Distributed Decomposition Over
Hyperspherical Domains
Aron Ahmadia1, David Keyes1, David Melville2, Alan Rosenbluth2, Kehan
Tian2
1 Department of Applied Physics and Applied Mathematics, Columbia University,
New York, NY 10027-6902, USA
2 IBM T.J. Watson Research Center, Yorktown Heights, New York 10598-0218,
USA
1 Abstract
We are motivated by an optimization problem arising in computational scaling
for optical lithography that reduces to finding the point of minimum radius
that lies outside of the union of a set of diamonds centered at the origin of
Euclidean space of arbitrary dimension. A decomposition of the feasible re-
gion into convex regions suggests a heuristic sampling approach to finding the
global minimum. We describe a technique for decomposing the surface of a hy-
persphere of arbitrary dimension, both exactly and approximately, into a spe-
cific number of regions of equal area and small diameter. The decomposition
generalizes to any problem posed on a spherical domain where regularity of
the decomposition is an important concern. We specifically consider a storage-
optimized decomposition and analyze its performance. We also show how the
decomposition can parallelize the sampling process by assigning each proces-
sor a subset of points on the hypersphere to sample. Finally, we describe a
freely available C++ software package that implements the storage-optimized
decomposition.
2 Global Optimization for Semiconductor Lithography
Mask Design
In the newly heralded field of computational scaling for optical semiconduc-
tor lithography (Singh [2007]), industrial scientists are now investigating a
global minimization formulation of the problem of mask design for optimizing
robust process windows (Rosenbluth et al. [2002], Rosenbluth et al. [2007]).
We consider the problem of forming an optimal mask design for the optical
printing of a given 2-D target image, which is considered as a set of sampled

2 A. J. Ahmadia, D. E. Keyes, D. O. Melville, A. E. Rosenbluth, K. Tian
target points that must be sufficiently illuminated for the image to print cor-
rectly. We minimize the sum of magnitudes of the set of nd exposure modes,
with each axis xi corresponding to an exposure mode, the magnitude for a
weighted sum of Fourier diffraction order amplitudes. In the space of the nd
exposure modes, each sample of the nh image features is represented as a nd-
dimensional diamond. The sampled image feature is sufficiently illuminated
by the set of exposure modes if their representative coordinate x ∈ Rnd lies
outside the diamond. Sufficient illumination of all sampled target features in
an exposure is achieved in all points outside of the union of the set of the
diamonds H. Although a global minimum is ideal, the goal of the problem
is to find good solutions in a reasonable amount of time. Additionally, global
minima that satisfy all the constraints may still be infeasible due to manufac-
turability considerations, so a good solution method will provide the global
minima and may provide a set of the best available local minima within each
orthant of the search space.
2.1 A Piecewise Linear Nonconvex Global Optimization Problem
The semiconductor lithography problem can be considered as the search for
the global minima of a piecewise linear optimization problem subject to piece-
wise linear nonconvex constraints:
minimize ‖x‖1
subject to Ai(x) · x ≥ bi
∀Ai ∈ H
(1)
We let P (Ai) be the set of constraint half-spaces describing the exterior
of diamond i. We define the discrete set of all (nr = nh ∗ 2nd) diamond
constraint half-spaces P (Ai) as Q. Later, we will use subsets of Q to define
convex partitions of the feasible set. We also define the jth principal axis
of diamond Ai as ci,j. We then note that each plane defining a half-space
connects the nd principal axes of the diamond. Ai(x) can be considered as a
function of the choices of signs for the vectors representing the principal axis
of the diamond. The choice of connection to the ’positive’ or ’negative’ end of
each principal axis uniquely determines the one of the 2nd planes in P (Ai).
We may enumerate a single plane Ai,k ∈ P (Ai) with a tuple, or ordered list
of nd signs si,k, with the sign of the jth element si,k(j) corresponding to the
ends of the principal axes ci,j it connects. We represent the concatenation
of length nd tuples si,k, corresponding to a choice of plane for each of the
nh diamonds, into a single tuple: s and note that a given tuple s refers to a
subset of Q such that a constraint plane is chosen from each diamond. Ai is
a member of the set of nh diamond polytopes in nd-dimensional space that x
must lie completely outside of, and the choice of half-space Ai(x) ∈ P (Ai) is
determined by the location of x with respect to the nd principal axes of the