The Quasi-reversibility Method to Numerically Solve an Inverse Source Problem for Hyperbolic Equations

Thuy T. Le, Loc H. Nguyen, Thi Phong Nguyen, William Powell

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17 Scopus citations


We propose a numerical method to solve an inverse source problem of computing the initial condition of hyperbolic equations from the measurements of Cauchy data. This problem arises in thermo- and photo-acoustic tomography in a bounded cavity, in which the reflection of the wave makes the widely-used approaches, such as the time reversal method, not applicable. In order to solve this inverse source problem, we approximate the solution to the hyperbolic equation by its Fourier series with respect to a special orthonormal basis of L2. Then, we derive a coupled system of elliptic equations for the corresponding Fourier coefficients. We solve it by the quasi-reversibility method. The desired initial condition follows. We rigorously prove the convergence of the quasi-reversibility method as the noise level tends to 0. Some numerical examples are provided. In addition, we numerically prove that the use of the special basic above is significant.

Original languageEnglish (US)
Article number90
JournalJournal of Scientific Computing
Issue number3
StatePublished - Jun 2021
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Software
  • Theoretical Computer Science
  • Numerical Analysis
  • General Engineering
  • Computational Theory and Mathematics
  • Computational Mathematics
  • Applied Mathematics


  • Hyperbolic equation
  • Inverse source problem
  • Orthonormal basis
  • Quasi-reversibility method


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