Fast solvers (Numerical Linear Algebra)

We present a new multilevel fast Fourier transform (MLFFT) method, and its application for the three dimensional (3D) structures capacitance extraction. The multilevel octree structure, multilevel interpolation/projection, and subdomain FFTs techniques are employed in the MLFFT. The distinctions between the MLFFT and the conventional FFT-based methods are discussed carefully. Numerical results demonstrate that the MLFFT is more efficient than the conventional FFT based methods especially when analyzing inhomogeneous problems.

We present a new multilevel matrix compression method (MLMCM) and its application to the analysis of scattering problems from three-dimensional (3-D) arbitrary shaped conductors. The compression is achieved without generating the full sub-blocks of the matrix by the rank-based method. Unlike the conventional rank-based method, incoming compression matrix U and outgoing compression matrix V are defined when coupling with a cluster of its far interaction groups. Only a small translation matrix D is redefined for every two coupling groups. The merits of the proposed method are: a. it is kernel function independent and can be applied to arbitrary complex media; b. it is more efficient than conventional rank-based methods. This paper shows numerical results to demonstrate the validity of the proposed method.

Gegenstand der vorliegenden Arbeit ist eine spezielle Klasse iterativer Verfahren zur Lösung, die auch als Verfahren des stärksten Abtiegs (methods of steepest decent) bezeichnet werden. Spätestens seit der Veröffentlichung von Cauchy vom 18. Oktober 1847 [Cauchy47] ist diese Form der Vorgehensweise in der Literatur bekannt. Kernphilosophie ist, daß statt der Gleichungsauflösung ein äquivalentes Ersatzproblem, die Minimierung einer differenzierbaren Funktion Φ, zur Aufgabenbearbeitung herangezogen
wird. Das oberste Ziel soll die Konstruktion von effizienten und robusten Verfahren sein, also von schnellen Lösern für eine große Aufgabenklasse.

We present two algorithms to compute system-specific polarizabilities and dispersion coefficients such that required memory and computational time scale linearly with increasing number of atoms in the unit cell for large systems. The first algorithm computes the atom-in-material (AIM) static polarizability tensors, force-field polarizabilities, and C 6 , C 8 , C 9 , C 10 dispersion coefficients using the MCLF method. The second algorithm computes the AIM polarizability tensors and C 6 coefficients using the TS-SCS method. Linear-scaling computational cost is achieved using a dipole interaction cutoff length function combined with iterative methods that avoid large dense matrix multiplies and large matrix inversions. For MCLF, Richardson extrapolation of the screening increments is used. For TS-SCS, a failproof conjugate residual (FCR) algorithm is introduced that solves any linear equation system having Hermitian coefficients matrix. These algorithms have mathematically provable stable convergence that resists round-off errors. We parallelized these methods to provide rapid computation on multi-core computers. Excellent parallelization efficiencies were obtained, and adding parallel processors does not significantly increase memory requirements. This enables system-specific polarizabilities and dispersion coefficients to be readily computed for materials containing millions of atoms in the unit cell. The largest example studied herein is an ice crystal containing >2 million atoms in the unit cell. For this material, the FCR algorithm solved a linear equation system containing >6 million rows, 7.57 billion interacting atom pairs, 45.4 billion stored non-negligible matrix components used in each large matrix-vector multiplication, and $19 million unknowns per frequency point (>300 million total unknowns).

This article uses the theory of 'forward error bound estimation' to show how rescaling the linear system affects the correspondence between the residual error in the preconditioned linear system and the solution error. Using examples of linear systems from models developed using the USGS GSFLOW package and the California State Department of Water Resources' Integrated Water Flow Model (IWFM), we observe that this error bound guides the choice of a practical measure for controlling the error in rescaled linear systems. It is found that forward error can be controlled in preconditioned GMRES by rescaling the linear system and normalizing the stopping tolerance. We implemented a preconditioned GMRES algorithm and benchmarked it against the Successive-Over-Relaxation (SOR) method. Improved error control reduces redundant iterations in the GMRES algorithm and results in overall simulation speedups as large as 7.7x. This research is expected to broadly impact groundwater modelers through the demonstration of a practical approach for setting the residual tolerance in line with the solution error tolerance.