Research Impact

My research focuses on the theoretical foundations of numerical algorithms—specifically Anderson Acceleration and algebraic splitting methods. Transitioning from theory to practice, these mathematical frameworks have been adopted by U.S. federal agencies to optimize solvers for fusion energy, exascale computing, and aerospace engineering. My work also underpins clinical tools for cardiovascular health and has been integrated into the capabilities of standard software infrastructures powering national supercomputing resources.

Engagements by U.S. National Laboratories & Federal Agencies

My research on Anderson Acceleration (AA) convergence theory has been relied upon by major U.S. Department of Energy (DOE) laboratories to solve mission-critical problems in fusion energy, national security, and high-performance computing (HPC).

Lawrence Livermore National Laboratory (LLNL)

Fusion Energy
1. - Application: Coupled Transport-Gyrokinetic Turbulence Simulations (Fusion Energy).
  - Impact: Researchers at LLNL adopted my theoretical heuristic to engineer the adaptive damping protocols for the Tango fusion code. The laboratory’s technical report cites my work (Evans et al., SINUM 2020) to optimize solver performance for high-stakes fusion simulations.
  - Reference: LLNL Technical Report LLNL-PROC-864392 | [Link to Report]
U.S. DOE Standard Software Infrastructure (SUNDIALS/KINSOL)

- - Application: Modernization of the KINSOL Nonlinear Solver.
  - Impact: My convergence analysis (SINUM 2020) provided the theoretical justification for new "varying damping and depth" capabilities released in the SUNDIALS/KINSOL suite. This software is the standard nonlinear solver infrastructure used across the DOE complex for large-scale scientific computing.
  - Reference: LLNL Technical Report LLNL-JRNL-2007253 | [Link to Report]

Oak Ridge National Laboratory (ORNL)

Exascale Astrophysics
- - Application: Neutrino Transport in Core-Collapse Supernovae.
  - Impact: My work (SINUM 2020) is cited as a foundational convergence result governing the solver behavior for exascale neutrino-transport simulations, a priority area for U.S. astrophysics research.
  - Reference: The Astrophysical Journal Supplement Series (2020) | [Link to Paper]
Artificial Intelligence (FedOSAA)
- - Application: FedOSAA (Federated Optimization).
  - Impact: In the DOE-funded method FedOSAA, my convergence-rate theorem (SINUM 2019 & 2020) provides the mathematical guarantee for accelerating fixed-point iterations in privacy-preserving distributed machine learning.
  - Reference: OSTI ID 3002238 | [Link to OSTI]

Biomedical & Clinical Hemodynamics

Application: Patient-Specific Cardiovascular Simulation & Diagnostics.
Impact: My algebraic splitting method (CMAME 2015) has been cited by the research group of Prof. Alessandro Veneziani (Emory University) to improve the stability and speed of Navier–Stokes solvers used in blood-flow simulations. This work supports NIH R01–funded projects focused on cardiovascular health.
Commercial Utilization: The solver frameworks are also utilized by the medical technology company Covanos Inc., ensuring the computational efficiency required for clinical-grade cardiovascular diagnostics.
Reference: Computer Methods in Applied Mechanics and Engineering | [Link to Paper]
Adopter: [Covanos Inc.] (Link to Company)

Standardization in Textbook (SIAM)

Book: Solving Nonlinear Equations with Iterative Methods: Solvers and Examples in Julia (C.T. Kelley, 2022).
Publisher: Society for Industrial and Applied Mathematics (SIAM).
Impact: My convergence results for Anderson Acceleration (SINUM 2019 & 2020) have been featured into the standard curriculum for nonlinear solvers. This textbook, authored by Prof. C.T. Kelley, utilizes my findings to define the modern theoretical understanding of acceleration methods for the next generation of applied mathematicians.
Link: [SIAM Bookstore]

Kelley's Book 2022

U.S. Air Force (AFOSR) – Aerospace & Hypersonics

Application: Fluid–Structure Interaction & Hypersonic Simulations.
Impact: In research funded by the Air Force Office of Scientific Research (AFOSR), my improved algebraic splitting method (CMAME 2017 & SISC 2017) is cited as a "new alternative" for pressure–velocity coupling. It serves as a benchmark for accuracy and stability in Computational Fluid Dynamics (CFD) solvers used for aerospace applications.
Reference: Mathematics of Computation (American Mathematical Society) | [Link to Paper]

Key References:

[CMAME 2015] L. Rebholz and M. Xiao. On reducing the splitting error... [View Paper on Research Page]
[SISC 2017] L. Rebholz and M. Xiao. Improved accuracy in algebraic splitting methods... [View Paper on Research Page]
[SINUM 2019] Pollock, S., Rebholz, L., & Xiao, M. Anderson-accelerated convergence of Picard iterations... [View Paper on Research Page]
[SINUM 2020] Evans, C., Pollock, S., Rebholz, L., & Xiao, M. A Proof that Anderson Acceleration... [View Paper on Research Page]

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