{"title":"Tractability of linear ill-posed problems in Hilbert space","authors":"Peter Mathé , Bernd Hofmann","doi":"10.1016/j.jco.2024.101867","DOIUrl":null,"url":null,"abstract":"<div><p>We introduce a notion of tractability for ill-posed operator equations in Hilbert space. For such operator equations the asymptotics of the best possible rate of reconstruction in terms of the underlying noise level is known in many cases. However, the relevant question is, which level of discretization, again driven by the noise level, is required in order to achieve this best possible accuracy. The proposed concept adapts the one from Information-based Complexity. Several examples indicate the relevance of this concept in the light of the curse of dimensionality.</p></div>","PeriodicalId":50227,"journal":{"name":"Journal of Complexity","volume":null,"pages":null},"PeriodicalIF":1.8000,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Complexity","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0885064X2400044X","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We introduce a notion of tractability for ill-posed operator equations in Hilbert space. For such operator equations the asymptotics of the best possible rate of reconstruction in terms of the underlying noise level is known in many cases. However, the relevant question is, which level of discretization, again driven by the noise level, is required in order to achieve this best possible accuracy. The proposed concept adapts the one from Information-based Complexity. Several examples indicate the relevance of this concept in the light of the curse of dimensionality.
期刊介绍：
The multidisciplinary Journal of Complexity publishes original research papers that contain substantial mathematical results on complexity as broadly conceived. Outstanding review papers will also be published. In the area of computational complexity, the focus is on complexity over the reals, with the emphasis on lower bounds and optimal algorithms. The Journal of Complexity also publishes articles that provide major new algorithms or make important progress on upper bounds. Other models of computation, such as the Turing machine model, are also of interest. Computational complexity results in a wide variety of areas are solicited.
Areas Include:
• Approximation theory
• Biomedical computing
• Compressed computing and sensing
• Computational finance
• Computational number theory
• Computational stochastics
• Control theory
• Cryptography
• Design of experiments
• Differential equations
• Discrete problems
• Distributed and parallel computation
• High and infinite-dimensional problems
• Information-based complexity
• Inverse and ill-posed problems
• Machine learning
• Markov chain Monte Carlo
• Monte Carlo and quasi-Monte Carlo
• Multivariate integration and approximation
• Noisy data
• Nonlinear and algebraic equations
• Numerical analysis
• Operator equations
• Optimization
• Quantum computing
• Scientific computation
• Tractability of multivariate problems
• Vision and image understanding.
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