Iterative solution of dense linear systems arising from the electrostatic integral equation in MEG.
Iterative solution of dense linear systems arising from the electrostatic integral equation in MEG.
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DOI：
10.1088/0031-9155/47/6/308
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发表时间：
2002-03
影响因子：
3.5
通讯作者：
Jussi Rahol;S. Tissari
中科院分区：
文献类型：
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作者：
Jussi Rahol;S. Tissari
研究脑磁图（MEG）中静电积分方程的边界元离散所产生的稠密线性方程组的迭代解。我们表明，现代迭代方法可以用来减少总的计算时间，避免了耗时的计算系数矩阵的LU分解。更重要的是，现代迭代方法使得有可能避免显式形成的系数矩阵时，需要使用大量的未知数。为了研究迭代求解器的收敛性，我们研究了系数矩阵的特征值分布。对于球，我们展示了如何的积分算子的特征值近似的系数矩阵的特征值时，配置和Galerkin方法被用作离散化方法。配点法直接逼近积分算子的特征值。Galerkin方法产生需要预处理的系数矩阵，以保持最佳收敛速度。使用ILU（0）预条件子，配点法和Galerkin法的迭代法都能快速收敛，且与离散点数无关。预条件子对总计算时间没有显著影响。
We study the iterative solution of dense linear systems that arise from boundary element discretizations of the electrostatic integral equation in magnetoencephalography (MEG). We show that modern iterative methods can be used to decrease the total computation time by avoiding the time-consuming computation of the LU decomposition of the coefficient matrix. More importantly, the modern iterative methods make it possible to avoid the explicit formation of the coefficient matrix which is needed when a large number of unknowns are used. To study the convergence of iterative solvers we examine the eigenvalue distributions of the coefficient matrices. For the sphere we show how the eigenvalues of the integral operator are approximated by the eigenvalues of the coefficient matrix when the collocation and Galerkin methods are used as discretization methods. The collocation method approximates the eigenvalues of the integral operator directly. The Galerkin method produces a coefficient matrix that needs to be preconditioned in order to maintain optimal convergence speed. With the ILU(0) preconditioner iterative methods converge fast and independent of the number of discretization points for both the collocation and Galerkin approaches. The preconditioner has no significant effect on the total computational time.