Model Reduction by Dirichlet-to-Neumann map for Molecular Dynamics and Quantum Mechanics
Open Access
- Author:
- Wu, Xiaojie
- Graduate Program:
- Mathematics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- July 10, 2018
- Committee Members:
- Xiantao Li, Dissertation Advisor/Co-Advisor
Xiantao Li, Committee Chair/Co-Chair
Ludmil Zikatanov, Committee Member
Wenrui Hao, Committee Member
Long-Qing Chen, Outside Member
Francesco Costanzo, Outside Member - Keywords:
- Molecular Dynamics
Absorbing boundary condition
Hartree-Fock method
Model reduction
Boundary element method
Wave propagation - Abstract:
- This dissertation studies model reduction techniques in the context of classical molecular dynamics and quantum mechanical models. Both the spatial and temporal reductions are considered. The model reduction is formulated as the Dirichlet-to-Neumman (DtN) map. In this dissertation, we restrict ourself into the systems where the number of degrees of freedoms is overwhelming, but the systems exihibit periodic structures, such as, lattice structure in crystalline solids or finite difference structure of the Laplace operator. In these special cases, we are able to take the advantage of the periodic structure, and evaluate the DtN map efficiently. The full model is reduced to a problem on the boundary of the computational domain. The main numerical tool used to compute the DtN map is the boundary element method. Since the lattice Green's function is computable for the system with the periodic structure, the boundary element method is practical. The boundary element method not only is crucial to static problems, where we seek the mechanical equilibirum, but also plays an important role in wave propagation problems. In wave propagation problems, the reduced model is the problem in a truncated domain with absorbing boundary conditions. The approximation of the dynamic DtN map provides such absorbing boundary condition. In this case, the stability of the reduced model must be ensured. The stability requirements will be presented. The same ansatz is applied to time-dependent Schr\"{o}dinger equation and the time-dependent Hartree-Fock equation. For continuous equations, we start from the discretized model by finite difference. The periodic structure of the nodal points are mathematically equivalent to the lattice structure. Lattice Green's function and the DtN map can be computed using a similar approach. Stable approximations will be presented. This approach can be naturally extended to the time-dependent Hartree-Fock equation when the potential is neglectable in the exterior region. The idea of model reduction is verified by several numerical experiments: fracture in atomistic model, phonons propagation in molecular dynamics, one-dimensional time-dependent Schr\"{o}dinger equation, and $^{16}$O+$^{16}$O colisions.