On preconditioning the treecode-accelerated boundary integral (TABI) Poisson–Boltzmann solver

We recently developed a treecode-accelerated boundary integral (TABI) solver for solving Poisson–Boltzmann (PB) equation [1]. The solver has combined advantages in accuracy, efficiency, memory, and parallelization as it applies a well-posed boundary integral formulation to circumvent many numerical difficulties associated with the PB equation and uses an O(Nlog⁡N) treecode to accelerate the GMRES iterative solver. However, as observed in our previous work [2], occasionally when the mesh generator produces low quality triangles, the number of GMRES iterations required to solve the discretized boundary integral equations Ax=b could be large. To address this issue, we design a preconditioning scheme using preconditioner matrix M such that M−1A has much improved condition while M−1z can be rapidly computed for any vector z. In this scheme, the matrix M carries the interactions between boundary elements on the same leaf only in the tree structure thus is block diagonal with many computational advantages. The sizes of the blocks in M are conveniently controlled by the treecode parameter N0, the maximum number of particles per leaf. The numerical results show that this new preconditioning scheme improves the TABI solver with significantly reduced iteration numbers and better accuracy, particularly for protein sets on which TABI solver previously converges slowly. In addition, this preconditioning scheme potentially can improve the condition number of various multipole method accelerated boundary elements solvers in scattering, fluids, elasticity, etc.