Adjoint krylov method speeds parameter inference in large linear systems
An Adjoint Krylov Approach to Parameter Inference for Affine Families of Linear Evolution Equations
Computational Engineering, Finance, and Science
Summary
Figuring out unknown parts of very big, complicated systems like chemical reactions or quantum processes is hard. The authors present a new way using something called an adjoint Krylov method that makes this problem easier and faster by smartly guessing and refining answers. Their approach reduces heavy calculations into manageable steps and runs efficiently even on a normal computer with limited memory. They tested it on different complex systems and showed it can accurately find the unknown parts needed to understand how these systems evolve.
What this means in practice
- •For quantum computing engineers: Efficiently tune parameters in Lindblad models describing quantum systems to improve simulation accuracy on limited hardware.
- •For computational chemists: Estimate reaction rates and parameters in chemical master equations using large-scale gradient computations with moderate memory usage.
Authors
Aditya Dendukuri, Shivkumar Chandrasekaran, Linda Petzold
Abstract
This paper introduces an adjoint Krylov method for parameter inference in large, sparse linear evolution equations whose system matrices depend affinely on unknown coefficients. We estimate these coefficients from discrete observations by constructing one forward and one adjoint Krylov approximation on each observation interval. Quadrature of the resulting bilinear form gives the full coefficient gradient through sparse matrix-vector products, without evaluating a separate Frechet derivative in every operator direction. We separate the gradient error into forward, adjoint, and quadrature terms and derive residual-based bounds for Markov and Lindblad generators. We also identify cases in which heuristic stopping estimates become small before the gradient has converged. With workspace reuse, we compute library gradients for a 12-qubit Lindblad generator with dimension 1.7 x 10^7 on a single machine with 16 GB of memory. Experiments on chemical master, Kuramoto-Sivashinsky, and Lindblad equations demonstrate accurate gradients and parameter recovery.