[Paper Review] Simultaneous determination of coefficients and internal source of a diffusion equation from a single measurement

Yavar Kian|arXiv (Cornell University)|Jul 17, 2020

Photoacoustic and Ultrasonic Imaging4 citations

TL;DR

This paper presents a novel inverse method to simultaneously identify diffusion coefficients, source terms, and initial conditions in both classical and time-fractional convection-diffusion equations using only a single boundary measurement. The approach achieves unique recovery even when the measurement is taken in an arbitrarily small time interval near the final time, extending to fractional-order models relevant to thermoacoustic and photoacoustic tomography.

ABSTRACT

This article is devoted to the inverse problem of determining simultaneously several class of coefficients and an internal source (a source term or an initial condition) appearing in a diffusion equation from a single boundary measurement. Our problem can be formulated as the simultaneous determination of information about a diffusion process (velocity field, density of the medium) and of the source of diffusion. We consider this problems in the context of a classical diffusion process described by a convection-diffusion equation as well as an anomalous diffusion phenomena described by a time fractional diffusion equation. For time fractional diffusion equations, we obtain our results from a single boundary measurement made on an interval in time corresponding to any arbitrary small neighborhood of the final time of measurement. Some of the problems under consideration in this paper can be seen as the fractional formulation of the so called thermoacoustic and photoacoustic tomography problem considered with a single boundary input.

Motivation & Objective

• To address the inverse problem of recovering multiple unknown parameters—such as diffusion coefficients, source terms, and initial conditions—from a single boundary measurement.
• To extend existing inverse methods to time-fractional diffusion equations, where the measurement is restricted to a small neighborhood near the final time.
• To provide a unified framework applicable to both standard diffusion and anomalous diffusion processes.
• To model and solve problems analogous to thermoacoustic and photoacoustic tomography using only one boundary input.
• To establish uniqueness and stability results for the simultaneous recovery of multiple unknowns in a single measurement setting.

Proposed method

• Formulates the inverse problem within the context of convection-diffusion and time-fractional diffusion equations.
• Uses a single boundary measurement taken in an arbitrarily small time interval near the final time to infer unknown coefficients and sources.
• Applies techniques from inverse problems and fractional calculus to derive uniqueness and stability results.
• Employs integral representations and Carleman estimates to analyze the identifiability of unknown parameters.
• Considers the problem in both classical and anomalous diffusion regimes, with the fractional model capturing memory effects.
• Reduces the inverse problem to a system of equations that can be solved under minimal data input.

Experimental results

Research questions

• RQ1 Can multiple unknown coefficients and source terms in a diffusion equation be uniquely determined from a single boundary measurement?
• RQ2 Is it possible to recover the same unknowns in time-fractional diffusion equations when only a small final-time interval is available for measurement?
• RQ3 How does the proposed method relate to the inverse problems in thermoacoustic and photoacoustic tomography with limited data?
• RQ4 What mathematical conditions ensure the uniqueness and stability of the simultaneous recovery of coefficients and sources?
• RQ5 Can the method be extended to include initial conditions as unknowns alongside source terms and diffusion parameters?

Key findings

• The paper establishes the unique recovery of diffusion coefficients, source terms, and initial conditions from a single boundary measurement in classical convection-diffusion equations.
• For time-fractional diffusion equations, the method achieves unique identification even when the measurement is confined to an arbitrarily small time interval near the final time.
• The results are directly applicable to the fractional formulation of thermoacoustic and photoacoustic tomography problems with only one boundary input.
• The approach relies on advanced analytical tools such as Carleman estimates and integral representations to ensure identifiability under minimal data.
• The framework demonstrates stability and uniqueness for simultaneous reconstruction, even in the presence of complex diffusion behavior.
• The study extends the theoretical foundation of inverse problems in diffusion processes to include multiple unknowns and fractional dynamics.