Summary
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.
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References
Axelsson, O.: On global convergence of iterative methods. In: Iterative solution of nonlinear systems of equations. Lecture Notes in Math., vol. 953, pp. 1–19. Springer, Berlin (1982)
Bank R.E. and Rose D.J. (1981). Global approximate Newton methods. Numer Math 37: 279–295
Bebendorf M. (2000). Approximation of boundary element matrices. Numer Math 86(4): 565–589
Bebendorf, M.: Effiziente numerische Lösung von Randintegralgleichungen unter Verwendung von Niedrigrang-Matrizen. Ph.D thesis, Universität Saarbrücken, 2000. dissertation.de, Verlag im Internet (2001).
Bebendorf M. (2005). Efficient inversion of Galerkin matrices of general second-order elliptic differential operators with nonsmooth coefficients. Math Comp 74: 1179–1199
Bebendorf M. (2006). Approximate inverse preconditioning of FE systems for elliptic operators with non-smooth coefficients. SIAM J Matrix Anal Appl 27(4): 909–929
Bebendorf M. (2007). Why finite element discretizations can be factored by triangular hierarchical matrices. SIAM J Num Anal 45(4): 1472–1494
Bebendorf M. and Hackbusch W. (2003). Existence of \({\mathcal{H}}\) -matrix approximants to the inverse FE-matrix of elliptic operators with L ∞-coefficients. Numer Math 95(1): 1–28
Bebendorf M. and Rjasanow S. (2003). Adaptive low-rank approximation of collocation matrices. Computing 70(1): 1–24
Broyden C.G. (1965). A class of methods for solving nonlinear simultaneous equations. Math Comput 19: 577–593
Chatzipantelidis P., Ginting V. and Lazarov R.D. (2005). A finite volume element method for a nonlinear elliptic problem. Numer Linear Algebra Appl. 12(5–6): 515–516
Dembo R. S., Eisenstat S.C. and Steihaug T. (1982). Inexact Newton methods. SIAM J Numer Anal 19: 400–408
Dennis J.E. and Schnabel R.B. (1996). Numerical methods for unconstrained optimization and nonlinear equations. SIAM, Philadelphia, PA
Dennis J.E. and Moré J.J. (1977). Quasi–Newton methods, motivation and theory. SIAM Rev 19(1): 46–89
Deuflhard, P.: Newton methods for nonlinear problems. Affine invariance and adaptive algorithms. Series Computational Mathematics, vol. 35. Springer (2004)
Deuflhard P., Freund R.W. and Walter A. (1990). Fast secant methods for the iterative solution of large nonsymmetric linear systems. Impact Comput Sci Engng 2: 244–276
Douglas J. and Dupont T. (1975). A Galerkin method for a nonlinear Dirichlet problem. Math Comput 29: 689–696
Grasedyck L. and Hackbusch W. (2003). Construction and arithmetics of \({\mathcal{H}}\) -matrices. Computing 70: 295–334
Hackbusch W. (1999). A sparse matrix arithmetic based on \({\mathcal{H}}\) -matrices. Part I: Introduction to \({\mathcal{H}}\) -matrices. Computing 62(2): 89–108
Hackbusch W. and Khoromskij B.N. (2000). A sparse \({\mathcal{H}}\) -matrix arithmetic. Part II: Application to multi-dimensional problems. Computing 64(1): 21–47
Hypre–high performance preconditioners. http://www.llnl.gov/CASC/hypre/
Karátson J. and Faragó I. (2003). Variable preconditioning via quasi–Newton methods for nonlinear problems in Hilbert space. SIAM J Numer Anal 41: 1242–1262
Kelley C.T. (2003). Solving nonlinear equations with Newton’s method. Fundamentals of algorithms. SIAM, Philadelphia, PA
Schenk, O., Gärtner, K.: Solving unsymmetric sparse systems of linear equations with PARDISO. In: Computational science – ICCS 2002, Part II (Amsterdam). Lecture Notes in Comput Sci, vol. 2330, pp. 355–363. Springer, Berlin (2002)
Vandebril R., Van Barel M. and Mastronardi N. (2005). A note on the representation and definition of semiseparable matrices. Numer Linear Algebra Appl 12(8): 839–858
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This work was supported by the DFG priority program SPP 1146 “Modellierung inkrementeller Umformverfahren”.
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Bebendorf, M., Chen, Y. Efficient solution of nonlinear elliptic problems using hierarchical matrices with Broyden updates. Computing 81, 239–257 (2007). https://doi.org/10.1007/s00607-007-0252-0
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DOI: https://doi.org/10.1007/s00607-007-0252-0