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Unique and Stable Recovery of Space-Variable Order in Multidimensional Subdiffusion

Jiho Hong, Bangti Jin, Yavar Kian

TL;DR

The paper tackles identifiability and stable recovery of a spatially variable fractional order $\alpha(x)$ in a subdiffusion model from boundary flux data. It develops a framework based on $L^r(\Omega)$ resolvent estimates, Laplace-domain solution representations, and precise asymptotic expansions of the Laplace-transformed Neumann data at $p=0$ and $p=1$ to prove uniqueness results for both piecewise-constant and general $\alpha$, along with semi-continuity and conditional Lipschitz stability results (the latter under monotonicity and full boundary data). The key contributions include extending one-dimensional identifiability results to multidimensional settings and providing quantitative stability estimates that link boundary flux measurements to the unknown variable-order function. The findings have potential impact for inverse problems in heterogeneous media exhibiting space-dependent anomalous diffusion and offer a pathway toward robust reconstruction algorithms under minimal measurement configurations.

Abstract

In this work we investigate the unique identifiability and stable recovery of a spatially dependent variable-order in the subdiffusion model from the boundary flux measurement. We establish several new unique identifiability results from the observation at one point on the boundary without / with the knowledge of medium properties, and a conditional Lipschitz stability estimate when the observation is available on the whole boundary. The analysis crucially employs resolvent estimates in the $L^r(Ω)$ ($r>d$) spaces, solution representation in the Laplace domain and novel asymptotic expansions of the Laplace transform of the boundary flux at $p= 0$ and $p=1$.

Unique and Stable Recovery of Space-Variable Order in Multidimensional Subdiffusion

TL;DR

The paper tackles identifiability and stable recovery of a spatially variable fractional order in a subdiffusion model from boundary flux data. It develops a framework based on resolvent estimates, Laplace-domain solution representations, and precise asymptotic expansions of the Laplace-transformed Neumann data at and to prove uniqueness results for both piecewise-constant and general , along with semi-continuity and conditional Lipschitz stability results (the latter under monotonicity and full boundary data). The key contributions include extending one-dimensional identifiability results to multidimensional settings and providing quantitative stability estimates that link boundary flux measurements to the unknown variable-order function. The findings have potential impact for inverse problems in heterogeneous media exhibiting space-dependent anomalous diffusion and offer a pathway toward robust reconstruction algorithms under minimal measurement configurations.

Abstract

In this work we investigate the unique identifiability and stable recovery of a spatially dependent variable-order in the subdiffusion model from the boundary flux measurement. We establish several new unique identifiability results from the observation at one point on the boundary without / with the knowledge of medium properties, and a conditional Lipschitz stability estimate when the observation is available on the whole boundary. The analysis crucially employs resolvent estimates in the () spaces, solution representation in the Laplace domain and novel asymptotic expansions of the Laplace transform of the boundary flux at and .

Paper Structure

This paper contains 14 sections, 15 theorems, 102 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results and discussions
  3. Uniqueness for piecewise-constant orders
  4. Uniqueness and stability in the generalized case
  5. Illustration of the theorems
  6. Preliminaries on the direct problem
  7. Well-posedness of the direct problem
  8. Asymptotic expansions
  9. Proof of main results
  10. Proof of Theorem \ref{['theorem:flux:lowfreq:asympt:onept']}
  11. Proof of Theorem \ref{['theorem:uniqueness:measfuns']}
  12. Proof of Theorem \ref{['theorem:stability']}
  13. Proof of Theorem \ref{['theorem:stability:fullbdydata']}
  14. Conclusion

Key Result

Theorem 2.1

For $i=1,2$, fix $\alpha^i$ satisfying Assumptions ass:alpha:basic and ass:stratified:structure with $n=n^i$, $\alpha_j=\alpha_j^i$, $\Omega_j=\Omega_j^i$, $j=0,\ldots,n^i$. Assume also that $\rho^i$, $\sigma^i$ and $q^i$ fulfill Assumption ass:coef and $g^i$ satisfies Assumption ass:g with $M=M^i$

Figures (1)

  • Figure 1: The contour plots of four variable orders $\alpha:B_1(0)\to[0.4,0.8]$ satisfying Assumption \ref{['ass:alpha:basic']}. The functions $\alpha^1$, $\alpha^3$ and $\alpha^4$ are piecewise constant functions satisfying Assumption \ref{['ass:stratified:structure']}, and there hold the monotonicity relation $\alpha^1\ge \alpha^2\ge \alpha^3$ almost everywhere. $\alpha^3$ and $\alpha^4$ satisfy the second assumption of part (b) of Theorem \ref{['theorem:flux:lowfreq:asympt:onept']}.

Theorems & Definitions (32)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.1
  • Theorem 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.1
  • ...and 22 more