Domain decomposition techniques in inexact Newton methods for efficient parallel performance.

In earlier research [5, 4] a domain decomposition technique [2, 3] was analysed and adapted for use within the Jacobi-Davidson method [1]. Jacobi-Davidson is an iterative method for computing approximate solutions of large scale eigenvalue problems. The parallel performance on distributed memory systems of this combined approach was studied before in [6] and more recently in [7]. The Jacobi-Davidson method is a special case of an inexact Newton methods applied to eigenvalue problems, which are mildy nonlinear. Focus of the current research is on the extension of the combined approach to a more general inexact Newton method for other, in first instance mildy, nonlinear problems. The research is both theoretical (analysis) and experimental (numerical experiments). The combined approach aims at good massively parallel behavior. Therefore, for a proof of concept, the use of HPC facilities with enough (corresponding to the current standards) CPU cores is an essential ingredient. (inexact) Newton methods are relevant if nonlinear problems have to be solved by computer. These nonlinear problems are certainly present in the top sectors Chemicals, Energy, High-tech Systems and Materials, Life Sciences, and Water. Therefore, a positive proof of concept may yield a nice scientific impact for these application areas. [1] Sleijpen, G.L.G., van der Vorst, H.A.: A Jacobi-Davidson iteration method for linear eigenvalue problems. SIAM J. Matrix Anal. Appl. 17, 401-425 (1996) [2] Tan, K.H., Borsboom, M.J.A.: On generalized Schwarz coupling applied to advection- dominated problems. Domain Decomposition Methods in Scientific and Engineering Computing, pp. 125-130. Amer. Math. Soc. (1994) [3] Tan, K.H.: Local coupling in domain decomposition. Ph.D. thesis, Utrecht University (1995) [4] Genseberger, M.: Domain decomposition in the Jacobi-Davidson method for eigenproblems. Ph.D. thesis, Utrecht University (2001) [5] Genseberger, M., Sleijpen, G.L.G., van der Vorst, H.A.: An optimized Schwarz method in the Jacobi-Davidson method for eigenvalue problems. Domain DecompositionMethods in Science and Engineering, pp. 289-296. UNAM (2003) - http://www.ddm.org/DD14/genseberger.pdf [6] Genseberger, M.: Improving the parallel performance of a domain decomposition preconditioning technique in the Jacobi-Davidson method for large scale eigenvalue problems. Applied Numerical Mathematics 60, 1083-1099 (2010) [7] Genseberger, M.: Practical aspects of domain decomposition in Jacobi-Davidson for parallel performance, Proceedings of the 21st International Conference on Domain Decomposition Methods (2014) - http://dd21.inria.fr/pdf/gensebergerm_contrib.pdf