[Paper Review] Global uniqueness in an inverse problem for time fractional diffusion equations

Yavar Kian, Lauri Oksanen+2 more|arXiv (Cornell University)|Jan 5, 2016

Numerical methods in inverse problems29 references3 citations

TL;DR

This paper establishes global uniqueness in an inverse problem for time-fractional diffusion equations on Riemannian manifolds, proving that the manifold structure (up to isometry), and smooth coefficients (conductivity, potential, and weight), can be uniquely determined from partial boundary measurements of the solution at a fixed time. The key result relies on analyzing the Dirichlet-to-Neumann map and boundary spectral data under minimal geometric assumptions.

ABSTRACT

Given $(M,g)$, a compact connected Riemannian manifold of dimension $d \geq 2$, with boundary $\partial M$, we consider an initial boundary value problem for a fractional diffusion equation on $(0,T) imes M$, $T>0$, with time-fractional Caputo derivative of order $α\in (0,1) \cup (1,2)$. We prove uniqueness in the inverse problem of determining the smooth manifold $(M,g)$ (up to an isometry), and various time-independent smooth coefficients appearing in this equation, from measurements of the solution on a subset of $\partial M$ at fixed time. In the "flat" case where $M$ is a compact subset of $\mathbb R^d$, two out the three coefficients $ρ$ (weight), $a$ (conductivity) and $q$ (potential) appearing in the equation $ρ\partial_t^αu- extrm{div}(a abla u)+ q u=0$ on $(0,T) imes Ω$ are recovered simultaneously.

Motivation & Objective

• To determine whether the Riemannian manifold structure and time-independent coefficients in a time-fractional diffusion equation can be uniquely recovered from partial boundary measurements.
• To investigate the extent to which the Dirichlet-to-Neumann map, defined on a subset of the boundary at a fixed time, determines the underlying geometric and physical parameters.
• To establish uniqueness results for the inverse problem in the case of fractional orders α ∈ (0,1) ∪ (1,2), extending classical results from parabolic and hyperbolic cases.
• To analyze the recovery of multiple coefficients—conductivity, potential, and weight—simultaneously from limited data, particularly in the flat case where M is a compact subset of R^d.
• To explore conditions under which the boundary spectral data, derived from the DN map, imply isometry of the manifolds and gauge equivalence of the coefficients.

Proposed method

• Formulates the initial boundary value problem for a time-fractional diffusion equation with Caputo derivative of order α ∈ (0,1) ∪ (1,2) on a compact Riemannian manifold (M,g) with boundary.
• Defines the partial Dirichlet-to-Neumann (DN) map using Dirichlet data supported on an open subset S_in of the boundary and measuring the Neumann trace on another subset S_out at a fixed time T_0.
• Establishes well-posedness of the IBVP in a suitable function space H_in,α,T_0, ensuring the DN map is bounded from this space into L^2(S_out).
• Analyzes the boundary spectral data (BSD) associated with the DN map, including eigenvalues and boundary traces of eigenfunctions of the elliptic operator A = -Δ_{g,μ} + q.
• Uses gauge equivalence of boundary spectral data to deduce isometry of the underlying manifolds and equivalence of coefficients under group transformations.
• Applies spectral inequality and geometric control conditions to reduce the problem to known results in hyperbolic inverse problems, particularly leveraging the hyperbolic DN map for α = 2.

Experimental results

Research questions

• RQ1 Can the Riemannian manifold (M,g) be uniquely determined (up to isometry) from the partial Dirichlet-to-Neumann map associated with a time-fractional diffusion equation?
• RQ2 To what extent can the smooth coefficients μ (weight), a (conductivity), and q (potential) be recovered simultaneously from partial boundary measurements?
• RQ3 Does the boundary spectral data derived from the DN map uniquely determine the manifold and coefficients under minimal geometric assumptions?
• RQ4 What conditions on the geometry of the manifold (e.g., spectral inequality or non-trapping) ensure uniqueness in the inverse problem for α ∈ (0,1) ∪ (1,2)?
• RQ5 How does the recovery of coefficients depend on the relative positions of the input (S_in) and output (S_out) subsets on the boundary?

Key findings

• The Riemannian manifold (M,g) is uniquely determined up to isometry from the partial Dirichlet-to-Neumann map when the coefficients μ and q are smooth and positive.
• In the flat case (M ⊂ R^d), the coefficients μ, a, and q can be recovered simultaneously from the DN map, provided the manifold is compact and the coefficients are smooth.
• When the boundary spectral data (BSD) are gauge equivalent, the underlying manifolds (M_k,g_k) are isometric, and the coefficients (μ_k,q_k) are related by a gauge transformation.
• Under the spectral inequality (5.5), the DN map for the hyperbolic case (α = 2) uniquely determines the manifold structure, and this result is extended to the fractional case via reduction.
• The condition that the boundary spectral data satisfy λ_{1,n} = λ_{2,n} and Θ_{1,n}(x,y) = Θ_{2,n}(x,y) for all n and (x,y) ∈ S_out × S_in implies isometry of the manifolds and gauge equivalence of the coefficients.
• The result holds even when S_in and S_out are disjoint, provided the spectral inequality (5.5) is satisfied, which is verified for non-trapping manifolds and under geometric control conditions.