SIAM Journal on Mathematical Analysis· 2026Q1
Identification of a Point Source in the Heat Equation from Sparse Boundary Measurements
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- Q1SCImago
- 2026year
Short summary
Researchers can uniquely identify the location and piecewise constant amplitude of a single heat source within a unit ball domain using sparse flux measurements on the boundary.
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Key points
- Unique recovery of a single heat source's location and amplitude is possible from sparse boundary flux data.
- The method applies to a unit ball domain (piecewise constant amplitude) and other smooth, bounded, simply connected domains (compactly supported amplitude).
- Mathematical tools used include Laplacian eigenfunctions, heat/Poisson kernels, and complex analysis.
- Numerical experiments demonstrate the practical feasibility of the recovery method.
AI-generated from the title and abstract; the full text is not read.
Abstract
Abstract. In this work we investigate the inverse problem of recovering one point source in the heat equation from sparse boundary measurement, i.e., the flux data at several points on the boundary. We prove the unique recovery of the location and piecewise constant in time amplitude when the domain is the unit ball in [Formula: see text] ([Formula: see text]) and the unique recovery of the location and compactly supported amplitude when the domain is simply connected, smooth, and bounded in [Formula: see text], under mild conditions on the observational points. The proof combines distinct analytical tools, including the representation of the flux data via Laplacian eigenfunctions on the unit ball, a detailed analysis of the properties of the heat and Poisson kernels, as well as methods drawn from complex analysis. Further, we present several numerical experiments to illustrate the feasibility of the recovery from sparse boundary data.
The authors' abstract, as published at the source. SIAM Journal on Mathematical Analysis, 2026 · DOI ↗
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Field: Mathematical Physics
Mathematical PhysicsMathematics