Basis Design for Numerical Homogenization

Definitions and mathematical statement

The supplied record concerns numerical homogenization: the construction of finite-dimensional approximation spaces for solutions of partial differential equations whose coefficients are rough or vary on multiple scales. A basis-design method selects basis functions $\phi_1,\ldots,\phi_n$ and the associated space \[ V_n=\operatorname{span}\{\phi_1,\ldots,\phi_n\}. \] The admissible function space, the meaning of roughness, and the admissible scale dependence are not specified in the record.

Source question. Can the identification of accurate basis elements for numerical homogenization of partial differential equations with rough or multiscale coefficients be facilitated by a general construction recipe or decision framework? In the supplied cleaned formulation, the question asks for a general method for constructing accurate finite-dimensional approximation bases for equations with arbitrary rough or multiscale coefficients.

To describe the missing accuracy specification, if a solution space $X$, a norm $\|\cdot\|_X$, and an exact solution $u\in X$ were fixed, one possible measure of the approximation capacity of $V_n\subseteq X$ would be \[ \inf_{v\in V_n}\|u-v\|_X. \] This expression is explanatory notation only: the source does not select this error criterion, prescribe an error bound, or specify whether accuracy concerns best approximation or the output of a numerical scheme.

Unresolved qualifications. The source does not specify the spatial domain, equation class, coefficient assumptions, forcing data, initial or boundary conditions, solution concept, approximation norm, dimension-versus-error requirement, computational cost, or permitted dependence of the basis on the coefficients and data. Consequently, it does not determine a fully quantified existence statement or performance guarantee. The word "arbitrary" cannot be interpreted here as a specified class of well-posed equations. The original wording permits a decision framework, while the cleaned wording emphasizes a construction method; neither wording supplies a universal optimality criterion.

Short English statement

The problem asks for a general recipe or decision framework for designing accurate finite-dimensional bases for numerical homogenization with rough or multiscale PDE coefficients. The equation class, admissible coefficients, boundary conditions, and accuracy requirements remain unspecified, so the record supports a methodological question requiring further hypotheses before a precise quantified claim can be stated.

Sources

• Local UnsolvedMath registry: supplied complete record with permanent ID $30002607$, title Basis Design for Numerical Homogenization, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The supplied proposed year is 2014; no individual proposer is identified.
• Supplied report citation: Computational Multiscale Methods (2015), Oberwolfach Reports, pp. 1625--1681, DOI 10.4171/owr/2014/30. https://doi.org/10.4171/owr/2014/30. The background attributes the problem to workshop open-problem discussions documented in Oberwolfach Reports. The citation's 2015 designation is retained alongside the supplied 2014 report identifier and proposed year; their relationship is not resolved in this draft.
• Supplied literature reference: Owhadi, Bayesian numerical homogenization. https://arxiv.org/abs/1406.6668.
• Supplied literature reference: Oberwolfach Report 30/2014. https://ems.press/content/serial-article-files/46520.
• Supplied literature assessment, checked 2026-08-21: the record labels the problem partially solved and states that Bayesian numerical homogenization supplies a general decision framework and derives optimal-recovery bases for broad integro-differential problems, while a universally optimal basis-design recipe for arbitrary multiscale operators is not established. This is supplied dated curation, described in the background as literature triage derived from the supplied OWR curation; it has not been independently verified here. Its time-sensitive status and cited claims require independent checking.
• The supplied statement status is corrected_verified. Its verification note reports checking against the official Oberwolfach report and says that the cleaned statement restores the source's hypotheses and notation while preserving its open questions. This is a supplied provenance assertion, not independent verification performed for this draft.