\chapter{Abstract}

The quest for new exotic materials lies in our theoretical understanding of strong correlations, which could only be sought via quantum emulators at large scale.
Optical lattices, being clean and easily tunable, become natural candidates.
This thesis is regarding numerical simulations of bosons and fermions in 3D optical lattices, which bridges the theory to the relevant experiments today as well as in the near future.
The first part is devoted to the various thermometry schemes which are made possible due to the state-of-art probing techniques in experiments. 
These schemes are quantitatively tested in this thesis to give realiable estimates of the thermodynamic temperature even in the presence of different experimental imperfections, therefore proving itself to be a feasible standard thermometer for bosonic optical lattices.
The second part is about the robust implementation of the QMC directed worm algorithm capable of scaling up to some 10 million sites with any lattice geometry, of which some manage to converge in merely a couple of hours.  
The current version is in fact one of the fastest, if not the fastest, in the world till date, specially designed for the needs of future experiments which will involve bigger bosonic clouds beyond parabolic trapping. 
Large-scale simulations involving large datasets are flexibly supported with chain automations in this new version.
The third part illustrates the use of density functional theory for fermionic shallow optical lattices.
We find that Stoner ferromagnetism, which was recently discovered to exist in fermionic gases, gets stablized by the optical lattice due to band structure effects.
At half-filling in the Hubbard model limit, the antiferromagnetic phase is recovered by density functional theory with the presence of a spin-density-wave gap that could be an indirect probe for antiferromagnetism in future experiments.
As a whole, the numerical simulations in this thesis are performed as numerically exact as possible, at least in some limit, with minimal assumptions, so as to ensure that our results here will remain relavant to both theoreticans and experimentalists in the coming future.