Publications and Preprints

• J. Guzmán, A. N. Hirani, B. Liu, P. Potu. Discrete Poincare and Bogovskii operators on cochains and Whitney forms. arXiv:2603.29018 [math.NA] (2026).
• C. Zhu, S. H. Christiansen, K. Hu, A. N. Hirani. Convergence and Stability of Discrete Exterior Calculus for the Hodge Laplace Problem in Two Dimensions. arXiv:2505.08966 [math.NA] (2025) [Accepted for publication in IMA Journal of Numerical Analysis.]
• B. Liu, C. Elkan, A. N. Hirani. Embeddings for Numerical Features Using tanh Activation. Proceedings of the Table Representation Learning Workshop at ACL 2025 (2025). Workshop associated with the 63rd Annual Meeting of the Association for Computational Linguistics. https://openreview.net/forum?id=eVOp3K6Pi6 (2025)
• M. D. Schubel, D. Berwick-Evans, A. N. Hirani. Averaging Property of Wedge Product and Naturality in Discrete Exterior Calculus. Advances in Computational Mathematics (2024) 10.1007/s10444-024-10179-8
• A. N. Hirani, A. T. S. Wan, N. Wojtalewicz. Conservative integrators for piecewise smooth systems with transversal dynamics. Journal of Computational Dynamics (2023) 10.3934/jcd.2023009.
• (UPDATED) D. Berwick-Evans, A. N. Hirani, and M. D. Schubel. Discrete vector bundles with connection. (Appeared originally in 2021 on arXiv as Discrete vector bundles with connection and Bianchi identity.) arXiv:2104.10277 [math.DG] (2026, 2023, 2021). A major update was posted in May 2026.
• M. Wang, P. Jagad, A. N. Hirani, R. Samtaney. Discrete exterior calculus discretization of two-phase incompressible Navier-Stokes equations with a conservative phase field method. Journal of Computational Physics. Vol. 488 (2023) 10.1016/j.jcp.2023.112245.
• A. N. Hirani, K. Kalyanaraman, H. Wang, S. Watts. Computing discrete harmonic differential forms in a given cohomology class using finite element exterior calculus. Computational Geometry: Theory and Applications 10.1016/j.comgeo.2022.101937 (2022).
• V. Karve and A. N. Hirani. GraphSAT - a decision problem connecting satisfiability and graph theory. arXiv:2105.11390 [math.CO] (2021), 31 pages.
• V. Karve, A. N. Hirani. The complete set of minimal simple graphs that support unsatisfiable 2-CNFs.Discrete Applied Mathematics Vol. 283 (2020), 123-132. 10.1016/j.dam.2019.12.017
• M. S. Mohamed, A. N. Hirani, R. Samtaney. Numerical convergence of discrete exterior calculus on arbitrary surface meshes. International Journal for Computational Methods in Engineering Science & Mechanics Vol. 19 (2018), 194-206. 10.1080/15502287.2018.1446196
• M. S. Mohamed, A. N. Hirani, R. Samtaney. Comparison of discrete Hodge star operators for surfaces. Computer-Aided Design Vol. 78 (2016), 118-125. 10.1016/j.cad.2016.05.002
• M. S. Mohamed, A. N. Hirani, R. Samtaney. Discrete exterior calculus discretization of incompressible Navier-Stokes equations over surface simplicial meshes. Journal of Computational Physics Vol. 312 (2016), 175-191.
  10.1016/j.jcp.2016.02.028
• A. Dziubek, G. Guidoboni, A. Harris, A. N. Hirani, E. Rusjan, W. Thistleton. Effect of ocular shape and vascular geometry on retinal hemodynamics: a computational model. Biomechanics and Modeling in Mechanobiology Vol. 15 (2016), 893-907. 10.1007/s10237-015-0731-8
• A. N. Hirani, K. B. Nakshatrala, J. H. Chaudhry. Numerical method for Darcy flow derived using Discrete Exterior Calculus. International Journal for Computational Methods in Engineering Science & Mechanics, Vol. 16 (2015), 151-169. 10.1080/15502287.2014.977500
• A. N. Hirani, K. Kalyanaraman, S. Watts. Graph Laplacians and Least Squares on Graphs. GABB’2015: Proceedings of Graph Algorithms Building Blocks workshop at the IEEE International Parallel and Distributed Processing Symposium, 812-821, 2015. 10.1109/IPDPSW.2015.73
• A. Demlow and A. N. Hirani. A posteriori error estimates for finite element exterior calculus: The de Rham complex . Foundations of Computational Mathematics Vol. 14 (2014), 1337-1371. 10.1007/s10208-014-9203-2.
• E. VanderZee, A. N. Hirani, D. Guoy, V. Zharnitsky, E. Ramos. Geometric and Combinatorial Properties of Well-Centered Triangulations in Three and Higher Dimensions. Computational Geometry: Theory and Applications, Vol. 46 (2013), 700-724. 10.1016/j.comgeo.2012.11.003,
• N. Bell and A. N. Hirani. PyDEC: Software and Algorithms for Discretization of Exterior Calculus. ACM Transactions on Mathematical Software, Vol. 39 (2012), 3:1-3:41. 10.1145/2382585.2382588
• A. N. Hirani, K. Kalyanaraman, and E. VanderZee Delaunay Hodge Star. Computer-Aided Design, Vol. 45 (2013), 540-544. 10.1016/j.cad.2012.10.038. Corrigendum 10.1016/j.cad.2017.09.005
• A. N. Hirani, K. Kalyanaraman, H. Wang, and S. Watts. Cohomologous Harmonic Cochains. arXiv:1012.2835 [cs.CG] (2011), 16 pages.
• A. N. Hirani, K. Kalyanaraman, and S. Watts. Least Squares Ranking on Graphs. arXiv:1011.1716 [cs.NA] (2011).
• A. N. Hirani and K. Kalyanaraman. Numerical Experiments for Darcy Flow on a Surface Using Mixed Exterior Calculus Methods. arXiv:1103.4865 [cs.NA] (2011).
• T. K. Dey, A. N. Hirani and B. Krishnamoorthy. Optimal Homologous Cycles, Total Unimodularity, and Linear Programming. SIAM Journal on Computing Vol. 40 (2011), 1026-1044. This is a journal version of the preliminary results that appeared in our STOC 2010 paper and contains additional results. 10.1137/100800245
• N. M. Dunfield and A. N. Hirani. The Least Spanning Area of a Knot and the Optimal Bounding Chain Problem. SoCG ’11: Proceedings of the 27th annual ACM symposium on Computational geometry (2011), 135–144. 10.1145/1998196.1998218
• T. K. Dey, A. N. Hirani and B. Krishnamoorthy. Optimal Homologous Cycles, Total Unimodularity, and Linear Programming. STOC '10 : 42nd ACM Symposium on Theory of Computing (2010), 221-230. 10.1145/1806689.1806721
• E. VanderZee, A. N. Hirani, D. Guoy, E. Ramos. Well-Centered Triangulation. SIAM Journal on Scientific Computing Vol. 31 (2010), 4497-4523. 10.1137/090748214
• E. VanderZee, A. N. Hirani, V. Zharnitsky, D. Guoy. A Dihedral Acute Triangulation of the Cube. Computational Geometry: Theory and Applications Vol. 43 (2010), 445-452. 10.1016/j.comgeo.2009.09.001
• A. Colombi, A. N. Hirani, B. F. Villac. Structure preserving approximations of conservative forces for application to small body dynamics. Journal of Guidance, Control, and Dynamics Vol. 32 (2009), 1847-1858. 10.2514/1.42067
• E. VanderZee, A. N. Hirani, D. Guoy. Triangulation of Simple 3D Shapes with Well-Centered Tetrahedra. Proceedings of the 17th International Meshing Roundtable (2008), 19-35. 10.1007/978-3-540-87921-3_2
• A. Colombi, A. N. Hirani, B. F. Villac. Structure preserving approximations of conservative forces for application to small body dynamics. Proceedings of AAS/AIAA Astrodynamics Specialist Conference (2008), Paper AIAA 2008-7205. PDF
• A. Colombi, A. N. Hirani, B. F. Villac. Adaptive Gravitational Force Representation for Fast Trajectory Propagation Near Small Bodies. Journal of Guidance, Control, and Dynamics Vol. 31 (2008), 1041-1051. 10.2514/1.32559
• J. Warren, S. Schaefer, A. N. Hirani, M. Desbrun. Barycentric Coordinates for Convex Sets. Advances in Computational Mathematics Vol. 27 (2007), 319-338. 10.1007/s10444-005-9008-6
• E. VanderZee, A. N. Hirani, D. Guoy, E. Ramos. Well-Centered Planar Triangulation - an Iterative Approach. Proceedings of the 16th International Meshing Roundtable (2007), 121-138. 10.1007/978-3-540-75103-8_7
• J. Jossey, A. N. Hirani. Equivalence Theorems in Numerical Analysis : Integration, Differentiation and Interpolation. arXiv:0709.4046 [math.NA] (2007).
• A. Colombi, A. N. Hirani, B. F. Villac. Efficient Gravity Field Computation for Trajectory Propagation Near Small Bodies. Proceedings of 17th AAS/AIAA Space Flight Mechanics Meeting (2007), Paper AAS 07-223. PDF
• A. N. Hirani, R. P. Russell. Approximations of Distant Retrograde Orbits for Mission Design. Proceedings of 16th AAS/AIAA Space Flight Mechanics Meeting (2006), Paper AAS 06-116 PDF
• T. Lam, A. N. Hirani, J. A. Kangas. Characteristics of Transfers to and Captures at Europa. Proceedings of 16th AAS/AIAA Space Flight Mechanics Meeting (2006), Paper AAS 06-188. PDF
• A. N. Hirani, M. W. Lo. Surface Structure of an Invariant Manifold of a Halo Orbit. Proceedings of 2005 AAS/AIAA Astrodynamics Specialists Conference (2005), Paper AAS 05-379. PDF
• M. Desbrun, A. N. Hirani, M. Leok, J. E. Marsden. Discrete Exterior Calculus. arXiv:math/0508341 [math.DG] (2005).
• M. Desbrun, A. N. Hirani, J. E. Marsden. Discrete Exterior Calculus for Variational Problems in Computer Graphics and Vision. (Invited paper). Proceedings of the 42nd IEEE Conference on Decision and Control (2003), 4902-4907. 10.1109/CDC.2003.1272393
• E. Grinspun, A. N. Hirani, M. Desbrun, P. Schröder. Discrete Shells. SCA '03: Proceedings of the 2003 ACM SIGGRAPH / Eurographics Symposium on Computer Animation (2003), 62-67. 10.5555/846276.846284
• Y. Tong, S. Lombeyda, A. N. Hirani, M. Desbrun. Discrete Multiscale Vector Field Decomposition. ACM Transactions on Graphics, Vol. 22 (2003), 445-452. 10.1145/882262.882290
• A. N. Hirani, J. E. Marsden, J. Arvo. Averaged Template Matching Equations. Proceedings of Energy Minimization Methods in Computer Vision and Pattern Recognition (EMMCVPR 2001), Springer Verlag Lecture Notes in Computer Science LNCS 2134, 2001, 528-543. 10.1007/3-540-44745-8_35
• A. N. Hirani, T. Totsuka. Combining Frequency and Spatial Domain Information for Fast Interactive Image Noise Removal. Proceedings of ACM SIGGRAPH 1996, 1996, 269-276. 10.1145/237170.237264
• A. N. Hirani, T. Totsuka, Dual Domain Interactive Image Restoration: Basic Algorithm., Proceedings of the IEEE International Conference on Image Processing, (1996), Vol. 1,, 797-800. 10.1109/ICIP.1996.559619
• A. N. Hirani, V. S. Subrahmanian, Algebraic Foundations of Logic Programming I: The Distributive Lattice of Logic Programs. Fundamenta Informaticae Vol. 13 (1990), 317-332. 10.3233/FI-1990-13306

Discrete Exterior Calculus (Ph.D. Thesis)

Discrete Exterior Calculus. California Institute of Technology, May 2003.

@phdthesis{Hirani2003,
	author = {Anil N. Hirani},
	doi = {10.7907/ZHY8-V329},
	month = {5},
	school = {California Institute of Technology},
	title = {Discrete Exterior Calculus},
	url = {https://resolver.caltech.edu/CaltechETD:etd-05202003-095403},
	year = {2003}}