[Paper Review] Unique determination of several coefficients in a fractional diffusion(-wave) equation by a single measurement

Yavar Kian, Zhiyuan Li+2 more|arXiv (Cornell University)|Jul 4, 2019

Numerical methods in inverse problems33 references4 citations

TL;DR

This paper establishes the unique determination of multiple coefficients—such as the fractional order, diffusion coefficient, convection field, and potential term—in a time-fractional diffusion(-wave) equation using a single Neumann boundary measurement on a partial boundary. The authors prove uniqueness for the coefficients via a novel argument combining complex geometric optics solutions and boundary integral identities, extending inverse problem theory to fractional PDEs with partial data.

ABSTRACT

We consider the inverse problem of determining different type of information about a diffusion process, described by ordinary or fractional diffusion equations stated on a bounded domain, like the density of the medium or the velocity field associated with the moving quantities from a single boundary measurement. This properties will be associated with some general class of time independent coefficients that we recover from a single Neumann boundary measurement, on some parts of the boundary, of the solution of our diffusion equation with a suitable boundary input, located on some parts of the boundary.

Motivation & Objective

• To address the inverse problem of identifying multiple physical coefficients in a time-fractional diffusion(-wave) equation from a single boundary measurement.
• To extend uniqueness results for coefficient inverse problems to fractional-order PDEs with partial boundary data.
• To establish the simultaneous recovery of the fractional order α, density ρ, diffusion coefficient a, convection field B, and potential c from one Neumann measurement.
• To generalize the analysis to the case of Riemannian manifolds, proving uniqueness of the manifold structure and metric from boundary data.

Proposed method

• Utilizes complex geometric optics solutions tailored to the time-fractional diffusion equation with Caputo derivatives.
• Employs a single Neumann boundary measurement on a subset of the boundary, combined with a Dirichlet input on another subset.
• Applies integral identities involving the solution and its normal derivative to derive relations between coefficients.
• Relies on the unique continuation property and spectral analysis of the associated elliptic operator on the manifold.
• Uses the method of stationary phase and asymptotic expansion techniques to extract coefficient information from boundary data.
• Reduces the inverse problem to a Dirichlet-to-Neumann map equivalence, leveraging results from hyperbolic inverse problems.

Experimental results

Research questions

• RQ1 Can multiple coefficients in a time-fractional diffusion equation be uniquely determined from a single Neumann boundary measurement?
• RQ2 Is it possible to recover the fractional order α, diffusion coefficient a, convection field B, and potential c simultaneously from partial boundary data?
• RQ3 Can the underlying Riemannian manifold structure and metric be uniquely reconstructed from boundary measurements of a fractional diffusion process?
• RQ4 How does the presence of partial data (on subsets of the boundary) affect the uniqueness of coefficient identification in fractional PDEs?

Key findings

• The fractional order α is uniquely determined from a single Neumann boundary measurement on a partial boundary.
• The coefficients ρ, a, B, and c in the fractional diffusion equation are uniquely identifiable from one boundary measurement under suitable regularity and positivity assumptions.
• The Riemannian manifold (M,g) and the potential c are uniquely determined up to isometry from the partial Dirichlet-to-Neumann map in the manifold case.
• The proof relies on the construction of complex geometric optics solutions and the use of spectral data derived from boundary measurements.
• The authors establish uniqueness even when the boundary measurement is restricted to a subset of the boundary, overcoming the limitations of classical full-boundary data assumptions.
• The results generalize previous uniqueness results for classical diffusion (α=1) to the fractional-order setting with partial data.