Convergence of Broyden’s Method in Banach Spaces

Convergence Estimates for Solution of Integral Equations with GMRES

“…In this section we illustrate this with a simple example. This example will also serve to motivate our Banach-Hilbert space setting, which, as was done in [9] in the context of Broyden's method, seeks to account for the successful application of an inner product (and hence Hilbert space) based algorithm to a problem naturally posed in a Banach space. A typical example of such a pairing of spaces is the solution of a problem posed in the Banach space of continuous functions on a compact set with GMRES and the L 2 scalar product.…”

“…Examples of sequences of strongly convergent-collectively compact maps arise in the approximate solution of linear second-kind Fredholm integral equations [3], [2], [13], elliptic partial dierential equations which are preconditioned by solvers for high order terms [9], [8], [20], [7], [6], radiative transport problems [24], [19], [15], and semiconductor device modeling [10], [11], [18]. In some of these papers, [8], [15], [13], GMRES is used as a coarse mesh solver in the context of a multilevel method and/or as an iterative This research was supported by National Science Foundation grants DMS-9122745, CCR-9102853, CCR-9400921, DMS-9321938, DMS-9020915, and DMS-9403224.…”

An Algorithm for Solving Boundary Value Problems

“…It was noted in [9] ). Numerical solutions of this problem were obtained by a shooting method (see [9,17]). …”

Local convergence of quasi-Newton methods under metric regularity

“…[3], [13], [15], [18], [22], [23], [25]. In some of these papers, such as [15] and [18], Y is a Banach space; in [15] X is also a Banach space having a continuous inner product, and with (27) being defined on the completion of X in the norm induced by this inner product.…”