Ill-posed problems with unbounded operators

  title={Ill-posed problems with unbounded operators},
  author={Alexander G. Ramm},
Let A be a linear, closed, densely defined unbounded operator in a Hilbert space. Assume that A is not boundedly invertible. If equation (1) Au = f is solvable, and ‖fδ−f‖ ≤ δ, then the following results are provided: problem Fδ(u) := ‖Au−fδ‖+α‖u‖ has a unique global minimizer uα,δ for any fδ, uα,δ = A∗(AA∗+ αI)fδ. There is a function α = α(δ), limδ→0 α(δ) = 0 such that limδ→0 ‖uα(δ),δ− y‖ = 0, where y is the unique minimal-norm solution to (1). A priori and a posteriori choices of α(δ) are… CONTINUE READING