Convergence of the Hilbert uniqueness method via Tikhonov regularization
A parabolic differential equation on \([0,1]\times [0,T]\) with Dirichlet data and homogeneous initial data of the following form is considered \[ {\partial y\over\partial t}-{\partial\over\partial x} \Biggl(a(x){\partial y\over\partial x}\Biggr)= 0, \] \[ y(0,x)= 0,\quad y(t,0)= \nu_0(t),\quad y(t,1)= \nu_1(t), \] with diffusion coefficient \(a(x)\in C^1(0,1)\), \(a(x)\geq \mu>0\). For this problem, the author identifies the Hilbert uniqueness method with the calculation of the pseudoinverse. Because of its ill-posedness, it is approximated by a regularized Hilbert uniqueness method which is identical with Tikhonov regularization. The author finds sufficient conditions for convergence and convergence rates. Numerical experiments are given.
scientific article; zbMATH DE number 1406369
scientific article