Non-dissipative and structure-preserving emulators via spherical optimization. (12th August 2022)

• Record Type: Journal Article
• Authors: Dai, Dihan; Epshteyn, Yekaterina; Narayan, Akil
• Is Part Of: Information and inference. Volume 12:Number 1(2023)
• Page Start: 494
• Page End: 523
• Publication Date: 2022-08-12
• Subjects: structure-preserving emulators -- high-order accuracy -- quadratic programming -- geodesic convex optimization; Mathematical models -- Periodicals; 519.605
• DOI: 10.1093/imaiai/iaac021
• Languages: English
• ISSNs: 2049-8764

Abstract

Abstract: Approximating a function with a finite series, e.g., involving polynomials or trigonometric functions, is a critical tool in computing and data analysis. The construction of such approximations via now-standard approaches like least squares or compressive sampling does not ensure that the approximation adheres to certain convex linear structural constraints, such as positivity or monotonicity. Existing approaches that ensure such structure are norm-dissipative and this can have a deleterious impact when applying these approaches, e.g., when numerical solving partial differential equations. We present a new framework that enforces via optimization such structure on approximations and is simultaneously norm-preserving. This results in a conceptually simple convex optimization problem on the sphere, but the feasible set for such problems can be very complex. We establish well-posedness of the optimization problem through results on spherical convexity and design several spherical-projection-based algorithms to numerically compute the solution. Finally, we demonstrate the effectiveness of this approach through several numerical examples.