Regularized versions of continuous newton's method and continuous modified newton's method under general source conditions
13 Nov 2008-Numerical Functional Analysis and Optimization (Taylor & Francis Group)-Vol. 29, pp 1140-1165
TL;DR: In this paper, the error analysis for the regularized modified Newton's method under a Holder-type source condition was carried out by replacing the monotonicity of F by a weaker assumption.
Abstract: Regularized versions of continuous analogues of Newton's method and modified Newton's method for obtaining approximate solutions to a nonlinear ill-posed operator equation of the form F(u) = f, where F is a monotone operator defined from a Hilbert space H into itself, have been studied in the literature. For such methods, error estimates are available only under Holder-type source conditions on the solution. In this paper, presenting the background materials systematically, we derive error estimates under a general source condition. For the special case of the regularized modified Newton's method under a Holder-type source condition, we also carry out error analysis by replacing the monotonicity of F by a weaker assumption. This analysis facilitates inclusion of certain examples of parameter identification problems, which was not possible otherwise. Moreover, an a priori stopping rule is considered when we have a noisy data f δ instead of f. This rule yields not only convergence of the regularized approxi...
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TL;DR: In this article, two approaches are proposed to speed up the airborne electromagnetic inversion process, moving footprint and direct solver methods, which can be used to speedup the process.
Abstract: Airborne electromagnetic (AEM) data processing and inversion has progressed from the early conductivity-depth imaging, to 1D inversion for a layered earth model, and most recently to 2D/3D inversions. The new processes have used methods for searching for solutions, applying constraints, and focusing images, such as Occam’s, laterally constrained, and holistic inversions. For airborne data imaging and 1D inversion, some algorithms are fast, whereas others have control over data misfit and the vertical or lateral smoothness. Beyond 1D algorithms, 2D/3D inversions are based on currently used methods such as isolated conductor models (which are good for highly conductive orebodies in resistive background) and techniques that discretized regions that are used for more complex structures and backgrounds. In the latter case, the computations are slow, so research is focusing on time-efficient computer algorithms for solving the equations such as Gauss-Newton, quasi-Newton, and nonlinear conjugate gradient algorithms. For the electromagnetic (EM) problem, the solution can be obtained in a more practical time frame if material that has minimal impact is ignored. Two approaches can be used to speedup the EM inversion process — the moving footprint and direct solver methods. We hope our work will to some extent help stimulate and focus the research in AEM inversion.
38 citations
TL;DR: In this article, a finite-volume (FV) method and a direct Gauss-Newton optimization was proposed to constrain the modeling volume to the AEM volume of influence (VOI) of a 3D source within the earth.
Abstract: We investigate an algorithm for 3D time-domain airborne electromagnetic (AEM) inversion based on the finite-volume (FV) method and direct Gauss-Newton optimization, where we obtain high efficiency by constraining the modeling volume to the AEM volume of influence (VOI) of a 3D source within the earth, rather than using the larger VOI of the AEM system. A half-space or layered earth is used to model the background field in the time domain, taking into account the transmitter waveform through convolution. Assuming that the 3D source of any secondary field detected at a survey point lies within the moving VOI of the airborne system, we conduct time-domain forward modeling and Jacobian calculation using an FV method within the 3D source VOI that requires a small number of cells for discretization. A local mesh and direct solver are shown to further speed up the computation. A synthetic isolated synclinal conductor inversion shows good agreement with the model geometry and provides a good fit to the data contaminated with noise. A synthetic multiple-body model inversion was also quite successful, showing that our algorithm is effective and about four times faster than inversion using the total-field method. Finally, we inverted GEOTEM data over the Lisheen deposit, where our inversion result was consistent with the published geology.
29 citations
TL;DR: The Lavrentiev regularization method and a modified Newton method for obtaining stable approximate solution to nonlinear ill-posed operator equations are considered and the algorithm is applied to numerical solution of the inverse gravimetry problem.
17 citations
TL;DR: It is shown that the adaptive scheme considered by Perverzev and Schock (2005) for choosing the regularization parameter can be effectively used here for obtaining an optimal order error estimate.
Abstract: Abstract In this paper, we consider an iterative method for the approximate solution of the nonlinear ill-posed operator equation Tx = y. The iteration procedure converges quadratically to the unique solution of the equation for the regularized approximation. It is known that (Tautanhahn (2002)) this solution converges to the solution of the given ill-posed operator equation. The convergence analysis and the stopping rule are based on a suitably constructed majorizing sequence. We show that the adaptive scheme considered by Perverzev and Schock (2005) for choosing the regularization parameter can be effectively used here for obtaining an optimal order error estimate.
9 citations
TL;DR: This work proposes multilevel augmentation methods for solving nonlinear ill-posed problems, involving monotone operators in the Hilbert space by using the Lavrentiev regularization method, and results are presented to illustrate the accuracy and efficiency of the proposed methods.
Abstract: We propose multilevel augmentation methods for solving nonlinear ill-posed problems, involving monotone operators in the Hilbert space by using the Lavrentiev regularization method. This leads to a fast solutions of the discrete regularization methods for the nonlinear ill-posed equations. The regularization parameter choice strategies considered by Pereverzev and Schock (2005) are introduced and the optimal convergence rates of the regularized solutions are obtained. Numerical results are presented to illustrate the accuracy and efficiency of the proposed methods.
5 citations
Cites background from "Regularized versions of continuous ..."
...However, it is shown in [25] that F ′(x0) with x0(t) = 1 for all t ∈ [0, 1], is not a non-negative operator....
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References
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TL;DR: Inverse problems have been studied in this article, where Tikhonov regularization of nonlinear problems has been applied to weighted polynomial minimization problems, and the Conjugate Gradient Method has been used for numerical realization.
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