Publications
A dynamic likelihood approach to filtering transport processes: advection-diffusion dynamics
Johannes Krotz,Juan M. Restrepo, Jorge Ramirez
Journal of Computational Physics
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.}}}
TBD
A hybrid Monte Carlo, discontinuous Galerkin method for linear kinetic transport equations
Johannes Krotz, Cory D. Hauck, Ryan G. McClarren
Journal of Computational Physics
We present a hybrid method for time-dependent particle transport problems that combines Monte Carlo (MC) estimation with deterministic solutions based on discrete ordinates. For spatial discretizations, the MC algorithm computes a piecewise constant solution and the discrete ordinates use bilinear discontinuous finite elements. From the hybridization of the problem, the resulting problem solved by Monte Carlo is scattering free, resulting in a simple, efficient solution procedure. Between time steps, we use a projection approach to “relabel” collided particles as uncollided particles. From a series of standard 2-D Cartesian test problems we observe that our hybrid method has improved accuracy and reduction in computational complexity of approximately an order of magnitude relative to standard discrete ordinates solutions.
@article{KROTZ_2024_HybridMCDG,
title = {A hybrid Monte Carlo, discontinuous Galerkin method for linear kinetic transport equations},
journal = {Journal of Computational Physics},
volume = {514},
pages = {113253},
year = {2024},
issn = {0021-9991},
doi = {https://doi.org/10.1016/j.jcp.2024.113253},
url = {https://www.sciencedirect.com/science/article/pii/S0021999124005011},
author = {Johannes Krotz and Cory D. Hauck and Ryan G. McClarren},
keywords = {Hybrid stochastic-deterministic method, Monte Carlo, Kinetic equations, Particle transport},
abstract = {We present a hybrid method for time-dependent particle transport problems that combines
Monte Carlo (MC) estimation with deterministic solutions based on discrete ordinates. For
spatial discretizations, the MC algorithm computes a piecewise constant solution and the
discrete ordinates use bilinear discontinuous finite elements. From the hybridization of
the problem, the resulting problem solved by Monte Carlo is scattering free, resulting in
a simple, efficient solution procedure. Between time steps, we use a projection approach to
“relabel” collided particles as uncollided particles. From a series of standard 2-D Cartesian
test problems we observe that our hybrid method has improved accuracy and reduction in
computational complexity of approximately an order of magnitude relative to standard discrete
ordinates solutions.}}
TBD
Variable resolution Poisson-disk sampling for meshing discrete fracture networks
Johannes Krotz, Matthew R. Sweeney, Carl W. Gable, Jeffrey D. Hyman, Juan M. Restrepo
Journal of Computational and Applied Mathematics
We present the near-Maximal Algorithm for Poisson-disk Sampling (nMAPS) to generate point distributions for variable resolution Delaunay triangular and tetrahedral meshes in two and three-dimensions, respectively. nMAPS consists of two principal stages. In the first stage, an initial point distribution is produced using a cell-based rejection algorithm. In the second stage, holes in the sample are detected using an efficient background grid and filled in to obtain a near-maximal covering. Extensive testing shows that nMAPS generates a variable resolution mesh in linear run time with the number of accepted points. We demonstrate nMAPS capabilities by meshing three-dimensional discrete fracture networks (DFN) and the surrounding volume. The discretized boundaries of the fractures, which are represented as planar polygons, are used as the seed of 2D-nMAPS to produce a conforming Delaunay triangulation. The combined mesh of the DFN is used as the seed for 3D-nMAPS, which produces conforming Delaunay tetrahedra surrounding the network. Under a set of conditions that naturally arise in maximal Poisson-disk samples and are satisfied by nMAPS, the two-dimensional Delaunay triangulations are guaranteed to only have well-behaved triangular faces. While nMAPS does not provide triangulation quality bounds in more than two dimensions, we found that low-quality tetrahedra in 3D are infrequent, can be readily detected and removed, and a high quality balanced mesh is produced.
@article{KROTZ_2022_PoissonDiskDFN,
title = {Variable resolution Poisson-disk sampling for meshing discrete fracture networks},
journal = {Journal of Computational and Applied Mathematics},
volume = {407},
pages = {114094},
year = {2022},
issn = {0377-0427},
doi = {https://doi.org/10.1016/j.cam.2022.114094},
url = {https://www.sciencedirect.com/science/article/pii/S0377042722000073},
author = {Johannes Krotz and Matthew R. Sweeney and Carl W. Gable and Jeffrey D. Hyman and Juan M. Restrepo},
keywords = {Discrete fracture network, Maximal Poisson-disk sampling, Mesh generation, Conforming Delaunay triangulation},
abstract = {We present the near-Maximal Algorithm for Poisson-disk Sampling (nMAPS) to generate point
distributions for variable resolution Delaunay triangular and tetrahedral meshes in two and
three-dimensions, respectively. nMAPS consists of two principal stages. In the first stage,
an initial point distribution is produced using a cell-based rejection algorithm. In the
second stage, holes in the sample are detected using an efficient background grid and filled
in to obtain a near-maximal covering. Extensive testing shows that nMAPS generates a variable
resolution mesh in linear run time with the number of accepted points. We demonstrate nMAPS
capabilities by meshing three-dimensional discrete fracture networks (DFN) and the surrounding
volume. The discretized boundaries of the fractures, which are represented as planar polygons, are
used as the seed of 2D-nMAPS to produce a conforming Delaunay triangulation. The combined mesh
of the DFN is used as the seed for 3D-nMAPS, which produces conforming Delaunay tetrahedra
surrounding the network. Under a set of conditions that naturally arise in maximal Poisson-disk
samples and are satisfied by nMAPS, the two-dimensional Delaunay triangulations are guaranteed
to only have well-behaved triangular faces. While nMAPS does not provide triangulation quality
bounds in more than two dimensions, we found that low-quality tetrahedra in 3D are infrequent,
can be readily detected and removed, and a high quality balanced mesh is produced.}}
TBD