Publications

Discrete differential geometry & geometry processing

Doctoral Thesis Geometry Driven Discretization of Differential Operators

Theo Braune

Discrete Exterior Calculus (DEC) has emerged as a key tool for the structure-preserving discretization of differential forms, proving vital in fields such as computational fluid dynamics, electromagnetism, and geometry processing. By maintaining the underlying geometric and topological properties of physical systems, DEC ensures numerical and physical fidelity. However, traditional DEC is restricted to scalar-valued forms. This limitation is significant, as many fundamental theories in physics and geometry, including general relativity, non-linear elasticity, and Yang-Mills gauge theo- ries, are naturally formulated using bundle-valued differential forms and their associated covariant operators. This thesis bridges that gap by extending the DEC framework to the bundle-valued setting. In the first part, we investigate the special case of connections on simplicial 2-manifolds, addressing the challenge of representing curvature and torsion consistently in a discrete environment. We introduce a modified discrete Levi-Civita connection and a corresponding discrete representation of the torsion 2-form, demonstrating their utility within a geometry processing pipeline. In the second part, we develop a general framework for discrete bundle-valued exterior calculus. A central contribution is the formalization of integration for bundle-valued forms by leveraging the interplay between connections and retractions. This theoretical foundation enables a rigor- ous discretization scheme that induces discrete bundle-valued forms directly from their smooth counterparts. Building on this integration theory, we introduce a novel discrete exterior covariant derivative. We demonstrate that this operator converges to its continuous counterpart under mesh refinement, overcoming limitations in prior research. Furthermore, we show that the Bianchi identities are satisfied in an exact combinatorial sense and hold true in the limit. In addition, we establish that core properties of the continuous theory, such as naturality and Whitney interpo- lation, are preserved within our framework. Finally, we demonstrate that the presented construc- tions are not limited to simplicial complexes, but extend to a broader class of non-simplicial cells.

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                                    @phdthesis{thesisTheoBraune,
                                    Author = {Theo F. Braune},
                                    School = {Institut Polytechnique de Paris},
                                    Title = {Geometry Driven Discretization of Differential Operators},
                                    Url = {https://theofbraune.github.io/blog/publications/projects/dissertation/thesis.pdf},
                                    Year = {2026}}

ACM Transactions on Graphics (SIGGRAPH 2026) Efficient Multiscale Lanczos Eigenpair Extraction

Theo Braune, Jeremie Dumas, Jean-Marc Thiery

Eigenpair extractions are crucial for various applications in geometry processing and graphics. State of the Art libraries like ARPACK or Spectra rely on the implicitly restarted Lanczos iteration to extract eigenpairs efficiently. However for some large scale problems they lack convergence speed and robustness. In this paper we present a simple multigrid extension to accelerate the convergence and robustness of the implicitly restarted Lanczos method, and we demonstrate the efficiency of our method on a variety of problems commonly found in geometry processing and graphics.

PDF

@article{BDT2026MultiscaleLanczos,
author = {Braune, Theo and Dumas, Jeremie and Thiery, Jean-Marc},
title = {Efficient Multiscale Lanczos Eigenpair Extraction},
year = {2026},
issue_date = {July 2026},
publisher = {Association for Computing Machinery},
address = {New York, NY, USA},
volume = {45},
number = {5},
url = {https://doi.org/10.1145/3811367},
doi = {10.1145/3811367},
abstract = {Eigenpair extractions are crucial for various applications in geometry processing
and graphics. State of the Art libraries like ARPACK or Spectra rely
on the implicitly restarted Lanczos iteration to extract eigenpairs efficiently.
However for some large scale problems they lack convergence speed and robustness.
In this paper we present a simple multigrid extension to accelerate
the convergence and robustness of the implicitly restarted Lanczos method,
and we demonstrate the efficiency of our method on a variety of problems
commonly found in geometry processing and graphics.},
journal = {ACM Trans. Graph.},
month = jul,
articleno = {5},
numpages = {14},
keywords = {Linear Algebra, Eigenpair extraction, Multigrid, Lanczos}
}

ACM Transactions on Graphics (SIGGRAPH 2025) Discrete Torsion of Connection Forms on Simplicial Meshes

Theo Braune*, Mark Gillespie*, Yiying Tong, Mathieu Desbrun

Discrete connections are a staple of vector field design and analysis on meshes, but the notion of torsion of a discrete connection has remained unstudied. This is all the more surprising as torsion is a crucial ingredient of the smooth theory, underlying the fundamental theorem of Riemannian geometry. We extend the existing geometry processing toolbox by developing a theory of torsion for discrete connections.

PDF Supplemental Notebook Video

@article{BGTD2025DiscreteTorsion,
author = {Braune, Theo and Gillespie, Mark and Tong, Yiying and Desbrun, Mathieu},
title = {Discrete Torsion of Connection Forms on Simplicial Meshes},
year = {2025},
issue_date = {August 2025},
publisher = {Association for Computing Machinery},
address = {New York, NY, USA},
volume = {44},
number = {4},
issn = {0730-0301},
url = {https://doi.org/10.1145/3731197},
doi = {10.1145/3731197},
abstract = {While discrete (metric) connections have become a staple of n-vector field design and analysis on simplicial meshes, the notion of torsion of a discrete connection has remained unstudied. This is all the more surprising as torsion is a crucial component in the fundamental theorem of Riemannian geometry, which introduces the existence and uniqueness of the Levi-Civita connection induced by the metric. In this paper, we extend the existing geometry processing toolbox by providing torsion control over discrete connections. Our approach consists in first introducing a new discrete Levi-Civita connection for a metric with locally-constant curvature to replace the hinge connection of a triangle mesh whose curvature is concentrated at singularities; from this reference connection, we define the discrete torsion of a connection to be the discrete dual 1-form by which a connection deviates from our discrete Levi-Civita connection. We discuss how the curvature and torsion of a discrete connection can then be controlled and assigned in a manner consistent with the continuous case. We also illustrate our approach through theoretical analysis and practical examples arising in vector and frame design.},
journal = {ACM Trans. Graph.},
month = jul,
articleno = {67},
numpages = {10},
keywords = {discrete metric connections, parallel transport, discrete torsion, discrete levi-civita connections}
}

arXiv preprint · 2024 A Discrete Exterior Calculus of Bundle-valued Forms

Theo Braune, Yiying Tong, François Gay-Balmaz, Mathieu Desbrun

The discretization of Cartan's exterior calculus of differential forms has been fruitful in a variety of theoretical and practical endeavors: from computational electromagnetics to the development of Finite-Element Exterior Calculus, the development of structure-preserving numerical tools satisfying exact discrete equivalents to Stokes' theorem or the de Rham complex for the exterior derivative have found numerous applications in computational physics. However, there has been a dearth of effort in establishing a more general discrete calculus for differential forms with values in vector bundles over a combinatorial manifold equipped with a connection. We propose a discretization of the exterior covariant derivative of bundle-valued differential forms that mimics its continuous counterpart and ensures numerical convergence with mesh refinement.

PDF Code Video Manim presentation

@misc{Braune2024FullDEC,
  author        = {Braune, Theo and Tong, Yiying and
                   Gay-Balmaz, Fran{\c{c}}ois and
                   Desbrun, Mathieu},
  title         = {A Discrete Exterior Calculus of
                   Bundle-valued Forms},
  year          = {2024},
  eprint        = {2406.05383},
  archivePrefix = {arXiv},
  primaryClass  = {math.NA},
}