Second order asymptotical regularization methods for inverse problems in partial differential equations

Abstract

We develop Second Order Asymptotical Regularization (SOAR) methods for solving inverse source problems in elliptic partial differential equations with both Dirichlet and Neumann boundary data. We show the convergence results of SOAR with the fixed damping parameter, as well as with a dynamic damping parameter, which is a continuous analog of Nesterov's acceleration method. Moreover, by using Morozov's discrepancy principle together with a newly developed total energy discrepancy principle, we prove that the approximate solution of SOAR weakly converges to an exact source function as the measurement noise goes to zero. A damped symplectic scheme, combined with the finite element method, is developed for the numerical implementation of SOAR, which yields a novel iterative regularization scheme for solving inverse source problems. Several numerical examples are given to show the accuracy and the acceleration effect of SOAR. A comparison with the state-of-the-art methods is also provided.

Inverse Problem for an Equation of the Reaction-Diffusion-Advection Type with Data on the Position of a Reaction Front: Features of the Solution in the Case of a Nonlinear Integral Equation in a Reduced Statement

“…If the size of the observation area is not large enough, the obtained solution can be used as a good initial approximation when solving the inverse problem in the full statement (see, for example, [38]). Some recent results concerning features of solving nonlinear inverse problems (see, for example, [47][48][49][50][51][52][53][54][55]), could be used to construct more efficient algorithms, including error estimation (see, for example, [56][57][58][59][60][61][62]).…”

On Some Features of the Numerical Solving of Coefficient Inverse Problems for an Equation of the Reaction-Diffusion-Advection-Type with Data on the Position of a Reaction Front

“…As prospects for the development of the proposed method, the implementation of methods for performing a posteriori estimation of the accuracy of the obtained solution should be noted (see, for example, [71][72][73][74][75][76][77][78][79][80][81]).…”

Determining the Coefficients of the Thermoelastic System from Boundary Information

“…Here we do not list all the references about these topics. We refer the reader to [34][35][36][37][38] for Maxwell's equations, to [39][40][41][42][43][44][45][46][47][48][49] for incompressible fluid, Schrödinger operator, elastic operator, and the related problems. For the studies about other types of Dirichlet-to-Neumann map, we also refer the reader to [3,[7][8][9]50,51] and references therein.…”