D. Calvetti, L. Reichel and Q. Zhang, Iterative solution methods for large linear discrete ill-posed problems, Appl. Comput. Control, Signals and Circuits, 1 (1998), to appear.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....of large scale discrete ill posed problems: Landweber iteration in x3.4.1, Conjugate Gradient on the normal equations (CGLS) in x3.4.2, the method of Bjorck, Grimme and Van Dooren [6] in x3.4.4, the method of Golub and von Matt in x3.4. 5 [26] and the method of Calvetti, Reichel and Zhang [8] [9] in x3.4.6. In x3.4.3 we discuss the preconditioning issue in the context of discrete ill posed problems. We recall that the problem we want to solve is that of recovering the solution of minkAx Gamma bk; x 2 IR n from the solution of minkAx Gamma bk; x 2 IR n , where A 2 IR m Thetan ....

....ensure that Delta is such that Delta kA y bk kA y bk, where b is the exact data vector. This requires knowledge on the norm of the unknown solution A y b of the unperturbed unconstrained problem. 33 3.4. 6 The method of Calvetti, Reichel and Zhang This method was presented in [8] and [9]. The method considers the problem of solving large ill conditioned systems of equations. The method is based on expressing the regularized solution x , which depends on the regularization parameter , as x = A)A y b; where (A) is a polynomial in A that can be regarded as a filter ....

D. Calvetti, L. Reichel, and Q. Zhang. Iterative solution methods for large linear discrete ill--posed problems. To appear in Applied and Computational Control, Signals and Systems I, ed. B.N. Datta, 1997.

....solution. The last two methods compute the regularized solution only, assuming that the regularization paramter is known. Another method has been presented recently for the solution of large scale discrete ill posed problems. We are referring to the method of Calvetti, Reichel and Zhang [7] [8] which has been used successfully for the restoration of images. Applications of this method to more general discrete ill posed problems have not been reported. 3.3.1 The method of Bjorck, Grimme and van Dooren (BGvD) The problem is to recover the solution of minkAx Gamma bk, x 2 IR n when ....

D. Calvetti, L. Reichel, and Q. Zhang. Iterative solution methods for large linear discrete ill--posed problems. To appear in Applied and Computational Control, Signals and Systems I, ed. B.N. Datta, 1997. 111

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D. Calvetti, L. Reichel and Q. Zhang, Iterative solution methods for large linear discrete ill-posed problems, Appl. Comput. Control, Signals and Circuits, 1 (1998), to appear.

....approximation of the Tikhonov filter function (10) On the other hand, for fixed, the Tikhonov filter function has singularities at the points t : i # in the complex plane, where i : # 1, while the exponential filter function (5) is analytic in the finite complex plane. We refer to [5] for further discussions on the filter function (10) Numerical examples that compare the filtering properties of the exponential and Tikhonov filter functions, as well as of the filter function defined by truncated singular value decomposition, are presented in [1] The present paper develops an ....

....quadrature rules (33) increases with m for each fixed j. Bounds for can be derived as follows. Express the function # (t) in [a, b] as a series of Chebyshev polynomials of the first kind for this interval. Bounds for the oe#cients in this expansion can be derived by techniques discussed in [5]. Since # (t) is analytic in the finite complex plane, the terms in the series converge to zero faster than geometrically. Druskin and Knizhnerman [6] show how the bounds for these terms can be used to bound the error #. This approach shows that the error decreases faster than geometrically ....

D. Calvetti, L. Reichel and Q. Zhang, Iterative solution methods for large linear discrete illposed problems, Applied and Computational Control, Signals and Circuits, 1 (1999), pp. 313--367.

....approximation of the minimal norm least squares solution x # of the consistent errorfree linear system of equations (1.3) by applying a few steps of an iterative solution method to the linear discrete ill posed problem with a contaminated right hand side (1. 1) This approach is discussed in [5, 8, 12, 15, 17] for linear discrete ill posed problems (1.1) with a symmetric matrix and in [3, 4, 6] for linear discrete ill posed problems with a nonsymmetric matrix. Here the iteration number can be thought of as the regularization parameter; see below. The performance of the Conjugate Gradient (CG) method ....

....(1.5) is a linear discrete ill posed problem. Iterative methods proposed in the literature for the solution of linear discrete ill posed problems with a symmetric matrix A determine iterates in the range of the matrix to ensure that the iterates are orthogonal to the null space of the matrix; see [7, 8, 12] for discussions. For instance, the Minimal Residual (MR) method by Paige and Saunders [19] is a popular iterative method for the solution of linear systems of equations with a symmetric indefinite nonsingular matrix. When applied to the solution of the discrete linear ill posed problem (1.1) with ....

D. Calvetti, L. Reichel and Q. Zhang, Iterative solution methods for large linear discrete illposed problems, Applied and Computational Control, Signals and Circuit, 1 (1999), pp. 313--367.

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D. Calvetti, L. Reichel and Q. Zhang, "Iterative solution methods for large linear discrete ill-posed problems," Applied Comput. Control, Signals and Circuits, 1, pp. 317-374, 1998.

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