Optimal SVD-Based Space–Time Focusing Signals

Definitions and mathematical statement

Consider a two-dimensional acoustic medium containing unknown point scatterers and an array of transducers. Write $S(\omega)$ for the frequency-domain scattering operator at frequency $\omega$, understood as mapping transducer input data to scattered-wave measurement data. The source does not specify its domain, codomain, or normalization. For a selected simple singular value of $S(\omega)$, let $\phi_{\mathrm{sym}}(\omega)$ denote the phase prescribed by the symmetric singular-value decomposition referred to in the source. The precise decomposition convention and phase prescription are not supplied.

Let $u_{\mathrm{sym}}$ denote the acoustic field produced by the associated superposition of time-harmonic input signals, and let $x_*$ denote the location of the selected scatterer. These symbols designate the source's proposed construction; its frequency range, spectral amplitudes, superposition convention, and association between a singular mode and a scatterer are not specified in the supplied record.

Source question. Does this phase choice produce an input signal whose resulting field $u_{\mathrm{sym}}$ optimally concentrates acoustic energy in space and time near $x_*$?

Unresolved formulation. The supplied assessment reports that the primary report did not identify a mathematical focusing functional, so optimality of the proposed signal is not established. A precise optimization statement requires a definition of acoustic energy concentration, a spatial neighborhood and temporal target or window, an admissible class of input signals, and constraints or normalization permitting comparison of signals. The acoustic propagation model and its boundary or radiation conditions also remain unspecified. Consequently, the question above preserves the supplied target but does not define a complete optimization problem or assert optimality.

Short English statement

The problem asks whether the phase selected by a symmetric singular-value decomposition, for a simple singular value of an acoustic scattering operator, yields optimal focusing near a selected point scatterer in both space and time. The supplied record leaves the focusing objective and admissible inputs undefined, so a precise optimality claim requires further formulation.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30002277$, source problem code catalogue entry, title: "Optimal SVD-Based Space--Time Focusing Signals." The supplied record belongs to the Oberwolfach Reports Open Problems collection and gives the proposed year as 2013; it names no proposer.
• Supplied primary report citation: Computational Electromagnetism and Acoustics (2013), Oberwolfach Reports, pp. 129--237, DOI 10.4171/owr/2013/03. https://doi.org/10.4171/owr/2013/03. The supplied background describes these reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature source, labeled "Cassier, selective focusing for time-dependent waves": https://www.fresnel.fr/perso/cassier/papers/Oberwolfach-report-2013.pdf. This label is retained as supplied attribution and does not identify Cassier as the proposer.
• Supplied literature source, labeled "original Oberwolfach report": https://doi.org/10.4171/owr/2013/03.
• Literature assessment checked on 2026-08-21: the supplied curation labels the problem open and reports that the primary report found no mathematical focusing functional, leaving optimality unestablished. This is supplied dated curation, not an independently verified literature assessment. The record also labels the statement corrected_verified and says it was checked against the official report; that verification claim is supplied metadata and has not been independently checked here.