Numerical multiscale methods to determine the coefficient in diffusion problems

Here we study the inverse problem of determining the highly oscillatory coefficient a^\varepsilon in some PDEs of the form u^\varepsilon_t - \nabla. (a^\varepsilon(x) \nabla u^\varepsilon)=۰, in a bounded domain \Omega \subset\mathbb{R}^d ; \varepsilon indicates the smallest characteristic wavelength in the problem (۰ < \varepsilon \ll ۱). Assume that g(t, x) is given input data for (t, x) \in (۰,T) \times\partial \Omega  and the associated output is the thermal flux a^\varepsilon(x)\nabla u(T_۰,x)\cdot n(x) measured on the boundary at a given time T_۰.  Due to the ill-posedness of the inverse problem, we reduce the dimension by seeking effective parameters. For the forward solver, we apply either analytic homogenization or some numerical multiscale methods such as the FE-HMM and LOD method.

کلیدواژه ها:

Heterogeneous multiscale method ، Homogenization ، Inverse problem ، Localized orthogonal decomposition method ، Parabolic partial differential equations