We present an algorithm for compressing the radiosity view factor model commonly used in radiation heat transfer and computer graphics. We use a format inspired by the hierarchical off-diagonal low rank format, where elements are recursively partitioned using a quadtree or octree and blocks are compressed using a sparse singular value decomposition -- the hierarchical matrix is assembled using dynamic programming. The motivating application is time-dependent thermal modeling on vast planetary surfaces, with a focus on permanently shadowed craters which receive energy through indirect irradiance. In this setting, shape models are comprised of a large number of triangular facets which conform to a rough surface. At each time step, a quadratic number of triangle-to-triangle scattered fluxes must be summed; that is, as the sun moves through the sky, we must solve the same view factor system of equations for a potentially unlimited number of time-varying righthand sides. We first conduct numerical experiments with a synthetic spherical cap-shaped crater, where the equilibrium temperature is analytically available. We also test our implementation with triangle meshes of planetary surfaces derived from digital elevation models recovered by orbiting spacecrafts. Our results indicate that the compressed view factor matrix can be assembled in quadratic time, which is comparable to the time it takes to assemble the full view matrix itself. Memory requirements during assembly are reduced by a large factor. Finally, for a range of compression tolerances, the size of the compressed view factor matrix and the speed of the resulting matrix vector product both scale linearly (as opposed to quadratically for the full matrix), resulting in orders of magnitude savings in processing time and memory space.

翻译：我们提出了一种用于压缩辐射传热与计算机图形学中常见辐射度视角因子模型的算法。采用分层对角低秩格式的启发式设计，通过四叉树或八叉树递归划分元素，并利用稀疏奇异值分解压缩块——借助动态规划组装分层矩阵。该研究面向持续阴影陨石坑（通过间接辐射接收能量）的时变热建模需求，涉及由大量粗糙表面三角形面片构成的行星表面模型。在每个时间步长中，需计算二次复杂度的三角形间散射通量总和：即随着太阳在天空中移动，需对潜在无限数量的时变右端项求解同一视角因子方程组。我们首先以合成球冠形陨石坑进行数值实验（其平衡温度存在解析解），同时基于轨道航天器获取的数字高程模型构建行星表面三角形网格进行测试。实验结果表明，压缩后的视角因子矩阵可在与完整矩阵组装时间相当的二次时间内完成构建，且构建过程中的内存需求大幅降低。最后，在多种压缩容差下，压缩视角因子矩阵的存储规模及矩阵-向量乘积运算速度均呈现线性缩放（相较于完整矩阵的二次复杂度），实现了处理时间与内存空间数量级的节约。