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A globally convergent Carleman-Picard method for an inverse initial-value problem for a nonlinear diffusive coagulation-fragmentation equation coagulation-fragmentation equation

Thuy T. Le, Minh-Binh Tran, Loc H. Nguyen

Abstract

We study an inverse initial-density problem for a nonlinear diffusive coagulation--fragmentation equation with known coagulation and fragmentation kernels. The objective is to recover the unknown initial particle-size distribution on a finite interval from time-dependent boundary observations of the solution and its size derivative. To solve this inverse problem, we develop a globally convergent numerical method based on a Legendre--exponential time reduction and a Carleman--Picard iteration. The time reduction transforms the original problem into a nonlinear coupled system for the spatial mode coefficients, while the Carleman weight and the corresponding Carleman estimate guaranty the global convergence of the Picard iteration without requiring a good initial guess. We prove the convergence of the proposed method and obtain a complete reconstruction procedure for the initial density. Numerical experiments with noisy boundary data demonstrate that the method yields accurate and stable reconstructions for several representative test profiles.

A globally convergent Carleman-Picard method for an inverse initial-value problem for a nonlinear diffusive coagulation-fragmentation equation coagulation-fragmentation equation

Abstract

We study an inverse initial-density problem for a nonlinear diffusive coagulation--fragmentation equation with known coagulation and fragmentation kernels. The objective is to recover the unknown initial particle-size distribution on a finite interval from time-dependent boundary observations of the solution and its size derivative. To solve this inverse problem, we develop a globally convergent numerical method based on a Legendre--exponential time reduction and a Carleman--Picard iteration. The time reduction transforms the original problem into a nonlinear coupled system for the spatial mode coefficients, while the Carleman weight and the corresponding Carleman estimate guaranty the global convergence of the Picard iteration without requiring a good initial guess. We prove the convergence of the proposed method and obtain a complete reconstruction procedure for the initial density. Numerical experiments with noisy boundary data demonstrate that the method yields accurate and stable reconstructions for several representative test profiles.
Paper Structure (12 sections, 8 theorems, 150 equations, 6 figures, 1 algorithm)

This paper contains 12 sections, 8 theorems, 150 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Carleman estimate, Legendre--exponential basis, and Lipschitz estimates
  3. A one-dimensional Carleman estimate
  4. The Legendre polynomial--exponential basis
  5. Some Lipschitz continuities
  6. Time-dimension reduction
  7. The Carleman--Picard iteration
  8. Numerical Experiments for the Carleman--Picard Method
  9. Generation of synthetic data by solving the forward problem
  10. Numerical Implementation
  11. Numerical reconstruction results
  12. Concluding remarks

Key Result

Lemma 1

There exists a constant $\beta_0>0$ such that for every $\beta\ge \beta_0$ there exists $\lambda_0>0$ (depending on $L$, $v_0$, and $\beta$) with the following property: for all $\lambda\ge \lambda_0$ and all $u\in C^2([0,L])$, one has the pointwise estimate for $v \in (0, L)$, where $C>0$ is independent of $u$ and $(\lambda,\beta)$, and the auxiliary function $U$ satisfies

Figures (6)

  • Figure 1: (a) The graph of the function $\varphi:\{15,\dots,45\}\to\mathbb{R}$ defined in \ref{['error_N']}, computed from Test 1 below. (b) Comparison between the forward solution $f(L,t)$ and its truncated Legendre-exponential expansion with $N=20$ over the interval $t\in[0,T]$. The blue curve represents $f(L,t)$, while the red markers denote its truncated expansion.
  • Figure 2: Test 1. The first row shows the results for the $5\%$ noise level, while the second row shows the results for the $10\%$ noise level. In each row, the left plot compares the true initial density and its reconstruction: the solid curve denotes the true function, and the dashed curve denotes the reconstructed one. The right plot shows the absolute consecutive error $\|f^{\mathrm{rec},(k+1)}-f^{\mathrm{rec},(k)}\|_{L^\infty(0,L)}$.
  • Figure 3: Test 2. The first row shows the results for the $5\%$ noise level, while the second row shows the results for the $10\%$ noise level. In each row, the left plot compares the true initial density and its reconstruction: the solid curve denotes the true function, and the dashed curve denotes the reconstructed one. The right plot shows the absolute consecutive error $\|f^{\mathrm{rec},(k+1)}-f^{\mathrm{rec},(k)}\|_{L^\infty(0,L)}$.
  • Figure 4: Test 3. The first row shows the results for the $5\%$ noise level, while the second row shows the results for the $10\%$ noise level. In each row, the left plot compares the true initial density and its reconstruction: the solid curve denotes the true function, and the dashed curve denotes the reconstructed one. The right plot shows the absolute consecutive error $\|f^{\mathrm{rec},(k+1)}-f^{\mathrm{rec},(k)}\|_{L^\infty(0,L)}$.
  • Figure 5: Test 4. The first row shows the results for the $5\%$ noise level, while the second row shows the results for the $10\%$ noise level. In each row, the left plot compares the true initial density and its reconstruction: the solid curve denotes the true function, and the dashed curve denotes the reconstructed one. The right plot shows the absolute consecutive error $\|f^{\mathrm{rec},(k+1)}-f^{\mathrm{rec},(k)}\|_{L^\infty(0,L)}$.
  • ...and 1 more figures

Theorems & Definitions (28)

  • Lemma 1: Carleman estimate in $1$D
  • Remark 1
  • Corollary 1: Integrated Carleman estimate
  • Proposition 1: See TrongElastic
  • Remark 2: The role of the weight $e^t$
  • Proposition 2
  • Remark 3
  • Definition 1: The projected collision operators
  • Remark 4
  • Remark 5: Exponential tail extension
  • ...and 18 more