Multiphysics applications are characterized by strongly-nonlinear coupled physical
mechanisms that produce a solution with a wide-range of length and time scales.
This implies that the transient problem is stiff, requiring small time steps for
explicit simulations, and that classical operator splitting methods can exhibit problems
with stability and accuracy. Fully-implicit methods are an attractive choice that can
often provide unconditionally-stable time integration techniques for these types of
systems. However, the linear systems arising from implicit integration can be challenging
to solve efficiently. Our approach is to apply a Newton-Krylov iteration. Key to the
scalability and efficiency of this technique is employing an effective preconditioner.
This project focuses on developing such a preconditioner with an initial target of resitive
magnetohydrodynamics. We develop approximate block factorizations for preconditioning these
systems. These methods segregate the linear operator into different sub-matrices based on the
components of the physics. These individual components are typically more amenable to black-box
AMG technology. The difficulty with this approach is that an effective approximation of the
physical coupling embodied in the Schur complement operator is required. For the Navier-Stokes
equations this has been a very active area of study. The approaches have included the basic
physics-based SIMPLE solution method to more sophisticated techniques based on commuting
arguments, such as the pressure-convection diffusion or least-squares commutator preconditioners.
The molecules of interest in computational biology naturally occur
in solution. For a typical molecular dynamics simulation it is
necessary to surround the solute with several layers of explicitly
represented water molecules. This explicit solvent environment can
increase the total atom count by an order of magnitude, greatly
increasing run times over the same system in vacuo.
For this project we focused on the primary bottleneck of molecular
dynamics, the electrostatic calculation. Our approach was to
solve the Poisson-Boltzmann Equation (PBE), a PDE that describes
the average electrostatic potential induced by the solvent around a
solute, using finite elements. This research explored novel
discretizations of the PBE using FOSLS
adaptive mesh refinement methods.
Potential of Mean Force
One problem which computational molecular biology is uniquely suited is
the determination and exploration of transition and folding pathways. These
provide insight into the intermediate conformations a protein (for example)
goes through to find its folded state. Such information would be useful if
one wanted to accelerate (or prevent) the transition of a protein. Alternatively,
knowing the energetic and entropic barriers along such a pathway may give
researchers greater understanding of the mechanism and function of a particular
This research explored techniques to compute the so-called free energy profile
along a prescribed transition pathway (or in our parlance a reaction coordinate).
The free energy profile, also known as the potential of mean force, expresses
the "sum" of the energetic and entropic barriers as a potential energy function
on the reaction coordinate. It is the potential energy corresponding to the
averaging of the forces on the molecule for each value of the reaction coordinate.
Our research in this area focuses on the development of two techniques for computing
the potential of mean force. These methods generalize two pre-existing methods.
The Weighted Residual Method (WRM) we developed
extends the thermodynamic integration method. In particular the WRM finds the
potential of mean force by forcing the residual to be orthogonal to a suitable basis
in an inner product naturally defined by molecular dynamics. The second method we
developed is based on maximum likelihood estimation (MLE) method. This technique
attempts to find the potential of mean force that would have most likely produced
the observations produced by a molecular dynamics simulation.