Scalable Multilevel Algorithms in Computational Sciences

点击翻译按钮获取中文摘要

Solvers for Partial Differential Equations (PDEs) are the backbone of much of scientific computing. In particular, they are the basis of Computational Fluid Dynamics (CFD), the modeling of liquid and gas flows. This project studies new, efficient methods for solving PDEs and implements those methods on modern high-performance parallel computers. These solvers are useful in areas other than their original CFD home - in particular, surprising applications to diverse areas such as image restoration and VLSI placement will be studied as well.Technically, this project will investigate efficient algebraic multiscale algorithms for elliptic and non-elliptic PDE and CFD problems on arbitrary unstructured meshes which are suitable for distributed and shared memory parallel computing architectures. In addition, it will study how these algorithms can be extended to other large scale non-PDE problems, including image restoration and VLSI placement problems. Three aspects of these multiscale algorithms will be emphasized in this work: (1) Issues arising from making these algorithms more algebraic (for ease of use) including robustness to anisotropy, jumps and oscillations in coefficients, homogenization, etc. (2) Extension of these algorithms from their normal elliptic setting to non-elliptic and more generally non-PDE, graph-based settings. (3) Performance on modern high performance computer architectures with particular attention paid to communication and cache memory latency. Particular attention will be placed on algorithms appropriate for solving discretization matrices arising from a variety of large scale scientific computing problems such as CFD for advection dominated problems, VLSI placement an image processing. The non-elliptic behavior of these practical problems renders the known multilevel theory inadequate and serves to motivate a balanced effort consisting of algorithmic development, theoretical analysis, and practical application.