On Ways To Improve Convergence Of Krylov Subspace Methods

"SNAP" atau "Solution by Null-space Approximation and Projection" ialah salah satu cara untuk menyelesaikan sistem linear apabila pekali matrik adalah besar dan "sparse". Objektifuya ialah Ulltuk mengatasi masalah penumpuan yang perlahan atau genangan yang disebabkan oleh nilai eigen yang kecil. Dissertasi ini bertujuan untuk menyediakan satu sorotan kritikal tentang kaedah "SNAP" yang dicadangkan pada 2006 oleh M. Illic, W. Turner dan Y. Saad. Dissertasi ini memfokuskan pada idea utama "SNAP" : algoritmanya, pembinaan Penghampiran Ruang nol hampiran dan dua algoritma yang dihasilkan oleh Illic. W. Turner dan Y. Saad iaitu SNAP-JD(m) dan 'Restarted SNAP-JD(m,kmax,l) "SNAP" or "Solution by Null-space Approximation and Projection" is one of the methods for solving linear system when the matrix coefficient is large and sparse. Its objective is to overcome the problem of slow convergence or stagnation which is caused by small eigenvalues. This dissertation is aimed at providing a critical review of the SNAP method which was proposed in 2006 by M. Illic, W. Turner and Y. Saad. The dissertation focused on the main idea of SNAP: the algorithm, the construction of Approximate Null Space and two algorithms generated by Illic, W. Turner and Y. Saad which are SNAP-JD(m) and Restarted SNAP-JD(m,kmax,l )

A partitioned model order reduction approach to rationalise computational expenses in multiscale fracture mechanics

We propose in this paper an adaptive reduced order modelling technique based on domain partitioning for parametric problems of fracture. We show that coupling domain decomposition and projection-based model order reduction permits to focus the numerical effort where it is most needed: around the zones where damage propagates. No \textit{a priori} knowledge of the damage pattern is required, the extraction of the corresponding spatial regions being based solely on algebra. The efficiency of the proposed approach is demonstrated numerically with an example relevant to engineering fracture.Comment: Submitted for publication in CMAM

Error analysis of coarse-grained kinetic Monte Carlo method

In this paper we investigate the approximation properties of the coarse-graining procedure applied to kinetic Monte Carlo simulations of lattice stochastic dynamics. We provide both analytical and numerical evidence that the hierarchy of the coarse models is built in a systematic way that allows for error control in both transient and long-time simulations. We demonstrate that the numerical accuracy of the CGMC algorithm as an approximation of stochastic lattice spin flip dynamics is of order two in terms of the coarse-graining ratio and that the natural small parameter is the coarse-graining ratio over the range of particle/particle interactions. The error estimate is shown to hold in the weak convergence sense. We employ the derived analytical results to guide CGMC algorithms and we demonstrate a CPU speed-up in demanding computational regimes that involve nucleation, phase transitions and metastability.Comment: 30 page

Reduced-Order Modeling for Flexible Spacecraft Deployment and Dynamics

The present work investigates reduced-order modeling for ultralight, packageable, and self-deployable spacecraft where reduced-order models (ROMs) are required to simulate deployment, structural dynamics during spacecraft maneuvers, and for real-time applications in trajectory optimization and control. In these contexts, ultralight, flexible spacecraft dynamics are characterized by geometrically nonlinear structural deformations combined with large rigid body motions. An approach based on proper orthogonal decomposition (POD), energy-conserving sampling and weighting (ECSW), and a floating frame of reference (FFR) is proposed to construct accurate and efficient ROMs. The proposed approach is then tested on a benchmark problem that involves geometrically nonlinear deformations, large rigid body motions, and strain energy release during dynamic snap-back, the last of which is analogous to the energy release during deployment. The resulting ROM for this benchmark problem is approximately 20% the size of the original full-order model with no appreciable loss of accuracy

High-performance model reduction procedures in multiscale simulations

Technological progress and discovery and mastery of increasingly sophisticated structural materials have been inexorably tied together since the dawn of history. In the present era — the so-called Space Age —-, the prevailing trend is to design and create new materials, or improved existing ones, by meticulously altering and controlling structural features that span across all types of length scales: the ultimate aim is to achieve macroscopic proper- ties (yield strength, ductility, toughness, fatigue limit . . . ) tailored to given practical applications. Research efforts in this aspect range in complexity from the creation of structures at the scale of single atoms and molecules — the realm of nanotechnology —, to the more mundane, to the average civil and mechanical engineers, development of structural materials by changing the composition, distribution, size and topology of their constituents at the microscopic/mesoscopic level (composite materials and porous metals, for instance).Postprint (published version