Mohammad Sina Nabizadeh

Vakonomic Fluids

Ritoban Roy-Chowdhury^*, Mohammad Sina Nabizadeh^*, Oliver Gross, Anthony Gruber^†, Albert Chern^† * Equal contribution † Equal senior authorship

Preprint

We introduce a novel structure-preserving simulation of incompressible fluid flow by discretizing it as a Hamiltonian system on a space of matrices, based on Koopman operator theory. This "matrix-hydrodynamics" approach preserves the geometric structure of the underlying physics, including Casimir invariants and a discrete version of Kelvin's circulation theorem, allowing it to capture intricate vortical structures while remaining numerically stable even on coarse meshes.

Preprint

Walk on Decomposed Subdomains: A Hybrid Monte Carlo-Deterministic Solver for Elliptic PDEs

Clément Jambon, Mohammad Sina Nabizadeh, Mina Konaković Luković

SIGGRAPH 2026 Best Paper Award

WoDS is a hybrid Monte Carlo-deterministic solver for elliptic PDEs on complex geometries. Our method decomposes the domain into simple subdomains, estimates local solution operators with Monte Carlo and couples them via a sparse linear system, delivering low-variance solutions orders of magnitude faster than pure Monte Carlo, without volumetric meshing.

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Schrödinger Bridges on Discretized Geometric Domains

Leticia Mattos Da Silva, Mohammad Sina Nabizadeh, Justin Solomon

SIGGRAPH 2026

We introduce a spatially discrete formulation of the Schrödinger bridge problem on meshes and grids that enables structure-preserving interpolation between probability distributions. We demonstrate our approach across mesh- and grid-based applications, including displacement interpolation, shape interpolation, and color histogram manipulation, highlighting its ability to achieve geometric fidelity with computational efficiency.

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Tangent Blow-Ups for Processing Non-Manifold Geometry

Alice Petrov, Mohammad Sina Nabizadeh, Ana Dodik, Justin Solomon

Symposium on Geometry Processing (SGP) 2026 Best Paper Award (Honorable Mention, top 1.5%)

We introduce the tangent blow-up, a representation inspired by algebraic geometry that restores structure at singularities by lifting to the product of the ambient space and the Grassmannian of tangent planes. We define discretized gradient, divergence, and Laplacian operators in the lifted domain and demonstrate geodesic computation, segmentation, surface parameterization, and curvature estimation on non-manifold geometry.

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Fluid Dynamics: From Geometric Formulations to Structure-preserving Simulations

Mohammad Sina Nabizadeh

Ph.D. Thesis, University of California San Diego

This dissertation develops a suite of geometric and structure-preserving methods for simulating incompressible fluid dynamics, beginning with foundational geometric formulations and culminating in structure-preserving numerical schemes.

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Fluid Implicit Particles on Coadjoint Orbits

Mohammad Sina Nabizadeh, Ritoban Roy-Chowdhury, Hang Yin, Ravi Ramamoorthi, Albert Chern

SIGGRAPH Asia 2024 Best Paper Award (Honorable Mention, top 1.5%)

We propose Coadjoint Orbit FLIP (CO-FLIP), a high order accurate, structure preserving fluid simulation method in the hybrid Eulerian-Lagrangian framework. Using a discrete Hamiltonian formulation we achieve energy and Casimir preservation; formally, the flow evolves on infinite-dimensional coadjoint orbits.

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Fluid Cohomology

Hang Yin, Mohammad Sina Nabizadeh, Baichuan Wu, Stephanie Wang, Albert Chern

SIGGRAPH 2023

We derive the dynamics of harmonic components for incompressible inviscid fluids, which has previously been overlooked in the vorticity-streamfunction formulation for Euler equations. We elucidate the physical laws associated with the new equations and show their importance in reproducing physically correct behaviors.

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Exterior Calculus in Graphics

Stephanie Wang, Mohammad Sina Nabizadeh, Albert Chern

SIGGRAPH 2023

This is a comprehensive course for graphics researchers to fill the gap between their undergraduate curriculum of multivariable calculus and cutting-edge graphics research. We use relevant examples from computer graphics to illustrate the theory of exterior calculus leading to advanced topics including continuum mechanics, fluid dynamics, and geometric optimizations.

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Wave Simulations in Infinite Spacetime

Chad McKell, Mohammad Sina Nabizadeh, Stephanie Wang, Albert Chern

Under Review

We present a method for solving the wave equation on the entire infinite domain using only finite computation time and memory. We perform wave simulations in infinite spacetime using a finite discretization of the bounded spacetime with no additional loss of accuracy introduced by the Kelvin transformation.

ArXiv Link

Covector Fluids

Mohammad Sina Nabizadeh, Stephanie Wang, Ravi Ramamoorthi, Albert Chern

SIGGRAPH 2022

We simulate Euler equations for inviscid fluids by using a novel formulation based on covectors. This new formulation is an entirely velocity-based method in the vein of advection-projection methods while emulating vortex methods. This allows for capturing more intricate vortex dynamics and reducing dissipation of energy with low computational overhead.

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Kelvin Transformations for Simulations on Infinite Domains

Mohammad Sina Nabizadeh, Ravi Ramamoorthi, Albert Chern

SIGGRAPH 2021

We solve infinite-domain simulation problems by applying the Kelvin transform. The transformation turns infinite-domain PDEs into bounded-domain ones via a domain inversion and a singularity factorization.

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