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The Carleman-Newton method to globally reconstruct the initial condition for nonlinear parabolic equations

https://doi.org/10.1016/j.cam.2024.115827

Abhishek, Anuj ; Le, Thuy T. ; Nguyen, Loc H. ; Khan, Taufiquar ( August 2024 , Journal of Computational and Applied Mathematics)

Free, publicly-accessible full text available August 1, 2025
The time dimensional reduction method to determine the initial conditions without the knowledge of damping coefficients

https://doi.org/10.1016/j.camwa.2024.03.038

Le, Thuy T ; Nguyen, Linh V ; Nguyen, Loc H ; Park, Hyunha ( July 2024 , Computers & Mathematics with Applications)

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
The Carleman convexification method for Hamilton-Jacobi equations

https://doi.org/10.1016/j.camwa.2024.02.021

Le, Huynh P.N. ; Le, Thuy T. ; Nguyen, Loc H. ( April 2024 , Computers & Mathematics with Applications)

Free, publicly-accessible full text available April 1, 2025
A Carleman-based numerical method for quasilinear elliptic equations with over-determined boundary data and applications

https://doi.org/10.1016/j.camwa.2022.08.032

Le, Thuy T. ; Nguyen, Loc H. ; Tran, Hung V. ( October 2022 , Computers mathematics with applications)

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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Effects of diffusional constraints on lifetime and selectivity in methanol-to-olefins catalysis on HSAPO-34

https://doi.org/10.1016/j.jcat.2018.10.031

Hwang, Andrew ; Le, Thuy T. ; Shi, Zhichen ; Dai, Heng ; Rimer, Jeffrey D. ; Bhan, Aditya ( January 2019 , Journal of Catalysis)

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