时域有限差分法的FPGA加速与舍入误差控制方法研究

Using Finite Difference Time Domain (FDTD) method to efficiently calculate the broadband electromagnetic scattering characteristics of electrically large target has important theoretical and practical significance. Currently, the common hardware acceleration methods in the context of computational electromagnetics are multi-core CPUs and GPU acceleration. On the other hand, in case of large-scale, long-term numerical simulation, accumulated rounding error may lead to unreliable calculation results. By analyzing the features of the FDTD method, this project aims to implement a hardware accelerator based on the FPGA platform. This method is expected to have significant advantages in speed and power consumption. In addition, this project proposes two approaches to reduce the rounding error with the aid of hardware architecture: (1) employ fused floating-point units to reduce the number of rounding operations; (2) employ fused floating-point units and Half-Unit-Biased floating point number to optimize hardware architecture, as a result, higher-precision floating-point arithmetic is supported by the identical hardware resources. At last, the project utilizes interval analysis method and FDTD accelerator to verify the proposed rounding error control method from both theoretical and experimental perspectives, which yields the reliable calculation results.

利用时域有限差分法（FDTD）快速、准确求解电大尺寸目标的宽带电磁散射特性具有重要的理论与工程价值。目前在计算电磁学领域里，常见的硬件加速方法主要是多CPU并行和GPU加速。另一方面，针对大规模、大尺度、长时程的数值仿真运算，累积舍入误差可能导致不可信的计算结果。本项目通过分析时域有限差分法的计算特点，实现一款基于FPGA平台的硬件加速器，该方案预计在速度和功耗上具有显著优势。此外，本项目提出两种方法在硬件架构上降低舍入误差：（1）通过融合式浮点运算单元减少舍入操作次数；（2）利用融合式浮点运算单元和半单元偏置浮点数优化硬件结构，在相同的硬件资源下支持更高精度的浮点运算。最后，本项目利用区间分析法和FDTD加速器，在理论和实验两个方面验证提出的舍入误差控制方法，确保计算结果可信。

FPGA技术已在通信,人工智能,工业控制等领域得到了广泛的应用。因此基于FPGA的加速设计已经成为当下非常重要的研究课题之一。舍入误差控制方法在矩阵分解、线性方程组求解等科学计算的优化中有着广泛的应用。主要研究内容包括时域有限差分法的硬件加速核设计，基于有限差分法的互连线寄生参数建模，混合精度计算与误差补偿算法。已完成时域有限差分法的FPGA加速设计，基于混合精度迭代优化的误差补偿算法设计，基于对数域的高能效半精度浮点运算系统设计，基于分式线性映射的匹配网络设计、调谐的理论体系构建和集成电路互连线寄生电容分析算法设计。已获得国家重点研发计划“高动态微光图像探测器件”，军科委创新特区重点项目“低噪声读出电路技术”以及大连市重点领域创新团队的进一步支持。研究成果已应用在集成电路互连线寄生电容的求解、大型稀疏矩阵方程求解、某时域有限差分法的并行求解器等方面，并且已取得较好效果。研究成果在集成电路设计，数值计算等科学领域具有一定的价值。