Boundary data completion: the method of boundary value problem factorization

Abstract

Abstract. We consider the following data completion problem for the Laplace equation in the cylindrical domain:smooth bounded open set and a > 0), limited by the faces Γ 0 = {0}×O and Γ a = {a}×O. The Neumann and Dirichlet boundary conditions are given on Γ 0 while no condition is given on Γ a . The completion data problem consists in recovering a boundary condition on Γ a . This problem has been known since Hadamard [12] to be ill-posed. The problem is set as an optimal control problem with a regularized cost function. To obtain directly an approximation of the missing data on Γ a we use the method of factorization of elliptic boundary value problems. This method allows to factorize a boundary value problem in the product of two parabolic problems. Here it is applied to the optimality system (i.e. jointly on the state and adjoint state equations).

An optimal quasi solution for the Cauchy problem for Laplace equation in the framework of inverse ECG

“…In the above equation, it is clear that ∇u is upper and lower bounded by a scalar multiple of (−∆) 1/4 Φ . In other words, a solution of the boundary data problem (1)-(4) exists if and only if E 1/2 (Γ) is the Hilbert space H 1/2 00 defined in [18] with inner product as in [2], all details can be reviewed at [16, chap 2]:…”

“…(12). That can also be done in the framework of the invariant embedding in [2], with the slightly difference that the embedding is taken from a to 0 instead of from 0 to a, and state u defined by (6)-(9) is the only one used. The boundary data problem (6)-(9) is embedded in a family of boundary data problem defined in the sub-domains Ω s = (s, a) × Γ.…”

“…Given a domain Ω and two separate parts Γ 0 and Γ 1 of its boundary ∂Ω, this problem, understood as the boundary value completion on Γ 1 , from given Cauchy data on Γ 0 , of an harmonic potential defined over Ω, has been widely studied because of its various medical and engineering applications [2,6,8,13]. Many examples in the literature show that this Cauchy problem can be approached as a linear inverse problem, severely ill-posed in the Hadamard sense [7,17].…”

“…• ζ † , ψ † , δ (1) , δ (2) ∈ V (550) such that any one of their Fourier coefficients in the {v 1 , v 2 ...} system is a pseudo random number with a normal standard distribution. Besides the error, we build a large sample of pseudo-random exact pairs (ζ † , ψ † ) in L 2 (Γ) × L 2 (Γ) to control the regularity on Φ † through its Fourier series.…”

“…• Φ † = (W A) p (−∆) − 1 4 ζ † , according with the cases described above. •φ = AΦ † + Bψ † + δ δ (1) δ (1) , andψ = ψ † + δ δ (2) δ (2) . 6.43584E-7 1.70329E-7 7.71689E-7 1.83486E-7 1E-3 0.003488645 0.000192304 0.004563972 0.000762609 R((W A) 3 (−∆) −1/4 ) 1E- 6 3.41411E-5 5.07517E-6 4.95877E-5 6.81315E-6 1E- 9 3.59424E-7 5.00185E-08 5.49233E-7 6.91165E-8 Table 1.…”