The discretization of certain integral equations, e.g., the first-kind Fredholm equation of Laplace's equation, leads to symmetric positive-definite linear systems, where the coefficient matrix is dense and often ill-conditioned. We introduce a new preconditioner based on a novel overlapping domain decomposition that can be combined efficiently with fast direct solvers. Empirically, we observe that the condition number of the preconditioned system is $O(1)$, independent of the problem size. Our domain decomposition is designed so that we can construct approximate factorizations of the subproblems efficiently. In particular, we apply the recursive skeletonization algorithm to subproblems associated with every subdomain. We present numerical results on problem sizes up to $16\,384^2$ in 2D and $256^3$ in 3D, which were solved in less than 16 hours and three hours, respectively, on an Intel Xeon Platinum 8280M.

翻译：某些整体方程式的离散化,例如,Laplace方程式的首种Fredholm方程式,导致对正-确定线性系统的对称正-确定线性系统,其中系数矩阵密度大,而且往往条件差。我们引入了一个新的先决条件,其基础是新颖的重叠域分解,可以与快速直接解决器有效结合。我们偶然地观察到,先决条件系统的条件号是O(1)美元,与问题大小无关。我们的域分解设计是为了使我们能够有效地构建子问题近似因数。特别是,我们对与每个子项相关的子项的子问题采用递归性骨架化算法。我们提出了问题大小的数字结果,问题大小在2D和3D中分别为16美元和256美元,分别在16小时和3小时内解决了Intel Xeon Platinum 8280M。