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  1. Free, publicly-accessible full text available August 1, 2025
  2. This paper aims to reconstruct the initial condition of a hyperbolic equation with an unknown damping coefficient. Our approach involves approximating the hyperbolic equation’s solution by its truncated Fourier expansion in the time domain and using the recently developed polynomial-exponential basis. This truncation process facilitates the elimination of the time variable, consequently, yielding a system of quasi-linear elliptic equations. To globally solve the system without needing an accurate initial guess, we employ the Carleman contraction principle. We provide several numerical examples to illustrate the efficacy of our method. The method not only delivers precise solutions but also showcases remarkable computational efficiency. 
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    Free, publicly-accessible full text available July 1, 2025
  3. Free, publicly-accessible full text available April 1, 2025
  4. We propose a globally convergent numerical method to compute solutions to a general class of quasi-linear PDEs with both Neumann and Dirichlet boundary conditions. Combining the quasi-reversibility method and a suitable Carleman weight function, we define a map of which fixed point is the solution to the PDE under consideration. To find this fixed point, we define a recursive sequence with an arbitrary initial term using the same manner as in the proof of the contraction principle. Applying a Carleman estimate, we show that the sequence above converges to the desired solution. On the other hand, we also show that our method delivers reliable solutions even when the given data are noisy. Numerical examples are presented. 
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  5. We propose a new iterative scheme to compute the numerical solution to an over-determined boundary value problem for a general quasilinear elliptic PDE. The main idea is to repeatedly solve its linearization by using the quasi-reversibility method with a suitable Carleman weight function. The presence of the Carleman weight function allows us to employ a Carleman estimate to prove the convergence of the sequence generated by the iterative scheme above to the desired solution. The convergence of the iteration is fast at an exponential rate without the need of an initial good guess. We apply this method to compute solutions to some general quasilinear elliptic equations and a large class of first-order Hamilton-Jacobi equations. Numerical results are presented. 
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