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Recovering the Fragmentation Rate in the Growth-Fragmentation Equation

Alvaro Almeida Gomez, Jorge Zubelli

TL;DR

This work tackles the inverse problem of recovering the fragmentation rate $B$ in growth-fragmentation models from noisy measurements of the stable profile $N$ and eigenvalue $\lambda$. The authors develop a Fourier-transform based regularization framework on locally compact groups to solve for $H = BN$ from $(k\mathcal{K} - I)H = \frac{d}{dx}(gN) + \lambda N$, accommodating general transport kernels beyond equal mitosis. They introduce several spectral-regularization schemes (spectral filtering, Tikhonov, quasi-reversibility, and Landweber) and prove error bounds, with the Landweber variant achieving the rate $O(\varepsilon^{\frac{2m}{2m+1}})$ for $m$-smooth $B$. Numerical experiments in the equal-mitosis setting illustrate stable recovery of $H$ and $B$ from noisy data, confirming the theoretical predictions and highlighting practical robustness. The proposed framework extends to a broad class of transport probabilities, with potential applications in biology and related fields.

Abstract

We consider the inverse problem of determining the fragmentation rate from noisy measurements in the growth-fragmentation equation. We use Fourier transform theory on locally compact groups to treat this problem for general fragmentation probabilities. We develop a regularization method based on spectral filtering, which allows us to deal with the inverse problem in weighted ${L}^2$ spaces. %Our approach regularizes the signal generated by differential operators in the frequency domain. As a result, we obtain a regularization method with error of order $O(\varepsilon^{\frac{2m}{2m+1}})$, where $\varepsilon$ is the noise level and $m>0$ is the {\em a priori} regularity order of the fragmentation rate.

Recovering the Fragmentation Rate in the Growth-Fragmentation Equation

TL;DR

This work tackles the inverse problem of recovering the fragmentation rate in growth-fragmentation models from noisy measurements of the stable profile and eigenvalue . The authors develop a Fourier-transform based regularization framework on locally compact groups to solve for from , accommodating general transport kernels beyond equal mitosis. They introduce several spectral-regularization schemes (spectral filtering, Tikhonov, quasi-reversibility, and Landweber) and prove error bounds, with the Landweber variant achieving the rate for -smooth . Numerical experiments in the equal-mitosis setting illustrate stable recovery of and from noisy data, confirming the theoretical predictions and highlighting practical robustness. The proposed framework extends to a broad class of transport probabilities, with potential applications in biology and related fields.

Abstract

We consider the inverse problem of determining the fragmentation rate from noisy measurements in the growth-fragmentation equation. We use Fourier transform theory on locally compact groups to treat this problem for general fragmentation probabilities. We develop a regularization method based on spectral filtering, which allows us to deal with the inverse problem in weighted spaces. %Our approach regularizes the signal generated by differential operators in the frequency domain. As a result, we obtain a regularization method with error of order , where is the noise level and is the {\em a priori} regularity order of the fragmentation rate.
Paper Structure (21 sections, 10 theorems, 77 equations, 4 figures)

This paper contains 21 sections, 10 theorems, 77 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fourier transform on fouriertrasnform
  4. L2spac
  5. The Fourier transform and the convolution operator
  6. The Fourier transform of fourierderi
  7. Invertibility of the operator Invofoperator
  8. Regularization of the inverse problem
  9. Regularization by spectral filtering
  10. Tikhonov filtering
  11. The quasi-reversibility Tikhonov filtering
  12. The Landweber‘s method
  13. The unbounded case
  14. Examples
  15. The equal-mitosis
  16. ...and 6 more sections

Key Result

Theorem 1

Suppose that $f\in L^1(\mathbb{R^{+}},\mu_{\rho}dx)$ and $\mathcal{F_{\rho}}f\in L^1(\mathbb{R},dx),$ then for a.e positive number $x$ we have

Figures (4)

  • Figure 1: Construction of $N$, solution of the direct problem for the equal- mitosis case.
  • Figure 2: Numerical reconstruction of $H$ and $B$, using the noise level $\varepsilon=10^{-2}$.
  • Figure 3: Numerical reconstruction of $H$ and $B$, using the noise level $\varepsilon=10^{-3}$.
  • Figure 4: Numerical error of the reconstruction of the functions $B$ and $H$ using the $L^2$ norm for several values of $\epsilon$. Note that errors are on a logarithmic scale

Theorems & Definitions (16)

  • Theorem : Inversion theorem rud
  • Theorem : Plancherel Theorem rud
  • Proposition 2.1
  • proof
  • Proposition 3.1
  • Theorem 4.1: Spectral regularization
  • proof
  • Theorem 4.2: Tikhonov regularization
  • proof
  • Theorem 4.3: Quasi-reversibility method
  • ...and 6 more