论文标题
在Calderón预处理上，用于对称脑电图前进问题的对称配方
On a Calderón preconditioner for the symmetric formulation of the electroencephalography forward problem without barycentric refinements
论文作者
论文摘要
我们提出了一种calderón预处理方案，用于对称脑电图（EEG）问题的对称配方，该方案可以治愈密集的离散化和高对比度分解。与针对脑电图问题提出的现有Calderón方案不同，它是无细化的，也就是说，静电积分运算符未通过在barycentry的双网格上定义的基本功能离散化。实际上，在预处理中，我们重复使用原始系统矩阵，从而减轻了计算负担。此外，提出的配方产生了线性方程式的对称，正定的系统系统，该系统允许应用结合梯度方法，与适用于非对称问题的其他Krylov子空间方法相比，一种迭代方法表现出较小的计算成本。数值结果证实了所提出的预处理技术对规范和现实情况的理论分析和证明。
We present a Calderón preconditioning scheme for the symmetric formulation of the forward electroencephalographic (EEG) problem that cures both the dense discretization and the high-contrast breakdown. Unlike existing Calderón schemes presented for the EEG problem, it is refinement-free, that is, the electrostatic integral operators are not discretized with basis functions defined on the barycentrically-refined dual mesh. In fact, in the preconditioner, we reuse the original system matrix thus reducing computational burden. Moreover, the proposed formulation gives rise to a symmetric, positive-definite system of linear equations, which allows the application of the conjugate gradient method, an iterative method that exhibits a smaller computational cost compared to other Krylov subspace methods applicable to non-symmetric problems. Numerical results corroborate the theoretical analysis and attest of the efficacy of the proposed preconditioning technique on both canonical and realistic scenarios.
