Sparse linear solvers or generalized deconvolution are fundamental to science and engineering, applied in partial differential equations (PDEs), scientific computing, computer vision, and beyond. Indirect solvers possess characteristics that make them undesirable as stable differentiable modules; existing direct solvers, though reliable, are too expensive to be adopted in neural architectures. We substantially accelerate direct sparse solvers by up to three orders of magnitude, violating common assumptions that direct solvers are too slow. We "condense" a sparse Laplacian matrix into a dense tensor, a compact data structure that batch-wise stores the Dirichlet-to-Neumann matrices, reducing the sparse solving to recursively merging pairs of dense matrices that are much smaller. The batched small dense systems are sliced and inverted in parallel to take advantage of dense GPU BLAS kernels, highly optimized in the era of deep learning. Our method is efficient, qualified as a strong zero-shot baseline for AI-based PDE solving, and a reliable differentiable module integrable into machine learning pipelines.

Computation of injective (or inversion-free) maps is a key task in geometry processing, physical simulation, and shape optimization. Despite being a longstanding problem, it remains challenging due to its highly nonconvex and combinatoric nature. We propose computation of variational quasi-harmonic maps to obtain smooth inversion-free maps. Our work is built on a key observation about inversion-free maps: A planar map is a diffeomorphism if and only if it is quasi-harmonic and satisfies a special Cauchy boundary condition. We hence equate the inversion-free mapping problem to an optimal control problem derived from our theoretical result, in which we search in the space of parameters that define an elliptic PDE. We show that this problem can be solved by minimizing within a family of functionals. Similarly, our discretized functionals admit exactly injective maps as the minimizers, empirically producing inversion-free discrete maps of triangle meshes. We design efficient numerical procedures for our problem that prioritize robust convergence paths. Experiments show that on challenging examples our methods can achieve up to orders of magnitude improvement over state-of-the-art, in terms of speed or quality. Moreover, we demonstrate how to optimize a generic energy in our framework while restricting to quasi-harmonic maps.

We propose quasi-harmonic weights for interpolating geometric data, which are orders of magnitude faster to compute than state-of-the-art. Currently, interpolation (or, skinning) weights are obtained by solving large-scale constrained optimization problems with explicit constraints to suppress oscillative patterns, yielding smooth weights only after a substantial amount of computation time. As an alternative, our weights are obtained as minima of an unconstrained problem that can be optimized quickly using straightforward numerical techniques. We consider weights that can be obtained as solutions to a parameterized family of second-order elliptic partial differential equations. By leveraging the maximum principle and careful parameterization, we pose weight computation as an inverse problem of recovering optimal anisotropic diffusivity tensors. In addition, we provide a customized ADAM solver that significantly reduces the number of gradient steps; our solver only requires inverting tens of linear systems that share the same sparsity pattern. Overall, our approach achieves orders of magnitude acceleration compared to previous methods, allowing weight computation in near real-time.

We propose using theDirichlet-to-Neumann operatoras an extrinsic alternative to the Laplacian for spectral geometry processing and shape analysis. Intrinsic approaches, usually based on the Laplace–Beltrami operator, cannot capture the spatial embedding of a shape up to rigid motion, and many previous extrinsic methods lack theoretical justification. Instead, we consider the Steklov eigenvalue problem, computing the spectrum of the Dirichlet-to-Neumann operator of a surface bounding a volume. A remarkable property of this operator is that it completely encodes volumetric geometry. We use the boundary element method (BEM) to discretize the operator, accelerated by hierarchical numerical schemes and preconditioning; this pipeline allows us to solve eigenvalue and linear problems on large-scale meshes despite the density of the Dirichlet-to-Neumann discretization. We further demonstrate that our operators naturally fit into existing frameworks for geometry processing, making a shift from intrinsic to extrinsic geometry as simple as substituting the Laplace–Beltrami operator with the Dirichlet-to-Neumann operator.