We present a variance reduction technique for Walk on Spheres (WoS) that solves elliptic partial differential equations (PDEs) by combining overlapping harmonic expansions of the solution, each estimated using unbiased Monte Carlo random walks. Our method supports Laplace and screened Poisson equations with Dirichlet, Neumann, and Robin boundary conditions in both 2D and 3D. By adaptively covering the domain with local expansion regions and reconstructing the solution inside each region using an infinite Fourier series of the harmonic function, our method achieves over an order of magnitude lower error than traditional pointwise WoS in equal time. While low-order truncations of the series typically introduce limited bias, we also introduce a stochastic truncation scheme that eliminates this bias in the reconstructed solution. Compared to recently developed caching algorithms for WoS, such as Boundary and Mean Value Caching, our approach yields solutions with lower error and fewer correlation artifacts.

Conceptual design is an essential stage in the design process, and its ultimate success largely depends on designers’ creativity. Both physical and digital prototypes are commonly adopted by designers to support ideation and creativity, providing intuitive perception and rapid iteration, respectively. In recent advancements, large-scale generation models are able to offer data-enabled creativity support by generating high-quality solutions comparable to human designers. This opens up an imaginary space for designers and brings new possibilities for design tools. In this study, we proposed a hybrid prototype method that synergistically combines physical models and generative artificial intelligence (AI) in the conceptual design stage. Correspondingly, we developed a hybrid prototype system to implement the proposed method. We conducted a comparative user study with 45 designers who completed a design task using the physical prototype method, standalone generative AI and the hybrid prototype method, respectively. Our results verified the effectiveness of the hybrid prototype method and investigated its mechanism in supporting creativity. Finally, we discussed the application value and optimisation space of the hybrid prototype method.