Greedy search of optimal approximate solutions

TL;DR

This paper introduces a greedy algorithm-based procedure for approximating solutions to parameter-dependent ill-posed problems, ensuring asymptotic optimality and providing a theoretical framework applicable to various problem types.

Contribution

It develops a novel greedy algorithm approach with Tychonoff regularization for solving ill-posed problems, offering a general theoretical framework and specific application to elliptic problems.

Findings

Algorithm preserves Kolmogorov approximation rates

Provides a-priori estimates via Tychonoff regularization

Applicable to a broad class of ill-posed problems

Abstract

In this paper we develop a procedure to deal with a family of parameter-dependent ill-posed problems, for which the exact solution in general does not exist. The original problems are relaxed by considering corresponding approximate ones, whose optimal solutions are well dfined, where the optimality is determined by the minimal norm requirement. The procedure is based upon greedy algorithms that preserve, at least asymptotically, Kolmogorov approximation rates. In order to provide a-priori estimates for the algorithm, a Tychonff-type regularization is applied, which adds an additional parameter to the model. The theory is developed in an abstract theoretical framework that allows its application to different kinds of problems. We present a specific example that considers a family of ill-posed elliptic problems. The required general assumptions in this case translate to rather natural uniform lower and upper bounds on coefficients of the considered operators.