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Determining initial conditions for nonlinear hyperbolic equations with time dimensional reduction and the Carleman contraction

Trong D. Dang, Loc H. Nguyen, Huong T. T. Vu

TL;DR

This work tackles the inverse problem of determining the initial condition $g(\boldsymbol{x})$ for a nonlinear, nonlocal hyperbolic equation by introducing a time dimensional reduction: the solution is approximated as $u(\boldsymbol{x},t)\approx\sum_{n=1}^N u_n(\boldsymbol{x})\Psi_n(t)$ with a polynomial-exponential basis $\{\Psi_n\}$, producing a $d$-dimensional quasi-linear elliptic system for $U=(u_1,\ldots,u_N)^T$. A Carleman-weighted contraction mapping $\Phi_{\lambda,\beta,\epsilon}$ is then constructed to solve this elliptic system globally, leveraging a Carleman estimate to guarantee convergence without requiring a good initial guess. Under Lipschitz conditions on the nonlinearity, the method yields a unique fixed point and a stability estimate that degrades gracefully with measurement noise. Numerical experiments in $2+1$ dimensions demonstrate accurate reconstructions of $g$ from boundary data with substantial speedups compared to traditional approaches, validating the approach for inverse source problems in thermo/photo-acoustic tomography and related settings.

Abstract

This paper aims to determine the initial conditions for quasi-linear hyperbolic equations that include nonlocal elements. We suggest a method where we approximate the solution of the hyperbolic equation by truncating its Fourier series in the time domain with a polynomial-exponential basis. This truncation effectively removes the time variable, transforming the problem into a system of quasi-linear elliptic equations. We refer to this technique as the "time dimensional reduction method." To numerically solve this system comprehensively without the need for an accurate initial estimate, we used the newly developed Carleman contraction principle. We show the efficiency of our method through various numerical examples. The time dimensional reduction method stands out not only for its precise solutions but also for its remarkable speed in computation.

Determining initial conditions for nonlinear hyperbolic equations with time dimensional reduction and the Carleman contraction

TL;DR

This work tackles the inverse problem of determining the initial condition for a nonlinear, nonlocal hyperbolic equation by introducing a time dimensional reduction: the solution is approximated as with a polynomial-exponential basis , producing a -dimensional quasi-linear elliptic system for . A Carleman-weighted contraction mapping is then constructed to solve this elliptic system globally, leveraging a Carleman estimate to guarantee convergence without requiring a good initial guess. Under Lipschitz conditions on the nonlinearity, the method yields a unique fixed point and a stability estimate that degrades gracefully with measurement noise. Numerical experiments in dimensions demonstrate accurate reconstructions of from boundary data with substantial speedups compared to traditional approaches, validating the approach for inverse source problems in thermo/photo-acoustic tomography and related settings.

Abstract

This paper aims to determine the initial conditions for quasi-linear hyperbolic equations that include nonlocal elements. We suggest a method where we approximate the solution of the hyperbolic equation by truncating its Fourier series in the time domain with a polynomial-exponential basis. This truncation effectively removes the time variable, transforming the problem into a system of quasi-linear elliptic equations. We refer to this technique as the "time dimensional reduction method." To numerically solve this system comprehensively without the need for an accurate initial estimate, we used the newly developed Carleman contraction principle. We show the efficiency of our method through various numerical examples. The time dimensional reduction method stands out not only for its precise solutions but also for its remarkable speed in computation.
Paper Structure (9 sections, 4 theorems, 60 equations, 4 figures, 3 algorithms)

This paper contains 9 sections, 4 theorems, 60 equations, 4 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. The time dimensional reduction model
  3. The Carleman contraction mapping
  4. The convergence of the Carleman contraction principle
  5. Numerical study
  6. Data generation
  7. The implementation for the inverse problem
  8. Numerical examples
  9. Concluding remarks

Key Result

Theorem 3.1

Assume Lipschitz and hence LipschitzF hold true. Then, there is a constant $\beta_0$ depending only on ${\bf M}$, $d$, $\Omega,$ and ${\bf x}_0$ such that for all $\beta \geq \beta_0$, we have for all $\lambda > \lambda_0$ where $\lambda_0 = \lambda_0({\bf M}, d, \Omega, d, {\bf x}_0, \beta)$ and $C = C({\bf M}, d, \Omega, d, {\bf x}_0, \beta)$ depending only on the listed parameters. Consequentl

Figures (4)

  • Figure 1: It is evident that when $N = 40$, the data $h({\bf x}^*, \cdot)$ is well approximated by cutting its Fourier series. The function $h$ in these figures is the data for Test 1 in this section. The point ${\bf x}^* = (-1, 0).$
  • Figure 2: True and numerical solutions of test 1. It is interesting mentioning that although the true solution has a high value (10) and the size of the "ellipse inclusion" is not small, our method can deliver a satisfactory solution without requesting a good initial guess. The error in computation occurs mostly at the boundary of the inclusion.
  • Figure 3: True and numerical solutions of test 2. Like in test 1, our method can deliver a satisfactory solution for test 2 without requesting a good initial guess. Also, the error in computation occurs mostly at the boundary of the inclusion.
  • Figure 4: True and numerical solutions of test 3. Like in tests 1 and 2, our method can deliver a satisfactory solution for test 2 without requesting a good initial guess. Also, the error in computation occurs mostly at the boundary of the inclusion.

Theorems & Definitions (12)

  • Remark 2.1
  • Remark 2.2: The significance of the polynomial-exponential basis
  • Remark 2.3
  • Theorem 3.1
  • Lemma 3.1: See Theorem 3.1 in LeLeNguyen:Arxiv2022
  • Corollary 3.1
  • proof
  • proof : Proof of Theorem \ref{['thm_contract']}
  • Remark 3.1
  • Theorem 4.1
  • ...and 2 more