Lazar M, Zuazua Iriondo E (2023)
Publication Language: English
Publication Status: Accepted
Publication Type: Journal article, Online publication
Future Publication Type: Journal article
Publication year: 2023
Book Volume: 8
Pages Range: 547-564
Article Number: 2
URI: https://dcn.nat.fau.eu/wp-content/uploads/greedy-solving-systems.pdf
Open Access Link: http://yokohamapublishers.jp/online2/oppafa/vol8/p547.html
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 defined, 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 Tychonoff-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.
APA:
Lazar, M., & Zuazua Iriondo, E. (2023). Greedy search of optimal approximate solutions. Pure and Applied Functional Analysis, 8, 547-564.
MLA:
Lazar, Martin, and Enrique Zuazua Iriondo. "Greedy search of optimal approximate solutions." Pure and Applied Functional Analysis 8 (2023): 547-564.
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