quantum mechanics, have transformation laws under changes of variables which allow one to design material parameters that steer waves around a hidden region, returning them to their original path on the far side. Not only are observers unaware of the contents of the hidden region, they are not even aware that something is being hidden; the object, which casts no shadow, is said to be cloaked. Proposals for, and even experimental implementations of, such cloaking devices have received the most attention, but other devices having striking effects on wave propagation, unseen in nature, are also possible. These designs are initially based on the transformation laws of the relevant PDEs, but due to the singular transformations needed for the desired effects, care needs to be taken in formulating and analyzing physically meaningful solutions. We recount the recent history of the subject and discuss some of the mathematical and physical issues involved. 1.
Circular resistor networks for electrical impedance tomography with partial boundary measurements.
"... Abstract. We introduce an algorithm for the numerical solution of electrical impedance tomography (EIT) in two dimensions, with partial boundary measurements. The algorithm is an extension of the one in [11, 49] for EIT with full boundary measurements. It is based on resistor networks that arise in ..."
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Cited by 5 (4 self)
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Abstract. We introduce an algorithm for the numerical solution of electrical impedance tomography (EIT) in two dimensions, with partial boundary measurements. The algorithm is an extension of the one in [11, 49] for EIT with full boundary measurements. It is based on resistor networks that arise in finite volume discretizations of the elliptic partial differential equation for the potential, on socalled optimal grids that are computed as part of the problem. The grids are adaptively refined near the boundary, where we measure and expect better resolution of the images. They can be used very efficiently in inversion, by defining a reconstruction mapping that is an approximate inverse of the forward map, and acts therefore as a preconditioner in any iterative scheme that solves the inverse problem via optimization. The main result in this paper is the construction of optimal grids for EIT with partial measurements by extremal quasiconformal (Teichmüller) transformations of the optimal grids for EIT with full boundary measurements. We present the algorithm for computing the reconstruction mapping on such grids, and we illustrate its performance with numerical simulations. The results show an interesting tradeoff between the resolution of the reconstruction in the domain of the solution and distortions due to artificial anisotropy induced by the distribution of the measurement points on the accessible boundary. 1.