Parallel Projection Methods And The Resolution Of Ill-posed Problems

In this paper, we consider a modification of the parallel projection method for solving overdetermined nonlinear systems of equations introduced recently by Diniz-Ehrhardt and Martínez [1]. This method is based on the classical Cimmino's algorithm for solving linear systems. The components of the function are divided into small blocks, as an attempt to correct the intrinsic ill-conditioning of the system, and the new iteration is a convex combination of the projections onto the linear manifolds defined by different blocks. The modification suggested here was motivated by the application of the method to the resolution of a nonlinear Fredholm first kind integral equation. We prove convergence results and we report numerical experiments. © 1993.2711124Diniz-Ehrhardt, Martínez, A parallel projection method for overdetermined nonlinear systems of equations (1993) Numerical Algorithms, , (to appear)Cimmino, Calcolo approssimato per le soluzioni dei sistemi di equazioni lineari (1938) La Ricerca Scientifica Ser II, 1, pp. 326-333. , Anno IVDe Pierro, Iusem, A simultaneous projection method for linear inequalities (1985) Linear Algebra and its Applications, 64, pp. 243-253dos Santos, A parallel subgradient projections method for the convex feasibility problem (1987) Journal of Computational and Applied Mathematics, 18, pp. 307-320Censor, Row-Action Methods for Huge and Sparse Systems and Their Applications (1981) SIAM Review, 23, pp. 444-466Santos, Iterative linear methods and regularization (1993) Ph.D. Dissertation, , Department of Applied Mathematics, University of CampinasElden, Algorithms for the regularization of ill-conditioned least problems (1977) BIT, 17, pp. 134-145Tikhonov, Arsenin, (1977) Solutions of Ill-Posed Problems, , John Wiley, New YorkVogel, A constrained least squares regularization method for nonlinear ill-posed problems (1990) SIAM Journal on Control and Optimization, 28 (1), pp. 34-49Ito, Künisch, On the Choice of the Regularization Parameter in Nonlinear Inverse Problems (1992) SIAM Journal on Optimization, 2, pp. 376-404Morozov, (1984) Methods for Solving Incorrectly Posed Problems, , Springer-Verlag, New YorkO'Sullivan, Wahba, A cross-validated Bayesian retrieval algorithm for nonlinear remote sensing experiments (1985) Journal of Computational Physics, 59, pp. 441-455Dennis, Schnabel, (1983) Numerical Methods for Unconstrained Optimization and Nonlinear Equations, , Prentice Hall, Englewood Cliffs, NJMartínez, Fixed-point quasi-Newton methods (1992) SIAM Journal on Numerical Analysis, 29, pp. 1413-143