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This 3 hour course provides a detailed overview of grid-free Monte Carlo methods for solving partial differential equations (PDEs) based on the walk on spheres (WoS) algorithm, with a special emphasis on problems with high geometric complexity. PDEs are a basic building block of models and algorithms used throughout science, engineering and visual computing. Yet despite decades of research, conventional PDE solvers struggle to capture the immense geometric complexity of the natural world. A perennial challenge is spatial discretization: traditionally, one must partition the domain into a high-quality volumetric mesh—a process that can be brittle, memory intensive, and difficult to parallelize. WoS makes a radical departure from this approach, by reformulating the problem in terms of recursive integral equations that can be estimated using the Monte Carlo method, eliminating the need for spatial discretization. Since these equations strongly resemble those found in light transport theory, one can leverage deep knowledge from Monte Carlo rendering to develop new PDE solvers that share many of its advantages: no meshing, trivial parallelism, and the ability to evaluate the solution at any point without solving a global system of equations. The course is divided into two parts. Part I will cover the basics of using WoS to solve fundamental PDEs like the Poisson equation. Topics include formulating the solution as an integral equation, generating samples via recursive random walks, and employing accelerated distance and ray intersection queries to efficiently handle complex geometries. Participants will also gain experience setting up demo applications involving data interpolation, heat transfer, and geometric optimization using the open-source “Zombie” library, which implements various grid-free Monte Carlo PDE solvers. Part II will feature a mini-panel of academic and industry contributors covering advanced topics including variance reduction, differentiable and multi-physics simulation, and applications in industrial design and robust geometry processing.
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This content will become publicly available on December 1, 2025
Characterizing the dispersion behavior of poly-atomic magnetic metamaterials
Abstract The propagation of magnetoinductive (MI) waves across magnetic metamaterials known as magnetoinductive waveguides (MIWs) has been an area of interest for many applications due to the flexible design and low-loss performance in challenging radio-frequency (RF) environments. Thus far, the dispersion behavior of MIWs has been limited to mono- and diatomic geometries. In this work, we present a recursive method to generate the dispersion equation for a general poly-atomic MIW. This recursive method greatly simplifies the ability to create closed-form dispersion equations for unique poly-atomic MIW geometries versus the previous method. To demonstrate, four MIW geometries that have been selected for their unique symmetries are analyzed using the recursive method. Using applicable simplifications brought on by the geometric symmetries, a closed-form dispersion equation is reported for each case. The equations are then validated numerically and show excellent agreement in all four cases. This work simultaneously aids in the further development of MIW theory and offers new avenues for MIW design in the presented dispersion equations.
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- Award ID(s):
- 2053318
- PAR ID:
- 10524060
- Publisher / Repository:
- Nature
- Date Published:
- Journal Name:
- Scientific Reports
- Volume:
- 14
- Issue:
- 1
- ISSN:
- 2045-2322
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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