H. E. Engl, M. Hanke, and A. Neubauer. Regularization of Inverse Problems. Kluwer Academic Publishers, 2000.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....parameters, and the solution and its derivatives the state variables. The forward problem is to compute the state variables given the system parameters and appropriate boundary conditions, which is a well posed problems. However in parameter identification, the problems is typically ill posed (see [5]) Partially supported by Chinese NSF grant 19731010 and the Knowledge Innovation Program of CAS For example, we consider the problem of identifying a distributed parameter q = q(x) in the one dimentional steady state diffusion equation in the form Gammar(qru) g; in (0; 1) 1) with ....

....F (q) Gammar(qr( Delta) Since u is the observation data, therefore, it may contain noise. Assume that the observed data can be expressed as u e = u e (3) with Gaussian noise e. Because of the ill posedness of the problem (1) some kind of regularization technique has to be applied (see [5, 13, 24]) Perhaps Tikhonov regularization method (see [9, 20] is the most well known method for dealing with such kind of problems. Given the regularization parameter ff 0, choose q 2 Q to solve the unconstrained minimization problem q2Q M [q] kF (q)u e Gamma gk ffkqk ; 4) where ....

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Engl H W, Hanke M and Neubauer 1996, Regularization of Inverse Problems, Dordrecht: Kluwer.

....There is a vast body of literature on regularization, much of which is devoted to determining the asymptotic behavior of the error in the optimally regularized solution as the error in the right hand side approaches zero. This literature has been admirably surveyed by Engl, Hanke, and Neubauer [4]. There is also a body of literature for well posed problems in which the asymptotic convergence of Krylov methods like MINRES are analyzed (see, for example, the recent book by Greenbaum [5] Our approach, which consists of leaving the error in b fixed and determining how the solution behaves ....

H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.

....(packet) thresholding method, which is a wavelet denoising method, is built into each iterative step so as to remove the noise from the original data. It also keeps the features of the original signal while denoising. In this sense, our method is more related to the Tikhonov least squares method [8] where a regularization operator is used to perturb the zeros of the convolution kernel and a penalty parameter is used to damp the highfrequency components for denoising. Since the least squares method penalizes the high frequency components of the original signal at the same rate as that of the ....

H. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.

....would be needed to deblur such an image. Significant computational resources would also be needed to estimate the unknown blur in each region. Furthermore, the nature of the blur function for highly defocused regions often leads to an unstable restoration problem in the presence of noise [4]. Dowski and Cathey [3] proposed the introduction of a cubic phase optical element at the pupil of an optical imaging system to present a more tractable problem for image restoration. The cubic phase approach allows a light efficient wide aperture implementation while creating a blur function ....

....are essential and direct methods are used. We suggest that in some applications, direct linear methods could be used to identify regions of interest in real time and more computationally intensive iterative algorithms could be subsequently applied in an attempt to improve restoration fidelity [4, 10]. A well designed incoherent imaging system can be treated as a linear shift invariant system. The image plane intensity can be expressed in terms of a convolution given by I(x; y) O(x; y) h(x; y) j(x; y) 1) where I(x; y) is the observed object intensity, O(x; y) is the unknown true ....

[Article contains additional citation context not shown here]

H. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers, Dordrecht, 1996.

....would be needed to deblur such an image. Significant computational resources would also be needed to estimate the unknown blur in each region. Furthermore, the nature of the blur function for highly defocused regions often leads to an unstable restoration problem in the presence of noise [4]. Dowski and Cathey [3] proposed the introduction of a cubic phase optical element at the pupil of an optical imaging system to present a more tractable problem for image restoration. The cubic phase approach allows a light efficient wide aperture implementation while creating a blur function ....

....are essential and direct methods are used. We suggest that in some applications, direct linear methods could be used to identify regions of interest in real time and more computationally intensive iterative algorithms could be subsequently applied in an attempt to improve restoration fidelity [4, 10]. A well designed incoherent imaging system can be treated as a linear shift invariant system. The image plane intensity can be expressed in terms of a convolution given by y) O(x,y) n(x, y) x, y) 1) where I(x, y) is the observed object intensity, O(x, y) is the unknown true object ....

[Article contains additional citation context not shown here]

H. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers, Dor- drecht, 1996.

....ill posed problems result in linear systems of the form: 1.1) g = Kf n, where K is a large ill conditioned matrix, n is a vector representing perturbations (such as noise) in the measured data, g, and the aim is to compute a good approximation to the unknown vector f. It is well known (cf. [5, 12]) that regularization is needed in order to avoid computing solutions that are corrupted by noise. Regularization can take many forms, such as Tikhonov regularization [9] truncated iterations [10] and truncated singular value decomposition [12] In this paper we assume an upper bound, A, on the ....

H. W. Engl, M. Hanke, and A. Neubauer. Regularization of Inverse Problems. Kluwer Academic Publishers, Dordrecht, 2000.

.... for parameter function estimation are based on an iteration within which the forward modeling problem is solved using the current parameters, the obtained solution is somehow used to update the parameters, and the process is repeated until the parameter function converges in some suitable norm [40]. For large scale inverse problems, inversion of the forward modeling problem within each iteration of the inverse problem generally requires some iterative algorithm in itself. Since, in principle, dozens if not hundreds of forward modeling solves are needed for a single data inversion, the ....

....basic geophysi1 cal inverse problem considered, then, is to construct a model of the conductivity profile under the terrain based on the knowledge of the applied electromagnetic sources and the measured responses. The corresponding mathematical problem is ill posed as is usual for inverse problems [40, 107]. The goal of this thesis is to derive fast solvers for time harmonic Maxwell s equations for low or moderate frequencies. Toward this goal, key hindrances in the forward problem are identified that point to analytic reformulations of the partial di#erential equations (PDEs) in terms of scalar ....

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H.W. Engl, M. Hanke, and A. Neubauer. Regularization of Inverse Problems. Kluwer, 1996.

....to expect noise in any measurement of currents and capacitances (or charge out ows) regularization methods have to be employed in order to compute a stable approximation of the solution. For a detailed discussion of this issue we refer to [2] In the simple case of Tikhonov regularization (cf. [10]) we have to minimize a functional of the form F full (C) j 1 I k j 1 Q kC C ; 12) where I and Q are the (noisy) measurements for current and charge out ow, C is an a priori guess for the doping pro le and is a positive real parameter. In the ....

Engl, H.W., Hanke, M., Neubauer, A., Regularization of Inverse Problems, Kluwer, Dordrecht, 1996.

....effectiveness of the procedures. 1 Introduction Let us consider the p Theta p system of linear equations Ax = b: If the matrix A is ill conditioned, a quite effective procedure is to regularize the system. Perhaps, the best known regularization method is due to Tikhonov [29, 30] see, also, [31, 32, 8]) which is based on the minimization of the quadratic functional J( x) kAx Gamma bk kHxk where 0 is a parameter, H a given q Theta p (q p) matrix and the symbol k Delta k denotes the Euclidean norm. Regularization consists of computing x = arg min J( x) Such a vector x is ....

H.W. Engl, M. Hanke, A. Neubauer, Regularization of Inverse Problems, Kluwer, Dordrecht, 1996.

....9 b (7) When the measurement locations are not on the finite element nodes, c is formulated as a interpolation matrix. Estimating physical motion parameters from displacement data is an ill posed problem due to the fact that physical parameters do not depend on the measurement data continuously [2]. To alleviate the ill posedness, a regularized output least squares function can be formulated by incorporating prior information about the parameter: T 2 ] 9 e c 2 ] 9 g b h e i j k l e n 2 ] g ] S 9 e i p (8) b h denotes measured displacement data, S represents prior knowledge about ....

....of gradient or Laplacian operators. A stable solution can be found by minimizing the objective function (8) using either an iterative method (quasi Newton) in the deterministic framework or using a search method in the stochastic framework such as the simulated annealing and the genetic algorithms [7, 2]. One of the difficulties in physical parameter recovery by minimizing the above regularization functional is that there exist large parameter discontinuities in the order of several magnitude, which tend to be over smoothed by the global smoothness constraint. In surface reconstruction [10] ....

H. W. Engl, M. Hanke, and A. Neubauer, "Regularization of inverse problems" Kluwer Academic Publishers, c1996.

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H. E. Engl, M. Hanke, and A. Neubauer. Regularization of Inverse Problems. Kluwer Academic Publishers, 2000.

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H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Kluwer, 1996.

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H.W. Engl, M. Hanke, and A. Neubauer. Regularization of Inverse Problems. Kluwer, 1996.

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Engl, H., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer Academic Publishers, Dordrecht (1996)

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H.W.Engl, M.Hanke, A.Neubauer, Regularization of Inverse Problems (Kluwer, Dordrecht, 1996).

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Engl H.W., Hanke M. and Neubauer A., Regularization of Inverse Problems, Kluwer Academic Publishers, 1996.

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H.W.Engl, M.Hanke, A.Neubauer, Regularization of Inverse Problems (Kluwer, Dordrecht, 1996).

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H.W.Engl, M.Hanke, A.Neubauer, Regularization of Inverse Problems (Kluwer, Dordrecht, 1996).

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H.W.Engl, M.Hanke, A Neubauer, Regularization of Inverse Problems (Kluwer, Dordrecht, 1996).

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H. Engl, M. Hanke, and A. Neubauer. Regularization of Inverse Problems. Kluwer Academic Publishers, Dordrecht, 1996.

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H.W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems (Kluwer, Dordrecht, 1996).

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H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.

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Engl H W, Hanke M, Neubauer et al. Regularization of Inverse Problems. Dordrecht: Kluwer, 1996.

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Engl H W, Hanke M and Neubauer 1996, Regularization of Inverse Problems, Dordrecht: Kluwer.

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H. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.

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