VAC (Variational approach for conformation dynamics)

Concept

• a trajectory with configurations ${x_1, …, x_T}$
• a set of basis functions defined on the space of configurations ${\chi_1(x), \dots, \chi_n(x)}$

we compute the two correlation matrices $C_0,C_\tau$

\[\begin{align*} c_{ij}(0) &= \langle \chi_i(x_t) \chi_j(x_t) \rangle_t,\\ c_{ij} (\tau) &= \langle \chi_i(x_t) \chi_j(x_{t+\tau}) \rangle_t \end{align*}\]

where $\langle\cdot\rangle_t$ is average over time $t$.

Then we solve the generalized eigenvalue problem

\[C_\tau r = C_0 r l(\tau)\]

where the eigenvalues $l(\tau)$ approximate the dominant eigenvalues of the Markov propagator or Markov backward propagator of the underlying dynamics.

The corresponding eigenfunction of the backward propagator is approximated by

\[\psi(x) = \sum_i r_i \chi_i(x)\]

Numerical examples

1. Metastable potential from a Gaussian stationary density
2. Ritz method with characteristic function (Markov state model)
3. Ritz method with a Hermite basis
4. Roothaan-Hall method with a Gaussian basis
5. Nonlinear optimization
6. Quartic potential

Paper

• Noé, Frank, and Feliks Nuske. “A variational approach to modeling slow processes in stochastic dynamical systems.” Multiscale Modeling & Simulation 11.2 (2013): 635-655. [BibTex]

Code

• origin : markovmodel/variational
• forked : tanaka-hiroki1989/variational
• markovmodel/PyEMMA