Recovery of a source term or a speed with one measurement and applications

2011/03/05 by Plamen Stefanov, Gunther Uhlmann, Günther Uhlmann +2 · 2 citations
Computer Science · Engineering · Mathematics · #35R30 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Stability and Controllability of Differential Equations #math.AP #msc:35R30

paper · pdf · doi:10.48550/arxiv.1103.1097

openalex publication_date 2011/03/05 · arxiv created 2011/03/06 · arxiv updated 2011/03/08 · openalex created_date 2025/10/27 · openalex updated_date 2026/07/28

Abstract

We study the problem of recovery the source a(t,x)F(x) in the wave equation in anisotropic medium with a known so that a(0,x)\not=0 with a single measurement. We use Carleman estimates combined with geometric arguments and give sharp conditions for uniqueness. We also study the non-linear problem of recovery the sound speed in the equation u_tt -c²(x)Δu =0 with one measurement. We give sharp conditions for stability, as well. An application to thermoacoustic tomography is also presented.

Citations

• A New Numerical Algorithm for Thermoacoustic and Photoacoustic Tomography with Variable Sound Speed
• On The Observability Inequalities for Exact Controllability of Wave Equations With Variable Coefficients
• Sharp Sufficient Conditions for the Observation, Control, and Stabilization of Waves from the Boundary

Cited by

• Recovery of Sturm-Liouville operators from partial boundary spectral data and applications
• Uniqueness and stability in determining the wave equation from a single passive boundary measurement

Related

• Uniqueness and stability in determining the wave equation from a single passive boundary measurement
• Determining both the source of a wave and its speed in a medium from boundary measurements
• Simultaneous Recovery of the Initial Source and Sound Speed for the Wave Equation under a Constitutive Constraint
• Fixed angle inverse scattering for sound speeds close to constant
• Instability of the linearized problem in multiwave tomography of recovery both the source and the speed