We present a Volume Integral formulation for the solution of large scale eddy-current problems coupled with low-rank approximation techniques. Two alternative approaches are introduced to map the problem unknowns into a subset of grid elements forming a base of global or mixed (global and local) cycles, respectively, and guarantee the well-posedness of the problem both in simply and multiply connected domains. The paper shows that the adoption of mixed cycles is computationally more efficient than global ones. In particular, integral formulations based on global cycles cannot be safely coupled with low-rank approximation techniques, which, however, are crucial to increase the size of the largest solvable problem, like the ones involving conducting structures in magnetic confinement fusion devices. The aim of this paper is to demonstrate how such bottleneck can be overcome by considering local and global cycles differently, on the basis of the cohomology theory. An improved, efficient, and robust algorithm for computing a base of global cycles is described in detail. In particular, the presented algorithm is able to almost minimize the cohomology basis length, i.e. the number of mesh edges forming such a basis, in order to allow an efficient solution of large scale problems. Furthermore, a novel and general method to handle global and local cycles together, in the context of low-rank approximated matrices, is shown to be efficient for the solution of large scale eddy-current problems in multiply connected domains. Along the manuscript, pseudo-codes are given, which clarify the proposed methods and help to implement them by Volume Integral Equation practitioners.

我们提出了一种体积积分公式，用于解决大规模涡流问题以及低秩逼近技术。引入了两种替代方法，将问题未知数映射到分别形成全局循环或混合（全局和局部）循环基础的网格元素子集中，并在简单连接域和多重连接域中保证问题的适定性。本文表明，采用混合循环比全局循环更有效。尤其是，基于全局循环的积分公式不能与低秩逼近技术安全地结合，但是，这对于增加最大可解决问题的大小至关重要，例如涉及磁约束聚变设备中导电结构的问题。本文的目的是证明在同调理论的基础上，如何通过不同地考虑局部和全局循环来克服这种瓶颈。详细描述了一种用于计算全局循环基数的改进，高效且健壮的算法。特别地，提出的算法能够几乎最小化同调基础长度，即形成这种基础的网格边缘的数量，以允许有效地解决大规模问题。此外，在低秩近似矩阵的情况下，一种新颖且通用的方法可以同时处理全局和局部循环，对于解决多重连接域中的大规模涡流问题，该方法是有效的。沿着手稿给出了伪代码，