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On inverse problems in multi-population aggregation models

Yuhan Li, Hongyu Liu, Catharine W. K. Lo

TL;DR

This work addresses inverse problems for nonlinear, non-local, multi-population aggregation models by aiming to identify diffusion, advection, and interaction kernels from partial observations while enforcing non-negativity. It introduces a transformative asymptotic method to recover the diffusion coefficient preceding the Laplacian in two dimensions and employs a high-order variation approach to guarantee positivity alongside CGO-based techniques for coefficient and kernel recovery. The authors establish unique identifiability results for the diffusion coefficients, advection terms, and integral kernels across two model formulations, with dedicated subsections detailing the recovery of each component and their corollaries. The results represent the first rigorous identifiability outcomes for inverse problems in finite multi-population biological systems, with potential implications for ecological and cellular dynamics modeled by non-local interactions.

Abstract

This paper focuses on inverse problems arising in studying multi-population aggregations. The goal is to reconstruct the diffusion coefficient, advection coefficient, and interaction kernels of the aggregation system, which characterize the dynamics of different populations. In the theoretical analysis of the physical setup, it is crucial to ensure non-negativity of solutions. To address this, we employ the high-order variation method and introduce modifications to the systems. Additionally, we propose a novel approach called transformative asymptotic technique that enables the recovery of the diffusion coefficient preceding the Laplace operator, presenting a pioneering method for this type of problems. Through these techniques, we offer comprehensive insights into the unique identifiability aspect of inverse problems associated with multi-population aggregation models.

On inverse problems in multi-population aggregation models

TL;DR

This work addresses inverse problems for nonlinear, non-local, multi-population aggregation models by aiming to identify diffusion, advection, and interaction kernels from partial observations while enforcing non-negativity. It introduces a transformative asymptotic method to recover the diffusion coefficient preceding the Laplacian in two dimensions and employs a high-order variation approach to guarantee positivity alongside CGO-based techniques for coefficient and kernel recovery. The authors establish unique identifiability results for the diffusion coefficients, advection terms, and integral kernels across two model formulations, with dedicated subsections detailing the recovery of each component and their corollaries. The results represent the first rigorous identifiability outcomes for inverse problems in finite multi-population biological systems, with potential implications for ecological and cellular dynamics modeled by non-local interactions.

Abstract

This paper focuses on inverse problems arising in studying multi-population aggregations. The goal is to reconstruct the diffusion coefficient, advection coefficient, and interaction kernels of the aggregation system, which characterize the dynamics of different populations. In the theoretical analysis of the physical setup, it is crucial to ensure non-negativity of solutions. To address this, we employ the high-order variation method and introduce modifications to the systems. Additionally, we propose a novel approach called transformative asymptotic technique that enables the recovery of the diffusion coefficient preceding the Laplace operator, presenting a pioneering method for this type of problems. Through these techniques, we offer comprehensive insights into the unique identifiability aspect of inverse problems associated with multi-population aggregation models.
Paper Structure (15 sections, 9 theorems, 115 equations)

This paper contains 15 sections, 9 theorems, 115 equations.

Table of Contents

  1. INTRODUCTION
  2. Problem setup and background
  3. Technical developments
  4. PRELIMINARIES AND STATEMENT OF MAIN RESULTS
  5. Notations and basic settings
  6. Well-posedness of the forward problem
  7. Statement of main results
  8. HIGH-ORDER VARIATION METHOD
  9. RECOVERY OF DIFFUSION COEFFICIENTS $\mathbf{d}$ in THEOREM \ref{['mainthm_DR']}
  10. RECOVERY OF THE ADVECTION COEFFICIENT $\pmb{\mu}$ AND THE NORMALIZATION CONSTANT OF $\mathbf{k}$ IN THEOREM \ref{['mainthm_Inte1']} AND COROLLARY \ref{['appthm_case1']}
  11. Unique recovery for the Theorem \ref{['mainthm_Inte1']}
  12. Application to the case of Corollary \ref{['appthm_case1']}
  13. RECOVERY OF THE ADVECTION COEFFICIENT $\pmb{\nu}$ AND THE INTEGRAL KERNEL $\mathbf{w}$ IN THEOREM \ref{['mainthm_Inte2']} AND COROLLARY \ref{['appthm_case2']}
  14. Unique recovery for the Theorem \ref{['mainthm_Inte2']}
  15. Application to the case of Corollary \ref{['appthm_case2']}

Key Result

Lemma 2.1

Consider the system where $\mathbf{d}=d I_N$ for constant $d$ and $N\times N$-identity matrix $I_N$. Assume: $(\mathbf{L1})$ for $p \geq 1,$ let $f_i(x) \in X_{p}:= C^0(\mathbb{T}^n)\cap L^{\infty} (\mathbb{T}^n)\cap L^{p} (\mathbb{T}^n)$ be non-negative for $\mathbf{f}=(f_1,\dots,f_N)$; $(\mathbf{L2})$$\mathbf{K} \in for a small ball $B_R(0)$ about the origin.

Theorems & Definitions (19)

  • Lemma 2.1: Theorem 5.1 of painter2023
  • Lemma 2.2: Corollary 5.1 of painter2023
  • Remark 2.3
  • Theorem 2.4
  • Corollary 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Remark 2.8
  • Corollary 2.9
  • Corollary 2.10
  • ...and 9 more