Efficient Numerical Transport between Smooth Densities

Definitions and mathematical statement

The supplied problem asks for an efficient numerical method for finding a diffeomorphism that transports one smooth density to another. The underlying space and the admissible densities are not specified, so the following gives the transport equation conditionally, without asserting a universal existence or algorithmic theorem.

In a formulation on a specified smooth manifold or smooth domain $M$, let $\mu_0$ and $\mu_1$ be smooth densities interpreted as measures. A diffeomorphism $\varphi\colon M\to M$ transports $\mu_0$ to $\mu_1$ if \[ \varphi_*\mu_0=\mu_1, \qquad (\varphi_*\mu_0)(B):=\mu_0\left(\varphi^{-1}(B)\right) \] for every measurable subset $B\subseteq M$. Equivalently, $\varphi^*\mu_1=\mu_0$. In particular, transport requires matching total masses, $\mu_0(M)=\mu_1(M)$; this compatibility condition is not explicitly stated in the source.

Numerical problem. Given two smooth densities in an appropriate specified setting admitting such a diffeomorphism, develop an efficient numerical procedure for computing or approximating a diffeomorphism satisfying the transport equation.

This remains a research question rather than a fully quantified algorithmic assertion: the source does not specify the dimension or geometry of $M$, compactness, boundary conditions, strict positivity versus possible vanishing of the densities, numerical input representation, approximation norm, error tolerance, convergence requirements, or a measure of computational efficiency. These choices, and hypotheses ensuring existence of the required diffeomorphism, remain unresolved qualifications of the formulation.

Short English statement

Develop an efficient numerical method to construct a smooth invertible map that carries one prescribed smooth density to another. The source leaves the geometric setting, existence hypotheses, boundary conditions, accuracy requirements, and meaning of efficiency unspecified.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30003095$, source problem code catalogue entry, title Efficient Numerical Transport between Smooth Densities. The supplied record assigns the problem to 2016 and names no individual proposer.
• Supplied source citation: Geometric Numerical Integration (2016), Oberwolfach Reports, pp. 869--948, DOI 10.4171/owr/2016/18. Supplied literature-source label: OWR 2016 report. https://doi.org/10.4171/owr/2016/18. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: Bauer--Joshi--Modin, DOI 10.1137/151006238. https://doi.org/10.1137/151006238. No further bibliographic details are supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and states that optimal information transport supplies efficient numerical algorithms for smooth density matching, while the broad numerical density-transport challenge is not solved in full generality. This is supplied dated curation derived from the OWR record, not an independently verified assessment. Its time-sensitive status and cited claims require independent verification.