Cubically convergent methods for selecting the regularization parameters in linear inverse problems

AbstractWe present three cubically convergent methods for choosing the regularization parameters in linear inverse problems. The detailed algorithms are given and the convergence rates are estimated. Our basic tools are Tikhonov regularization and Morozov's discrepancy principle. We prove that, in comparison with the standard Newton method, the computational costs for our cubically convergent methods are nearly the same, but the number of iteration steps is even less. Numerical experiments for an elliptic boundary value problem illustrate the efficiency of the proposed algorithms

Elastic-Net Regularization: Error estimates and Active Set Methods

This paper investigates theoretical properties and efficient numerical algorithms for the so-called elastic-net regularization originating from statistics, which enforces simultaneously l^1 and l^2 regularization. The stability of the minimizer and its consistency are studied, and convergence rates for both a priori and a posteriori parameter choice rules are established. Two iterative numerical algorithms of active set type are proposed, and their convergence properties are discussed. Numerical results are presented to illustrate the features of the functional and algorithms

A velocity tracking approach for the Data Assimilation problem in blood flow simulations

preprintSeveral advances have been made in Data Assimilation techniques applied to blood flow modeling. Typically,idealized boundary conditions, only verified in straight parts of the vessel, are assumed. We present ageneral approach, based on a Dirichlet boundary control problem, that may potentially be used in differentparts of the arterial system. The relevance of this method appears when computational reconstructions ofthe 3D domains, prone to be considered sufficiently extended, are either not possible, or desirable, due tocomputational costs. Based on taking a fully unknown velocity profile as the control, the approach uses adiscretize then optimize methodology to solve the control problem numerically. The methodology is appliedto a realistic 3D geometry representing a brain aneurysm. The results show that this DA approach may bepreferable to a pressure control strategy, and that it can significantly improve the accuracy associated totypical solutions obtained using idealized velocity profilesinfo:eu-repo/semantics/submittedVersio

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