基于配点策略的新型间断小波Galerkin法复杂GPR模型正演

Discontinuous Galerkin method (DGM) can easily work with unstructured grids. It has the capacity to deal with discontinuous interfaces among different electric bodies. It also can be easily paralleled on cluster. But it has the disadvantages of using more unknowns and more memory. In order to further improve the performance of the DGM algorithm, this project introduces the new collocation strategy into the wavelet collocation method, which can control the increasing or decreasing of the grid node densities by using wavelet coefficient thresholds. Then, it uses the automatic Delaunay unstructured grid generation technique to simplify the generation of grids. Then, the so-called second generation wavelet is constructed by lifting scheme. It also uses the shape function of the discontinuous wavelet Galerkin method (DWGM) algorithm. Combined with the posterior error estimation algorithm, the grid size and the wavelet order and wavelet scale are adaptively adjusted where the small and high order wavelet units are used in the strong singularity region and the smooth region adopts large grid and low order wavelet units, so as to achieve the algorithm hp adaptiveity. When the selected wavelet has tightly supported orthogonality, the obtained coefficient matrix is sparse, and the computational complexity can be greatly reduced by threshold operation. Combined with the parallel technology of CUDA architecture platform, we aim to improve the efficiency of DWGM. The new algorithm will have broad applications in geophysical fields.

间断Galerkin法能与非结构化网格结合，具有灵活处理电性参数分界面的能力，易于实现并行运算，其缺点是具有更多的未知数，内存占用更多。为了进一步提升该算法的性能，本项目引入小波配点法中的配点策略，以小波系数阈值控制网格节点的增减，构建Delaunay非结构化网格，从而简化网格节点的生成方式，解决网格疏密自动调控的难题。然后，采用提升方案构造第二代小波，并将它作为间断小波Galerkin法(DWGM)的形函数，结合后验误差估计，实现小波阶数和小波尺度由粗及细动态调节，达到在梯度变化大的区域用小网格、高阶小波单元，光滑区域用大网格、低阶小波单元，实现hp自适应。当选用的小波具有紧支撑正交特性时，所获得的系数矩阵是稀疏的，通过阈值运算，可以大大减少计算量。结合CUDA架构平台的并行技术，提高DWGM的效率。可以预见，该新算法必将具有广阔的应用前景。

本项目通过构建了复杂探地雷达模型，应用间断Galerkin方法进行了GPR正演，取得了一系列成果： (1) 推导了DGTD算法的迭代公式，并在边界处采用UPML吸收边界条件。(2) 将自适应多尺度小波配点策略(AWCM)应用到GPR正演中，具有多尺度分解和快速变换的特点。(3) 将DGTD算法与其他传统方法进行对比，分析了不同数值流量、时间离散格式、网格与基函数对计算精度和效率的影响。(4) 应用GPU并行加速策略，大大提高了计算效率。(5)推导了三维间断Galerkin算法公式，并利用三维GPR模型验证了其准确性与实用性。(6) 已发表论文22篇，其中SCI检索19篇，培养毕业硕士研究生8名，博士研究生1名，申请发明专利8项，出版专著2本，参加国际会议6次，申请人圆满地完成了各项研究内容，成果指标也超额完成。