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Backward problem for a degenerate viscous Hamilton-Jacobi equation: stability and numerical identification

S. E. Chorfi, A. Habbal, M. Jahid, L. Maniar, A. Ratnani

Abstract

This work is devoted to the analysis of the backward problem for a viscous Hamilton-Jacobi equation with degenerate diffusion and a general Hamiltonian that is not necessarily quadratic. First, we focus on linear degenerate parabolic equations in the nondivergence setting. We prove the conditional stability of the backward problem using Carleman estimates. Then, by a linearization technique, we prove similar results for the nonlinear viscous Hamilton-Jacobi equation. Regarding numerical identification, we first investigate the linear degenerate equation with noisy data using the adjoint state method, combined with a Conjugate Gradient algorithm, to solve the associated minimization problem. Finally, the numerical identification for the nonlinear viscous Hamilton-Jacobi equation is investigated by the Van Cittert iteration. Numerical tests are presented to show the performance of the proposed algorithms.

Backward problem for a degenerate viscous Hamilton-Jacobi equation: stability and numerical identification

Abstract

This work is devoted to the analysis of the backward problem for a viscous Hamilton-Jacobi equation with degenerate diffusion and a general Hamiltonian that is not necessarily quadratic. First, we focus on linear degenerate parabolic equations in the nondivergence setting. We prove the conditional stability of the backward problem using Carleman estimates. Then, by a linearization technique, we prove similar results for the nonlinear viscous Hamilton-Jacobi equation. Regarding numerical identification, we first investigate the linear degenerate equation with noisy data using the adjoint state method, combined with a Conjugate Gradient algorithm, to solve the associated minimization problem. Finally, the numerical identification for the nonlinear viscous Hamilton-Jacobi equation is investigated by the Van Cittert iteration. Numerical tests are presented to show the performance of the proposed algorithms.
Paper Structure (18 sections, 15 theorems, 155 equations, 7 figures, 2 tables, 2 algorithms)

This paper contains 18 sections, 15 theorems, 155 equations, 7 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Linear degenerate parabolic equation
  3. Degenerate equations in the nondivergence setting
  4. Wellposedness
  5. Carleman estimate
  6. Stability of the backward problem
  7. Degenerate viscous Hamilton–Jacobi equation
  8. Identification for linear degenerate parabolic problem
  9. Fréchet differentiability of the Tikhonov functional
  10. Existence and uniqueness
  11. Numerical identification for Degenerate viscous Hamilton-Jacobi equation
  12. Linear degenerate parabolic equation
  13. Discretization of the problem
  14. Numerical experiments and results
  15. Degenerate viscous Hamilton–Jacobi equation
  16. ...and 3 more sections

Key Result

Lemma 2.1

Given $(u,v) \in H^2_{\frac{1}{a}}(0,1) \times H^1_{\frac{1}{a}}(0,1)$, we have

Figures (7)

  • Figure 1: Recovered initial data: (a) for multiple noise levels, (b) iterative recovery for $p=1\%$
  • Figure 2: Decrease of the functional during the Conjugate Gradient process
  • Figure 3: Recovered initial data: (a) for multiple noise levels, (b) iterative recovery for $p=1\%$
  • Figure 4: Decrease of the functional during the Conjugate Gradient process
  • Figure 5: (Bottom) Exact and recovered initial data after $12$ iterations from noise-free data. (Top) Residual error in terms of iterations
  • ...and 2 more figures

Theorems & Definitions (33)

  • Lemma 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • proof
  • Theorem 2.6
  • proof
  • Remark 3.1
  • ...and 23 more