@article{le_le_nguyen_2024, title={The Carleman convexification method for Hamilton-Jacobi equations}, volume={159}, ISSN={["1873-7668"]}, url={https://doi.org/10.1016/j.camwa.2024.02.021}, DOI={10.1016/j.camwa.2024.02.021}, abstractNote={We propose a new globally convergent numerical method to compute solution Hamilton-Jacobi equations defined in Rd, d≥1, on a truncated bounded domain. This method is named the Carleman convexification method. By Carleman convexification, we mean that we use a Carleman weight function to convexify the conventional least squares mismatch functional. We will prove a new version of the convexification theorem guaranteeing that the mismatch functional involving the Carleman weight function is strictly convex and, therefore, has a unique minimizer. Moreover, a consequence of our convexification theorem guarantees that the minimizer of the Carleman weighted mismatch functional is an approximation of the viscosity solution we want to compute. Some numerical results in 1D and 2D will be presented.}, journal={COMPUTERS & MATHEMATICS WITH APPLICATIONS}, author={Le, Huynh P. N. and Le, Thuy T. and Nguyen, Loc H.}, year={2024}, month={Apr}, pages={173–185} }
 @article{abhishek_le_nguyen_khan_2024, title={The Carleman-Newton method to globally reconstruct the initial condition for nonlinear parabolic equations}, volume={445}, ISSN={["1879-1778"]}, url={https://doi.org/10.1016/j.cam.2024.115827}, DOI={10.1016/j.cam.2024.115827}, abstractNote={We propose to combine the Carleman estimate and the Newton method to solve an inverse problem for nonlinear parabolic equations from lateral boundary data. The stability of this inverse problem for determination of initial condition is conditionally logarithmic. Hence, numerical results due to the conventional least squares optimization might not be reliable. In order to enhance the stability, we approximate this problem by truncating the high frequency terms of the Fourier series that represents the solution to the governing equation. By this, we derive a system of nonlinear elliptic PDEs whose solution consists of Fourier coefficients of the solution to the parabolic governing equation. We solve this system by the Carleman-Newton method. The Carleman-Newton method is a newly developed algorithm to solve nonlinear PDEs. The strength of the Carleman-Newton method includes (1) no good initial guess is required and (2) the computational cost is not expensive. These features are rigorously proved. Having the solutions to this system in hand, we can directly compute the solution to the proposed inverse problem. Some numerical examples are displayed.}, journal={JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS}, author={Abhishek, Anuj and Le, Thuy T. and Nguyen, Loc H. and Khan, Taufiquar}, year={2024}, month={Aug} }
 @article{hao_le_nguyen_2024, title={The Fourier-based dimensional reduction method for solving a nonlinear inverse heat conduction problem with limited boundary data}, volume={128}, ISSN={["1878-7274"]}, DOI={10.1016/j.cnsns.2023.107679}, abstractNote={The objective of this article is to introduce a novel technique for computing numerical solutions to the nonlinear inverse heat conduction problem. This involves solving nonlinear parabolic equations with Cauchy data provided on one side Γ of the boundary of the computational domain Ω. The key step of our proposed method is the truncation of the Fourier series of the solution to the governing equation. The truncation technique enables us to derive a system of 1D ordinary differential equations. Then, we employ the well-known Runge–Kutta method to solve this system, which aids in addressing the nonlinearity and the lack of data on ∂Ω∖Γ. This new approach is called the Fourier-based dimensional reduction method. By converting the high-dimensional problem into a 1D problem, we achieve exceptional computational speed. Numerical results are provided to support the effectiveness of our approach.}, journal={COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION}, author={Hao, Dinh-Nho and Le, Thuy T. and Nguyen, Loc H.}, year={2024}, month={Jan} }
 @article{le_nguyen_nguyen_park_2024, title={The time dimensional reduction method to determine the initial conditions without the knowledge of damping coefficients}, url={https://doi.org/10.1016/j.camwa.2024.03.038}, DOI={10.1016/j.camwa.2024.03.038}, journal={Computers & Mathematics with Applications}, author={Le, Thuy T. and Nguyen, Linh V. and Nguyen, Loc H. and Park, Hyunha}, year={2024}, month={Jul} }
 @article{abney_le_nguyen_peters_2023, title={A Carleman-Picard approach for reconstructing zero-order coefficients in parabolic equations with limited data}, url={https://arxiv.org/abs/2309.14599}, DOI={10.48550/ARXIV.2309.14599}, abstractNote={We propose a globally convergent computational technique for the nonlinear inverse problem of reconstructing the zero-order coefficient in a parabolic equation using partial boundary data. This technique is called the"reduced dimensional method". Initially, we use the polynomial-exponential basis to approximate the inverse problem as a system of 1D nonlinear equations. We then employ a Picard iteration based on the quasi-reversibility method and a Carleman weight function. We will rigorously prove that the sequence derived from this iteration converges to the accurate solution for that 1D system without requesting a good initial guess of the true solution. The key tool for the proof is a Carleman estimate. We will also show some numerical examples.}, publisher={arXiv}, author={Abney, Ray and Le, Thuy T. and Nguyen, Loc H. and Peters, Cam}, year={2023} }
 @article{nguyen_le_nguyen_klibanov_2023, title={Numerical differentiation by the polynomial-exponential basis}, url={https://arxiv.org/abs/2304.05909}, DOI={10.48550/ARXIV.2304.05909}, abstractNote={Our objective is to calculate the derivatives of data corrupted by noise. This is a challenging task as even small amounts of noise can result in significant errors in the computation. This is mainly due to the randomness of the noise, which can result in high-frequency fluctuations. To overcome this challenge, we suggest an approach that involves approximating the data by eliminating high-frequency terms from the Fourier expansion of the given data with respect to the polynomial-exponential basis. This truncation method helps to regularize the issue, while the use of the polynomial-exponential basis ensures accuracy in the computation. We demonstrate the effectiveness of our approach through numerical examples in one and two dimensions.}, publisher={arXiv}, author={Nguyen, Phuong M. and Le, Thuy T. and Nguyen, Loc H. and Klibanov, Michael V.}, year={2023} }
 @article{le_khoa_klibanov_nguyen_bidney_astratov_2023, title={Numerical verification of the convexification method for a frequency-dependent inverse scattering problem with experimental data}, url={https://arxiv.org/abs/2306.00761}, DOI={10.48550/ARXIV.2306.00761}, abstractNote={The reconstruction of physical properties of a medium from boundary measurements, known as inverse scattering problems, presents significant challenges. The present study aims to validate a newly developed convexification method for a 3D coefficient inverse problem in the case of buried unknown objects in a sandbox, using experimental data collected by a microwave scattering facility at The University of North Carolina at Charlotte. Our study considers the formulation of a coupled quasilinear elliptic system based on multiple frequencies. The system can be solved by minimizing a weighted Tikhonov-like functional, which forms our convexification method. Theoretical results related to the convexification are also revisited in this work.}, publisher={arXiv}, author={Le, Thuy and Khoa, Vo Anh and Klibanov, Michael Victor and Nguyen, Loc Hoang and Bidney, Grant and Astratov, Vasily}, year={2023} }
 @article{dinh-nho_le_nguyen_2023, title={The dimensional reduction method for solving a nonlinear inverse heat conduction problem with limited boundary data}, url={https://arxiv.org/abs/2305.19528}, DOI={10.48550/ARXIV.2305.19528}, abstractNote={The objective of this article is to introduce a novel technique for computing numerical solutions to the nonlinear inverse heat conduction problem. This involves solving nonlinear parabolic equations with Cauchy data provided on one side $\Gamma$ of the boundary of the computational domain $\Omega$. The key step of our proposed method is the truncation of the Fourier series of the solution to the governing equation. The truncation technique enables us to derive a system of 1D ordinary differential equations. Then, we employ the well-known Runge-Kutta method to solve this system, which aids in addressing the nonlinearity and the lack of data on $\partial \Omega \setmunus \Gamma$. This new approach is called the dimensional reduction method. By converting the high-dimensional problem into a 1D problem, we achieve exceptional computational speed. Numerical results are provided to support the effectiveness of our approach.}, publisher={arXiv}, author={Dinh-Nho, H`ao and Le, Thuy T. and Nguyen, Loc H.}, year={2023} }
 @article{le_nguyen_nguyen_park_2023, title={The time dimensional reduction method to determine the initial conditions without the knowledge of damping coefficients}, url={https://arxiv.org/abs/2308.13152}, DOI={10.48550/ARXIV.2308.13152}, abstractNote={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 a 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.}, publisher={arXiv}, author={Le, Thuy T. and Nguyen, Linh V. and Nguyen, Loc H. and Park, Hyunha}, year={2023} }
 @article{le_nguyen_tran_2022, title={A Carleman-based numerical method for quasilinear elliptic equations with over-determined boundary data and applications}, volume={125}, url={https://doi.org/10.1016/j.camwa.2022.08.032}, DOI={10.1016/j.camwa.2022.08.032}, abstractNote={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.}, journal={Computers & Mathematics with Applications}, publisher={Elsevier BV}, author={Le, Thuy T. and Nguyen, Loc H. and Tran, Hung V.}, year={2022}, month={Nov}, pages={13–24} }
 @article{le_nguyen_2022, title={A convergent numerical method to recover the initial condition of nonlinear parabolic equations from lateral Cauchy data}, volume={30}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-85085658625&partnerID=MN8TOARS}, DOI={10.1515/jiip-2020-0028}, abstractNote={Abstract}, number={2}, journal={Journal of Inverse and Ill-Posed Problems}, publisher={Walter de Gruyter GmbH}, author={Le, Thuy Thi Thu and Nguyen, Loc Hoang}, year={2022}, pages={265–286} }
 @article{le_klibanov_nguyen_sullivan_nguyen_2022, title={Carleman contraction mapping for a 1D inverse scattering problem with experimental time-dependent data}, volume={38}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-85126525729&partnerID=MN8TOARS}, DOI={10.1088/1361-6420/ac50b8}, abstractNote={Abstract}, number={4}, journal={Inverse Problems}, author={Le, T.T. and Klibanov, M.V. and Nguyen, L.H. and Sullivan, A. and Nguyen, L.}, year={2022} }
 @article{le_2022, title={Global reconstruction of initial conditions of nonlinear parabolic equations via the Carleman-contraction method}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-85131640139&partnerID=MN8TOARS}, DOI={10.48550/arXiv.2205.10648}, abstractNote={We propose a global convergent numerical method to reconstruct the initial condition of a nonlinear parabolic equation from the measurement of both Dirichlet and Neumann data on the boundary of a bounded domain. The first step in our method is to derive, from the nonlinear governing parabolic equation, a nonlinear systems of elliptic partial differential equations (PDEs) whose solution yields directly the solution of the inverse source problem. We then establish a contraction mapping-like iterative scheme to solve this system. The convergence of this iterative scheme is rigorously proved by employing a Carleman estimate and the argument in the proof of the traditional contraction mapping principle. This convergence is fast in both theoretical and numerical senses. Moreover, our method, unlike the methods based on optimization, does not require a good initial guess of the true solution. Numerical examples are presented to verify these results.}, journal={arXiv}, author={Le, T.T.}, year={2022} }
 @article{le_le_nguyen_2022, title={The Carleman convexification method for Hamilton-Jacobi equations on the whole space}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-85133374996&partnerID=MN8TOARS}, DOI={10.48550/arXiv.2206.09824}, abstractNote={We propose a new globally convergent numerical method to solve Hamilton-Jacobi equations in $\mathbb{R}^d$, $d \geq 1$. This method is named as the Carleman convexification method. By Carleman convexification, we mean that we use a Carleman weight function to convexify the conventional least squares mismatch functional. We will prove a new version of the convexification theorem guaranteeing that the mismatch functional involving the Carleman weight function is strictly convex and, therefore, has a unique minimizer. Moreover, a consequence of our convexification theorem guarantees that the minimizer of the Carleman weighted mismatch functional is an approximation of the viscosity solution we want to compute. Some numerical results in 1D and 2D will be presented.}, journal={arXiv}, author={Le, H.P.N. and Le, T.T. and Nguyen, L.H.}, year={2022} }
 @article{abhishek_le_nguyen_khan_2022, title={The Carleman-Newton method to globally reconstruct a source term for nonlinear parabolic equation}, url={https://arxiv.org/abs/2209.08011}, DOI={10.48550/ARXIV.2209.08011}, abstractNote={We propose to combine the Carleman estimate and the Newton method to solve an inverse source problem for nonlinear parabolic equations from lateral boundary data. The stability of this inverse source problem is conditionally logarithmic. Hence, numerical results due to the conventional least squares optimization might not be reliable. In order to enhance the stability, we approximate this problem by truncating the high frequency terms of the Fourier series that represents the solution to the governing equation. By this, we derive a system of nonlinear elliptic PDEs whose solution consists of Fourier coefficients of the solution to the parabolic governing equation. We solve this system by the Carleman-Newton method. The Carleman-Newton method is a newly developed algorithm to solve nonlinear PDEs. The strength of the Carleman-Newton method includes (1) no good initial guess is required and (2) the computational cost is not expensive. These features are rigorously proved. Having the solutions to this system in hand, we can directly compute the solution to the proposed inverse problem. Some numerical examples are displayed.}, publisher={arXiv}, author={Abhishek, Anuj and Le, Thuy and Nguyen, Loc and Khan, Taufiquar}, year={2022} }
 @article{le_nguyen_2022, title={The Gradient Descent Method for the Convexification to Solve Boundary Value Problems of Quasi-Linear PDEs and a Coefficient Inverse Problem}, volume={91}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-85129123569&partnerID=MN8TOARS}, DOI={10.1007/s10915-022-01846-3}, abstractNote={We study the global convergence of the gradient descent method of the minimization of strictly convex functionals on an open and bounded set of a Hilbert space. Such results are unknown for this type of sets, unlike the case of the entire Hilbert space. Then, we use our result to establish a general framework to numerically solve boundary value problems for quasi-linear partial differential equations with noisy Cauchy data. The procedure involves the use of Carleman weight functions to convexify a cost functional arising from the given boundary value problem and thus to ensure the convergence of the gradient descent method above. We prove the global convergence of the method as the noise tends to 0. The convergence rate is Lipschitz. Next, we apply this method to solve a highly nonlinear and severely ill-posed coefficient inverse problem, which is the so-called back scattering inverse problem. This problem has many real-world applications. Numerical examples are presented.}, number={3}, journal={Journal of Scientific Computing}, publisher={Springer Science and Business Media LLC}, author={Le, Thuy T. and Nguyen, Loc}, year={2022} }
 @article{klibanov_le_nguyen_sullivan_nguyen_2021, title={Convexification-based globally convergent numerical method for a 1D coefficient inverse problem with experimental data}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-85106104061&partnerID=MN8TOARS}, journal={arXiv}, author={Klibanov, M.V. and Le, T.T. and Nguyen, L.H. and Sullivan, A. and Nguyen, L.}, year={2021} }
 @article{le_nguyen_nguyen_powell_2021, title={The Quasi-reversibility Method to Numerically Solve an Inverse Source Problem for Hyperbolic Equations}, volume={87}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-85105425284&partnerID=MN8TOARS}, DOI={10.1007/s10915-021-01501-3}, abstractNote={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 $$L^2$$ . 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.}, number={3}, journal={Journal of Scientific Computing}, author={Le, T.T. and Nguyen, L.H. and Nguyen, T.-P. and Powell, W.}, year={2021} }
 @article{klibanov_le_nguyen_2020, title={Numerical solution of a linearized travel time tomography problem with incomplete data}, volume={42}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-85093113145&partnerID=MN8TOARS}, DOI={10.1137/19M1299487}, abstractNote={We propose a new numerical method to solve the linearized problem of the travel time tomography with incomplete data. Our method is based on the technique of the truncation of the Fourier series wi...}, number={5}, journal={SIAM Journal on Scientific Computing}, publisher={Society for Industrial & Applied Mathematics (SIAM)}, author={Klibanov, Michael V. and Le, Thuy T. and Nguyen, Loc H.}, year={2020}, pages={B1173–B1192} }