ACMD Seminar: Analytic solutions to nonlinear ODEs via spectral power series

Rebecca Morrison
Assistant Professor, Computer Science, University of Colorado - Boulder

Tuesday, May 6, 2025, 3:00-4:00 PM ET (1:00-2:00 PM MT)

 In-person at Boulder 1-1107 with VTC to Gaithersburg Bldg. 101 LR-D*
Online at: Zoom Gov (email seminar chairs for link to talk)

Abstract: Solutions to most nonlinear ordinary differential equations (ODEs) rely on numerical solvers, but this gives little insight into the nature of the trajectories and is relatively expensive to compute. In this talk, we derive analytic solutions to a class of nonlinear, homogeneous ODEs with linear and quadratic terms on the right-hand side. We formulate a spectral power series (SPS) expansion of each state variable, whose function depends on the eigenvalues of the linearized system, and solve for the coefficients using some linear algebra and combinatorics. The SPS method overlaps with approaches using the eigendecomposition of the Koopman operator. While Koopman theory is powerful and quite general, the SPS method is so far restricted to a specific class of systems, but the proof is constructive and we can also show convergence. Various experiments exhibit quickly decaying coefficients, such that a good approximation to the true solution consists of just a few terms.

Bio: Rebecca Morrison is an assistant professor in the Computer Science Department at CU Boulder. Before joining CU, she was a postdoc in the AeroAstro Department at MIT, received her PhD in Computational Science, Engineering, and Mathematics at UT Austin, and completed a BA in Physics at Scripps College. Rebecca’s research identifies and exploits sparse structures (sparse graphical models or sparse dynamics) in nonlinear, interacting, multi-physics, and/or non-Gaussian systems. She also works on a variety of topics within the fields of calibration, validation, and uncertainty quantification, and has experience with applications including combustion, epidemiology, climate, satellites and reentry vehicles, and space weather. Rebecca currently serves as Program Director for the SIAM Activity Group on Data Science; contributes to computational libraries for Bayesian methods, uncertainty quantification, and epidemiology; and received the Johnson & Johnson Women in STEM2D Award in Mathematics.

Host: Zach Grey

Note: This talk will be recorded to provide access to NIST staff and associates who could not be present to the time of the seminar. The recording will be made available in the Math channel on NISTube, which is accessible only on the NIST internal network. This recording could be released to the public through a Freedom of Information Act (FOIA) request. Do not discuss or visually present any sensitive (CUI/PII/BII) material. Ensure that no inappropriate material or any minors are contained within the background of any recording. (To facilitate this, we request that cameras of attendees are muted except when asking questions.)

*Safety Precaution: The hallway leading from the Courtyard to the exit closest to B-111 and B-113 will be used by contractors to move debris, machinery, and other supplies, as well as will be heavily trafficked by the contractors throughout the process. Be aware of the safety precautions posted during this time.

Note: Visitors from outside NIST must contact Meliza Lane at least 24 hours in advance.