A polynomial dimension-dependence analysis of Bramble--Pasciak--Xu preconditioners
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915658031169536 |
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| author | Jiang, Boou Park, Jongho Xu, Jinchao |
| author_facet | Jiang, Boou Park, Jongho Xu, Jinchao |
| contents | We investigate the dimension dependence of Bramble--Pasciak--Xu (BPX) preconditioners for high-dimensional partial differential equations and establish that the condition numbers of BPX-preconditioned systems grow only polynomially with the spatial dimension. Our analysis requires a careful derivation of the dimension dependence of several fundamental tools in the theory of finite element methods, including the elliptic regularity, Bramble--Hilbert lemma, trace inequalities, and inverse inequalities. We further introduce a new quasi-interpolation operator into finite element spaces, a variant of the classical Scott--Zhang interpolation, whose associated constants scale polynomially with the dimension. Building on these ingredients, we prove a multilevel norm equivalence theorem and derive a BPX preconditioner with explicit polynomial bounds on its dimensional dependence. This result has notable implications for emerging quantum computing methodologies: recent studies indicate that polynomial dependence of BPX preconditioners on dimension can yield exponential speedups for quantum-algorithmic approaches over their classical counterparts. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_06166 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A polynomial dimension-dependence analysis of Bramble--Pasciak--Xu preconditioners Jiang, Boou Park, Jongho Xu, Jinchao Numerical Analysis 65N55, 65N30, 65F08 We investigate the dimension dependence of Bramble--Pasciak--Xu (BPX) preconditioners for high-dimensional partial differential equations and establish that the condition numbers of BPX-preconditioned systems grow only polynomially with the spatial dimension. Our analysis requires a careful derivation of the dimension dependence of several fundamental tools in the theory of finite element methods, including the elliptic regularity, Bramble--Hilbert lemma, trace inequalities, and inverse inequalities. We further introduce a new quasi-interpolation operator into finite element spaces, a variant of the classical Scott--Zhang interpolation, whose associated constants scale polynomially with the dimension. Building on these ingredients, we prove a multilevel norm equivalence theorem and derive a BPX preconditioner with explicit polynomial bounds on its dimensional dependence. This result has notable implications for emerging quantum computing methodologies: recent studies indicate that polynomial dependence of BPX preconditioners on dimension can yield exponential speedups for quantum-algorithmic approaches over their classical counterparts. |
| title | A polynomial dimension-dependence analysis of Bramble--Pasciak--Xu preconditioners |
| topic | Numerical Analysis 65N55, 65N30, 65F08 |
| url | https://arxiv.org/abs/2512.06166 |