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Regularized Reconstruction of Scalar Parameters in Subdiffusion with Memory via a Nonlocal Observation

Andrii Hulianytskyi, Sergei Pereverzyev, Sergii Siryk, Nataliya Vasylyeva

TL;DR

The paper addresses the inverse problem of identifying scalar parameters in two- and multi-term time-fractional operators governing subdiffusion with memory, using a nonlocal observation $\psi(t)=\int_{\Omega}u(x,t)\,dx$. It develops explicit reconstruction formulas for the leading fractional orders $\nu_1$ and $\nu_2$ (and extensions to $\nu_i$ in $M$-term operators), proves uniqueness and stability, and proposes a computational algorithm based on Tikhonov regularization with a quasi-optimality principle to handle noisy data. The approach is grounded in fractional Hölder spaces, Volterra-type reformulations, and Mittag-Leffler calculus, and it yields regularity results for the state $u$ and concrete procedures to recover additional coefficients such as $\rho_i$ when needed. The results are relevant for practical modeling in biological transport (e.g., oxygen distribution) and demonstrate effective parameter recovery from discrete, noisy measurements, highlighting the method’s applicability to nonautonomous subdiffusion with memory.

Abstract

In the paper, we propose an analytical and numerical approach to identify scalar parameters (coefficients, orders of fractional derivatives) in the multi-term fractional differential operator in time, $\mathbf{D}_t$. To this end, we analyze inverse problems with an additional nonlocal observation related to a linear subdiffusion equation $\mathbf{D}_{t}u-\mathcal{L}_{1}u-\mathcal{K}*\mathcal{L}_{2}u=g(x,t),$ where $\mathcal{L}_{i}$ are the second order elliptic operators with time-dependent coefficients, $\mathcal{K}$ is a summable memory kernel, and $g$ is an external force. Under certain assumptions on the given data in the model, we derive explicit formulas for unknown parameters. Moreover, we discuss the issues concerning to the uniqueness and the stability in these inverse problems. At last, by employing the Tikhonov regularization scheme with the quasi-optimality approach, we give a computational algorithm to recover the scalar parameters from a noisy discrete measurement and demonstrate the effectiveness (in practice) of the proposed technique via several numerical tests.

Regularized Reconstruction of Scalar Parameters in Subdiffusion with Memory via a Nonlocal Observation

TL;DR

The paper addresses the inverse problem of identifying scalar parameters in two- and multi-term time-fractional operators governing subdiffusion with memory, using a nonlocal observation . It develops explicit reconstruction formulas for the leading fractional orders and (and extensions to in -term operators), proves uniqueness and stability, and proposes a computational algorithm based on Tikhonov regularization with a quasi-optimality principle to handle noisy data. The approach is grounded in fractional Hölder spaces, Volterra-type reformulations, and Mittag-Leffler calculus, and it yields regularity results for the state and concrete procedures to recover additional coefficients such as when needed. The results are relevant for practical modeling in biological transport (e.g., oxygen distribution) and demonstrate effective parameter recovery from discrete, noisy measurements, highlighting the method’s applicability to nonautonomous subdiffusion with memory.

Abstract

In the paper, we propose an analytical and numerical approach to identify scalar parameters (coefficients, orders of fractional derivatives) in the multi-term fractional differential operator in time, . To this end, we analyze inverse problems with an additional nonlocal observation related to a linear subdiffusion equation where are the second order elliptic operators with time-dependent coefficients, is a summable memory kernel, and is an external force. Under certain assumptions on the given data in the model, we derive explicit formulas for unknown parameters. Moreover, we discuss the issues concerning to the uniqueness and the stability in these inverse problems. At last, by employing the Tikhonov regularization scheme with the quasi-optimality approach, we give a computational algorithm to recover the scalar parameters from a noisy discrete measurement and demonstrate the effectiveness (in practice) of the proposed technique via several numerical tests.

Paper Structure

This paper contains 18 sections, 17 theorems, 204 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Main Results: Reconstruction of the triples $(\nu_1,\nu_{2},u)$ and $(\nu_1,\nu_{i^{*}},u)$
  3. Functional setting
  4. General assumptions in the model
  5. Statement of the main results
  6. Proof of Theorem \ref{['t3.1']}
  7. Auxiliary results
  8. Solvability of \ref{['2.2']}-\ref{['2.4']}
  9. Uniqueness of a solution in \ref{['2.2']}-\ref{['2.4']}
  10. Proof of Lemma \ref{['l3.2']}
  11. Reconstruction of $(\nu_{1},\nu_{i^{*}},\rho_{i^{*}},u)$ by the Measurement $\psi$
  12. Influence of Noisy Data on Computation of Orders of Fractional Derivatives
  13. The bound of $\Delta_{2}$ in the case of the I type FDO
  14. The estimate of $\Delta_{2}$ in the case of the II type FDO
  15. Numerical Regularized Reconstruction of Scalar Parameters
  16. ...and 3 more sections

Key Result

Theorem 2.1

Let positive $T$ be arbitrary but finite, $\mathfrak{C}_{0}\neq 0$ and $\rho_{2}(t)\neq 0$ for any $t\in[0,t^{*}]$. Under assumptions h1-h6, the inverse problem 2.2-2.4 has a unique solution $(\nu_{1},\nu_{2},u)$. Besides, $\nu_{1}$ and $\nu_{2}$ are successively computed via formulas: and with $\lambda\in(0,1);$ while the function $u$ is a unique solution of 2.2-2.3, which has the regularity

Theorems & Definitions (31)

  • Theorem 2.1
  • Theorem 2.2
  • Lemma 2.1
  • Proposition 3.1
  • proof
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Remark 3.1
  • Lemma 3.3
  • ...and 21 more