Efficient solution of nonlinear elliptic problems using hierarchical matrices with Broyden updates
The numerical solution of nonlinear problems is usually connected with Newton's method. Due to its computational cost, variants (so-called inexact and quasi-Newton methods) have been developed in which the arising inverse of the Jacobian is replaced by an approximation. In this article we present a new approach which is based on Broyden updates. This method does not require to store the update history since the updates are explicitly added to the matrix. In addition to updating the inverse we introduce a method which constructs updates of the LU decomposition. To this end, we present an algorithm for the efficient multiplication of hierarchical and semi-separable matrices. Since an approximate LU decomposition of finite element stiffness matrices can be efficiently computed in the set of hierarchical matrices, the complexity of the proposed method scales almost linearly. Numerical examples demonstrate the effectiveness of this new approach.
hierarchical matrices broyden updates quasi-newton methods nonlinear elliptic equations