Conjugate gradient type methods for ill-posed problems
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Louis and R. Plato, The instability of some gradient methods for ill-posed problems, Numer. Engl, M.
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Hanke and A. Neubauer, Regularization of Inverse Problems , 1st ed. Kluwer, Dordrecht, Publisher Moscow University, Moscow, in Russian. Gilyazov, Regularizing conjugate-direction methods, Comput. Longman, Harlow, Hanke, The minimal error method is a regularization method, Proc. Springer, New York, Nemirovskii, The regularizing properties of the adjoint gradient method in ill-posed problems, U.
Nemirovskii and B. Polyak, Iterative methods for solving linear ill-posed problems under precise information I, Moscow Univ. Plato, Optimal algorithms for linear ill-posed problems yield regularization methods, Numer.
Plato, The method of conjugate residuals for solving the Galerkin equations associated with symmetric positive semidefinite ill-posed problems, SIAM J. We consider an ill-posed deconvolution problem from astronomical imaging with a given noise-contaminated observation, and an approximately known convolution kernel.
The limitations of the mathematical model and the shape of the kernel function motivate and legitimate a further approximation of the convolution operator by one that is self-adjoint.
This simplifies the reconstruction problem substantially because the efficient conjugate gradient method can now be used for an iterative computation of a regularized approximation of the true unblurred image. Since the constructed self-adjoint operator fails to be positive definite, a symmetric indefinite conjugate gradient technique, called MR-II is used to avoid a breakdown of the iteration.
The traveltime t between a source and a receiver is computed through the line integral of slowness, along a ray path:. This model representation leads to the relation. We consider a first-order truncated Taylor series approximation around s 0 :. The matrix G k contains elements gij that correspond to the distances that the i-th ray travels in the j-th block. This matrix can be constructed with a ray tracing algorithm 1 1. Let us denote by tobs the observed traveltime.
By applying the expansion 2.
Conjugate Gradient Type Methods for Ill-Posed Problems - Martin Hanke - Google Books
This is called a least squares problem, which can be formally stated as finding the minimum of the following objective function:. Least-squares solutions very often do not provide good results and sometimes they do not even exist. In this work, we have considered linear tomography, where the geometry of the rays does not depend on the distribution of velocities 7.
- The conjugate gradient method for linear ill-posed problems with operator perturbations.
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In other words, the rays are straight, and there is no need to do further ray tracing, nor to update the vector of estimated slowness. In general, this approximation is valid only for small velocity or slowness contrasts. We consider the classical conjugate gradient method for normal equations to solve the least-squares linear system 2. Given a initial guess s0 and a maximum number of iterations kmax,.
The matrix GTG is not explicitly computed, since it usually has a higher number of nonzero entries and a higher condition number than G Monte-Carlo GCV 15 :. Recalling the penalization of the first and second derivatives of the estimated model parameters proposed in 20 , we modify the minimal product criterion from 3 by introducing an additional term that penalizes large oscillations in the current iterate:. We consider the model shown in Figure 1 left , which is described as a non symmetric anticlinal with tectonic origin.
Such situation has great relevance in oil exploration, since it has structural traps with folds that could accumulate hydrocarbons. In the following we evaluate the following relative mean square errors: the error of the perturbed data with respect to the observed data, the calculated traveltimes with respect to the perturbed data, and the estimated velocities and slownesses with respect to the true model.
These errors are defined as follows:.
In analogy with 15 , the error is normalized so that its scale is comparable with the scale of the stopping criterion functions. The iterative scheme 2. Table 1 shows the number of iterations and the mean square errors of each inversion. The estimated models are satisfactory, allowing the identification location and velocity of the different layers, including the target reservoir.
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We considered the solution of an ill-conditioned least-squares problem arising from the discretization of a model for linear seismic tomography, which is a tool with wide application in reservoir geophysics. The discrete problem was presented as a system of linear equations, which we solved by the standard conjugate gradient method regularized by stopping in an early stage.
In the numerical experiments with a synthetic geological model, we perturbed the observed data with different levels of noise. The estimated model parameters were all satisfactory, although the resolution in the reconstructed velocity tomograms were compromised when the noise level was higher.
We expect that the effectiveness of the stopping criteria would be better in the absence of noise or for low noise if an appropriate preconditioning of the linear system is employed. We have also performed numerical experiments with conjugate gradient methods with Tikhonov regularization and noticed that, also in this case, a large number of iterations is needed for convergence.
Some of the preconditioners proposed in the literature that might be appropriate are based on the Fast Fourier Transform 4 and on the Algebraic Reconstruction Technique This motivates further testing and an error analysis in future works. Oliveira and A. Digital ray tracing in two-dimensional refractive fields. Journal of the Acoustic Society of America, 72 5 ,