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This paper concerns a one-phase inverse Stefan problem in one-dimensional space. The problem is ill-posed in the sense that the solution does not depend continuously on the data. We also consider noisy data for this problem. As such, we first regularize the proposed problem by the discrete mollification method. We apply the integration matrix using Bernstein basis polynomials for the discrete mollification method. Through this method, the execution time was gradually decreased. We then extend the space marching algorithm for solving our problem. Moreover, proofs of stability and convergence of the process are given. Finally, the results of this paper have been illustrated and examined by some numerical examples. Numerical examples confirm the efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

This paper is concerned with the inverse scattering problem which aims to determine the spatially distributed dielectric constant coefficient of the 2D Helmholtz equation from multifrequency backscatter data associated with a single direction of the incident plane wave. We propose a globally convergent convexification numerical algorithm to solve this nonlinear and ill-posed inverse problem. The key advantage of our method over conventional optimization approaches is that it does not require a good first guess about the solution. First, we eliminate the coefficient from the Helmholtz equation using a change of variables. Next, using a truncated expansion with respect to a special Fourier basis, we approximately reformulate the inverse problem as a system of quasilinear elliptic PDEs, which can be numerically solved by a weighted quasi-reversibility approach. The cost functional for the weighted quasi-reversibility method is constructed as a Tikhonov-like functional that involves a Carleman Weight Function. Our numerical study shows that, using a version of the gradient descent method, one can find the minimizer of this Tikhonov-like functional without any advanced a priori knowledge about it. [ABSTRACT FROM AUTHOR]

This paper studies an inverse problem of recovering an unknown diffusion coefficient in a parabolic equation. We adopt a total variation regularization method to deal with the ill-posedness. This method has the advantage to solve problems that the solution is non-smooth or discontinuous. By transforming the problem into an optimal control problem, we derive a necessary condition of the control functional. Through some prior estimates of the direct problem, the uniqueness and stability of the minimizer are obtained. In the numerical part, a Gauss–Jacobi iteration scheme is used to deal with the non-linear term. Some numerical examples are presented to illustrate the performance of the proposed algorithm. [ABSTRACT FROM AUTHOR]

This article presents a computational analysis of the adjoint problem approach for parabolic inverse coefficient problems based on boundary measured data. The proposed coarse-fine grid algorithm constructed on the basis of this approach is an effective computational tool for the numerical solution of inverse coefficient problems with various Neumann or/and Dirichlet type measured output data. In the previous Part I paper it was shown that the ill-posedness also depends on where Neumann and Dirichlet conditions are given: in the direct problem or as an output data. Based on integral identities relating solutions of direct problems to appropriate adjoint problems solutions, a coarse-fine grid algorithm for parabolic coefficient identification problems is constructed. It is shown that use of a coarse grid for the interpolation of the unknown coefficient and a fine grid for the numerical solution of the well-posed forward and backward parabolic problems guarantees an optimal compromise between the accuracy and stability in numerically solving the inverse problems. The efficiency and applicability of this method is demonstrated on various numerical examples with noisy free and noisy data. [ABSTRACT FROM AUTHOR]