Adaptive Runge–Kutta regularization for a Cauchy problem of a modified Helmholtz equation

Abstract

In this paper, we investigate the Cauchy problem for the modified Helmholtz equation. We consider the data completion problem in a bounded cylindrical domain on which the Neumann and the Dirichlet conditions are given in a part of the boundary. Since this problem is ill-posed, we reformulate it as an optimal control problem with an appropriate cost function. The method of factorization of boundary value problems is used to immediately obtain an approximation of the missing boundary data. In order to regularize this problem, we firstly scrutinize two classical regularizations for the cost function. Then we propose a new numerical regularization named “adaptive Runge–Kutta regularization”, which does not require any penalization term. Finally, we compare them numerically.

Decoupled matrix Riccati differential equations approach for robust boundary data completion in time-fractional diffusion problems

“…Then, it is necessary to use some regularization techniques [18] according to the resolution in the presence of noisy observations. We set forward the regularization of the optimal control problem performed in [18] by considering the cost function…”

“…In this section, we propose an innovative numerical approach for solving the completion of missing data through the boundary factorization method using the anadromic scheme detailed in [18] with the regularization given in subsection 3.4. For simplicity we consider Ω := (0, a) × (0, b), a and b are positive real numbers, see Figure 2:…”

“…After this discretization, our goal is to solve the data completion problem at each node time t k using the factorization method introduced in [18]: We split the problem (2.1) into two well-posed sub-problems with mixed boundary conditions. For one of them (problem (2.2)), (Dirichlet, Neumann) conditions are enforced on (Γ m , Γ u ), and (Neumann,Dirichlet) conditions for the second problem (2.3):…”

“…As is known, the fractional diffusion equation is ill-posed since the solution does not continue with respect to the Cauchy data [12]. Then, it is necessary to use some regularization techniques [18] according to the resolution in the presence of noisy observations. We set forward the regularization of the optimal control problem performed in [18] by considering the cost function…”

“…This research aims to present a novel mathematical formulation for missing data reconstruction without the use of interpolation techniques. To do this, we employ a őnite difference scheme for temporally discretizing the Caputo fractional derivative introduced in [41] to produce a spatial completion data problem at each time node, which is solved by using the factorization method introduced in [2,15] where the authors solved the completion data problem for the Laplace equation and developed in [18] for the modiőed Helmholtz equation. This method allows transforming the problem obtained at each node of time into a decoupled system of őrst-order differential equa- as expounded in the works [4,32].…”