Local spectral approximation of unbounded operators: non-asymptotic and unified error quantification for subspace methods
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866909980116910080 |
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| author | Stroschein, Timothy |
| author_facet | Stroschein, Timothy |
| contents | We introduce a framework for subspace methods which approximate the spectra of self-adjoint, unbounded operators in a local region. Using the projection-valued measure, we derive integrated spectral inequalities that also apply to unbounded operators. Our framework is non-asymptotic, gap-independent, and enables a unified error quantification of numerical routines subject to multiple error sources. Furthermore, we formalize the class of methods applicable to our framework, and establish a rigorous foundation for dimension detection in the presence of noise as solution to frequent numerical artifacts such as spectral pollution. The practical relevance of this non-asymptotic analysis is substantiated by its recent application to sampled prolate filter diagonalization, where it successfully predicted a sharp accuracy transition linking spectral density to the minimal observation time required to decompose a signal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_07513 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Local spectral approximation of unbounded operators: non-asymptotic and unified error quantification for subspace methods Stroschein, Timothy Numerical Analysis Mathematical Physics Spectral Theory 65F15, 65J10, 47A75, 47B25, 47A58 G.1.2; G.1.3 We introduce a framework for subspace methods which approximate the spectra of self-adjoint, unbounded operators in a local region. Using the projection-valued measure, we derive integrated spectral inequalities that also apply to unbounded operators. Our framework is non-asymptotic, gap-independent, and enables a unified error quantification of numerical routines subject to multiple error sources. Furthermore, we formalize the class of methods applicable to our framework, and establish a rigorous foundation for dimension detection in the presence of noise as solution to frequent numerical artifacts such as spectral pollution. The practical relevance of this non-asymptotic analysis is substantiated by its recent application to sampled prolate filter diagonalization, where it successfully predicted a sharp accuracy transition linking spectral density to the minimal observation time required to decompose a signal. |
| title | Local spectral approximation of unbounded operators: non-asymptotic and unified error quantification for subspace methods |
| topic | Numerical Analysis Mathematical Physics Spectral Theory 65F15, 65J10, 47A75, 47B25, 47A58 G.1.2; G.1.3 |
| url | https://arxiv.org/abs/2505.07513 |