# Integrated Multiscale Modeling of Molecular Computing Devices Page: 3 of 8

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Results

We now detail the results stated in the summary.

1. We have constructed an approximation of the free space Green's function for the Helmholtz equation in

such a way that the application of the resulting operator is split between the spatial and the Fourier do-

mains, similar to Ewald method for evaluating lattice sums. Furthermore, we developed new quadratures

appropriate for integration in the disk in the Fourier domain to account for the singularity of the Green's

function. The contribution in the spatial domain requires a convolution with a small number of Gaussians

with negative exponents and positive weights which fits well into our existing fast algorithm. The combina-

tion of approximation and new quadratures yields a fast algorithm for computing volumetric convolutions

with this oscillatory Green's function in dimensions two and three. The algorithmic complexity effectively

scales as O(kd log k + C(log E-)d), where E is selected accuracy, k is the number of wavelengths in the

problem, d is the dimension, and C is a constant. Importantly, the algorithm maintains its efficiency when

applied to functions with singularities. We note that our approximation differs from that used in the Fast

Multipole Method and, as k -> 0, makes a smooth transition to the free space Green's function for the

Poisson equation. The results have been published in [5].

2. We have developed a fast and accurate algorithm for computing convolutions with the quasi-periodic

Helmholtz Green's function in two and three dimensions. This Green's function is defined by a conditionally

convergent (lattice) sum. Although the main idea of our approach is similar to that of Ewald summation,

in contrast to Ewald's method we use different means to interpret the sum. This provides an additional

insight and a fast algorithm for its application. As in Ewald's method, we split the sum into two, one in the

spatial domain and the other in the Fourier domain and both with exponential decay in their corresponding

domains. In each domain we describe a method for fast convolution with combined algorithmic complexity

O(kd log k), where k is the number of wavelengths in the problem and d is the dimension.

Due to the exponential decay of the terms in both in the spatial and Fourier domains, we truncate them

for a finite by arbitrary accuracy. As a result, our algorithm maintains its performance when applied to

functions with singularities. We approximate the terms in the spatial domain as a sum of Gaussians having

negative exponents and positive weights. Thus, to compute a convolution, we may use the multiresolution

algorithm developed previously (and, in part, supported by this grant). In the Fourier domain, we use the

Unequally Spaced Fast Fourier Transform (USFFT) to compute convolutions with the terms of Fourier

sum. Our algorithm is applicable to volumetric problems and has computational complexity 0(4d log n).

We note that in the Fourier domain we use quadratures for bandlimited functions. The results have been

published in [4].

3. We have developed a multiresolution representation of a class of integral operators satisfying boundary

conditions on simple domains and constructed fast algorithms for their application. We have also elucidated

some delicate theoretical issues related to the construction of periodic Green's functions for Poisson's

equation.

By applying the method of images to the non-standard form of the free space operator, we obtain lattice

sums that converge absolutely on all scales, except possibly on the coarsest scale. On the coarsest scale the

lattice sums may be only conditionally convergent and, thus, allow for some freedom in their definition.

We use the limit of square partial sums as a definition of the limit and obtain a systematic, simple

approach to the construction (in any dimension) of periodized operators with sparse non-standard forms.

We illustrate the results on several examples in dimensions one and three: the Hilbert transform, the

projector on divergence free functions, the non-oscillatory Helmholtz Green's function and the Poisson

operator. Remarkably, the limit of square partial sums yields a periodic Poisson Green's function which is

not a convolution. Using a short sum of decaying Gaussians to approximate periodic Green's functions, we

arrive at fast algorithms for their application. We further show that the results obtained for operators with

periodic boundary conditions extend to operators with Dirichlet, Neumann, or mixed boundary conditions.

The results are available electronically [3].

4. We have implemented improvements to the algorithm based on separated multiresolution representations

of Green's functions for the Poisson kernel and other non-oscillatory kernels and/or potentials. The speed

up is about 3-5 times depending on the order of the multiwavelet basis (the more significant acceleration is

for higher orders). This code is being used in the construction of solutions to the multiparticle Schrbdinger

equation, see [2, 6, 7].

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Beylkin, Gregory. Integrated Multiscale Modeling of Molecular Computing Devices, report, March 23, 2012; United States. (digital.library.unt.edu/ark:/67531/metadc840610/m1/3/: accessed September 25, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.