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Research projects
Fast multipole method
The fast multipole method is a technique to calculate the interaction between N particles, e.g. electrostatic forces, in O(N) operations. This has application in N-body problems, molecular dynamics, and the boundary element method, for example to solve the Poisson equation or Maxwell's equations. A naive approach requires N^2 operations and is therefore intractable for large N. Several multipole formulations were developed. Recently, a fast multipole method for acoustic based on a Fourier basis was created. The conventional approach uses spherical harmonics; a Fourier basis is preferred since it allows using fast Fourier transforms which are extremely efficient and widely available.
A kernel independent fast multipole method was created using a combination of Chebyshev interpolation and singular value decomposition. Approximation using Chebyshev polynomials have spectral accuracy and nearly satisfy the minmax property, that they achieve a given maximum error uniformly using the lowest order approximation. The method was extended to periodic boundary conditions which requires correctly computing conditionally convergent series.
A technique was also developed to improve the parallel efficiency of the fast multipole method by reducing the amount of communication between processors. This was achieved for the 1/r kernel using an expansion of the far field using plane waves. The problem can then be shown to involve a large number of embarrassingly parallel computational steps. The method can be shown to scale significantly faster than particle mesh Ewald for example which is based on 3D fast Fourier transforms.
Bio-molecular modeling
Professor Darve has developed several numerical methods to study mechanical properties of proteins. This includes:
- a new approach to compute free energy called the adaptive biasing force,
- new time integrators with multiple time steps and a multiscale spatial decomposition, e.g. Asynchronous Variational Integrators,
- novel techniques to compute electrostatic forces on parallel computers called the fast plane wave algorithm
- efficient numerical methods to calculate generalized Langevin equations (with memory) and Fokker-Planck equations.
In collaboration with the NASA Ames Research Center, Darve modeled the insertion of an amphipathic peptide in a membrane. This is a precursor step in forming an ion channel through the membrane.
Caption: Amphipathic helix inserted in a mimetic membrane
Molecular modeling can also be applied to understanding the mechanical properties of bundles of microtubules found in sensory neurons of nematodes (C. elegans) and the structure of ion channels in touch receptor neurons. This is in collaboration with M. Goodman (School of Medicine, Stanford). These microtubules are responsible for amplifying mechanical forces applied to the surface of the neuron. Upon touch to the surface of the skin, the ion channels in the membrane of touch receptor neurons open to let ions flow in and out of the cell.
Caption: Acid-sensing ion channel; this protein is similar to the ion channel in the touch receptor neurons. The top part represents the outer domain of the protein while the bottom part forms the transmembrane region.
Caption: The pore opens to let ions flow into the cell. Finding the structure of the open state for this channel protein is an active area of research.
Caption: Molecular model of the protein inside the membrane used in the molecular dynamics simulations.
Streaming computing
A new generation of processors with multi- or many-cores, e.g. multi-core Intel, GPU, Cell, is about to revolutionize the way scientific computing is done. These processors can perform operations 10 to 100 times faster than a conventional single core processor. However these processors are very difficult to program. New algorithms and programming environments are therefore needed.
Professor Darve developed code and algorithms to perform particle simulations and solve fluid dynamics equations of graphics cards. He is currently developing a domain specific language to solve partial differential equations on parallel platforms from clusters, multi-core processors to graphics cards. This is in collaboration with the computer science department at Stanford.
Microfluidics
Charged particles such as DNA flowing in micro or nano size channels have a complex dynamics due to electrophoresis and electroosmosis flows. In particle particle will have strong hydrodynamic and electrostatic interactions with nearby particles or walls. Several electrokinetic phenomena were studied such as the Induced Charge Electrophoresis which was shown to stabilize sedimenting systems.
