AI 中文总结
研究针对亥姆霍兹方程离散化产生的密集线性系统模拟受限问题，提出改进分层矩阵压缩鲁棒性的方法，通过新准则等提高压缩可靠性，经测试解决早期收敛问题，内存占用不变，还应用于生物医学模型，证实加速大规模模拟的可行性。
AI 中文摘要
亥姆霍兹方程的积分公式离散化会产生密集线性系统，限制了大规模或高频声学模型的模拟。快速算法如分层矩阵压缩可减少内存占用，但常用的自适应交叉近似存在早期收敛问题。本文提出新的对角收敛准则、枢轴策略的附加矩阵元素、扩展的可容许条件和持续收敛检查，以提高分层矩阵压缩的鲁棒性。在各种离散化体积和边界积分算子上测试，结果表明该方法成功压缩所有基准矩阵，解决了标准算法的早期收敛问题，且内存占用相同，复杂度分析显示在恒定频率下随网格细化呈对数线性内存缩放，还成功应用于经颅超声传播模型，证实了其在生物医学应用中加速大规模高分辨率网格模拟的可行性。
英文摘要
Discretizing integral formulations of the Helmholtz equation yields dense linear systems. Hence, simulating acoustic models at larger scales or higher frequencies is typically constrained by memory capacity. Fast algorithms, such as hierarchical matrix compression, reduce the memory footprint substantially while controlling the approximation error in matrix-vector multiplications. However, the commonly used Adaptive Cross Approximation suffers from early-convergence problems, where the iterative construction of low-rank decompositions stops before reaching the targeted error tolerance. This failure arises when the error estimator does not capture significant components of the matrix structure under partial pivoting. This manuscript proposes a new diagonal convergence criterion, additional matrix elements for the pivoting strategy, an extended admissibility condition, and a sustained convergence check to improve the robustness of hierarchical matrix compression. These modifications improve compression reliability without increasing memory. We tested our compression strategy on various discretized volume and boundary integral operators. The computational results show that our approach successfully compresses all benchmark matrices within predefined tolerances, thereby resolving the early-convergence issues encountered in standard algorithms. This robust matrix compression was achieved at the same memory footprint as alternative compression strategies. Furthermore, a complexity analysis shows log-linear memory scaling with mesh refinement at constant frequency. Finally, we successfully applied our robust matrix compression algorithm to a coupled system of volume and boundary integral operators that models transcranial ultrasound propagation. This confirms the feasibility of our robust algorithm to accelerate large-scale simulations with high-resolution meshes in a biomedical application.
