Spherical Harmonics (SH) are fundamental mathematical tools for compactly representing low-frequency spherical functions in computer graphics. Evaluating products of SH-represented functions is a core operation in many rendering algorithms and often becomes a major computational bottleneck. Existing SH product methods face two key challenges: insufficient computational efficiency, and the inflexible requirement that all input functions share the same SH order. To overcome these limitations, we introduce the Generalized Spherical Harmonics Product , a novel formulation that supports inputs with different SH orders and allows flexible truncation of the output. We further propose an efficient and practical computation approach based on Spherical Grids (SphGrid). We establish explicit bidirectional conversions between SphGrid representations and SH coefficients, thus enabling SphGrid to serve as an intermediate representation. Under this framework, SH product computation is reduced to simple point-wise multiplications on the grid. We then derive sufficient conditions on the grid resolution to guarantee exact computation. Using lower resolutions yields an approximate variant that provides a favorable trade-off between accuracy and efficiency. Our approach is not only more general but also demonstrates superior performance even for the specialized case of identical input orders. In particular, our accurate method achieves a 3.5–6.0× speedup over the state-of-the-art method in a practical setting for graphics, while our approximate variant yields lower errors and a 2.5–6.5× speedup over the state-of-the-art approximate method. We rigorously analyze the correctness and efficiency of our method and demonstrate its effectiveness in real-time rendering applications.

Grid-free Monte Carlo methods are capable of solving Poisson equations on highly complex domains. However, existing methods operate solely in the primal domain and can converge slowly due to high variance. Inspired by gradient-domain rendering, we introduce a gradient-domain framework for Poisson problems. Specifically, we devise a new Monte Carlo estimator that directly targets differences of the solution between spatially varying query locations. Further, we adopt state-of-the-art reconstruction techniques originated in gradient-domain rendering to allow efficient reconstruction of the solutions without incurring additional bias. We demonstrate the effectiveness of our technique by comparing solutions obtained using our method and several state-of-the-art baselines.