Multiscale Preconditioning of BEM Equations for Electrostatic Systems

ECS-9521001 Levin 这个项目的目标是创建稀疏近似的完全填充的边界元矩阵，其中一个使用积分公式，如电荷模拟或表面电荷模拟。 这些众所周知的数值方案被用来找到电场和附近的高压设备，如保护环，绝缘子串，变压器。 稀疏近似由系数矩阵的相似性变换形成。 这里使用的一个的成本是最优阶N。 尽管如此，使用稀疏矩阵进行计算的好处通常并不能证明创建它的成本是合理的，除非问题有多个右侧（即，想要模拟多个激励模式）。 这个项目将试图量化基于小波的相似性变换的优点，并提出实际的方法，他们可以使用。 该提案表明，对于具有数千个未知数的大型问题，由于磁盘访问的急剧减少，在小波基中工作可能会有直接的优势;变换的成本可能会比以前想象的更快地恢复。 此外，有吸引人的方式经济地创建逆矩阵的稀疏近似。 这些是迭代求解器（如广义共轭梯度和GMRES）的最佳预条件。 预处理器可以显著减少实现特定容差所需的迭代次数。 因此，稀疏近似不仅可以减少每次迭代的成本（通过稀疏化线性地），还可以减少迭代次数。 研究不同的预处理方案构成了该提案的第二个目标。 当然，小波基提供的最有趣的可能性是创建一个稀疏的系数矩阵，而不遭受相似性变换的代价。 该项目的第三部分将致力于艾德将这些基础集成到一个以前存在的边界元包，并直接产生稀疏系统。 3

ECS-9521001 Levin The objective of this project is to create sparse approximations of the fully populated boundary element matrices that one obtains using integral formulations like charge simulation or surface charge simulation. These well known numerical schemes are used to find the electric field on and near high voltage devices like guard rings, insulator strings, and transformers. The sparse approximations are formed by a similarity transform of the coefficient matrix. The cost of the one employed here is of optimal order N . Nonetheless, the benefits of computing with a sparse matrix typically do not justify the costs of creating it unless the problem has multiple right hand sides (that is, one wants to simulate multiple excitation modes). This project will attempt to quantify the advantage of wavelet-based similarity transforms and suggest practical ways in which they can be employed. The proposal suggests that for large problems with thousands of unknowns there may be an immediate advantage in working in the wavelet basis because of the dramatic reduction in disk access; the costs of the transform might be recovered more quickly than previously thought. Additionally, there are attractive ways of economically creating sparse approximations of the inverse matrix. These are the best kinds preconditioners for iterative solvers like generalized conjugate gradients and GMRES. Preconditioners can dramatically reduce the number of iterations required to achieve a particular tolerance. So, not only do sparse approximations reduce the cost of each iteration (linearly by the sparsification), they can be used to reduce the number of iterations as well. Investigating different preconditioning alternatives constitutes the second objective of the proposal. Of course the most interesting possibility offered by wavelet bases is that of creating a sparse coefficient matrix without suffering the expense of the similarity transformation. The third part of the project will be devot ed to integrating these bases into a previously existing boundary element package and producing the sparse systems directly. 3