Sparse Direct Solvers for Unassembled Hyper-Matrices

当前可用的稀疏直接求解器背后的基本假设，即孤立的单个求解应该花费最少的时间，并不代表自适应有限元方法设置中的最优解。在那里，可以在发生局部细化时更新当前细化解的中间结果（以分解的形式）。该项目有两项创新：允许应用程序将有关细化的信息传递给直接求解器库的接口，以及允许稀疏矩阵及其分解以非组装格式存储的非组装超矩阵（unassemble HyperMatrix），这有助于更新矩阵及其分解。换句话说，本研究基于继承域的细化历史中的消除树的思想，并存储在所有级别上未组装的元素矩阵，仅在分解需要时组装它们。由于结果求解器在元素矩阵方面起作用，因此密集的子矩阵自然暴露为分解的一部分，这将允许像3级BLAS这样的高性能内核得到充分利用。研究的重点包括API的设计，将求解器最佳地集成到应用程序中，UnassembledHyperMatrix基础设施的开发，以及稳定性和复杂性的研究，特别是不确定情况下的旋转策略的设计。计算仿真已成为理论和实验两种传统方法在科学上的平等伙伴。对于许多应用，从汽车振动的模拟到飞机雷达信号的计算，有限元方法都是首选的计算方法。这些方法中最好的是使用自适应近似来平衡精度和解决方案所需的时间。毫无疑问，大部分时间都花在求解线性方程组上。在自适应应用程序的特定设置中，减少这一解决时间是本研究的重点。这个项目的创新之处在于将线性系统以一种新的格式存储，即unassemble hypermatrix，这使得过去的计算在适应发生时可以被重用。与这一理论创新携手并进的是接口的开发，该接口允许应用程序表达对求解器库的适应。新方法有可能大幅降低与适应相关的解决方案的成本。

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.