Numerical multiscale methods to determine the coefficient in diffusion problems

Here we study the inverse problem of determining the highly oscillatory coefficient a^ε in some PDEs of the form u^ε_t − ∇.(a^ε(x)∇u^ε) = 0, in a bounded domain Ω ⊂ ℝ^d; ε indicates the smallest characteristic wavelength in the problem (0 < ε ≪ 1). Assume that g(t, x) is given input data for (t, x) ∈ (0, T) × ∂Ω and the associated output is the thermal flux a^ε(x)∇u(T0, x) · n(x) measured on the boundary at a given time T₀. 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.