Dr. Brian Kha Tran

I am a Marc Kac Postdoctoral Fellow in Applied Mathematics at Los Alamos National Laboratory. I am a member of the Theoretical Division T-5 Applied Math and Plasma Physics group, mentored by Ben S. Southworth. My research focuses on structure-preserving discretization and time integration methods for differential equations arising in physics and engineering, such as adjoint systems and radiation hydrodynamics, with applications to control, optimization, and mathematical physics. 

Papers

• Properties of Adjoint Systems for Evolutionary PDEs (with B. S. Southworth and M. Leok), Journal of Nonlinear Science, submitted, 2024. arXiv.

• Order Conditions for Nonlinearly Partitioned Runge-Kutta Methods (with B. S. Southworth and T. Buvoli), SIAM Journal on Numerical Analysis, submitted, 2024. arXiv.

• Type II Hamiltonian Lie Group Variational Integrators with Applications to Geometric Adjoint Sensitivity Analysis (with M. Leok), IMA Journal of Numerical Analysis, submitted, 2023. arXiv.

• Geometric Methods for Adjoint Systems (with M. Leok), Journal of Nonlinear Science, 34:25 (75 pages), 2024. DOI. arXiv. 

• Variational Structures in Cochain Projection Based Variational Discretizations of Lagrangian PDEs (with M. Leok), Journal of Nonlinear Science, under revision, 2022. arXiv. 

• Multisymplectic Hamiltonian Variational Integrators (with M. Leok), International Journal of Computer Mathematics (Special Issue on Geometric Numerical Integration, Twenty-Five Years Later), 99(1), 113-157, 2022. DOI. arXiv. 

Thesis

• Geometric Variational Integrators for Multisymplectic PDEs and Adjoint Systems, University of California San Diego, 2023. Advisor: Prof. Melvin Leok. eScholarship.

Talks and Presentations

• Variational structures in cochain projection based discretization of classical field theories. ICIAM 2023, Minisymosium on "Finite element complexes for structure-preservation in continuum mechanics. Waseda University. 2023 (Upcoming).

• Type II Hamiltonian Lie Group Variational Integrators for Geometric Adjoint Sensitivity Analysis. Foundations of Computational Mathematics 2023, Workshop in "Geometric Integration and Computational Mathematics". Sorbonne University. 2023. 

• Type II Hamiltonian Lie Group Variational Integrators for Geometric Adjoint Sensitivity Analysis. LANL Theoretical Division Seminar. Los Alamos, NM. 2023. 

• Geoemtric Methods for Adjoint Systems. JMM 2023, AMS Contributed Paper Session on Topics in Analysis and Control Theory. Boston, MA. 2023. 

• Geometric Integration of Multisymplectic Hamiltonian PDEs. Trends in Soliton Dynamics and Singularity Formation for Nonlinear Dispersive PDEs. Texas A&M University. 2022.

• Geometric Methods for Adjoint Systems. Computational Geometric Mechanics Research Seminar, UCSD. 2022. 

• Geometric Integration of Adjoint DAE Systems. UCSD Center for Computational Mathematics Seminar. 2022. 

• Geometric Methods for Adjoint Systems. Symmetries, Invariants, and their Applications. Dalhousie University. 2022. (Poster presentation)

• Adjoint Systems via Dirac Structures. XVI Young Researchers Workshop in Geometry, Mechanics and Control. Centre de Recerca Matemàtica, UAB. 2021.

• Multisymplectic Variational Integrators for Hamiltonian PDEs. Zoom for Thought Seminar, UCSD Mathematics. 2021.

• Variational Discretizations of Lagrangian and Hamiltonian Field Theories. Advancement to Candidacy, UCSD Mathematics. 2021.

• Multisymplectic Hamiltonian Variational Integrators. UCSD Center for Computational Mathematics Seminar. 2021.

• Discrete Multisymplectic and Variational Principles of the FEM in Lagrangian Field Theory. UCSD Center for Computational Mathematics Seminar. 2020.

• Structure-Preserving Discretizations of Classical Field Theories. UCSD Center for Computational Mathematics Seminar. 2019.

Service

• Reviewer for the Journal of Computational Dynamics.

• At UCSD, I organized the Computational Geometric Mechanics Research Seminar.

Conferences Attended

• Scale Bridging Meeting and Workshop. Los Alamos National Laboratory. 2024.

• Structured Machine Learning and Time-Stepping for Dynamical Systems. Banff International Research Station, Banff (Hybrid). 2024.

• ICIAM 2023. Waseda University. 2023.

• Foundations of Computational Mathematics 2023. Sorbonne University. 2023.

• Joint Mathematics Meetings 2023. Boston, MA. 2023.

• Trends in Soliton Dynamics and Singularity Formation for Nonlinear Dispersive PDEs. Texas A&M University. 2022.

• Symmetries, Invariants, and their Applications (A Celebration of Peter Olver's 70th Birthday). Dalhousie University. 2022.

• Critical Issues in Mathematics Education 2022: Initiating, Sustaining, and Researching Mathematics Department Transformation of Introductory Courses for STEM Majors. MSRI, Berkeley (Hybrid). 2022.

• Holistic Design of Time-Dependent PDE Discretizations. ICERM, Brown University. 2022.

• XVI Young Researchers Workshop in Geometry, Mechanics and Control. Centre de Recerca Matemàtica, UAB. 2021.

• MSRI Summer Graduate School: Gauge Theory in Geometry and Topology. MSRI, Berkeley (Virtual). 2021.

• Flash GAMP (Geometric Algorithms and Methods in Physics). Virtual. 2021.

• Gone Fishing: Integrable Systems and Group Actions. Georgia Southern University (Virtual). 2020.

• Structure-Preserving Geometric Discretization of Physical Systems. Princeton. 2020.

• Structure Preservation and General Relativity. University of Cambridge. 2019.

• Chicago Summer School in Analysis. University of Chicago. 2016.

• PIMS – NSF Undergraduate Workshop on Supersymmetry. University of British Columbia, Vancouver. 2016.

Expository Writing

• CUDA Parallelization of a 2D PIC Code (with S. Tran). Final Project for Phys 244 Parallel Computation, Spring 2021.

• Group Actions on Multisymplectic Manifolds. Final Paper for Phys 220 Group Theoretical Methods in Physics, Fall 2020.

• Application of the BRST Method to Geometric Quantization. Final Paper for Phys 215C Quantum Field Theory III, Spring 2020.

• Multisymplectic Geometry and Classical Field Theories. Final Paper for Math 257 Symplectic Geometry, Winter 2020.

Undergraduate Research

• Honors Thesis (see 2016-2017) under Prof. Michael Holst and Prof. Melvin Leok. UCSD Center for Computational Mathematics. 2016-2018.

• Research Assistant under Dr. Fred Spada. UCSD Center for Memory and Recording Research. 2016-2018. 

• Research Assistant under Prof. Brian Keating. UCSD Cosmology Group. 2015-2016.