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Multi-physics Preconditioners

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.

Relevant Publications:

Implicit Solvation

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 and goal-oriented adaptive mesh refinement methods.

Relevant Publications:

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 molecule.

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.

Relevant Publications:

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Eric C. Cyr

E-mail: Eric C. Cyr
(505)844-0360 (Phone)

Mailing address
Sandia National Laboratories
P.O. Box 5800, MS 1320
Albuquerque, NM 87185-1320

Useful links
University of Illinois
Clemson University