# <span class=strong>Reception and Poster Session</span><br><br/><br/><b>Poster submissions welcome from all participants</b><br><br/><br/><a<br/><br/>href=/visitor-folder/contents/workshop.html#poster><b>Instructions</b></a>

**Directional fast multipole method for electromagnetics**

Paul Tsuji (The University of Texas at Austin)

No Abstract**Frechet differentiation of boundary integral representations**

Michael Epton (The Boeing Company)

The general form of the boundary integral representation formula

for a first order linear system in convervation law form is

reviewed.

The Frechet derivative of this representation is then studied

with the aid

of the parametric derivative formulae for the normal boundary

element,

n ds (resp. n dS or n dV) for boundaries in R^{2}

(resp. R^{3}or

R^{4}). It

is shown how the proper handling of the surface variation

enables the

recovery of the obvious representation formula the parametric

derivative

of the field vector.**A massively parallel butterfly algorithm for applying Fourier**

integral operators

Jack Poulson (The University of Texas at Austin)

In August of 2008, Candes, Demanet, and Ying introduced a fast

algorithm for applying Fourier integral operators of the form

∫ e^{iΦ(x,k)}f(k)dk, where Φ(x,k) is the so-called

phase function and is required to be real-valued and linear in

the second argument. Using a resolution of 1/N in each

dimension, the transform of arbitrary sources in the unit

d-dimensional cube of the frequency domain may be naively

evaluated over the unit cube of the spatial domain with

computational complexity O(N^{2d}). The algorithm of Candes

et al. yields the near-optimal complexity

O(N^{d}log_{2}(N)).

The contribution of the author is a new method for

parallelizing the above fast algorithm on distributed-memory

machines. The method assumes only a power-of-two number of

processes and has been shown to strong scale up to thousands of

cores of Blue Gene/P with 90% efficiency even for extremely

small problems. This is achieved by carefully peeling off

partitions of the frequency domain and applying them to the

spatial domain as the butterfly algorithm progresses. The

result is that, using 2^{P}processes in d dimensions, the

only communication required by each process is P

reduce-scatter summations over a team of 2^{d}processes.

Results are presented for a generalized Radon transform in

two-dimensions, though the implementation has been shown to

work in arbitrary dimensions. The source code is available at

href=http://code.google.com/p/butterflyfio>http://code.google.com/p/butterflyfio.**AFMPB: An adaptive fast multipole Poisson-Boltzmann solver for calculating electrostatics in biomolecular systems**

Benzhuo Lu (Chinese Academy of Sciences)

Poisson-Boltzmann (PB) electrostatics is a well established model in biophysics, however its application to the study of large scale biosystem dynamics such as the protein-protein encounter is still limited by the efficiency and memory constraints of existing numerical techniques. In this poster, we present an efficient and accurate scheme which incorporates recently developed novel numerical techniques to further enhance the present computational ability. In particular, a boundary integral equation approach is applied to discretize the linearized PB (LPB) equation; the resulting integral formulas are well conditioned and are extended to systems with arbitrary number of biomolecules; a robust meshing method is developed for molecular surface meshing; the solution process is accelerated by the Krylov subspace methods and an adaptive new version of fast multipole method (FMM). The Adaptive Fast Multipole Poisson-Boltzmann (AFMPB) solver is released under open source license agreement, and the meshing tool TMSmesh will also be released.**Second kind integral equations for the first kind Dirichlet**

problem of the biharmonic equation in three dimensions

Shidong Jiang (New Jersey Institute of Technology)

No Abstract**Fast integral equation methods for the modified Helmholtz equation**

Bryan Quaife (Simon Fraser University)**Treecode accelerated electrostatic calculation in implicit solvated biomolecular systems**

Weihua Geng (University of Michigan)

Poisson-Boltzmann (PB) equation based implicit solvent model can greatly reduce the computational cost in simulating solvated biomolecular systems by applying the mean field approximation in permittivities and capturing the mobile ions with Boltzmann distribution. However, solving PB equation suffers many numerical difficulties ranging from discontinuous permittivities and electrostatic field across the dielectric interface, the infinite boundary conditions, and charge singularities. In this project, we provide an efficient and accurate numerical algorithm, which adopts a well-conditioned boundary integral equation to handle these difficulties while accelerates the Krylov subspace based iterative methods such as GMRES with treecode. This Cartesian coordinates based treecode is an O(N*log(N)) scheme with properties of easy implementation, efficient memory usage O(N) , and straightforward parallelization. Benchmark testing results on Kirkwood sphere plus simulations of biomoleculars in various sizes are provided to demonstrate the accuracy, efficiency and robustness of the present methods.**Cartesian treecodes**

Robert Krasny (University of Michigan)

We present recent work on treecode algorithms for computing integrals

that arise in particle simulations. Our approach uses a Cartesian

Taylor series for the far-field expansion and is suitable for a

variety of kernels including multiquadric radial basis functions, the

screened Coulomb potential, and a regularized Biot-Savart kernel.

Sample results are shown.**Periodic density functional theory solver using multiresolution**

analysis with MADNESS

William Thornton (University of Tennessee)

Joint work with Robert J. Harrison and George Fann.

We report the first all electron adaptive refinement multiresolution band structure solver for crystalline systems. Most current software implementations of density functional theory for crystalline systems either use plane waves or a split representation such as linear augmented plane waves (LAPW) or linear muffin tin orbitals (LMTO) for their basis set. MADNESS, on the other hand, uses a fully numerical basis set which resides on an adaptive mesh. This choice has introduced numerous challenges in the implementation of a solid state band structure code. Instead of explicitly diagonalizing the one-particle Hamiltonian to update the single particle orbitals, we reformulate the problem into a bound state Helmholtz integral equation. Another challenge has been implementing the bound state Helmholtz Green's function and the Coulomb Green's function to include a lattice sum over the periodic images in the unit cell. To ensure timely convergence, we wrap the main Kohn-Sham loop inside a Krylov accelerated inexact Newton solver. We present all of these items and some preliminary results in our poster.