Sparse Direct Solvers for Unassembled Hyper-Matrices

未组装超矩阵的稀疏直接求解器

• 批准号: 0625917
• 负责人: Victor Eijkhout
• 金额: $ 39.67万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-09-01 至 2010-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0625917&HistoricalAwards=false
• 关键词: Sparse; Direct; Solvers; Unassembled; Hyper

The underlying assumption behind currently available sparse directsolvers, namely that an individual solve in isolation should takeminimal time, does not represent an optimal solution in the setting ofan adaptive finite element method. There, intermediate results, inthe form of a factorization, from the solution of a current refinementcan be updated when a local refinement occurs. Two innovationsunderlie the project: an interface that allows applications to passinformation about refinement to the direct solver library and a datastructure, the Unassembled HyperMatrix, that allows sparse matricesand their factorization to be stored in an unassembled format thatfacilitates the updating of the matrix and its factorization. Inother words, this research is based on the ideas of inheriting theelimination tree from the refinement history of the domain and storingelement matrices unassembled on all levels, assembling them only whennecessary for the factorization. Since the resulting solver functionsfully in terms of element matrices, dense submatrices are naturallyexposed as part of the factorization which will allow high-performancekernels like the level-3 BLAS to be fully exploited.The focus of the research includes the design of an API that optimally integrates thesolver into applications, the development of the UnassembledHyperMatrix infrastructure, and an investigation of stability andcomplexity, in particular the design of pivoting strategies for theindefinite case.Computational simulation has joined the two traditional methods inscience, theory and experiment, as an equal partner. For manyapplications, ranging from the similation of vibration in anautomobile to the computation of the radar signature an airplane onradar, finite element methods are the computational method of choice.The best of these methods use adaptive approximations to balanceaccuracy against the time required for the solution. Invariably, mosttime is spent in the solution of a system of linear equations. It isthe reduction of this solution time, in the specific setting of anadaptive application, that is the focus of thisresearch. The innovation that is the basis of this project consistsin storing the linear system in a new format, the UnassembledHyperMatrix, that allows much of a past computation to be reused whenan adaptation occurs. Hand in hand with this theoretical innovationgoes the development of an interface that allows an application toexpress adaptation to the solver library. The new approach has thepotential for reducing the cost of a solution related to an adaptationsubstantially.

当前可用的稀疏指导底座背后的基本假设，即单独求解应分离的时间，并不代表OFAN自适应有限元方法的最佳解决方案。 在那里，当局部细化发生时，以当前细化的溶液更新当前的修补剂的溶液中的中间结果。 在项目中进行的两个创新：一个允许应用程序将改进的信息传递到直接求解器库和数据构建的未组装的超泥浆，允许稀疏的矩阵及其因子化以无配合的格式存储，从而更新了矩阵和其分解。 说词，这项研究基于从域和存储矩阵的完善历史上遗传灭气树的思想，并在各个层面上都没有组装，只有在进行分解时才组装它们。 Since the resulting solver functionsfully in terms of element matrices, dense submatrices are naturallyexposed as part of the factorization which will allow high-performancekernels like the level-3 BLAS to be fully exploited.The focus of the research includes the design of an API that optimally integrates thesolver into applications, the development of the UnassembledHyperMatrix infrastructure, and an investigation of stability andcomplexity,特别是，对于case.com的设计策略的设计已成为同等伙伴的两种传统方法，理论和实验加入了两种传统方法。 对于许多应用程序，从对Anautomobile中的振动的模拟到雷达签名的计算，飞机Onradar飞机的计算是选择的计算方法。这些方法中的最佳方法使用自适应近似值到解决方案所需时间所需的balanceacecuracy。 通常，大部分时间都花在线性方程系统的解决方案中。 在Anadaptive应用程序的特定设置中，这是此解决方案的减少，这是本研究的重点。 作为该项目的基础的创新，以新的格式将线性系统组成，即非组装Hypermatrix，在发生适应时，可以重复使用过去的大部分计算。 与这种理论创新汇总，可以开发接口，该界面允许应用程序表达对求解器库的适应。 新方法具有降低与适应性逐步化相关的解决方案成本的潜力。