Publications

We derive a general expression for the streaming term in radiative transport equations and other transport problems when formulated in curvilinear coordinates, emphasizing coordinate systems adapted to the geometry of the domain and the directional dependence of particle transport. By parametrizing the angular variable using a local orthonormal frame, we express directional derivatives in terms of curvature-related quantities that reflect the geometry of underlying spatial manifolds. Our formulation highlights how the interaction between coordinate choices and curvature influences the streaming operator, offering geometric interpretations of its components. The resulting framework offers intuitive insight into when and how angular dependence can be simplified and may guide the selection of coordinate systems that balance analytical tractability and computational efficiency.

@misc{krotz2025curvilinearcoordinatescurvatureradiative,
      title={Curvilinear coordinates and curvature in radiative transport}, 
      author={Johannes Krotz and Ryan G. McClarren},
      year={2025},
      eprint={2508.20852},
      archivePrefix={arXiv},
      primaryClass={math.NA},
      url={https://arxiv.org/abs/2508.20852}, 
}

When scientists simulate how light, radiation, or particles move through a medium — whether inside a star, in the Earth's atmosphere, or in a nuclear reactor — they use transport equations. At the heart of these equations is a deceptively simple piece: the streaming term, which tracks how particles move forward in space and direction.

This sounds straightforward, but there's a twist. Particles don't always move through nice, flat, box-shaped domains. Often the geometry is curved — like spherical shells around a planet, cylindrical reactors, or more irregular shapes. When the geometry is curved, the math describing particle directions also bends. Choosing the right coordinate system can make the problem much easier — or much harder.

The Core Idea: Curvature Meets Transport

This work tackles the streaming term when the coordinates themselves are curved — so-called curvilinear coordinates. Instead of sticking with the usual Cartesian grid, we allow coordinates that better match the geometry of the problem (spheres, cylinders, ellipses, translating surfaces, etc.).

The key contribution is a unified way to write the directional derivative (how particles stream) in terms of curvature quantities — intuitive geometric measures of how surfaces bend and how directions twist around each other. In plain terms:

• The math of particle transport isn't just about position and direction.
• It's also about how the shape of space (flat vs. curved) changes the apparent motion.
• By expressing the streaming operator via curvatures, we see how geometry and physics interact.

Why This Matters

Traditionally, deriving the streaming term in a curved system requires tedious, case-by-case geometry. Each new coordinate choice (spherical, cylindrical, elliptical…) brings its own messy formulas. This framework offers a common language based on curvature, which helps in three ways:

• Simplification: it tells you when certain terms vanish in a given geometry, so the equations get simpler.
• Intuition: instead of pages of indices, you reason with "is the surface flat, curved, or twisting?"
• Practical guidance: it informs coordinate choices that balance tractability and accuracy.

Putting It Into Practice

The paper walks through familiar cases — cylindrical and spherical setups — and also more involved choices like ellipsoids or translating graphs. The general recipe shows:

• Cylinders: several terms drop out because the relevant surfaces are flat and directions don't twist.
• Spheres: curvature naturally appears with clean, symmetric contributions.
• Ellipsoids / translating surfaces: complexity increases, but the curvature view still identifies which pieces matter and why.

Takeaway

The contribution isn't a new solver; it's a geometric perspective that makes derivations clearer and decisions more informed. By seeing transport through the lens of curvature, researchers can:

• avoid unnecessary terms,
• understand the effect of geometry on streaming, and
• choose coordinate systems that make the physics — and the computation — more manageable.

It's a modest but useful step: connecting particle transport — a cornerstone of many physical simulations — more directly with the language of geometry.

A Bayesian data assimilation scheme is formulated for advection-dominated advective and diffusive evolutionary problems, based upon the Dynamic Likelihood (DLF) approach to filtering. The DLF was developed specifically for hyperbolic problems –waves–, and in this paper, it is extended via a split step formulation, to handle advection-diffusion problems. In the dynamic likelihood approach, observations and their statistics are used to propagate probabilities along characteristics, evolving the likelihood in time. The estimate posterior thus inherits phase information. For advection-diffusion the advective part of the time evolution is handled on the basis of observations alone, while the diffusive part is informed through the model as well as observations. We expect, and indeed show here, that in advection-dominated problems, the DLF approach produces better estimates than other assimilation approaches, particularly when the observations are sparse and have low uncertainty. The added computational expense of the method is cubic in the total number of observations over time, which is on the same order of magnitude as a standard Kalman filter and can be mitigated by bounding the number of forward propagated observations, discarding the least informative data.

@article{KROTZ_2025_DynamicLikelihoodFilter,
title = {A dynamic likelihood approach to filtering transport processes: advection-diffusion dynamics},
journal = {Journal of Computational Physics},
volume = {536},
pages = {114089},
year = {2025},
issn = {0021-9991},
doi = {https://doi.org/10.1016/j.jcp.2025.114089},
url = {https://www.sciencedirect.com/science/article/pii/S0021999125003729},
author = {Johannes Krotz and Juan M. Restrepo and Jorge Ramirez},
keywords = {Data assimilation, Bayesian estimation, Dynamic likelihood filter, 
Advection-diffusion, Transport, Kalman filter},
abstract = {A Bayesian data assimilation scheme is formulated for advection-dominated advective 
        and diffusive evolutionary problems, based upon the Dynamic Likelihood (DLF) approach to 
        filtering. The DLF was developed specifically for hyperbolic problems –waves–, and in this
        paper, it is extended via a split step formulation, to handle advection-diffusion problems. 
        In the dynamic likelihood approach, observations and their statistics are used to propagate 
        probabilities along characteristics, evolving the likelihood in time. The estimate posterior
        thus inherits phase information. For advection-diffusion the advective part of the time 
        evolution is handled on the basis of observations alone, while the diffusive part is informed 
        through the model as well as observations. We expect, and indeed show here, that in 
        advection-dominated problems, the DLF approach produces better estimates than other 
        assimilation approaches, particularly when the observations are sparse and have low 
        uncertainty. The added computational expense of the method is cubic in the total number of 
        observations over time, which is on the same order of magnitude as a standard Kalman filter 
        and can be mitigated by bounding the number of forward propagated observations, discarding 
        the least informative data.}}}
        

Imagine you’re trying to follow the spread of smoke in the air, or how pollutants drift and mix in a river. These processes are driven by two forces: advection (movement along currents, carrying things downstream or with the wind) and diffusion (spreading and blurring out over time). Scientists want to predict these processes as accurately as possible — but the problem is, we never have perfect information. We rely on models (our best guess of the physics) and observations (data from sensors). Blending these two is called data assimilation.

The classic tool for this job is the Kalman Filter, which acts like a smart balance: constantly correcting a model with incoming data. It works well when observations are plentiful and evenly spaced. But in the real world, data is often sparse (only a few sensors, not everywhere all the time) — and this is exactly where traditional methods begin to struggle.

The Core Idea: A Dynamic Likelihood

This paper develops the Dynamic Likelihood Filter (DLF) for advection–diffusion problems. The key innovation is that instead of waiting passively for the next measurement, the DLF projects existing observations forward in time along the natural flow of the system. In other words, it doesn’t just use a measurement once and discard it. It “carries” that information along the path particles or pollutants would naturally follow, generating pseudo-observations at new times and places where sensors may not exist.

For the advective part of the process (the bulk flow), these pseudo-observations are especially powerful. For the diffusive part (the spreading and smoothing), the DLF blends in predictions from the model itself. Together, this allows the filter to maintain not only the amount of what is being transported, but also its location and phase — something traditional methods tend to lose quickly as diffusion blurs the signal.

Why This Matters

The DLF is designed for the kinds of real-world challenges that matter most:

• Sparse but reliable data: a few high-quality sensors instead of dense networks.
• Advection-dominated systems: where currents, winds, or flows move things much more than they diffuse.
• Uncertain models: situations where we know the equations but don’t capture every detail of reality.

These situations arise in meteorology (weather forecasting with limited stations), environmental monitoring (tracking contaminants in rivers or the atmosphere), and engineering (any process where materials or signals are transported through space and time).

What the Results Show

Numerical experiments in the paper show that the DLF consistently outperforms the standard Kalman Filter when advection dominates and data is sparse. It:

• Produces more accurate estimates of both intensity and location.
• Corrects errors in initial conditions more quickly.
• Handles uncertain or slightly wrong models better, staying anchored to reality.

There is a tradeoff: the DLF requires more computation. But the paper demonstrates practical ways to manage this, such as discarding “stale” pseudo-observations once they no longer add value. This keeps the method efficient while retaining its accuracy advantage.

The Bigger Picture

Originally, the Dynamic Likelihood idea was built for wave problems. Extending it to advection–diffusion, one of the most common forms of transport in nature and engineering, shows its broader potential. By projecting observations dynamically, the DLF squeezes more information out of every measurement. It makes data assimilation smarter, more resilient to sparse data, and better at tracking where things actually are.

In short: The Dynamic Likelihood Filter lets us do more with less. It turns limited observations into a continuous source of information, improving forecasts in fields from weather and climate science to pollution monitoring and engineering transport problems.