17,138 research outputs found
Discrete logarithm computations over finite fields using Reed-Solomon codes
Cheng and Wan have related the decoding of Reed-Solomon codes to the
computation of discrete logarithms over finite fields, with the aim of proving
the hardness of their decoding. In this work, we experiment with solving the
discrete logarithm over GF(q^h) using Reed-Solomon decoding. For fixed h and q
going to infinity, we introduce an algorithm (RSDL) needing O (h! q^2)
operations over GF(q), operating on a q x q matrix with (h+2) q non-zero
coefficients. We give faster variants including an incremental version and
another one that uses auxiliary finite fields that need not be subfields of
GF(q^h); this variant is very practical for moderate values of q and h. We
include some numerical results of our first implementations
Distributed-memory large deformation diffeomorphic 3D image registration
We present a parallel distributed-memory algorithm for large deformation
diffeomorphic registration of volumetric images that produces large isochoric
deformations (locally volume preserving). Image registration is a key
technology in medical image analysis. Our algorithm uses a partial differential
equation constrained optimal control formulation. Finding the optimal
deformation map requires the solution of a highly nonlinear problem that
involves pseudo-differential operators, biharmonic operators, and pure
advection operators both forward and back- ward in time. A key issue is the
time to solution, which poses the demand for efficient optimization methods as
well as an effective utilization of high performance computing resources. To
address this problem we use a preconditioned, inexact, Gauss-Newton- Krylov
solver. Our algorithm integrates several components: a spectral discretization
in space, a semi-Lagrangian formulation in time, analytic adjoints, different
regularization functionals (including volume-preserving ones), a spectral
preconditioner, a highly optimized distributed Fast Fourier Transform, and a
cubic interpolation scheme for the semi-Lagrangian time-stepping. We
demonstrate the scalability of our algorithm on images with resolution of up to
on the "Maverick" and "Stampede" systems at the Texas Advanced
Computing Center (TACC). The critical problem in the medical imaging
application domain is strong scaling, that is, solving registration problems of
a moderate size of ---a typical resolution for medical images. We are
able to solve the registration problem for images of this size in less than
five seconds on 64 x86 nodes of TACC's "Maverick" system.Comment: accepted for publication at SC16 in Salt Lake City, Utah, USA;
November 201
Probing nonlinear adiabatic paths with a universal integrator
We apply a flexible numerical integrator to the simulation of adiabatic
quantum computation with nonlinear paths. We find that a nonlinear path may
significantly improve the performance of adiabatic algorithms versus the
conventional straight-line interpolations. The employed integrator is suitable
for solving the time-dependent Schr\"odinger equation for any qubit
Hamiltonian. Its flexible storage format significantly reduces cost for storage
and matrix-vector multiplication in comparison to common sparse matrix schemes.Comment: 8 pages, 6 figure
Adiabatic optimization without local minima
Several previous works have investigated the circumstances under which
quantum adiabatic optimization algorithms can tunnel out of local energy minima
that trap simulated annealing or other classical local search algorithms. Here
we investigate the even more basic question of whether adiabatic optimization
algorithms always succeed in polynomial time for trivial optimization problems
in which there are no local energy minima other than the global minimum.
Surprisingly, we find a counterexample in which the potential is a single basin
on a graph, but the eigenvalue gap is exponentially small as a function of the
number of vertices. In this counterexample, the ground state wavefunction
consists of two "lobes" separated by a region of exponentially small amplitude.
Conversely, we prove if the ground state wavefunction is single-peaked then the
eigenvalue gap scales at worst as one over the square of the number of
vertices.Comment: 20 pages, 1 figure. Journal versio
- …