We introduce a spatially discrete formulation of the Schrödinger bridge problem on meshes and grids that enables structure-preserving and scalable interpolation between probability distributions. Our approach builds on the duality between entropy-regularized optimal transport and the log-heat equation, deriving a discrete theory that is compatible with mesh-based finite element discretizations. The resulting Sinkhorn algorithm alternates application of the heat kernel with multiplicative updates to enforce marginal constraints. Compared to interpolation via Wasserstein barycenters, our formulation produces sharper interpolants for a given level of regularization and enforces exact endpoint marginals, in addition to enjoying faster computation. It also scales to high-resolution meshes and finer temporal discretizations, avoiding the prohibitive cost of directly discretizing dynamical transport. We demonstrate our approach across mesh- and grid-based applications, including displacement interpolation, shape interpolation, and color histogram manipulation, highlighting its ability to achieve geometric fidelity with computational efficiency.

We introduce a framework for solving a class of parabolic partial differential equations on triangle mesh surfaces, including the Hamilton-Jacobi equation and the Fokker-Planck equation. PDE in this class often have nonlinear or stiff terms that cannot be resolved with standard methods on curved triangle meshes. To address this challenge, we leverage a splitting integrator combined with a convex optimization step to solve these PDE. Our machinery can be used to compute entropic approximation of optimal transport distances on geometric domains, overcoming the numerical limitations of the state-of-the-art method. In addition, we demonstrate the versatility of our method on a number of linear and nonlinear PDE that appear in diffusion and front propagation tasks in geometry processing.

Drawing a direct analogy with the well-studied vibration or elastic modes, we introduce an object’s fracture modes , which constitute its preferred or most natural ways of breaking. We formulate a sparsified eigenvalue problem, which we solve iteratively to obtain the n lowest-energy modes. These can be precomputed for a given shape to obtain a prefracture pattern that can substitute the state of the art for realtime applications at no runtime cost but significantly greater realism. Furthermore, any realtime impact can be projected onto our modes to obtain impact-dependent fracture patterns without the need for any online crack propagation simulation. We not only introduce this theoretically novel concept, but also show its fundamental and practical advantages in a diverse set of examples and contexts.