  We propose inverse problems of crack propagation using the phase-field
models. First, we study the crack propagation in an inhomogeneous media in
which fracture toughness varies in space. Using the two phase-field models
based on different surface energy functionals, we perform simulations of the
crack propagation and show that the $J$-integral reflects the effective
inhomogeneous toughness. Then, we formulate regression problems to estimate
space-dependent fracture toughness from the crack path. Our method successfully
estimates the positions and magnitude of tougher regions. We also demonstrate
that our method works for different geometry of inhomogeneity.
