  This PhD thesis conducts a focused study of strongly correlated materials
with localized electron orbitals. We have studied two real materials (LuNiO$_3$
and VO$_2$) and one model system, i.e., the Anderson impurity model. The thesis
is divided into two parts.
  In the first part, we use the DFT+$U$ method, to calculate the equilibrium
phase transitions of LuNiO$_3$ as an example of rare-earth nickelates under a
substrate-induced strain, and the nonequilibrium phase transitions of VO$_2$ as
an example of narrow-gap Mott insulators under laser-pulse-induced
photoexcitations. The effect of adding $U$ is manifested in both materials as
the change of band structure in response to the change of orbital occupancies
of electrons, i.e., the soft band effect. This effect brings about competitions
of electrons between different orbitals and gives rise to multiple metastable
states.
  In the second part, we go beyond band theory and study how we can use the
density matrix renormalization group (DMRG) method based on matrix product
states (MPS) to perform real-time evolutions of the Anderson impurity model,
towards the goal of building a real-time impurity solver for the nonequilibrium
dynamical mean-field theory (DMFT). We study both the quenched and periodically
driven single-impurity Anderson models (SIAM) and have obtained some complexity
results based on a non-standard implementation of DMRG (the 4-MPS method) we
developed in the thesis. We find in the star geometry of the SIAM that the
ordering of the bath orbitals in the MPS affects dramatically the increase of
entanglement entropy and thus the computational complexity of the simulation.
