Existence of optimal maps in the reflector-type problems

In this paper, we consider probability measures $\mu$ and $\nu$ on a $d$--dimensional sphere in \Rd, d \geq 1, and cost functions of the form c(\x,\y)=l(\frac{|\x-\y|^2}{2}) that generalize those arising in geometric optics where $l(t)=-\log t.$ We prove that if $\mu$ and $\nu$ vanish on $(d-1)$--rectifiable sets, if $|l'(t)|>0,$ $\lim_{t\to 0^+}l(t)=+\infty,$ and $g(t):=t(2-t)(l'(t))^2$ is monotone then there exists a unique optimal map $T_o$ that transports $\mu$ onto $\nu,$ where optimality is measured against $c.$ Furthermore, \inf_{\x}|T_o\x-\x|>0. Our approach is based on direct variational arguments. In the special case when $l(t)=-\log t,$ existence of optimal maps on the sphere was obtained earlier by Glimm-Oliker and independently by X.-J. Wang under more restrictive assumptions. In these studies, it was assumed that either $\mu$ and $\nu$ are absolutely continuous with respect to the $d$--dimensional Haussdorff measure, or they have disjoint supports. Another aspect of interest in this work is that it is in contrast with a result by Gangbo-McCann who proved that when $l(t)=t$ then existence of an optimal map fails when $\mu$ and $\nu$ are supported by Jordan surfaces

A rigorous analysis using optimal transport theory for a two-reflector design problem with a point source

We consider the following geometric optics problem: Construct a system of two reflectors which transforms a spherical wavefront generated by a point source into a beam of parallel rays. This beam has a prescribed intensity distribution. We give a rigorous analysis of this problem. The reflectors we construct are (parts of) the boundaries of convex sets. We prove existence of solutions for a large class of input data and give a uniqueness result. To the author's knowledge, this is the first time that a rigorous mathematical analysis of this problem is given. The approach is based on optimal transportation theory. It yields a practical algorithm for finding the reflectors. Namely, the problem is equivalent to a constrained linear optimization problem.Comment: 5 Figures - pdf files attached to submission, but not shown in manuscrip

Optical design of two-reflector systems, the Monge-Kantorovich mass transfer problem and Fermat's principle

It is shown that the problem of designing a two-reflector system transforming a plane wave front with given intensity into an output plane front with prescribed output intensity can be formulated and solved as the Monge-Kantorovich mass transfer problem.Comment: 25 pages, 2 figure

DiBELLA: Distributed long read to long read alignment

We present a parallel algorithm and scalable implementation for genome analysis, specifically the problem of finding overlaps and alignments for data from "third generation" long read sequencers [29]. While long sequences of DNA offer enormous advantages for biological analysis and insight, current long read sequencing instruments have high error rates and therefore require different approaches to analysis than their short read counterparts. Our work focuses on an efficient distributed-memory parallelization of an accurate single-node algorithm for overlapping and aligning long reads. We achieve scalability of this irregular algorithm by addressing the competing issues of increasing parallelism, minimizing communication, constraining the memory footprint, and ensuring good load balance. The resulting application, diBELLA, is the first distributed memory overlapper and aligner specifically designed for long reads and parallel scalability. We describe and present analyses for high level design trade-offs and conduct an extensive empirical analysis that compares performance characteristics across state-of-the-art HPC systems as well as a commercial cloud architectures, highlighting the advantages of state-of-the-art network technologies

Extreme Scale De Novo Metagenome Assembly

Metagenome assembly is the process of transforming a set of short, overlapping, and potentially erroneous DNA segments from environmental samples into the accurate representation of the underlying microbiomes's genomes. State-of-the-art tools require big shared memory machines and cannot handle contemporary metagenome datasets that exceed Terabytes in size. In this paper, we introduce the MetaHipMer pipeline, a high-quality and high-performance metagenome assembler that employs an iterative de Bruijn graph approach. MetaHipMer leverages a specialized scaffolding algorithm that produces long scaffolds and accommodates the idiosyncrasies of metagenomes. MetaHipMer is end-to-end parallelized using the Unified Parallel C language and therefore can run seamlessly on shared and distributed-memory systems. Experimental results show that MetaHipMer matches or outperforms the state-of-the-art tools in terms of accuracy. Moreover, MetaHipMer scales efficiently to large concurrencies and is able to assemble previously intractable grand challenge metagenomes. We demonstrate the unprecedented capability of MetaHipMer by computing the first full assembly of the Twitchell Wetlands dataset, consisting of 7.5 billion reads - size 2.6 TBytes.Comment: Accepted to SC1

Communication-Avoiding Optimization Methods for Distributed Massive-Scale Sparse Inverse Covariance Estimation

Across a variety of scientific disciplines, sparse inverse covariance estimation is a popular tool for capturing the underlying dependency relationships in multivariate data. Unfortunately, most estimators are not scalable enough to handle the sizes of modern high-dimensional data sets (often on the order of terabytes), and assume Gaussian samples. To address these deficiencies, we introduce HP-CONCORD, a highly scalable optimization method for estimating a sparse inverse covariance matrix based on a regularized pseudolikelihood framework, without assuming Gaussianity. Our parallel proximal gradient method uses a novel communication-avoiding linear algebra algorithm and runs across a multi-node cluster with up to 1k nodes (24k cores), achieving parallel scalability on problems with up to ~819 billion parameters (1.28 million dimensions); even on a single node, HP-CONCORD demonstrates scalability, outperforming a state-of-the-art method. We also use HP-CONCORD to estimate the underlying dependency structure of the brain from fMRI data, and use the result to identify functional regions automatically. The results show good agreement with a clustering from the neuroscience literature.Comment: Main paper: 15 pages, appendix: 24 page

Regimes of stability of accelerator modes

The phase diagram of a simple area-preserving map, which was motivated by the quantum dynamics of cold atoms, is explored analytically and numerically. Periodic orbits of a given winding ratio are found to exist within wedge-shaped regions in the phase diagrams, which are analogous to the Arnol'd tongues which have been extensively studied for a variety of dynamical systems, mostly dissipative ones. A rich variety of bifurcations of various types are observed, as well as period doubling cascades. Stability of periodic orbits is analyzed in detail.Comment: Submitted to Physica