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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 ..."
Abstract

Cited by 40 (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.
A Multigrid Approach For Minimizing A Nonlinear Functional For Digital Image Matching
 Computing
, 2000
"... In this paper, we consider a multigrid application in digital image processing. Here, the problem is to find a map, which transforms an image T into another image R such that the grey level of the different images are nearly equal in every pictureelement. This problem arises in the investigation of ..."
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Cited by 13 (2 self)
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In this paper, we consider a multigrid application in digital image processing. Here, the problem is to find a map, which transforms an image T into another image R such that the grey level of the different images are nearly equal in every pictureelement. This problem arises in the investigation of human brains. The complete inverse problem is ill posed in the sense of Hadamard and nonlinear, so the numerical solution is quite difficult. We solve the inverse problem by a Landweber iteration. In each minimization step an approximate solution for the linearized problem is computed with a multigrid method as an inner iteration. Finally, we present some experimental results for synthetic and real images.
TOWARDS AN ALGEBRAIC MULTIGRID METHOD FOR TOMOGRAPHIC IMAGE RECONSTRUCTION – IMPROVING CONVERGENCE OF ART
"... Abstract. In this paper we introduce a multigrid method for sparse, possibly rankdeficient and inconsistent least squares problems arising in the context of tomographic image reconstruction. The key idea is to construct a suitable AMG method using the Kaczmarz algorithm as smoother. We first present ..."
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Abstract. In this paper we introduce a multigrid method for sparse, possibly rankdeficient and inconsistent least squares problems arising in the context of tomographic image reconstruction. The key idea is to construct a suitable AMG method using the Kaczmarz algorithm as smoother. We first present some theoretical results about the correction step and then show by our numerical experiments that we are able to reduce the computational time to achieve the same accuracy by using the multigrid method instead of the standard Kaczmarz algorithm. 1
Adaptive Grid Optical Tomography
"... Imagebased modeling of semitransparent, dynamic phenomena is a challenging task. We present an optical tomography method that uses an adaptive grid for the reconstruction of a threedimensional density function from its projections. The proposed method is applied to reconstruct thin smoke and flam ..."
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Imagebased modeling of semitransparent, dynamic phenomena is a challenging task. We present an optical tomography method that uses an adaptive grid for the reconstruction of a threedimensional density function from its projections. The proposed method is applied to reconstruct thin smoke and flames volumetrically from synchronized multivideo recordings. Our adaptive reconstruction algorithm computes a timevarying volumetric model, that enables the photorealistical rendering of the recorded phenomena from arbitrary viewpoints. In contrast to previous approaches we sample the underlying unknown, threedimensional density function adaptively which enables us to achieve a higher effective resolution of the reconstructed models. Categories and Subject Descriptors (according to ACM CCS):
Summary
"... We present two different examples of image applications where exploiting structured matrices seems attractive, either for reducing the high complexity of the reconstruction problem or because a shift invariance naturally occurs in the mathematical description. We discuss how structures can be incorp ..."
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We present two different examples of image applications where exploiting structured matrices seems attractive, either for reducing the high complexity of the reconstruction problem or because a shift invariance naturally occurs in the mathematical description. We discuss how structures can be incorporated in the models and we point out some new open problems arising in these contexts. 1
Multigrid Tomographic Inversion with Variable Resolution Data and Image Spaces
"... A multigrid inversion approach that uses variable resolutions of both data space and image space is proposed. Since computational complexity of inverse problems typically increases with a larger number of unknown image pixels and a larger number of measurements, the proposed algorithm further reduce ..."
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A multigrid inversion approach that uses variable resolutions of both data space and image space is proposed. Since computational complexity of inverse problems typically increases with a larger number of unknown image pixels and a larger number of measurements, the proposed algorithm further reduces the computation relative to conventional multigrid approaches, which change only the image space resolution at coarse scales. The advantage is particularly important for datarich applications, where data resolutions may differ for different scales. Applications of the approach to Bayesian reconstruction algorithms in transmission and emission tomography with a generalized Gaussian Markov random field image prior are presented, both with a Poisson noise model and with a quadratic data term. Simulation results indicate that the proposed multigrid approach results in significant improvement in convergence speed compared to the fixedgrid iterative coordinate descent (ICD) method and a multigrid method with fixed data resolution. Index Terms Multigrid algorithms, multiresolution, inverse problems, image reconstruction, computed tomography,
Scaling and Gradual Refinement in Grid and Regularization Parameters for Nonlinear Inverse Problems
, 2000
"... This paper considers problems of distributed parameter estimation from data measurements on solutions of dierential equations. A nonlinear least squares functional is minimized to approximately recover the sought parameter function (i.e., the model). This functional consists of a data tting term, ..."
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This paper considers problems of distributed parameter estimation from data measurements on solutions of dierential equations. A nonlinear least squares functional is minimized to approximately recover the sought parameter function (i.e., the model). This functional consists of a data tting term, involving the solution of a nite volume or nite element discretization of the forward dierential equation, and a Tikhonovtype regularization term, involving the discretization of a mix of model derivatives. The grid spacing of the model discretization, as well as the relative weight of the entire regularization term, aect the sort of regularization achieved. We investigate a number of questions arising regarding their relationship, including the degree of nonlinearity of the least squares functional. We also investigate the correct scaling of the regularization matrix, where we rigorously associate the practice of using unscaled regularization matrices with approximations of...