Reconstructing constant point heat sources from few temperature sensors: sparse selection and spatial resolution

Abstract

We investigate the recovery of an unknown number of stationary point heat sources, together with their locations and constant positive strengths, from temperature histories recorded by a few sensors. The reconstruction combines candidate selection by orthogonal matching pursuit or least angle regression with continuous constrained fitting. A noise-based discrepancy rule selects the smallest accepted source count among those tested. The analysis distinguishes source-count selection, individual-parameter recovery, and aggregate estimation. For a close pair, the local single-source residual is quadratic in separation when the projected curvature is nonzero, whereas a compensating unit parameter direction produces a linearized response bounded by a constant times the cube of separation. This difference explains why a pair may be distinguishable from a single source while its individual parameters remain poorly determined. A conditional stability bound controls errors in total strength and the strength-weighted center, and a local Gaussian approximation relates single-source rejection to noise and separation. Numerical experiments assess sensor configuration, temporal sampling, noise, separation, and initialization, with additional tests using independently generated finite-plate data and a three-dimensional model. Correct source-count selection need not imply accurate individual parameters, whereas unresolved pairs can retain accurate aggregate estimates.