  We formulate a constructive theory of noncommutative Landau-Ginzburg models
mirror to symplectic manifolds based on Lagrangian Floer theory. The
construction comes with a natural functor from the Fukaya category to the
category of matrix factorizations of the constructed Landau-Ginzburg model. As
applications, it is applied to elliptic orbifolds, punctured Riemann surfaces
and certain non-compact Calabi-Yau threefolds to construct their mirrors and
functors. In particular it recovers and strengthens several interesting results
of Etingof-Ginzburg, Bocklandt and Smith, and gives a unified understanding of
their results in terms of mirror symmetry and symplectic geometry. As an
interesting application, we construct an explicit global deformation
quantization of an affine del Pezzo surface as a noncommutative mirror to an
elliptic orbifold.
