31,645 research outputs found
Solution of the inverse scattering problem by T-matrix completion. II. Simulations
This is Part II of the paper series on data-compatible T-matrix completion
(DCTMC), which is a method for solving nonlinear inverse problems. Part I of
the series contains theory and here we present simulations for inverse
scattering of scalar waves. The underlying mathematical model is the scalar
wave equation and the object function that is reconstructed is the medium
susceptibility. The simulations are relevant to ultrasound tomographic imaging
and seismic tomography. It is shown that DCTMC is a viable method for solving
strongly nonlinear inverse problems with large data sets. It provides not only
the overall shape of the object but the quantitative contrast, which can
correspond, for instance, to the variable speed of sound in the imaged medium.Comment: This is Part II of a paper series. Part I contains theory and is
available at arXiv:1401.3319 [math-ph]. Accepted in this form to Phys. Rev.
Barrier Frank-Wolfe for Marginal Inference
We introduce a globally-convergent algorithm for optimizing the
tree-reweighted (TRW) variational objective over the marginal polytope. The
algorithm is based on the conditional gradient method (Frank-Wolfe) and moves
pseudomarginals within the marginal polytope through repeated maximum a
posteriori (MAP) calls. This modular structure enables us to leverage black-box
MAP solvers (both exact and approximate) for variational inference, and obtains
more accurate results than tree-reweighted algorithms that optimize over the
local consistency relaxation. Theoretically, we bound the sub-optimality for
the proposed algorithm despite the TRW objective having unbounded gradients at
the boundary of the marginal polytope. Empirically, we demonstrate the
increased quality of results found by tightening the relaxation over the
marginal polytope as well as the spanning tree polytope on synthetic and
real-world instances.Comment: 25 pages, 12 figures, To appear in Neural Information Processing
Systems (NIPS) 2015, Corrected reference and cleaned up bibliograph
Decomposition Methods for Large Scale LP Decoding
When binary linear error-correcting codes are used over symmetric channels, a
relaxed version of the maximum likelihood decoding problem can be stated as a
linear program (LP). This LP decoder can be used to decode error-correcting
codes at bit-error-rates comparable to state-of-the-art belief propagation (BP)
decoders, but with significantly stronger theoretical guarantees. However, LP
decoding when implemented with standard LP solvers does not easily scale to the
block lengths of modern error correcting codes. In this paper we draw on
decomposition methods from optimization theory, specifically the Alternating
Directions Method of Multipliers (ADMM), to develop efficient distributed
algorithms for LP decoding.
The key enabling technical result is a "two-slice" characterization of the
geometry of the parity polytope, which is the convex hull of all codewords of a
single parity check code. This new characterization simplifies the
representation of points in the polytope. Using this simplification, we develop
an efficient algorithm for Euclidean norm projection onto the parity polytope.
This projection is required by ADMM and allows us to use LP decoding, with all
its theoretical guarantees, to decode large-scale error correcting codes
efficiently.
We present numerical results for LDPC codes of lengths more than 1000. The
waterfall region of LP decoding is seen to initiate at a slightly higher
signal-to-noise ratio than for sum-product BP, however an error floor is not
observed for LP decoding, which is not the case for BP. Our implementation of
LP decoding using ADMM executes as fast as our baseline sum-product BP decoder,
is fully parallelizable, and can be seen to implement a type of message-passing
with a particularly simple schedule.Comment: 35 pages, 11 figures. An early version of this work appeared at the
49th Annual Allerton Conference, September 2011. This version to appear in
IEEE Transactions on Information Theor
How to mesh up Ewald sums (I): A theoretical and numerical comparison of various particle mesh routines
Standard Ewald sums, which calculate e.g. the electrostatic energy or the
force in periodically closed systems of charged particles, can be efficiently
speeded up by the use of the Fast Fourier Transformation (FFT). In this article
we investigate three algorithms for the FFT-accelerated Ewald sum, which
attracted a widespread attention, namely, the so-called
particle-particle-particle-mesh (P3M), particle mesh Ewald (PME) and smooth PME
method. We present a unified view of the underlying techniques and the various
ingredients which comprise those routines. Additionally, we offer detailed
accuracy measurements, which shed some light on the influence of several tuning
parameters and also show that the existing methods -- although similar in
spirit -- exhibit remarkable differences in accuracy. We propose combinations
of the individual components, mostly relying on the P3M approach, which we
regard as most flexible.Comment: 18 pages, 8 figures included, revtex styl
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