From Walks to Skeletons: A Reusable Transport Architecture for Interactive Monte Carlo PDE Solvers

Yunfeng Liao; Yucheng Wu; Xiucheng Li

Home/Archive/arXiv

arXiv · 2610.36931

From Walks to Skeletons: A Reusable Transport Architecture for Interactive Monte Carlo PDE Solvers

Yunfeng Liao·Yucheng Wu·Xiucheng Li

View original source ↗Read source document ↗Export metadata ↓

Abstract

Walk-on-Spheres (WoS) enables mesh-free solutions of elliptic PDEs, but conventional methods repeatedly discard sampled trajectories, limiting their efficiency in interactive applications. We introduce reusable transport skeletons, a persistent representation that organizes stochastic walks into a shared computational graph. Our framework enables efficient reuse across spatial queries and boundary conditions, while supporting progressive refinement, structure-preserving compression, and local adaptation to dynamic geometry. It further enables interactive editing of transport between selected regions. Experiments demonstrate substantial computational savings while preserving solution accuracy. By transforming transient stochastic trajectories into reusable transport structures, our method provides a flexible foundation for interactive Monte Carlo PDE solving.

Explore related subjects

cs.GR cs.NA math.NA

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Yunfeng Liao, Yucheng Wu, Xiucheng Li. 2026-10-27. From Walks to Skeletons: A Reusable Transport Architecture for Interactive Monte Carlo PDE Solvers. https://arxiv.org/abs/2610.36931

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Total Deformation-based MPM for Perfectly Elastic Solid Simulation with Strong Deformations

We present a new MPM-based solver for perfectly elastic solids, which can recover from strong deformations and naturally allows contact handling. Typical MPM solvers suffer from numerical fracture, where objects break apart even when no plasticity is included in the elasticity model. Previous approaches for perfectly elastic solids solve this by computing forces on a fixed grid in the reference configuration. However, this makes them incompatible with standard collision handling methods. We instead propose a method for deformable solid simulation without numerical fracture, that still computes elastic forces on the current grid. Our method can easily include grid-based collision handling, as commonly used in MPM frameworks. Additionally, we compute the deformation directly from the particle positions and a fixed reference grid, without any updates based on velocity gradients. This facilitates the inclusion of particle-level collision handling and zero-energy mode suppression, which counteracts artifacts from non-linear deformations. As a result, we can efficiently simulate complex scenarios of elastic objects undergoing large deformations, including contact.

cs.GR↗

arXiv2026

DomeGS: Perceptually Consistent Background Completion for Unbounded 3D Gaussian Splatting in Virtual Reality

The growing need for high-quality 3D content remains among the most prevalent challenges in virtual reality (VR). Recently, novel view synthesis techniques have enabled rapid content creation, with 3D Gaussian Splatting (3DGS) emerging as one of the most promising methods for generating high-fidelity 3D scenes. However, using 3DGS to reconstruct unbounded outdoor scenes can leave horizons incomplete due to missing data or a lack of distant geometric constraints. Such incompleteness can break immersion, compromising VR's core characteristic. A typical workaround in traditional VR systems is a textured cube (skybox) or sphere (skydome) surrounding the environment to create the illusion of a distant background. However, these approaches struggle to blend seamlessly with the reconstructed 3DGS geometry. To address these issues, we introduce the DomeGS framework, which incorporates the skydome concept not as a flat background shell, but as a part of the reconstructed Gaussian Splatting 3D scene. By integrating the horizon and sky into the 3DGS representation, our method provides a unified, scene-consistent environment bridging foregrounds and backgrounds for greater realism. Experimental evaluation with N=36 users demonstrates that DomeGS significantly increases spatial presence (p=0.042) and realism (p=0.042) compared to the skybox baseline.

cs.GR↗

arXiv2026

GaussianSpline: Adaptive Bézier Splines for Highly Dynamic 4D Gaussian Splatting

We present GaussianSpline, a dynamic 4D Gaussian Splatting method that models per-Gaussian motion using adaptive, non-uniform piecewise cubic Bézier splines. Unlike approaches based on global deformations, fixed-order polynomials, or uniformly parameterized splines, GaussianSpline jointly learns the Bézier control points and the temporal partition of each Gaussian's trajectory. This adaptive representation allocates greater temporal capacity to highly dynamic regions while maintaining compact trajectories for near-static regions, enabling high-fidelity reconstruction of complex and rapidly changing motion. To jointly achieve globally smooth motion and local adaptivity, we enforce $G^1$ continuity between consecutive Bézier segments using a reflection-based constraint on their control points. This produces smooth, fully differentiable trajectories while preserving the flexibility of non-uniform temporal parameterization. Experiments on the Neural 3D Video benchmark, the Google Immersive dataset, as well as on synthetic scenes demonstrate that GaussianSpline substantially improves reconstruction quality in scenes with highly non-uniform and challenging dynamics, while maintaining competitive performance on general dynamic scenes compared with state-of-the-art 4D Gaussian Splatting methods.

cs.GR↗