Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Carleman Contraction Mapping Method for a Coefficient Inverse Problem of the Epidemiology

Michael V. Klibanov, Trung Truong

TL;DR

The paper tackles a coefficient inverse problem for a spatial–temporal SIR-type parabolic system with unknown infection and recovery rates and velocity fields, using incomplete boundary data. It introduces the Carleman contraction mapping method (CCMM), combining a Carleman-weighted quadratic functional with a time-Volterra reformulation to obtain a globally convergent iterative scheme that decouples the unknown coefficients from the transformed system. Rigorous convergence analysis shows strong convexity of the weighted functionals and global convergence of both gradient projection and gradient descent schemes, with stability estimates under data noise. Numerical experiments in a 2D setting demonstrate robust recovery of $β(\mathbf{x})$ and $γ(\mathbf{x})$ for various shapes and noise levels, validating the method's potential for real-time epidemic monitoring and calibration.

Abstract

It is proposed to monitor spatial and temporal spreads of epidemics via solution of a Coefficient Inverse Problem for a system of three coupled nonlinear parabolic equations. To solve this problem numerically, a version of the so-called Carleman contraction mapping method is developed for this problem. On each iteration, a linear problem with the incomplete lateral Cauchy data is solved by the weighted Quasi-Reversibility Method, where the weight is the Carleman Weight Function. This is the function, which is involved as the weight in the Carleman estimate for the corresponding parabolic operator. Convergence analysis ensures the global convergence of this procedure. Numerical results demonstrate an accurate performance of this technique for noisy data.

The Carleman Contraction Mapping Method for a Coefficient Inverse Problem of the Epidemiology

TL;DR

The paper tackles a coefficient inverse problem for a spatial–temporal SIR-type parabolic system with unknown infection and recovery rates and velocity fields, using incomplete boundary data. It introduces the Carleman contraction mapping method (CCMM), combining a Carleman-weighted quadratic functional with a time-Volterra reformulation to obtain a globally convergent iterative scheme that decouples the unknown coefficients from the transformed system. Rigorous convergence analysis shows strong convexity of the weighted functionals and global convergence of both gradient projection and gradient descent schemes, with stability estimates under data noise. Numerical experiments in a 2D setting demonstrate robust recovery of and for various shapes and noise levels, validating the method's potential for real-time epidemic monitoring and calibration.

Abstract

It is proposed to monitor spatial and temporal spreads of epidemics via solution of a Coefficient Inverse Problem for a system of three coupled nonlinear parabolic equations. To solve this problem numerically, a version of the so-called Carleman contraction mapping method is developed for this problem. On each iteration, a linear problem with the incomplete lateral Cauchy data is solved by the weighted Quasi-Reversibility Method, where the weight is the Carleman Weight Function. This is the function, which is involved as the weight in the Carleman estimate for the corresponding parabolic operator. Convergence analysis ensures the global convergence of this procedure. Numerical results demonstrate an accurate performance of this technique for noisy data.

Paper Structure

This paper contains 20 sections, 9 theorems, 123 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Statements of Forward and Inverse Problems
  3. Transformation
  4. Two Estimates
  5. Numerical Method for Problem (\ref{['3.11']})-(\ref{['3.14']})
  6. Sets
  7. The sequence of quadratic functionals
  8. The functional number 0
  9. The functional number $n\geq 1$
  10. Some Properties of the Functional $J_{n,\lambda ,\xi }\left( W\right)$
  11. Strong convexity of the functional $J_{n,\lambda ,\xi }\left( W\right)$ on the set $\overline{B\left( M\right) }$
  12. Global convergence of the gradient projection method
  13. Global convergence of the gradient descent method
  14. Convergence of Minimizers to the True Solution
  15. Introducing noise in the input data
  16. ...and 5 more sections

Key Result

Theorem 4.1

Let $d>0$ be the number in (2.7). There exists a sufficiently large number $\lambda _{0}=\lambda _{0}\left( Q_{T},d\right) \geq 1$ and a number $C=C\left( Q_{T},d\right) >0,$ both depending only on listed parameters, such that the following Carleman estimate holds:

Figures (5)

  • Figure 1: Experiment with different values of $\lambda$. We choose $\lambda =5$ as an optimal value of $\lambda .$ This value is used in all other tests.
  • Figure 2: Reconstruction of $\gamma$ (letter "A") at different noise levels.
  • Figure 3: Reconstruction of $\beta$ (letter "M") at different noise levels.
  • Figure 4: Reconstruction of $\gamma$ (letter "$\Omega$") and $\beta$ (letter "B") with values 0.4 and 0.6 respectively inside these shapes and 0.1 outside at 2% of the noise.
  • Figure 5: Reconstruction of $\gamma$ (letter “ $\Omega$” ) and $\beta$ (letter “ B” ) with values 0.8 and 1 respectively inside these shapes and 0.1 outside at 2% of the noise.

Theorems & Definitions (15)

  • Remark 1.1
  • Theorem 4.1: Carleman estimate for the operator $\partial _{t}-d\Delta$ - Theorem 9.4.1 in KL
  • Theorem 4.2: An estimate for the Volterra integral operator
  • Remark 5.1
  • Theorem 6.1
  • Remark 6.2
  • Proof 1: Proof of \ref{['Theorem 6.1']}
  • Theorem 6.3
  • Theorem 6.4
  • Theorem 7.1: convergence for noisy data
  • ...and 5 more