Identification of the anti-diffusion coefficient for the linear Kuramoto-Sivashinsky equation
The direct problem is written as \[ u_{t}+ (\sigma(x)u_{xx})_{xx} + \gamma(x) u_{xx} = f\ \ ((x,t) \in (0, L)\times (0,T)),\ \] \[ u(0,t) = 0,\ u(L, t) = 0,\ u_{x}(0, t) = 0, \ u_{x}(L, t) = 0,\ u(x, 0)=u_{0}(x). \] The corresponding inverse problem is to recover the unknown coefficient \(\gamma(x)\) using that data \(u(x,T)=m(x)\). This inverse problem is reduced to the optimization problem \[ \min_{\gamma\in M}\int_{0}^{L}|u(x,T)-m(x)|^{2}\,dx + \alpha\|\gamma\|_{W_{2}^{1}(0,L)}^{2}, \] where \(M=\{\gamma\in W_{2}^{1}(0,L): \ \|\gamma\|_{W_{2}^{1}(0,L)}\leq \eta \}\) and \(\alpha,\eta \) are positive parameters. It is proven that a solution to this optimization problem is unique and continuously depends on the data. The numerical algorithm for solving the optimization problem is described and the results of numerical experiments are presented.
scientific article; zbMATH DE number 7315637
scientific article