  We study the Holstein model of spinless fermions, which at half-filling
exhibits a quantum phase transition from a metallic Tomonaga-Luttinger liquid
phase to an insulating charge-density-wave (CDW) phase at a critical
electron-phonon coupling strength. In our work, we focus on the real-time
evolution starting from two different types of initial states that are CDW
ordered: (i) ideal CDW states with and without additional phonons in the system
and (ii) correlated ground states in the CDW phase. We identify the mechanism
for CDW melting in the ensuing real-time dynamics and show that it strongly
depends on the type of initial state. We focus on the far-from-equilibrium
regime and emphasize the role of electron-phonon coupling rather than dominant
electronic correlations, thus complementing a previous study of photo-induced
CDW melting [H. Hashimoto and S. Ishihara, Phys. Rev. B 96, 035154 (2017)]. The
numerical simulations are performed by means of matrix-product-state based
methods with a local basis optimization (LBO). Within these techniques, one
rotates the local (bosonic) Hilbert spaces adaptively into an optimized basis
that can then be truncated while still maintaining a high precision. In this
work, we extend the time-evolving block decimation (TEBD) algorithm with LBO,
previously applied to single-polaron dynamics, to a half-filled system. We
demonstrate that in some parameter regimes, a conventional TEBD method without
LBO would fail. Furthermore, we introduce and use a ground-state density-matrix
renormalization group method for electron-phonon systems using local basis
optimization. In our examples, we account for up to $M_{\rm ph} = 40$ bare
phonons per site by working with $O(10)$ optimal phonon modes.
