By making use of tools from convex analysis, we formulate an inexact NewtonLandweber iteration method for solving nonlinear inverse problems in Banach spaces. The method consists of two components: an outer Newton iteration and an inner scheme. The inner scheme provides increments by applying Landweber iteration with non-smooth uniformly convex penalty terms to local linearized equations. The o...

We consider inverse iteration-based eigensolvers, which require at each step solving an “inner” linear system. We assume that this linear system is solved by some (preconditioned) Krylov subspace method. In this framework, several approaches are possible, which differ by the linear system to be solved and/or the way the preconditioner is used. This includes methods such as inexact shift-and-inv...

In our papers [Inverse Problems, 15, 309-327,1999] and [Numer. Math., 88, 347-365, 2001] we proposed algorithm REGINN being an inexact Newton iteration for the stable solution of nonlinear ill-posed problems. REGINN consists of two components: the outer iteration, which is a Newton iteration stopped by the discrepancy principle, and an inner iteration, which computes the Newton correction by so...

Convergence results are provided for inexact inverse subspace iteration applied to the problem of finding the invariant subspace associated with a small number of eigenvalues of a large sparse matrix. These results are illustrated by the use of block-GMRES as the iterative solver. The costs of the inexact solves are measured by the number of inner iterations needed by the iterative solver at ea...

The identification of instability in large-scale dynamical systems caused by Hopf bifurcation is difficult because of the problem of identifying the rightmost pair of complex eigenvalues of large sparse generalized eigenvalue problems. A new method developed in [Meerbergen and Spence, SIAM J. Matrix Anal. Appl., 31 (2010), pp. 19821999] avoids this computation, instead performing an inverse ite...

In this paper we consider the computation of an eigenvalue and corresponding eigenvector of a large sparse Hermitian positive definite matrix using inexact inverse iteration with a fixed shift. For such problems the large sparse linear systems arising at each iteration are often solved approximately by means of symmetrically preconditioned MINRES. We consider preconditioners based on the incomp...

In this paper, we consider a generalized inexact Newton-Landweber iteration to solve nonlinear ill-posed inverse problems in Banach spaces, where the forward operator might not be Gâteaux differentiable. The method is designed with non-smooth convex penalty terms, including L1-like and total variation-like functionals, capture special features of solutions such as sparsity piecewise constancy. ...

In this paper, we study an inexact inverse iteration with inner-outer iterations for solving the generalized eigenvalue problem Ax = Bx; and analyze how the accuracy in the inner iterations aaects the convergence of the outer iterations. By considering a special stopping criterion depending on a threshold parameter, we show that the outer iteration converges linearly with the threshold paramete...