Non-Dissipative and Structure-Preserving Emulators via Spherical Optimization

TL;DR

This paper introduces a novel convex optimization framework on the sphere that constructs structure-preserving, norm-preserving function approximations, addressing limitations of existing dissipative methods in data analysis and PDE solving.

Contribution

It develops a new spherical optimization approach that enforces structural constraints while preserving norms, with proven well-posedness and efficient algorithms.

Findings

Effective structure-preserving approximations demonstrated

Algorithms successfully compute solutions in complex feasible sets

Numerical examples validate the approach's advantages

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.