Randomized Sketching Algorithms for Low-Memory Dynamic Optimization

Journal Article · Mon May 10 00:00:00 EDT 2021 · SIAM Journal on Optimization

DOI:https://doi.org/10.1137/19m1272561· OSTI ID:1784618

Muthukumar, Ramchandran ^[1]; Kouri, Drew P. ^[2]; Udell, Madeleine ^[1]

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1. Cornell Univ., Ithaca, NY (United States)
2. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

This paper develops a novel limited-memory method to solve dynamic optimization problems. The memory requirements for such problems often present a major obstacle, particularly for problems with PDE constraints such as optimal flow control, full waveform inversion, and optical tomography. In these problems, PDE constraints uniquely determine the state of a physical system for a given control; the goal is to find the value of the control that minimizes an objective. While the control is often low dimensional, the state is typically more expensive to store. This paper suggests using randomized matrix approximation to compress the state as it is generated and shows how to use the compressed state to reliably solve the original dynamic optimization problem. Concretely, the compressed state is used to compute approximate gradients and to apply the Hessian to vectors. The approximation error in these quantities is controlled by the target rank of the sketch. This approximate first- and second-order information can readily be used in any optimization algorithm. As an example, we develop a sketched trust-region method that adaptively chooses the target rank using a posteriori error information and provably converges to a stationary point of the original problem. Numerical experiments with the sketched trust-region method show promising performance on challenging problems such as the optimal control of an advection-reaction-diffusion equation and the optimal control of fluid flow past a cylinder.

• Research Organization: Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)
• Sponsoring Organization: USDOE National Nuclear Security Administration (NNSA)
• Grant/Contract Number: AC04-94AL85000
• OSTI ID: 1784618
• Report Number(s): SAND--2019-13974J; 681517
• Journal Information: SIAM Journal on Optimization, Journal Name: SIAM Journal on Optimization Journal Issue: 2 Vol. 31; ISSN 1052-6234
• Publisher: Society for Industrial and Applied Mathematics (SIAM)
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
Navier—Stokes equations
PDE-constrained optimization
adaptivity
adjoint equation
flow control
matrix approximation
randomized algorithm
single-pass algorithm
sketch18 ing
trust-region method