First, the well-posedness and numerical simulation of PDEs involving pseudo-differential operators are considered.

One example is the water wave equation. Lack of rigorous analysis of its time discretization inhibits the design of more efficient algorithms. First, a model simplified from the water wave equation of infinite depth is considered. This model preserves two main properties of the water wave equation: non-locality and hyperbolicity. Systematic stability studies of the fully discrete approximation of such systems is also complemented. As a result, an optimal time discretization strategy is provided in the form of a modified CFL condition, i.e. $\Delta t=O(\sqrt{\Delta x})$. Meanwhile, the energy stable property is established for certain explicit Runge-Kutta methods.

Another example is the vectorial Peierls-Nabarro model. First, a non-local scalar Ginzburg-Landau equation with an anisotropic positive singular kernel is derived from the original model. We first prove that minimizers of the PN energy for this reduced scalar problem exist. We also prove that these minimizers are smooth 1D profiles. Then a De Giorgi-type conjecture of single-variable symmetry for both minimizers and layer solutions is established. The proof of this De Giorgi-type conjecture relies on a delicate spectral analysis which is especially powerful for nonlocal pseudo-differential operators with strong maximal principle.

The second problem is the Online principal component analysis (PCA). It has been an efficient tool in practice to reduce dimension. However, convergence properties of the corresponding ODE (the deterministic flow) are still not fully unknown. A new technique is developed to determine stable manifolds of the ODE. This technique analyzes the rank of the initial datum. Using this technique, we derived the explicit expression of the stable manifolds. As a consequence, exponential convergence to stable equilibrium points was proved. The success of this new technique should be attributed to the semi-decoupling property of the SGA method: iteration of previous components does not depend on that of later ones.

The convergence property of the discrete algorithm is also considered. The algorithm is viewed as a stochastic process on the parameter space and semi-group. First, the discrete algorithm can be viewed as a semigroup: $S^k\varphi=\mathbb{E}[\varphi(\mathbf W(k))]$. Second, stochastic differential equations (SDEs) are constructed on the Stiefel manifold, i.e. the diffusion approximation, to approximate the semigroup. By weak convergence, the algorithm is 'close to' the SDEs. Finally, reversibility of the SDEs to prove long time convergence.

The third problem is the rigorous verification of the linear response theory (LRT). This theory considers the response of a system at equilibrium to external perturbations. It assumes that eh response is a linear functional of the perturbation. In particular, Langevin dynamics is considered. Equivalent conditions of reversibility of Langevin dynamics are proposed. Then, utilizing the ergodicity of Fokker-Planck equations, the response functional is rigorously verified for both over-damped and under-damped Langevin dynamics. As a corollary, the Green-Kubo relation is also verified.