Regularization of the continuation problem for elliptic equations

Abstract

We investigate the continuation problem for the elliptic equation. The continuation problem is formulated in operator form Aq D f . The singular values of the operator A are presented and analyzed for the continuation problem for the Helmholtz equation. Results of numerical experiments are presented.

A Modification of Gradient Descent Method for Solving Coefficient Inverse Problem for Acoustics Equations

“…The same holds for Approach 3 and the 10% noise level. Such situation is not rare for inverse and ill-posed problems and leads us to the fact that the iteration number should be considered as the regularization parameter [49][50][51]. We plan to address the question of the optimal number of iterations depending on the noise level in further work.…”

Solution of the Ill-Posed Cauchy Problem for Systems of Elliptic Type of the First Order

“…Based on [6,7,[12][13][14] we have constructed the Carleman matrix and based on it the approximate solution of the Cauchy problem for the matrix factorization of the Helmholtz equation. Boundary value problems, as well as numerical solutions of some problems, are considered in [30][31][32][33][34][35][36][37][38][39]. When solving correct problems, sometimes, it is not possible to find the value of the vector function on the entire boundary.…”

The Plane Waves Method for Numerical Boundary Identification

“…Traditional numerical methods, in conjunction with an appropriately chosen regularization/stabilization method, have been employed to solve inverse problems associated with Helmholtz-type equations, such as the finite-difference method (FDM) [4,5], the finite element method (FEM) [25,26] and the boundary element method (BEM) [39,40], respectively. Both the FDM and the FEM require the discretization of the domain of interest which is time consuming and tedious, especially for complicated geometries.…”