A non-local boundary value problem method for the Cauchy problem for elliptic equations

Abstract

Let H be a Hilbert space with norm • , A : D(A) ⊂ H → H a positive definite, self-adjoint operator with compact inverse on H, and T and given positive numbers. The ill-posed Cauchy problem for elliptic equationswith a 1 being given and α > 0 the regularization parameter. A priori and a posteriori parameter choice rules are suggested which yield order-optimal regularization methods. Numerical results based on the boundary element method are presented and discussed.

Solving a Cauchy problem for a 3D elliptic PDE with variable coefficients by a quasi-boundary-value method

“…General stability estimates are given in, e.g., [3,37]; see also [16,34]. In the case of a rectangular geometry, more concrete results can be found [24].…”

“…The Cauchy problem for an elliptic equation is a classical ill-posed problem and occurs in several important applications, such as inverse scattering [17,37], electrical impedance tomography [10], optical tomography [7], and thermal engineering [23]. The topic is treated in several monographs [16,36,37,41,42,45], and in numerous papers, see [1,3,6,8,9,18,24,25,34,47,48,51,57,61] and the references therein. Even if some of the theoretical investigations are quite general, the numerical procedures proposed are typically for the two dimensional case and often only valid for the problem with constant coefficients.…”

An improved non-local boundary value problem method for a cauchy problem of the Laplace equation

“…It should be mentioned that the method of separation of variables is used to give the expression of solution, so the method proposed in this paper can be extended to solve the Cauchy problem of Laplace equation in cylindrical domain like doing in [7]. But it can not be applied in more general geometries, this is a limit of non-local boundary value problem method.…”

“…In [7], Hào-Duc-Lesnic used this method to solve the following Cauchy problem for elliptic equations in a cylindrical domain…”

“…In this paper, we will present an improved non-local boundary value problem method to construct stable approximate solutions to the problems (1.2) and (1.3). Our used method has a little difference with one in [7]. Here we replace the initial conditions u(…”