Efficient solution of nonlinear elliptic problems using hierarchical matrices with Broyden updates

• Published in: Computing Vol. 81; no. 4; pp. 239 - 257
• Main Authors: BEBENDORF, M, CHEN, Y
• Format: Journal Article
• Language: English
• Published: Wien Springer 01.12.2007; Springer Nature B.V
• Subjects: Algorithmics. Computability. Computer arithmetics; Applied sciences; Approximation; Computer science; control theory; systems; Decomposition; Exact sciences and technology; Global analysis, analysis on manifolds; Mathematics; Methods; Methods of scientific computing (including symbolic computation, algebraic computation); Nonlinear equations; Numerical analysis; Numerical analysis. Scientific computation; Sciences and techniques of general use; Studies; Theoretical computing; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Multiplication; 34B15; 65F05. quasi-Newton methods; Broyden updates; Approximation; Quasi Newton method; Decomposition method; Nonlinear problems; nonlinear elliptic equations; Algorithm; Partial differential equation; Finite element method; 65F50; 90C53; Numerical analysis; Elliptic equation; hierarchical matrices; Scientific computation; Numerical solution; Newton method; 65N30
• ISSN: 0010-485X; 1436-5057
• DOI: 10.1007/s00607-007-0252-0

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. [PUBLICATION ABSTRACT]