Convergence of Spectral Estimators for Diffusion Tensors

Definitions and mathematical statement

Let $S$ denote the unknown matrix-valued diffusion coefficient and let $\mu$ denote the invariant density mentioned in the supplied record. The spectral relations supplied for the pairs $(\nu_k,u_k)$ are \[ \operatorname{div}(S\nabla u_k)=\nu_k\mu u_k, \qquad \text{with Neumann boundary conditions}. \] The spatial domain, dimension, index range, coefficient class, and interpretation of these equations are unspecified. In particular, the record describes $(\nu_k,u_k)$ as noisy observations while also writing the equations as exact equalities; it does not distinguish exact eigenpairs from their noisy observations. The meaning of the Neumann condition, including whether it prescribes a normal derivative or a conormal flux, is also unspecified.

Intended convergence question. Establish convergence to $S$ of the proposed spectral reconstruction estimator obtained by fitting these relations to noisy eigenpair observations, subject to the hypotheses and limiting observation regime intended by the source.

Unresolved qualifications. The supplied record gives no definition of the estimator, fitting functional, admissible coefficient class, regularization procedure, or parameter choices. It specifies neither a noise model nor an asymptotic regime, topology, or probabilistic mode of convergence. It also does not state assumptions ensuring that the observations identify $S$. Consequently, the intended question can be recorded, but a fully quantified convergence assertion cannot be recovered from the supplied evidence. No particular estimator, boundary operator, or convergence theorem is selected in this draft.

Short English statement

The intended problem asks whether fitting spectral equations to noisy eigenpair data consistently recovers a matrix-valued diffusion coefficient $S$. The estimator, hypotheses, noise regime, and meaning of convergence are missing, so the supplied record does not determine a precise convergence theorem. Its open status is a supplied, dated assessment requiring independent review.

Sources

• Local UnsolvedMath registry: data/unsolvedmath/problems.json, permanent ID $30003508$, title Convergence of Spectral Estimators for Diffusion Tensors, source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. No individual proposer is identified in the supplied record.
• Supplied report citation: Computational Inverse Problems for Partial Differential Equations (2018), Oberwolfach Reports, pp. 1463--1549, DOI 10.4171/owr/2017/24. Supplied source URL and literature-source link, labelled "original OWR report": https://doi.org/10.4171/owr/2017/24. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The record separately gives the proposed year as 2017; the relationship between this year and the citation's 2018 date is not explained.
• Literature assessment dated 2026-08-21: supplied dated curation, not independently verified. It labels the problem open and states that the report explicitly poses convergence analysis for its spectral estimator of a matrix-valued diffusion coefficient from noisy eigenpairs as open. It further states that later convergence results found concern different inverse-problem observation models. The original report is the only supplied literature-source link; no references identifying those later results are supplied. The background expressly calls for independent verification of time-sensitive status and cited claims.
• The registry supplies the statement-status label corrected_verified, with an explanation that mathematical content was preserved while notation, typography, and sentence structure were repaired for publication. This is supplied editorial metadata, not independent mathematical verification of this draft.